INTRODUCTION.

The calculus of fluxions, the most important addition that ever was made to the fabric of abstract science, was invented in the latter part of the seventeenth century. Considered as a purely intellectual speculation, it far transcends any theory that had been previously known in pure mathematics; and it is not less estimable by reason of its applicability to physical knowledge. By its powers as an instrument of analysis, the science of geometry has been exceedingly enlarged; and the elegant speculations of the ancient geometers now go but a little way into the vast field of knowledge which has been laid open by the fluxional calculus, the sublime discoveries of Apollonius and Archimedes being merely easy deductions from more extended theories. It has, further, rendered most efficient aid in the investigation of the laws of nature, more especially in establishing the true theory of the motions of the heavenly bodies, which, without its assistance, could not have attained its present high position in the scale of human knowledge.

It may easily be supposed that the invention of the calculus was not at once fully revealed to the world. Its principles being inherent in the nature of geometrical magnitude, they would of course be gradually disclosed with the advance of mathematical science.

As long as the geometers of antiquity confined their speculations to the comparison of figures bounded by straight lines, there was no difficulty. The axioms and the mental process of conceiving the magnitudes compared to be placed one upon another, was sufficient to establish the equality of triangles in the cases when these could be brought entirely to coincide. The principle of superposition might have been carried further in the elements, so as to prove the equality of parallelograms, which are the complements of those about the diameter of a parallelogram; and of parallelograms having equal bases and altitudes; and the equality of the square on the hypotenuse of a right angled triangle to the squares on its sides. In fact, the doctrines of plane geometry were sufficient to prove that any two equal rectilineal figures admit always of being decomposed into spaces, such that, corresponding to every space in one of the figures, there may be found a space exactly equal and similar to it in the other figure; so that in this way any two equal rectilineal figures, however dissimilar, may yet, by division, be brought to coincide.

The principle of superposition, which was sufficient for the comparison of rectilineal plane figures, one with another, could not, however, be extended to the comparison of rectilineal with curvilineal, or of curvilineal with curvilineal figures. In treating of these, the ancients found it necessary to introduce new notions and modes of demonstration into geometry; and the difficulty which occurred in reasoning about curvilineal figures gave rise to the method of exhaustions, the most refined and important of their geometrical theories.

The fundamental principle of the method of exhaustions is this very simple proposition, that the difference of any two unequal quantities, by which the greater exceeds the less, may be added to itself so often, as to exceed any finite quantity of the same kind. On this the geometers of antiquity founded their propositions concerning curvilineal figures. By its aid they found that polygons may be inscribed in circles, which shall differ from them by less than any assignable space; and hence they might infer that the circles have to each other the same ratio as similar polygons inscribed in them, namely, the squares of their diameters. Although in this way they could assure themselves of the truth of the propositions, yet as this kind of proof was of a looser nature than that by which they established their more elementary doctrines, they fortified it against all cavil by their usual mode of synthetic demonstration, proving that the square of the diameter of one circle was to that of another as the first circle to a space that was neither less nor greater than the second, and therefore exactly equal to it. By a like process they proved that a cone is one third of a cylinder of the same base and altitude; and from the demonstrations which they gave of these propositions, it might be concluded in general, that when two variable quantities which have a constant ratio approach to two determined quantities, so as to differ at last from them by less than any assignable space, the same constant ratio is also the ratio of the determined quantities, which are the limits of the variable quantities.

The cautious way in which the ancients proceeded in finally establishing the truth of their propositions concerning the proportions of solids, could be of very little use in their original discovery. Indeed we may reasonably suppose, that in the first instance they arrived at a knowledge of their truth by a much shorter road, probably in a way not very different from that followed by the earlier geometers in modern times, who have extended their theories. But whatever path they followed, it led them to some truths of the highest importance at the time they were first known. After ages of fruitless labour in attempting to square the circle, the fine discoveries of Archimedes, that the parabola was two thirds of its circumscribing parallelogram, and the sphere two thirds of its circumscribing cylinder, and its surface four times that of one of its great circles, must have been received by his contemporaries with high satisfaction.

In the beginning of the seventeenth century, the writings of Archimedes greatly engaged the attention of the principal cultivators of mathematical science, who soon seized the spirit of his method, which was altogether distinct from the unwieldy machinery of the ancient mode of synthetic demonstration. He had only considered such solids as could be generated by the revolution of conic sections about an axis, and he had not even exhausted all the cases of these. The celebrated Kepler greatly extended this theory, in a treatise which he composed on the mensuration of round solids. These he conceived to be formed by the revolution of the conic sections about any ordinate, or a tangent to the curve at the vertex, or, in short, about any line whatever within or without the curve. In this way he described about ninety new solids, and proposed them as problems for the consideration of his contemporaries; but of these he resolved only a few of the most simple. In this work he first introduced the name and notion of infinity into the language of geometry. He considered the circle as composed of an infinite number of triangles, having their vertex at the centre, and forming the circumference by their bases. The cone he regarded as composed of an infinite number of pyramids, resting on the infinitely small triangles which formed its circular base, and having their vertices at its vertex; and he supposed a cylinder of the same base and altitude as the cone to be made up of prisms having the The bold extension which Kepler thus gave to the cautious language of geometry drew the attention of his contemporaries, some of whom eagerly followed in his steps. Cavalieri, a friend and disciple of Galileo, embodied the new views in his geometry of indivisibles, the theory of which he possessed in 1629, although it was not published until 1635. In this work, lines were considered as made up of points, surfaces as composed of lines, and solids as formed of surfaces. Cavalieri was exceedingly careful to verify his method, comparing its results with those which had been established by the ancient geometrical mode of demonstration; and, assured by their agreement, he ventured boldly in his new career. His principles were attacked, but he defended them by showing that they might be translated into those of Archimedes, from which they only differed in brevity of expression; his surfaces and lines being in fact merely the thin solids and small inscribed and circumscribed triangles of the ancient geometer, and their number being supposed so great that the difference between them and the figure about which they were described was less than any assignable quantity.

The new views proposed by Kepler were taken up in France by Roberval, who had carefully studied the writings of Archimedes, and formed a theory by which he could resolve problems concerning curvilinear figures. Instead of supposing lines to be made up of points, and surfaces of lines, as had been assumed by Cavalieri, he regarded a line as composed of an infinite number of infinitely short lines, a surface as composed of an infinite number of infinitely narrow surfaces, and so on; an assumption which at bottom was the same as that of the Italian geometer, but capable of being wrought up into a theory differing less from the spirit of the ancient geometry. Roberval concealed his views, employing them secretly as the means of invention in geometry, and expecting in this way to get the start of his contemporaries in reputation as a geometer. In the meanwhile the work of Cavalieri appeared and frustrated his selfish expectations.

The facility which the introduction of the notion of infinitely small quantities gave in the extension of geometry, induced all the mathematicians of that day to employ it in their researches. In the doctrine of curve lines it was applied to the theory of tangents by Roberval, Descartes, and Fermat, with great success. So very near had this last-mentioned geometer approached to the fluxional or differential calculus, that Laplace says he ought to be reputed its true inventor. Laplace was no doubt a great authority; but the facts on which he formed his opinion have always been generally known, and no one before him has attempted to reverse the decision of the age in which the calculus was invented. The justice of the decision has been subjected to a scrutiny by men of the highest powers of mind, who, differing in many of their opinions, have yet agreed in ascribing the invention to others.

The problem of drawing tangents to curves is indeed closely connected with the calculus, and at the period when this was about to be revealed that problem was much agitated, and successfully resolved by Slusius, Wallis, Barrow, and others. The writings and discoveries of Descartes exerted a most beneficial influence on the new geometry, particularly by his happy application of algebra to geometry. Before his time geometrical problems had been resolved by algebra, and a rich mine of discovery thereby opened. But he was the first who thought of applying algebraic formulae to express the nature of curve lines; an invention of vast importance to both branches of mathematical science.

Another most important benefit that resulted from the new views which Descartes brought forward, was the introduction of variable quantity. The early analysts distinguished quantities chiefly from the circumstance of their being known or unknown; and the main object of inquiry with them was to discover the chain of reasoning by which they might pass from the given quantities to the determination of those required. The view which Descartes took of geometrical magnitude brought the relations of quantities forward as a principal subject of contemplation. A new subject of discussion was brought into mathematical science, namely, the changes which take place in quantities connected by an invariable relation, when one of them is supposed to change its value, and pass from one state of magnitude to another. The science of geometry owes its elegance and the variety of its doctrines to the combined effect of magnitude and position, as they influence the subjects under consideration; and, in addition to these, the doctrine of curve lines has introduced a new element, namely, the property of varying in magnitude, as an attribute of quantity. This property of indefinitude, in respect of greatness, although it may appear a trivial affection of quantity, has, however, led to the most important consequences; for from this, combined with other principles, have sprung the fluxional calculus and other mathematical theories, which, originating in the ordinary conceptions of the human mind, have engaged the attention of men endowed with the highest powers of intellect for more than a century past.

In the progress of time the notions of infinite quantity introduced by Kepler, Cavalieri, and others, into mathematics, became familiar to geometers, who perceived the immense value of the improvement, and saw that, by due care in its application, there was no risk of its leading into error. Dr Wallis, in the year 1655, gave an admirable specimen of this new analysis in his Arithmetic of Infinites; a work which contained the first traces of the algebraic analysis applied to the quadrature of curvilinear spaces. Pursuing the views of Cavalieri, he reduced the problem of finding the areas of curves to the summation of the powers of an arithmetical series, consisting of an infinite number of terms, or rather the ratio of the mean of all the terms to the last term. He also showed by the same principles, that certain curve lines might be rectified, or that straight lines might be found to which they were exactly equal; a remark that was shortly afterwards verified by William Neil, a young mathematician of that period, who, at the age of nineteen, showed that the length of the semicubical parabola might be found in finite terms. This was the first curve that was rectified, and the second was the cycloid, a discovery due to Sir Christopher Wren.

Thus the principles of the fluxional calculus were at this period coming gradually into view. Their importance was seen and understood, and in substance, although without an appropriate notation, they were employed in extending the dominion of geometry; it might therefore be supposed that in the course of time they would have been gradually formed into a system, just in the way the science of algebra advanced from the rude form it had when first introduced into Italy, to that degree of compactness and perfection which it acquired by the successive improvements of Vieta, Harriot, and Descartes. But, in the case of the fluxional calculus, a happy combination of circumstances anticipated the slow operation of time. Newton and Leibnitz, two men endowed with the highest powers of mind, appeared nearly at the same time in the field of discovery. The genius of each shone with resplendent lustre, even amidst the blaze of intellect which enlightened the latter period of the seventeenth century, and which has rendered it for ever memorable in the annals of hu- man knowledge. They both turned their attention to this branch of mathematical science, and, as is now generally admitted, each, independently of the other, collected the undigested principles into a system, and, by a powerful effort, performed at once what the united labour of ordinary minds would have hardly accomplished in a century.

In ascribing to each of these great men the full honour due to the merit of the invention of the calculus, it is proper to add, that this is a question which at one time divided the opinions of the learned world, and gave rise to a controversy which was agitated with great keenness for almost a whole century. There never could be any doubt as to Newton being the inventor of the calculus of fluxions. He has been acknowledged by all to have been the inventor, and the first inventor; but the question strongly contested has been, whether Leibnitz, without knowing what had been done by Newton, invented his differential calculus by the force of his own mind, or borrowed it from the fluxional calculus, with which at bottom it is identical.

In the course of this controversy it was clearly established by the British mathematicians, who almost to a man ranged themselves on the side of their illustrious countryman, that, as far back as the year 1669, Newton, then in his twenty-sixth year, and a fellow of Trinity College, communicated to Dr Barrow a tract, entitled De Analyti per Aequationes Numero Terminorum Infinitas. In this manuscript the method of fluxions was first indicated; and rules deduced from it given for the quadrature of curves, and various other matters to which the calculus is applicable, were distinctly specified. From this tract, which was sent by Dr Barrow to Collins, the secretary of the Royal Society, it is established beyond all doubt that Newton was then in full possession of his method; and indeed it is certain from other evidence that he must have had it as far back as 1666. The important discoveries made by Newton were communicated without reserve by the secretaries of the Royal Society to mathematicians on the Continent, as well as to those in Britain. In this way it might be generally known that Newton possessed a most valuable instrument of research, by which he had been led to these results, although its nature was too obscurely indicated to give any assistance in divining its precise form.

Leibnitz appears to have been later than Newton in beginning his career of discovery in the mathematics. In the year 1672, being then in London, he communicated to some members of the Royal Society what he supposed to be discoveries relating to the differences of numbers. It was, however, shown to him that the same subject had been discussed long before, by Mouton, a French geometer. He then appears for the first time to have turned his attention to infinite series. Afterwards, on his return to Germany in 1674, he announced to Mr Oldenburgh, then secretary of the Royal Society of London, that he possessed very general analytical methods, by which he had found theorems of great importance relating to the quadrature of the circle, by means of series. In answer, the secretary informed him that Newton and James Gregory had also discovered methods which gave the quadrature of curves, and which extended to the circle.

In the year 1676 Newton addressed a letter to the secretary of the Royal Society, which was made known to Leibnitz. This was the first direct communication between these two great men. It contained the binomial theorem, which Newton had known in the year 1669, and a variety of other matters relating to infinite series and quadratures; but nothing directly relating to the method of fluxions. At this time Newton entertained no jealousy of Leibnitz, for he speaks of him in the letter with great respect. In a second letter Newton explained at his request the way in which he had found the binomial theorem, and he adverted also to his own calculus, explaining its nature under the form of an anagram of transposed letters, which was expressed thus, Gaccadai3eff713B9n4o49r71a9f12ox. It is not easy to see what certain information Leibnitz could draw from this assemblage of characters; yet such was the effect of national prejudice in biasing the minds of the British mathematicians, that Raphson, in his History of Fluxions, a work dedicated to the Royal Society of London, expressly asserts that Leibnitz deciphered the anagram, and found it to be this sentence, Data equaciones fluentes quotunque quantitates involvente inventire fluxiones, et vice versa. "Having any given equation involving never so many flowing quantities, to find the fluxions, and vice versa." We believe there was not the least ground for this assertion; and from such a specimen of want of candour, it is easy to infer that there was very little chance of Leibnitz getting justice done him by the English mathematicians.

In the month of June 1677 Leibnitz sent to Mr Oldenburgh, to be communicated to Newton, a letter containing the first essays of a method which extended to all that could be done by that of Newton. The death of Oldenburgh, which happened soon afterwards, put an end to the correspondence, and it was not until the year 1684 that Leibnitz made his discovery public, it being then inserted in the Leipsic Acts; so that whilst Newton's claim to the priority of the discovery must be admitted by all, it is not less certain that Leibnitz was the first to give the full benefit of the calculus to the world; for Newton's Method of Fluxions only became generally known in the year 1687, by the publication of his Philosophia Naturalis Principia.

It is certain that Leibnitz enjoyed unchallenged for fifteen years the honour of being the inventor of his calculus; even Newton himself rendered him that justice in the first edition of his immortal work. Subsequently, however, a foreign mathematician, Fatio de Duillier, piqued, it is said, by having been omitted in an enumeration of eminent geometers by Leibnitz, declared, in a treatise on the line of shortest descent, printed at London in 1699, that he was obliged by the undeniable evidence of things to acknowledge Newton not only as the first, but as by many years the first inventor of this calculus; from whom, whether Leibnitz, the second inventor, borrowed anything or not, he would rather they who had seen Newton's letters and other manuscripts should judge than himself.

The insinuation contained in this passage of Fatio's book could not but be very offensive to Leibnitz, who inserted an animated reply in the Leipsic Acts of 1700, and complained to the Royal Society of London of the injustice done him; and here the affair rested for a time; but it was revived some years afterwards. When Newton's treatise on the quadrature of curves, and his enumeration of lines of the third order, was published, the Leipsic journalists gave an unfavourable review of the work. Amongst other things, after a brief exposition of the nature of fluxions, they added, that Newton uses and has always used fluxions for the differences of Leibnitz, just as Fabri had substituted in his synopsis of geometry motion instead of the indivisibles of Cavalieri. This most unjust accusation excited great indignation in the minds of the British mathematicians, one of whom, Keill, Savilian professor of astronomy at Oxford, in a paper inserted in the Philosophical Transactions of the year 1708, affirmed that Newton was the first inventor of the calculus, and that Leibnitz, in publishing it in the Leipsic Acts, had merely changed the name and the notation. Leibnitz, thus directly charg- Introductory with having taken his calculus from that of Newton, addressed a letter to Mr. afterwards Sir Hans Sloane, the secretary of the Royal Society, in which he required that Keill should retract his accusation. Keill however refused to do this, and in answer addressed a letter to the secretary, in which he professed to show, not only that Newton had preceded Leibnitz in the invention, but that he had given so many indications of his calculus that its nature might easily be understood by any man of ordinary understanding. This letter was sent to Leibnitz, who addressed a second letter to the Royal Society, requiring that they should stop these reproaches of Keill, saying that he was too young a man to know what had passed between him and Newton.

The society, thus appealed to as a judge, appointed a committee to examine the old letters, papers, and documents which had passed between the mathematicians on the subject. The report of this committee does not appear altogether satisfactory. "We take the proper question to be," say the reporters, "not who invented this or that method, but who was the first inventor of the method; and we believe those who have reputed Leibnitz the first inventor, knew little or nothing of the correspondence between Mr Collins and Mr Oldenburg long before, nor of Mr Newton having that method about fifteen years before Mr Leibnitz began to publish his in the Acta Eruditorum of Leipzig; for which reason we reckon Mr Newton the first inventor, and are of opinion that Mr Keill, in asserting the same, has been nowise injurious to Mr Leibnitz." This declares nothing on the only point about which there could be any doubt, namely, whether Leibnitz had formed his calculus entirely by the power of his own genius, or had availed himself of a knowledge of Newton's invention. The judgment satisfied neither party. Keill wished to establish against Leibnitz the charge of plagiarism, but was disappointed; and Leibnitz found in the decision grounds of complaint against the Royal Society. In a communication to the Abbé Conti, a Venetian nobleman and a common friend of his and Newton, then in England, he accused the English nation of desiring to be considered as almost the only inventors. He said it did not appear, as had been well observed by Bernoulli, that Newton possessed before him the infinitesimal characteristic and algorithm, although it would have been easy to have found it if he had been so disposed, just as it would have been easy for Apollonius to have discovered the theory of curve lines of Descartes. He complained that his opponents had attacked his candour by forced and unfounded interpretations of his words. The question was the calculus of differences, and his opponents made it turn entirely upon series, where Newton had no difficulty in going before him. He maintained that he had found for himself a general method for series, and that he had no need of Newton's method of extractions. There were other grounds of complaint, such as, that only extracts from letters had been given in the Commercium Epistolicum, whereas the entire letters should have been published. Throughout the whole, which is a postscript to a letter, he manifests much chagrin; and he concludes with proposing this problem, intended, he says, to feel the pulse of the English analysts: To find a line which shall cut perpendicularly all curves of a determinate series, and of the same kind; for example, all hyperbolas, which shall have the same vertex and centre. This he desires the abbé to propose as from himself, or a friend.

In the abbé's reply to this letter he informed Leibnitz that he had delayed writing until he could at the same time send the answer which Newton had given to his postscript. He tells him that he had read the Commercium Epistolicum, and the originals of the letters which it contained, and, on the whole, he inferred that, setting aside matters foreign to the dispute, the only question at issue was, whether Newton had found the Calculus of Infinitesimals, or Fluxions, before, or after him? The abbé says, you published it first, it is true; but you admit that Newton allowed much to transpire in the letters which he wrote to Oldenburgh and others. This has been proved at length in the Commercium and its extract; what is your answer? This is what the public requires, to form a correct opinion on the matter. The abbé further informs Leibnitz that his friends believe he ought to give an answer, if not to Keill, at least to Newton himself, who, in the letter which the abbé had received from him to be communicated to Leibnitz, had challenged him in express terms. In a spirit of friendship and good sense, he farther adds, that he wishes to see a good understanding between them, that the public will profit but little by their disputes, and that the additions to knowledge which they prevent are a pure loss to posterity. He moreover tells him that the king, George I., had desired to be informed of all that had passed between Newton and him. From this we may judge what an interest was excited in the public mind by a dispute which, even at this day, few could perfectly understand. The answer which Leibnitz gave to the abbé's friendly advice was, that he had put his letters, and Newton's, with his own answers to them, into the hands of Mr Remond at Paris, with a view to have a neutral and impartial opinion as to the matters in dispute. The whole were then to be submitted to the Abbé Varignon and other members of the Royal Academy of Sciences, and afterwards sent to England. In his answers he disavowed the sense which the English mathematicians had extracted from the assertion of the Leipzig journalists, namely, that Newton employed and always had employed fluxions for the differentials of Leibnitz, which they understood to mean that Newton had availed himself of Leibnitz's method, only substituting fluxions for differentials. It was from this passage that the English had declared Leibnitz to be the aggressor, in having abetted an attack of Newton; but Leibnitz declared the English interpretation to be full of malignity towards him, and that the sentence in the Leipzig Acts (Pro differentiis Leibnitianis D. Newtonus exhibet semperque adhibuit fluxiones), meant that Newton had employed fluxions after having seen his differences, and also before he had seen them.

We believe enough has been detailed to show that, besides a desire to arrive at truth, there was infused into this controversy the jealousy of two rivals for the glory of a great discovery; and, in addition, the spirit of partizanship of their respective admirers, and that natural feeling which induces men of different countries to exalt their own nation, although sometimes at the expense of their neighbours. It is not wonderful, then, that at the time this controversy was at its height, men were greatly divided in their opinions.

It is now impossible to obtain absolute certainty as to the question, whether Leibnitz derived any aid from the discoveries previously made by Newton, which, to a certain extent, were made known to the scientific world. It may therefore be some satisfaction, in the want of perfect evidence, to have the judgment of Montucla, the historian of the mathematics, on this subject. "In regard to the principle of fluxions," says he, "there are only three places of the Commercium Epistolicum which treat of it with sufficient clearness to prove that Newton had found it before Leibnitz, but I believe too obscurely to take from him the merit of the discovery. One of these is a letter from Newton to Oldenburgh, who had remarked to him that Slusius and Gregory had found a very simple method of tangents. Newton answered that he conjectured what it was, and he gave an example, which in effect the same thing as these two geometers had found. He added that it was but a particular case, or rather a corollary to a method much more general, by which tangents to all sorts of curves, whether geometrical or mechanical, might be found without laborious calculation. He repeated the same thing, without explaining himself further, in his second letter, which has been already mentioned, and he concealed the principle under transposed letters. The only writing in which Newton had allowed any thing of his method to transpire is his Analysis per Equationes numero Terminorum infinitas. He there briefly and obscurely reveals his principle of fluxions. It cannot be denied that the principle and method were there indicated; but it is not certain that Leibnitz had ever seen the passage. His adversaries have never said that it had been communicated to him by letter; they have merely thrown out a suspicion that Collins might have communicated it to him in his second journey to London. Indeed the suspicion is not altogether void of probability, insomuch as Leibnitz admitted that in his interview with the secretary he had seen a part of his epistolary correspondence. I believe, however, that it would be rash to pronounce upon this. There might have been grounds for suspicion if Leibnitz had merely given a few essays of his new calculus; but when we consider the improvements he made, as appears by the number of pieces he contributed to the Leipsic Acts, it seems probable that he owed his calculus to his own genius, and his efforts to divine the nature of a method by which Newton had made so many fine discoveries. This is the more probable, as from the method of tangents of Barrow to the differential calculus, the step is not too great to have been made by a power of mind of which Leibnitz had given so many proofs.

If it had been true, as the British mathematicians alleged, that Leibnitz had taken his calculus from that of Newton, it would have been a very remarkable circumstance that such a long period should have elapsed between his first putting forth his pretensions to the honour of the invention, and the denial of Newton and his friends. This is a thing uncommon in the history of contested claims to discovery, when both the claimants are alive and all the facts known. It is also remarkable that Newton, while in the prime of life, did not of his own accord bring forward a charge of plagiarism against his contemporary. On the contrary, in the first edition of his Principia, which appeared in 1687, he expressly admitted Leibnitz's claim; and this admission is found even in the edition of the Principia published in 1713. It is now well known, that before the period of the controversy between Newton and Leibnitz, the mind of the former had at one time been in a state which rendered him unfit to draw correct conclusions from the ordinary incidents of life; witness his letters to Mr Pepys and Mr Locke in September 1693 (Brewster's Life of Newton). At this time he seems to have laboured under great bodily indisposition, from which he afterwards in appearance recovered; indeed it may be doubted whether his temper was so equal and mild in the latter period of his life as has been represented by his biographers. A collection of letters which passed between Flamsteed the astronomer and Abraham Sharp the celebrated calculator, which have lately come into the possession of the Astronomical Society of London, prove that Newton, in conjunction with Halley, opposed Flamsteed in the publication of his Historia Coelestis, a work of the highest merit, and the result of the labour of his life, with a pertinacity of purpose which is by no means an amiable trait in his character.

It is certain that Newton entertained towards Leibnitz a very different feeling at one period of his life, from that which he in the end evinced. In the edition of the Principia 1713, and those which preceded it, he says, "Ten years ago, in a correspondence with Mr Leibnitz, I informed him that I had a method of determining maxima and minima, of drawing tangents, and resolving similar problems, which applied alike to rational and irrational terms, but concealed it under transposed letters. He answered that he had found a like method, which he communicated, and which differed from mine only in the terms and the signs, and in the way of conceiving quantities to be generated." This concession was liberal, and worthy of Newton. However, in the edition of 1720, he retracted this act of justice, having struck out the passage, as Montucla assures us, with his own hand. The historian of the mathematics, however, finds an excuse for this harshness in the treatment which Newton had received from Leibnitz's friends.

The circumstance of the calculus having had two distinct origins, had the inconvenient effect of giving it two different forms and names. Newton at different times used different notations before he finally adopted that which has been followed by the English mathematicians. Leibnitz and his followers employed one somewhat different; and each sect pertinaciously adhered to the example set by its leader. There was a still greater difference in the view which each party took of quantity, and the way in which it may be generated. Newton conceived geometrical magnitudes to be generated by continued motion, a line by a point in motion, a surface by a moving line, a solid by a moving surface, an angle by a line turning about a point, and so on. It is evident, that by assuming geometrical quantities as the representatives of time, force, and whatever can be expressed by number, these can be conceived to increase or decrease according to the same laws as their geometrical representatives. His calculus therefore consisted of two parts:

1st, Supposing two quantities to have a given relation to each other (for example, the one to be always equal to the square of the other), and the rate of increase of one of them at any instant of time to be known; to find the rate of increase of the other at the same instant. These rates of increase were called the fluxions of the quantities; and the rules for their determination constituted the direct method of fluxions.

2d, The second part was the reverse problem, in which the relation between the rates of increase of two quantities which depended the one on the other, being given, it was required to discover the relation of the quantities. This was the inverse method.

Leibnitz at first supposed quantity to increase by the addition of some indefinite portion of a quantity of the same kind, and his differentials were quantities proportional to the instantaneous changes in the greatness of the quantities thus generated. He afterwards found it shorter to introduce directly these instantaneous changes into his calculus under the name of infinitely little differences (infinitum petites). These however he did not regard as absolute zeros, but only as not comparable to finite magnitudes. His calculus had, like Newton's, two parts, the differential calculus, which gave rules for deducting the relation of the differentials of quantities from that of the quantities themselves, and the integral calculus, which resolved the reverse problem, or discovered the relation of the quantities when the relation of their differentials was known. This corresponded to the inverse method of fluxions, as the differential calculus corresponded to the direct method.

There is this distinction, then, between the two methods: Newton's fluxions are any finite quantities which have to each other the ratio of the velocities, or degree of quickness with which the quantities are generated. The differentials of Leibnitz are infinitely small quantities, a creation of the mind more removed from ordinary apprehension than the finite representatives of fluxions. This distinction has often been urged in favour of the superiority of Newton's calculus. On the other hand, Newton has introduced time into his calculus. Now this is an element foreign to geometry, and belonging to mechanics, a quite distinct branch of science. The metaphysical distinctions between the two forms of the calculus have had but little influence on its progress. Whichever of the two forms is chosen, the rules are the same; and so also are the difficulties to be overcome.

Some time elapsed before the newly invented calculus was actively employed. At length Leibnitz, to rouse the attention of geometers, proposed in 1687 this problem: To determine the curve along which a heavy body must descend, so as to approach by equal distances in equal times to a horizontal plane. Huygens was the first to resolve the problem. He showed what was the nature of the curve; but he did not give his demonstration. James Bernoulli also resolved the problem by the differential calculus, and published his analysis in the Leipzig Acts. His younger brother John united himself in a close friendship with Leibnitz, and continued throughout life his coadjutor and staunch defender. He made the calculus known in France, where he resided for a time, and gave lessons to the Marquis de l'Hôpital. Part of these form the Analyse des infiniment Petits, a work which bears the name of the marquis, with but a slight acknowledgment in the preface of the share which his preceptor had in its composition. Indeed it was almost entirely his; and of this injury Bernoulli complained justly, but privately, in his correspondence with Leibnitz. The remaining lessons, which teach the integral calculus, are given in the collection of Bernoulli's works, published in 1742. The whole form an excellent exposition of the principles of the calculus, and contain some of its finest applications.

The continental mathematicians who were remarkable for their skill in the differential calculus were at first few in number. They might be nearly all included in the names of Leibnitz himself, James and John Bernoulli, De l'Hôpital, and Varignon. They were, however, remarkably active, particularly the Bernoullis, whose writings have contributed powerfully to its advancement.

The inventor of the fluxional calculus seems to have taken but little interest in its extension. His treatise on quadratures did not appear till 1706; and his work on fluxions was not published in his life time, but appeared in 1736, nine years after his death. He had however fully availed himself of its power in the composition of his Principia, without revealing to the full extent the nature of the instrument with which he wrought, but establishing the truth of his propositions by synthetic demonstration, after the manner of the ancient geometers. Probably this was done in deference to the taste of that time; for a contemporary writer, Hermann, in following his example in the composition of his Phoronomia, assigned to Leibnitz as a reason, that the geometrical method was likely to be better understood than the analytical in Italy. This way of proceeding is now, however, disused by the best writers, and replaced by the more legitimate and undisguised use of the calculus.

The extensive views which the new calculus opened up, afforded the means of resolving problems which had baffled the skill of the earlier mathematicians, and suggested many new ones, which without its aid would have been quite intractable. The cycloid and the catenary, curves which had eluded the penetrating mind of Galileo, were more fully comprehended; many new ones were suggested, and the cultivators of the calculus challenged each other to an investigation of their properties. There is an elegant class of problems which relate to the greatest and least values of quantities. Some of these, called problems of maxima and minima, can be resolved by the ordinary geometry and algebra; but there is a class which lies beyond the dominion of these, and to it belongs the solid of least resistance, the curve of swiftest descent, and, in general, the problems called isoperimetrical, one of the simplest of which is to find the nature of the line which, being given in length, shall comprehend the greatest possible space. These formed the subject of a warm but honourable contest between the two Bernoullis, brothers; and their investigation laid the foundations of that branch of the calculus, the highest, which has been formed into a theory by the labours of Euler and Lagrange, under the name of the Calculus of Variations.

Among the early cultivators of the calculus in Britain we may reckon Cotes, the friend of Newton, who died at the early age of thirty-four (his Harmonia Mensuraria, and other writings, indicate a genius of the highest order), De Moivre, Taylor, Craig, David Gregory, and Stirling. It must, however, be confessed, that the continental school in the course of time advanced before that of Britain, as well in the number of its disciples, as in their high mathematical genius and the importance of their discoveries. John Bernoulli had two sons, Nicolas and Daniel, who rivalled their father's skill in the applications of the calculus; and these had as condisciples Hermann and Euler. This last mathematician has carried the subject to a point of perfection far beyond what it had attained in the hands of those that had gone before him.

Amongst the means by which the improvement of the calculus has been promoted, we may reckon the problems which passed between the British and continental mathematicians as challenges to each other, in answering which Newton himself condescended sometimes to enter the lists. The doctrine of infinite series, a branch of the mathematics of English origin, gives powerful aid to the calculus in its greatest difficulties; but, legitimately, its assistance ought not to be called in if the problem admit of being resolved in finite terms. The English sometimes failed in the observance of this most proper condition of a good solution, and thus gave rise to reproaches from their opponents, which in at least one instance were not unjustly incurred by Newton himself. In the course of this warfare, Keill, the champion of the English, proposed as a challenge to John Bernoulli, to determine the nature of the curve which a projectile describes in a medium resisting as the square of the velocity. Bernoulli soon resolved the problem, not only in the case proposed, but also when the resistance was as any power whatever of the velocity. He then proposed to put his solution into the hands of some confidential person in London, provided that Keill would deliver his solution to the same person. Keill, however, had upon trial not been able to resolve the problem, and therefore preserved a profound silence, and made himself, by his boasting and failure, quite ridiculous. The triumph of Bernoulli was complete, and he did not miss the opportunity of bestowing severe censure on his humbled adversary, recollecting, we may suppose, the treatment which his deceased and lamented friend Leibnitz had received from him and his friends.

As another proof of the spirit and rivalry of the supporters of the two analytical theories, we may mention that Brooke Taylor proposed a problem in the integral calculus to all geometers not English, and sent it to Mr Montmort to be communicated to the foreign geometers. Notwithstanding the general terms of the challenge, it was well understood to be particularly aimed at John Bernoulli, who in return offered to wager fifty Louis that he would resolve the problem, and to stake fifty more that he would propose a problem which he himself could resolve, but which Taylor could not. The English mathematician did not think it prudent to accept the offered condition. Indeed it was a complaint against the English geometers, that although they made no scruple of trying to puzzle foreigners with difficulties, they rarely responded to the counter challenges proposed to themselves.

Perhaps there never was a considerable invention or discovery which had not to encounter opposition; sometimes from the slowness with which the human mind yields to the force of truth when opposed to long-received opinions, and sometimes from less excusable causes, such as mistaken or interested views, or mere jealousy excited by the fame of the inventor. The new calculus had in the very outset its opponents, such as the Abbé de Cetelan, a zealous Cartesian, who declared that it would be better to extend the principles of the Cartesian geometry than to seek for new methods; and this was said in the preface of a book composed on the principles, somewhat disguised, of the very calculus of which he was an opponent. It had another adversary in Nieuwentit, a man who had written some tolerable works on morality and religion, but who had but small pretensions to be regarded as a geometer. Cetelan was satisfactorily answered by De l'Hôpital, as was Nieuwentit by Leibnitz, and afterwards by Bernoulli and Hermann, who proved that this adversary of the calculus really did not know what he opposed.

The calculus had a more formidable enemy in M. Rolle, a skilful algebraist and indefatigable calculator, but a man full of confidence in his own notions, rash in forming his opinions, and jealous of the inventions of others. He attacked the certainty of its principles, and he attempted to show that its conclusions were at variance with those obtained by methods previously known, which were acknowledged to be correct. His attack was repelled by Varignon, who completely obviated the objections to the truth of the principles, and further showed that the supposed discrepancy between its conclusions and those obtained by other methods, were mistakes he had committed from haste and inadvertence. These disputes occupied the French Academy a considerable part of the year 1701. The members were chiefly geometers considerably advanced in years, who had been long accustomed to other methods, and were therefore not much disposed to receive new doctrines. Some took no part in the dispute, yet were not sorry to see a storm raised against a theory for which they had no great liking, and took no means to allay it; others, more under the influence of their passions and prejudices, yielded to these, and declared open war against it. In this state of things the best course was supposed to be that of hearing all which could be said for and against the calculus. The academy was long involved in the dispute. Rolle brought forward objection upon objection; and although Varignon continually obviated them, yet the former always claimed the victory. In the end the dispute degenerated into a real quarrel, and commissioners were appointed to decide on it. These were Père Gouye, MM. Cassini and de la Hire, who, however, pronounced no judgment; but the public opinion, or at least the opinion of geometers, was in favour of Varignon. The first controversy thus ended, or at least was suspended, for want of a decision from the commission; but Rolle, the champion of the opponent of the calculus, soon renewed hostilities. Its defence was next taken up by M. Saurin. The ground of attack was the indefinite form which the calculus gives for the subtangent of a curve at the point where two branches intersect each other, and which in this case is expressed by the fraction \( \frac{0}{0} \).

Saurin's answer was satisfactory; but Rolle, intrenched in masses of calculation, obstinately maintained the combat. The academy was again appealed to. The Abbé Bignon, who conducted its affairs, undertook to decide the controversy, with the assistance of MM. Gallois and De la Hire, two judges by no means favourable to Saurin. They gave no absolute judgment; but recommended to Rolle to conform more strictly to the rules of the academy, and to Saurin to forgive the proceedings of his adversary. Rolle suffered in the estimation of competent judges; he, however, afterwards did justice to the calculus, by acknowledging that he had done wrong in opposing it, and admitting that he had been urged forward by the instigation of malevolent persons, one of whom was the Abbé Gallois; and his demise in 1707 accordingly put an end to the controversy.

In England the Newtonian calculus had to sustain an attack on its principles from a writer of first-rate talents, Berkeley, bishop of Cloyne. The circumstances which led to it are curious. Mr Addison had given the bishop an account of the behaviour of their common friend, Dr Garth, in his last illness, which was highly displeasing to these two advocates of revealed religion; for when Addison began to discourse with Garth on a future state, "Surely," said the latter, "I have no reason to believe these trifles, when my friend Dr Halley, who has dealt so much in demonstration, has assured me that the doctrines of Christianity are incomprehensible, and religion itself an imposture." The bishop therefore took up arms against Halley, and in the year 1734 addressed to him, as an infidel, a discourse called The Analyst, the object of which was to prove that mathematicians acted inconsistently in objecting to mysteries in faith, seeing that they did not hesitate to admit much greater mysteries, and even falsehoods, in their own science; and he chose the principles of the doctrine of Fluxions, as laid down by Newton and adopted by his followers, to prove the truth of his proposition. It may be supposed that so able a writer would not fail to make an impression on the public mind. But the mathematicians were not slow in coming forward in defence of the doctrines of their chief. In the same year came out a tract with this title: "Geometry No Friend to Infidelity, or a Defence of Sir I. Newton and the British Mathematicians, by Philalethes Cantabrigiensis," supposed to be Dr Jurin. As usual the attack was renewed, and again repelled by the same hand. Other defences of Newton appeared, one of the best of which was from the pen of Benjamin Robins; it was entitled A Discourse concerning the Nature and Certainty of Sir Isaac Newton's Method of Fluxions, and of Prime and Ultimate Ratios. But the most important result of this controversy was A Treatise on Fluxions from the pen of Colin MacLaurin, professor of mathematics in the university of Edinburgh, printed in 1742. This defence of the principles of Newton's views, expressly intended to obviate all objections, is quite satisfactory. The treatise is indeed considerably prolix; but this was a consequence of the circumstances in which it was composed. It however contains a great deal more than a mere theory of fluxions. We have in it some of the finest applications of analysis to the principal problems which had been agitated amongst geometers from the invention of the calculus to the time in which the work appeared.

By the middle of the century the calculus proceeded with a rapid march of improvement. Euler in 1744 greatly enriched it by his solution of the Isoperimetric Problem (Solutio problematis isoperimetrii latissimo sensu accepti), a theory which he afterwards wrought up into the Calculus of Variations, and which has been still further improved by Lagrange. Brooke Taylor, by his Methodus Incrementorum, had extended the foundations, and even proceeded a great way in the structure, of a kindred calculus, that of finite differences. Stirling had followed in the same path in his Methodus Differentialis; and Euler had opened a new field for discovery in the calculus of partial differences, in which D'Alembert followed his steps, and carried the subject further. The same period abounded in good writers. In this country Demoivre, Simpson, Landen, and Waring, stand among the foremost in the list for the originality of their views; on the continent Ricatti, Clairaut, Fontaine, and others, followed in the train of Euler and the Bernoullis.

It is a curious fact in the history of mathematics, that there are instances of ladies who have applied their talents to its study and improvement with much success. Hypatia, the daughter of the ancient geometer Theon, is one notable instance. Unfortunately we have no vestiges of her writings by which we might form an estimate of her proficiency in science which at first sight seems not to have many attractions for the female mind. We have another instance in an Italian lady, Maria Gaetana Agnesi, who was actually professor of mathematics in the University of Bologna in the year 1748; and how well she was qualified for the office appears from her Analytical Institutions, a work of great excellence on algebra, the theory of curve lines, and the differential calculus, composed for the instruction of the youth of Italy. This work is so excellent, that Bossut, a French mathematician of great eminence, translated the part on the differential calculus into French, and incorporated it with his Course of Mathematics, as the best treatise he could find on the subject. There is also an English translation by the Rev. John Colson, Lucasian professor of mathematics in the university of Cambridge. Her countryman Frisi, a most competent judge, mentions her in enumerating the mathematicians of Italy, and bestows great praise on her work, calling it opus nitidissimum, ingeniosissimum, et maximum certe opus quod hacenus ex feminine aliquis calamo producit. At the present time we have another admirable instance of a lady who has surmounted the difficulties of the calculus. Mrs Somerville, in her work entitled Mechanism of the Heavens, London, 1831, has enriched English literature with a treatise on physical astronomy, in which the different branches of the calculus are combined with the most refined theories of mechanics. Her book does her infinite credit, and indeed is highly honourable to the whole female sex.

Geometers have differed in opinion as to the best way of working up the principles of the calculus into a system. Newton, as has already been stated, employed the theory of motion as the means of connecting its doctrines with the principles of the ordinary algebraic analysis. Leibnitz, again, with the same view, conceived quantity as passing from one degree of magnitude to another by the continual accretion of infinitely small parts. The mind finds no great difficulty in distinctly apprehending the subject in its simplest state either way. Objections have, however, been taken to both, and attempts made to substitute a better. Euler considered the infinitely small quantities of Leibnitz as absolute zeros, that have to each other ratios derived from those of the vanishing quantities which they replace. D'Alembert proposed to make the basis of the calculus the consideration of the ratios of the limits of the quantities. An English mathematician, Landen, has substituted for the Newtonian method of fluxions another purely analytical. His views are contained in a work entitled The Residual Analysis, a new branch of the Algebraic Art, by John Landen, 1764. Lastly, Lagrange, in the Memoirs of the Berlin Academy for 1772, proposed to make the calculus altogether independent of the consideration of infinity, and to rest on principles purely analytical, thus connecting it with the doctrines of the ordinary algebra. He has since realised and extended his views in his Théorie des Fonctions Analytiques, also in his Leçons sur les Calcul des Fonctions; works which, from their excellence and the celebrity of their author, have formed a new era in the history of the calculus.

The twofold origin of the calculus, besides placing its principles on different foundations, gave it also two different forms of notation; and in this state it continued during the whole of the eighteenth century, to the great inconvenience of mathematical students, and, we may add, to the hindrance of the progress of science. A change, however, has taken place within the last twenty years; during that period some British geometers, imbued with the mathematics of the continent, adopted also its notation in preference to that of the followers of Newton, and employed it publicly in their researches. This spirit of innovation in time took possession of the minds of the junior members of the university of Cambridge, and now we may say that it has completely supplanted the native notation. We have retained the name of Fluxions, because, from the changes which have taken place in the way of treating the subject since the days of Leibnitz and Newton, it seems to be as proper as the term differential. It is otherwise with the notation. This is consecrated by its having been employed in the writings of Euler, D'Alembert, Lagrange, and other great masters in mathematical science; and besides, it has advantages over the other in point of symmetry and compactness, and the established reputation of a whole century of useful service. It is true, the late illustrious Lagrange laid it aside in his attempt above mentioned to new-model the calculus; but subsequent writers, in extracting what was excellent from his works, have in general invested his views with the ordinary notation of the continent. In the following treatise we shall conform to this beneficial alteration.

Writers on the Infinitesimal Calculus, before the Invention of the Method of Fluxions.

Kepler, Nova Stereometria Doliorum Vinariorum...1615 Cavalieri, Geometria Indivisibilium..............1635 ——— Exercitationes Geometricae Sex.............1647 Roberval, Traité des Indivisibles. Mem. de l'Acad. des Sciences.................................1666 Descartes, in his Letters and Geometry, book 2d, and his mathematical works.......................1637 Torricelli, De Sphaera, et Solidis Spheralibus....1644 Gregory St Vincent, Opus Geometricum de Quadratura Circuli..................................1647 Wallis, Arithmetica Infinitorum....................1655 Fermat, Opera Varia Mathematica...............1679 Mercator, Logarithmo-Technia.....................1668 James Gregory, Vera Circuli et Hyperbolic Quadra- tur..................................................1668 Brouncker, Squaring the Hyperbola, Phil. Trans. London.............................................1668 Huygens, Opera; Horologium Oscillatorium...1673, 1724 Barrow, Lectiones Geometricae....................1670 Slusius, Tangents to all Geometrical Curves, Phil. Trans................................................1672 Wren, Rectification of the Cycloid, ibid........1673 Ismael Bullialdus, Arithmetica Infinitorum, libri vii.1682 Leibnitz, Quadrature of the Circle, Leips. Acts and Phil. Trans.....................................1682 Viviani, Exerc. Math. de Formatione et Mensura For- nicum..............................................1692 Writers on the Fluxionary or Differential Calculus.

Newton, De Analysis per Æquationes numero terminorum Infinitas, circulated in manuscript in 1669, and printed in the Commercium Epistolicum, 1712. Also contained in a volume edited by W. Jones, and entitled Analysis per Quantitatum Series, Fluxiones, ac Differentias.

— Principia, lib. ii. sect. iii. lemma 2.

— Tractatus de Quadratura Curvarum, published with his Optics.

— The Method of Fluxions.

Leibnitz, Nova Methodus pro maximis et minimis itemque tangentibus. Leipsic Acts.

James Bernoulli, Analysis Magni Problematis Isoperimetrii; Acta Erud. See also his Works.

John Bernoulli, Acta Erud. Leip.; also Mémoires de l'Acad. des Sciences.

Leibnitz and Bernoulli, Commercium Phil. et Math. 2 vols. 4to.

Demeoivre, Use of Fluxions in Geometric Problems, in the Phil. Trans.

De l'Hôpital, Analyse des Infiniment Petits.

David Gregory, On the Catenary, Phil. Trans.

Carré, Méthode pour la Mésure des Surfaces.

Craig, Solid of least Resistance and Curve of quickest Descent, Phil. Trans.

— De Calculo Fluentium.

Cheyne, Fluxionum Methodus Inversa.

Harris, Lexicum Technicum.

Hayes, A Treatise on Fluxions.

Ditton, Institution of Fluxions.

Manfredi, De Construct. Equat. Diff. primi gradus.

Taylor, Methodus Incrementorum.

Cotes, Harmonia Mensurarum.

— Estimatio errorum in mixta Mathesi.

Riccati, Animad. in Equat. differentiales, secundi gradus, Acta Erud.

Clarke, An Institution of Fluxions.

Hermann, De Constructione Equationum Differentialium, Comm. Petrop.

Fontenelle, Éléments de la Géométrie de l'Infini.

Clairaut, Determinatio Curve ejusdem Differentialis, Acta Erud.

Demeoivre, Miscellanea Analytica.

Stone, Method of Fluxions.

Hodgson, The Doctrine of Fluxions.

Didier, Le Calcul Différentiel.

Simpson, A Treatise on Fluxions.

Maclaurin, A Treatise on Fluxions.

Emerson, The Doctrine of Fluxions.

Domenico Agnesi, Instituzioni Analitiche.

French translation of the same work by Cousin.

An English translation by Colson.

Euler, Methodus Inveniendi Lineas Curvas maximi minime proprietate gaudentes.

— Introductio in Analysis Infinitorum.

— Institutiones Calculi Differentialis.

— Institutiones Calculi Integralis.

The titles of Euler's academical memoirs, of which there is a very great number on the Differential Calculus, are given in the edition of his Calculus Differentialis, printed at Pavia in 1787, and occupy thirty-two quarto pages.

Walmealy, Analyse des Mésures des Rapport et des Angles.

Stirling, Methodus Differentialis.

Bougainville, Traité du Calcul Intégral.

Landen, Mathematical Lucubrations.

— The Residual Analysis.

Landen, Mathematical Memoirs.

Saunderson, Method of Fluxions.

Lyons, Treatise of Fluxions.

Kæstner, Separatio Indeterminat. in Æquat. Diff.

D'Alembert, Opuscules Mathématiques.

Robins, Mathematical Tracts.

Waring, Miscellanea Analytica.

— Meditations Analytice.

Condorcet, Du Calcul Intégral.

Le Seur et Jacquier, Éléments du Calcul Intégral.

Lexell, Methodus Integrandi Æq. Diff. Comm. Petrop.

Fontaine, Traité du Calcul Diff. et Intégral.

Giannelli, De Fluxionibus et earum Usu.

Cousin, Leçons du Calcul Différentiel, &c.

La Place, L'Usage du Calcul aux Différences partielles, Mem. de l'Acad.

Condorcet, De Integ. cujusdam Æquationis, Comm. de Bouon.

Malfiti, Delle formale differentiali la cui integrazione dipende dalla rettificazione delle sezioni coniche.

Paoli, Memoria sul' equazione a differenze finite e parziali.

Monge, Sur le Calcul Intégral des Equat. aux Diff. partielles, Mem. de l'Acad.

Charles, Recherches sur le Calcul Intégral, Mem. de l'Acad.

L'Huillier, Exposition des Principes de Calcul Supérieur.

— Principiorum Calculi Diff. et Integ.

Mascheroni, Annotationes ad Cal. Integ. Euleri.

Tabiescen, Principia atque Historia Calculi Diff. et Integ. nec non Methodi Fluxionum.

Lagrange, Calcul des Variations, Mts. Taur. vol. ii.

— Théorie des Fonctions Analytiques.

2d edit.

— Leçons sur le Calcul des Fonctions, 2d edit.

The titles of his Memoirs, inserted in Academical Collections, are given in his Mécanique Analytique, second edition.

Vince, Principles of Fluxions.

Carnot, Réflexions sur la Métaphysique du Calcul Infinitesimal.

Bossut Traité du Calcul Différentiel.

Lacroix, Traité du Calcul Différentiel et du Calcul Intégral, first edit. 1797, second.

Abrogast, Du Calcul des Dérivations.

The preceding catalogue contains the names of the principal improvers and cultivators of the calculus, from its first invention to the end of the last century. However, it does not contain all their writings; to have enumerated these would have extended the catalogue to too great a length. Almost all the improvements of the calculus were first given in the form of academical memoirs, from which they have been drawn, and incorporated into the systems that have from time to time been published. The great repertories in which it is contained are the writings of Newton, Cotes, Demeoivre, the Bernoullis, Maclaurin, Simpson, Euler, D'Alembert, Lagrange, Legendre, La Place, Monge, Poisson, Gauss, Ampère, Ivory, &c. From these have been constructed the treatises of De l'Hôpital, Bougainville, Le Seur and Jacquier, Cousin, and others, which were the most complete at the different periods when they appeared. At this time the most extensive treatise is the second edition of Lacroix, which came out between 1810 and 1819. In the present century there have been published in France treatises by Garnier, Du Bourguet, Bouchariat, Cauchy, and others which we have not seen. In Britain we have had treatises by Woodhouse, Deauly, Lardner, Jephson, Young, Thomson, and others. The university of Cambridge, long infertile, now teems with treatises on the subject, such as those of Brown, Coddington, Earnshaw, Hymer, Jarret, Millar, Myers, and Peacock. There are, besides, treatises on the calculus in courses of mathematics, such as the Elementi d'Algebra of Paoli, the Cours des Mathematiques of Bezout, a like work by Fran- ceur, &c. There have also been works constructed for the benefit of students, which may be used with any treatise on the subject, such as the excellent collection of examples by Peacock, Herschel, and Babbage, which was intended as an accompaniment to an English translation of an abridgment of Lacroix's treatise; and the integral tables of Hirsch, which have been translated from the original German into English.

There is a branch of the calculus of great interest, which of late has engaged, and will probably long engage, the attention of mathematicians; we mean the subject of Elliptic Transcendents. This theory originated in the discovery of a very remarkable property of the ellipse and hyperbola by Fagnano, an Italian mathematician, who first showed that arcs of these curves can be assigned, of which the difference is expressible in finite algebraic terms. This most interesting subject has been discussed by the following writers:

Fagnano, Produzioni Mathematiche, tom. ii. p. 336...1750 Euler, Calculus Integralis, tom. i. sect. ii. and tom. iii. supp. Bossut, Leipsic Acts, 1754; Mém. de Math. et de Phys. tom. iii. Landen, Lond. Phil. Trans. 1775; and Mathematical Lucubrations.........................1780 Legendre, Mém. de l'Acad. des Sciences, 1786; Mémoire sur les Transcendantes Elliptiques, 1792. (There is an English translation of this memoir in Leyburn's Mathematical Repository, vols. ii. and iii. new series.) Exercices de Calcul Integral, 3 vols.; Schumacher's Journal, Nos. 123, 127, 130. Traité des Fonctions Elliptiques..............1825 Miscell. Taur. tom. iv.; Théorie des Fonct. Analytiques, 2d edit. Ivory, New Series for the Rectification of the Ellipse, Trans. R. S. Edin. vol. iv.; Fagnani's Theorem made more general, Leyburn's Math. Repository, vol. i. new series; On the Theory of the Elliptic Transcendents, Trans. R. S. Lond. 1831. Wallace, Formula for the Rectification of an Ellipse, &c., Trans. R. S. Edin. vol. v. Woodhouse, Integration of Certain Differentials, Trans. R. S. Lond. 1804. Brinkley, Demonstration of Fagnani's Theorem, Trans. Irish Acad. vol. ix. Abel, J. Crell's Journal, vol. ii. Jacobi Fundamenta Novae Theorie Functionum Ellipticarum...............................1829 Plana, Memoir read in the Turin Academy............1828

PART I.

DIRECT METHOD OF FLUXIONS, OR DIFFERENTIAL CALCULUS.

1. In the application of algebra to the theory of curve lines, some of the quantities under consideration are conceived as having always the same magnitude as the parameter of a parabola, and the axes of an ellipse or hyperbola; others again are indefinite in respect of magnitude, and may have any number of particular values; such are the co-ordinates at any point in a curve. This difference in the nature of the quantities has equally place in various theories of the pure and mixed mathematics, and it naturally suggests the division of all quantities into two kinds; namely, such as are constant, and such as are variable.

A constant quantity is that which is supposed to have always the same value. A variable quantity, again, is that which may change its value by increasing or decreasing, so that in passing from one degree of magnitude to another it will have had in succession every possible intermediate magnitude.

Thus, in a circle, the radius is a constant quantity, and any arc of the circle, also its cosine, sine, tangent, secant, &c. are variable quantities. In the ellipse, the axes or semiaxes are considered as constant; and the co-ordinates to any point in the curve, and in general any lines or spaces or angles which in the same ellipse admit of different values, are variable. So again in the parabola, the parameter is constant, and the co-ordinates to any point in the curve, also any arc of the curve, and the space contained by that arc and the co-ordinates, are to be regarded as variable.

In what follows, we shall in general call constant quantities simply constants, and variable quantities variables.

It is usual to denote constants by the letters \(a, b, c\), &c. towards the beginning of the alphabet, and variables by the letters \(x, y, z\), &c. towards the end.

2. One quantity is said to be a function of another when they are so related that the latter being supposed to change its value, the former also changes its value. Thus, in geometry, the radius of a circle being constant, the cosine, the sine, the tangent, the secant, &c. are functions of the arc. On the other hand, the arc may be considered as a function of any one of these quantities. In mechanics, the force by which a body is urged forward, the space it has passed over, and its velocity at every instant, may all be regarded as functions of the time.

In this algebraic expression,

\[ y = \frac{a + x}{a - x}, \]

in which \(a\) is a constant and \(x\) a variable, the quantity \(y\) is regarded as a function of \(x\), which is called the independent variable. Here \(y\) is expressed by \(x\); but by resolving the equation, \(x\) may be expressed by \(y\) thus:

\[ x = \frac{a(y - 1)}{y + 1}; \]

the quantity \(x\) is now a function of \(y\). In either case, the constants \(a\) and 1 are not considered. In these examples,

\[ u = a + bx + cx^2, \] \[ u = \sqrt{(a^2 + bx + x^2)}, \] \[ u = \frac{a + bx}{c + x^2}, \]

\(u\) is a function of \(x\).

A quantity may be a function of several independent variable quantities. Thus, in the expression

\[ u = ax^2 + bxy + cy^2, \]

\(u\) is a function of the independent variables \(x\) and \(y\).

The variety of forms of functions of a variable is endless. They may, however, be resolved into a few elementary functions such as these,

\[ x^n, \quad a^x, \quad \log x, \quad \sin x, \quad \cos x. \]

The first of these, \(x^n\), in which \(n\) is constant, and all that can be formed from it by a finite number of the eleme- tary operations of algebra, are called algebraic functions; for example, this,

\[ u = ax^3 + bx^2 - cx \]

which is formed by addition, subtraction, multiplication, division, and the extraction of the square root. These two, \(a^x\), log \(x\), which cannot be expressed by a finite number of terms composed of powers of \(x\), are called transcendental functions, and the remaining two, sin \(x\), cos \(x\), and such as may be formed from them, viz. tan \(x\), sec \(x\), &c., are called trigonometrical, also angular and circular functions.

In the last example, the value of the function \(u\) may be immediately found, if the value of \(x\) be known. Such functions are called explicit. There are functions, however, which require to be separated from the variable by the resolution of equations, or other means, before their value can be found; as in this example,

\[ u = \frac{u + x}{u - x}, \]

in which \(u\) is supposed to be a function of \(x\). In this case, before the value of \(u\), corresponding to a given value of \(x\), can be found, a quadratic equation must be resolved. Such a function is called an implicit function of \(x\). On the other hand, \(x\) is an implicit function of \(u\). By the resolution of an equation we find

\[ u = \frac{1}{2} \left( x + \frac{\sqrt{x^4 + 6x^2 + 1}}{x} \right); \]

\(u\) is now an explicit function of \(x\).

The early algebraists indicated the square root of a quantity by placing the letter \(r\) before it; thus \(rx\) meant the square root of \(x\). The letter was afterwards changed into the sign \(\sqrt{\cdot}\), and the square root of \(x\) was conveniently expressed thus, \(\sqrt{x}\). In like manner, it is convenient to have a symbol which shall express generally a function of a variable, and accordingly it is now usual to express any function of a variable \(x\) by the symbol \(fx\) or \(f(x)\); also by \(Fx\) or \(fx\). Here the letters \(f\), or \(F\), or \(f\), are considered merely as the abbreviation of the word function, and not as a co-efficient to the letter \(x\). In this way we include all these expressions,

\[ u = x^n, \quad u = a^x, \quad u = \log x, \quad u = \cos x, \quad u = \sin x, \]

in one general expression, viz. \(u = fx\) or \(u = f(x)\).

When a function depends on two variable quantities \(x\), \(y\), which are independent of each other, such as this expression,

\[ u = ax^3 + bxy + cy^3, \]

we shall express it thus,

\[ u = f(x, y). \]

Here we take no notice of the constant quantity \(a, b, c\). A like notation may be applied when three independent variables enter a function.

3. The manner in which we propose to treat this subject requires that the reader should distinctly understand what is meant by a limit to the value of a variable function, or a variable ratio. In the elements of geometry it is shown that the area of a circle is less than that of any regular polygon described about it; and that the greater the number of sides of the polygon, the more nearly is its area equal to that of the circle; so that any space being given, however small, a polygon may be described about the circle, which shall differ from it by less than that space.

Suppose now a series of polygons to be described about a circle, each having the number of its sides greater than that which preceded it, the double for example. These will approach continually to an equality with the circle as a limit, which the series, however far continued, can never absolutely reach in respect of magnitude, but from which some term, and all that follow it, may differ by less than any space assignable.

In like manner, the circle is the limit of the area of all regular polygons inscribed in it; and the circumference of the circle is the limit of the perimeters of regular polygons described in it or about it.

In our article Algebra (sect. 266) it has been shown, that, putting \(a\) for an arc of a circle, the fraction \(\frac{a}{\sin a}\) is always greater than an unit, but that \(\frac{a}{\tan a}\) is always less; farther, that the arc being supposed to decrease continually, these fractions approach to each other, and may differ by less than any assignable quantity; therefore unity, which is always between them, is their common limit. Hence it appears, that of these three quantities, an arc, its sine, and its tangent, the ratio of any two is that of equality. In the same article it has been shown that the arc \(a\) is the limit of the fractions \(n \tan \frac{a}{n}\) and \(n \sin \frac{a}{n}\), supposing the number \(n\) to be increased continually.

The geometrical series

\[ 1 + x + x^2 + x^3 + \ldots + x^{n-1} \]

is equivalent to the function \(\frac{1-x^n}{1-x}\) (Algebra, sect. 56); when \(x\) is less than 1, and \(n\) is infinitely great, then \(x^n = 0\), and the function \(= \frac{1}{1-x}\); therefore, in this case, the sum of any finite number of terms of the series will always be less than \(\frac{1}{1-x}\), but will approach to it as the number of terms increases, and may differ from it by less than any assignable quantity; hence it follows that the function \(\frac{1}{1-x}\) is the limit of the sum of the series.

By the binomial theorem

\[ \left(1 + \frac{x}{n}\right)^n = 1 + \frac{x}{1} + \frac{x^2}{1 \cdot 2} \left(1 - \frac{1}{n}\right) + \frac{x^3}{1 \cdot 2 \cdot 3} \left(1 - \frac{1}{n}\right) \left(1 - \frac{2}{n}\right) + \ldots + \frac{x^n}{1 \cdot 2 \cdot 3 \cdot \ldots \cdot n} + \ldots \]

This is true, whatever be the numerical value of \(n\); but suppose \(n\) to be a large number, then the factors \(1 - \frac{1}{n}, 1 - \frac{2}{n}, \ldots\) will differ but little from unity; and they will differ the less as \(n\) is greater. Suppose now \(n\) to be indefinitely great, then these factors may be accounted each equal to an unit; and, therefore,

\[ \left(1 + \frac{x}{n}\right)^n = 1 + \frac{x}{1} + \frac{x^2}{1 \cdot 2} + \frac{x^3}{1 \cdot 2 \cdot 3} + \ldots + \frac{x^n}{1 \cdot 2 \cdot 3 \cdot \ldots \cdot n} + \ldots \]

It has been shown in algebra (sect. 177) that the second member of this equation is the development of the function \(e^x\), where \(e\) denotes the number 2·7182818, viz. the base of Napier's logarithms; therefore, if \(n\) be supposed to increase continually, the expression \(\left(1 + \frac{x}{n}\right)^n\) will approach to \(e^x\) as its limit; and making \(x = 1\), the limit of \(\left(1 + \frac{1}{n}\right)^n = e = 2·7182818\), a constant quantity.

4. Since the value of a function depends essentially on that of the variable, which is its basis, any change in the value of the latter will produce a corresponding change in the value of the former. First, let the function be \( u = x^2 \), and let us suppose that the variable changes its value, being increased by a quantity \( h \), and thereby becoming \( x + h \); let \( u' \) denote the corresponding value of \( u \), so that we have

\[ u' = (x + h)^2 = x^2 + 2xh + h^2 = u + 2xh + h^2; \]

hence \( u' - u = 2xh + h^2 \).

It thus appears that, \( h \) denoting the increment of the variable \( x \), the corresponding increment of the function, viz.

\[ u' - u = 2xh + h^2. \]

The algebraic expression for a ratio is a fraction whose numerator is the antecedent, and denominator the consequent. In the present case, the ratio of the increment of the function to that of its variable is

\[ \frac{u' - u}{h} = 2x + h. \]

As another example, let the function be \( u = x^3 \); then, supposing, as before, that the variable \( x \) is increased by the quantity \( h \), and, putting \( u' \) for the new value of the function, we have

\[ u' = (x + h)^3 = x^3 + 3x^2h + 3xh^2 + h^3; \]

\[ \frac{u' - u}{h} = 3x^2 + 3xh + h^2. \]

In this case we see that when \( x \) has increased to \( x + h \), the function \( u \) has become

\[ u + 2xh + 3xh^2 + h^3, \]

having received the increment \( 3x^2h + 3xh^2 + h^3 \), a quantity composed of the integer powers of \( h \). Farther, it appears that the expression for the ratio of the increments is

\[ 3x^2 + 3xh + h^2, \]

of which the first term \( 3x^2 \) is entirely independent of the value of \( h \), the increment of \( x \).

When the function is \( u = x^n \), we have, putting, as in the former examples, \( h \) for the increment of the variable, and \( u' \) for the new value of the function,

\[ u' = u + nx^{n-1}h + nx^{n-2}h^2 + \cdots + h^n; \]

\[ \frac{u' - u}{h} = nx^{n-1} + nx^{n-2}h + \cdots + h^{n-1}. \]

From these examples it appears, that when \( x \) changes its value and becomes \( x + h \), then the new value of the function

\[ u = x^n \text{ becomes } u' = u + nx^{n-1}h + nx^{n-2}h^2 + \cdots + h^n; \]

and in general that

\[ u = x^n \text{ becomes } u' = u + nx^{n-1}h + nx^{n-2}h^2 + \cdots + h^n; \]

the new value of the function being composed of a series of terms, the first of which is its original value, and the following terms, integer powers of the increment \( h \), multiplied by \( p, q, r, \ldots \), certain other functions of \( x \), the forms of which depend on the original function, whence they have been derived.

Farther, it appears that

\[ \text{when } u = x^2, \text{ then } \frac{u' - u}{h} = 2x + h; \]

\[ \text{when } u = x^3, \text{ then } \frac{u' - u}{h} = 3x^2 + 3xh + h^2; \]

\[ \text{when } u = x^4, \text{ then } \frac{u' - u}{h} = 4x^3 + 6x^2h + 4xh^2 + h^3; \]

and, in general, that when \( x = u^n \), \( n \) being any whole number, then

\[ \frac{u' - u}{h} = p + qh + rh^2 + sh^3 + \cdots. \]

Thus it appears that the expression for \( \frac{u' - u}{h} \), the ratio of the increment of the function \( x^n \) to that of its variable \( x \), may be resolved into two parts, one, \( p \), which is independent of the increment, and another, \( qh + rh^2 + sh^3 + \cdots \), or

\[ h(q + rh + sh^2 + \cdots), \]

which, having \( h \) the increment as a factor, must decrease with it, and, by giving a sufficiently small value to \( h \), may become less than any assignable quantity, so that the first term \( p \) is the limit of the ratio.

Let us consider the complex function

\[ u = a + bx + cx^2. \]

When \( x \) becomes \( x + h \), \( u \) becomes

\[ u' = a + b(x + h) + c(x + h)^2 = a + bx + cx^2 + (b + 2cx)h + ch^2, \]

hence \( u' - u = (b + 2cx)h + ch^2 \),

\[ \frac{u' - u}{h} = b + 2cx + ch. \]

In this case the limit of the ratio is \( b + 2cx \). By a like examination of particular functions it will be found that they have all a common property, that is,

If \( u \) be any function of a variable \( x \), and if in that function \( x \) be supposed to change its value and become \( x + h \); the corresponding new value of the function will be

\[ u' = u + ph + qh^2 + rh^3 + \cdots, \]

and the ratio of the increments of the function and its variable,

\[ \frac{u' - u}{u} = p + qh + rh^2 + \cdots, \]

an expression which, when \( h \) is supposed to decrease continually, has for its limit the first term \( p \); and this limit will be different for different functions, there being such a connection between the function and its limit, that the one may be found from the other.

This property of the increments of a function and its variable suggests an important and extensive analytical theory, which will consist of two parts.

I. Any function of a variable quantity being proposed, to determine the limit of the ratio of the corresponding increments of the function and its variable.

II. On the other hand, having given the ratio of the increments, to find the function from which it has been derived.

These two inquiries constitute at bottom the Direct and Inverse Method of Fluxions of Newton, the Differential and Integral Calculus of Leibnitz, and the Theory of Functions of Lagrange. Our present subject is the direct method of fluxions, or its equivalent, the differential calculus.

5. We have seen (4), that by the transition of a function, for example, \( u = x^2 \), from one state of magnitude to another, by a change in the magnitude of its variable \( x \), it acquires the new value \( u' = u + 2xh + h^2 \). The whole difference \( u' - u \) between its new and first values is \( 2xh + h^2 \). It is the first term of this difference, viz. \( 2xh \), that constitutes what is called the differential of the function from which it has been derived; and the determination of this first term for any function is the object of this first part of our calculus.

In like manner, in the function \( u = x^3 \), which, when \( x \) becomes \( x + h \), changes to \( u' = u + 3x^2h + 3xh^2 + h^3 \), the whole difference between the two states of magnitude of the function is \( 3x^2h + 3xh^2 + h^3 \), and its differential is \( 3x^2h \). And the whole difference between the first and succeeding value of the function \( u = x^4 \) is \( 4x^3h + 6x^2h^2 + 4xh^3 + h^4 \), and its differential, is \( 4x^3h \). And, in general, whatever be the nature of the function \( u \), if \( x + h \) be substituted in it instead of \( x \), and the expression thus formed be expanded into a series,

\[ u + ph + qh^2 + rh^3 + sh^4 + \cdots, \] the complete difference between the two states of the function is

\[ ph + qh^2 + rh^3 + sh^4 + \ldots \]

and its differential is the first term \( ph \).

In conformity with this definition, the differential of the variable \( x \) itself will be its increment \( h \), which, because of its use in generating the differential of the function, has been designated by a peculiar symbol, viz. \( dx \); here the letter \( d \) is to be understood as a characteristic, not as a co-efficient. The letter \( d \) is also prefixed to the symbol for a function to denote its differential. Thus, supposing \( u = x^2 \), we have \( du = 2x dx \), an expression which means that the differential of the function \( u \) is equal to the differential of the variable multiplied by \( 2x \) as a co-efficient; so also, if \( u = x^3 \), the equation \( du = 3x^2 dx \) means that the differential of \( u \) is equal to the differential of \( x \) multiplied by \( 3x^2 \).

6. Since, when \( u = x^2 \), then \( du = 2x dx \), this last expression may otherwise stand thus, \( \frac{du}{dx} = 2x \); under this form \( 2x \) is the co-efficient in the expression for the differential of \( u \), and it is on that account called the differential co-efficient of the function \( u \) or \( x^2 \). In like manner, when \( u = x^3 \), then \( du = 3x^2 dx \) and \( \frac{du}{dx} = 3x^2 \); in this case \( 3x^2 \) is the differential co-efficient of the function \( u = x^3 \). The general symbol for the differential co-efficient of any function \( u \) is \( \frac{du}{dx} \), and the new value of \( u \) being \( u + ph + qh^2 + \ldots \), we have \( \frac{du}{dx} = p \), where \( p \) is some function of \( x \), which depends on the form of the function \( u \), and is deducible from it when its form is known.

To denote the differential of an expression formed in any way, we prefix the symbol \( d \) to the expression. Thus \( d[(a + x)(b - x^2)] \) denotes the differential of a function produced by multiplying the factors \( a + x \) and \( b - x^2 \), and \( d[(a + x)(b - x^2)] \) serves to denote its differential co-efficient.

The definition which has been given of a differential suggests immediately this rule for finding the differential of a function of a single variable.

Substitute \( x + h \) in the function instead of \( x \); expand the expression thus formed into a series of terms composed of integer powers of \( h \), and take the term which contains the first power of \( h \) for the differential, exchanging the letter \( h \) for the symbol \( dx \).

Thus, let the function be \( u = ax + bx^2 \), in which \( a \) and \( b \) are constants, then, putting \( x + h \) for \( x \), and \( u' \) for the new value of \( u \), we have

\[ \begin{align*} u' &= a(x + h) + b(x + h)^2 \\ &= ax + bx^2 + (a + 2bx)h + bh^2 \\ &= u + (a + 2bx)h + bh^2. \end{align*} \]

Here \( du \), the differential of the function, is \((a + 2bx)h\), or \((a + 2bx)dx\), and \( \frac{du}{dx} = a + 2bx \) the differential co-efficient.

7. The rule given in last article, for determining the differential of a function, supposes that we have methods by which any function may be expanded into a series of terms, into each of which some integer power of the increment \( h \) enters only as a factor. It is sufficient for our purpose, however, if we can find that term of the series which contains the simple power of \( h \); and this can in general be found easier than the general development, which indeed may be the very thing to be investigated by the calculus. Now we have seen (4) that the co-efficient of this term is the expression for the limit of the ratio of the corresponding increments of a function and its variable, and the same quantity is the differential co-efficient of the function (6), therefore the determination of the differential of a function is the same thing as the determination of the limit of the ratio of the increments.

Let the function be \( u = \frac{a^x}{x} \); then \( u' = \frac{a^x}{x + h} \), and

\[ \begin{align*} u' - u &= \frac{a^x}{x + h} - \frac{a^x}{x} = \frac{-a^xh}{x(x + h)} \\ \text{and the limit of the ratio of the increments is} \\ \frac{u' - u}{h} &= \frac{-a^x}{x(x + h)}. \end{align*} \]

Now, without expanding the expression for the limit, it is obvious that as \( h \) decreases, that expression approaches to \( -\frac{a^x}{x^2} \), which is therefore \( \frac{du}{dx} \), the differential co-efficient of the function; hence \( du = -\frac{a^x dx}{x^2} \). In like manner, \( u \) being the general expression for any function, and \( u' = u + ph + qh^2 + \ldots \), its expanded value when \( x + h \) is put instead of \( x \), we have

\[ \begin{align*} \frac{u' - u}{h} &= p + qh + \ldots \\ \text{and limit of} \frac{u' - u}{h} &= p = \frac{du}{dx} \text{ and } du = pdx. \end{align*} \]

According to this view of the subject, the differential calculus is the finding of the ratios of the simultaneous increments of a function and of the variable on which it depends.

The process of calculation by which the differential of any function \( u \) is found, may be regarded as a particular operation performed on quantity, analogous to the elementary operations of algebra; and it may, like these, be distinguished by a particular name accordingly. The result of the process being the differential, the process itself is called differentiation, and to perform the process on a function is to differentiate the function.

8. It is material to observe, that the limit of the ratio of the increments of a function and its variable are the very same, whichever of the two be considered as a function of the other. Thus, \( u \) being a function of \( x \), when \( x \) becomes \( x + h \), \( u \) becomes \( u + ph + qh^2 + rh^3 + \ldots \), \( p, q, r, \ldots \) being functions of \( x \), deducible from the function \( u \).

Let us put

\[ k = ph + qh^2 + rh^3 + \ldots \]

so that the contemporaneous increments of \( x \) and \( u \) are \( h \) and \( k \). Now, from the value of \( k \), by the reversion of series (Algebra, art. 163), we find

\[ h = \frac{1}{p}k - \frac{q}{p^2}k^2 + \ldots \]

So that, regarding \( x \) as a function of \( u \), when \( u \) becomes \( u + h \), \( x \) becomes

\[ \frac{1}{p}k - \frac{q}{p^2}k^2 + \ldots \]

and the general expression for the ratio of the increments is

\[ \frac{h}{k} = \frac{1}{p} - \frac{q}{p^2}k + \ldots \]

which has for its limit \( \frac{1}{p} \), and the limit of the ratio \( \frac{k}{h} \) is, as before, \( = p \).

9. It is easy to see that two equal functions must have equal differentials; for whatever be the value of the va- Direct Method.

Variable on which they depend, it must necessarily happen that the respective changes they receive in consequence of the change which is attributed to the variable must also be equal. Thus, if \( u \) and \( v \) be two functions, such that \( u = v \), whatever may be the value of \( x \); and if, when \( x \) becomes \( x + h \), then \( u \) becomes \( u' \), and \( v \) becomes \( v' \), we shall have \( u' - u = v' - v \), and

\[ \frac{u' - u}{h} = \frac{v' - v}{h}. \]

If then \( p \) and \( q \) denote the limits of these ratios, \( p = q \) and \( pdx = qdx \), that is, \( du = dv \).

From this it follows, that under whatever form a function appears, its differential is the very same quantity. For example, the differential of \( x^2 + ax + a^2 \) will be identical with the differential of \( (x + a)(x - ax + a^2) \), its equal.

The converse of this proposition is not generally true, and we should be wrong in affirming that two equal differentials belong to equal functions. For, let \( u = a + bx \) be a function of \( x \), then, substituting \( x + h \) for \( x \), and putting \( u' \) for the new value of \( u \), we have \( u' = a + bx + bh = u + bh \), and

\[ \frac{u' - u}{h} = b. \]

We see here that the constant \( a \) does not enter into the limit of the ratio of the increments, which would therefore be the very same for the function \( u = bx \); hence it follows that the differential \( bdx \) belongs alike to \( a + bx \), and to \( bx \). Thus it appears, that in differentiating any function whatever, all the constant quantities combined with it, either by addition or subtraction, disappear. With respect to those which are connected by multiplication and division, they enter the result as co-efficients.

10. The differential of any function of a variable \( x \) may be found by the general methods indicated in article 6; but it is convenient to have rules adapted to particular cases.

Let \( r \) and \( s \) be two functions of a variable \( x \). It is proposed to investigate a rule for finding the differential of \( u = rs \), their product.

Suppose that by the substitution of \( x + h \) for \( x \) in the functions \( r \) and \( s \), and their expansion, they become

\[ r' = r + pr + qh + \ldots, \quad s' = s + ps + qh + \ldots, \]

In these expressions, \( p, q, \ldots \) represent functions of \( x \), derived from \( r \); and \( p', q', \ldots \) other functions of \( x \) derived from \( s \). Corresponding to these let \( u' \) denote \( r's' \), the new value of the product \( rs = u \). By actual multiplication we find

\[ u' = r's' = rs + (rp' + sp)h + (rq + pp')h^2 + \ldots. \]

Hence, putting \( u \) for \( rs \), also transposing and dividing by \( h \), there is got

\[ \frac{u' - u}{h} = rp' + sp + (rq' + pp' + sq)h + \ldots. \]

The terms \( rp' \) and \( sp \) in the second member of this equation are functions of \( x \), which are independent of the function \( h \); the following terms are all multiplied by \( h \), therefore they decrease and vanish with it. So that

\[ \text{limit of ratio } \frac{u' - u}{h} = rp' + sp. \]

Now \( p = \text{limit of } \frac{r' - r}{h} \), and \( q = \text{limit of } \frac{s' - s}{h} \).

Instead of the limits of the ratios, let us put the differential co-efficients of the functions \( u, r, s \) (art. 6), and we find

\[ \frac{du}{dx} = r \frac{ds}{dx} + s \frac{dr}{dx}, \]

and \( du = rds + sdr \).

Hence we have this rule,

To find the differential of the product of two functions, multiply the differential of each by the differential of the other function, and add the products.

11. Since, when \( u = rs \), we have \( du = sdr + rds \), it follows that

\[ \frac{du}{u} = \frac{dr}{r} + \frac{ds}{s}. \]

If we suppose \( u = rte \), the product of three factors, by putting \( s = te \), we have \( u = rs \), and

\[ \frac{du}{u} = \frac{dr}{r} + \frac{ds}{s}; \]

but since \( s = te \), we have for the same reason

\[ \frac{ds}{s} = \frac{dt}{t} + \frac{dv}{v}; \]

therefore \( \frac{du}{u} = \frac{dr}{r} + \frac{dt}{t} + \frac{dv}{v} \).

By writing the product \( rst \) instead of \( u \), there is got, after proper reduction,

\[ du = trds + rdt + rtdv. \]

In general, If a function be the product of any number of functions of a variable, its differential is the sum of the products obtained by multiplying the differential of each by the product of all the other functions.

The same rule may also be briefly expressed in symbols, thus: whatever be the number of functions \( r, s, t, v \),

\[ d(rstv) = rstv \left\{ \frac{dr}{r} + \frac{ds}{s} + \frac{dt}{t} + \frac{dv}{v} \right\}. \]

12. To find the differential of a fraction whose numerator and denominator are functions of a variable \( x \); let \( u = \frac{r}{s} \), then \( r = us \), and

\[ dr = uds + sdu, \]

and, putting for \( u \) its value \( \frac{r}{s} \),

\[ dr = \frac{rds}{s} + sdu; \]

therefore, \( du = \frac{sdr - rds}{s^2} \).

Hence we have this rule: To find the differential of a fraction; from the differential of the numerator multiplied by the denominator subtract the differential of the denominator multiplied by the numerator, and divide by the square of the denominator.

The rule for the differential of \( u = \frac{r}{s} \) may be also symmetrically expressed thus:

\[ \frac{du}{u} = \frac{dr}{r} - \frac{ds}{s}; \]

and in general, if \( u = \frac{rst}{vy} \), then

\[ \frac{du}{u} = \frac{dr}{r} + \frac{ds}{s} + \frac{dt}{t} - \frac{dv}{v} - \frac{dy}{y}. \]

For, by what has been shown, it appears that when \( u = \frac{rst}{vy} \), then

\[ \frac{du}{u} = d(rst) - d(vy). \]

Now \( d(rst) = \frac{dr}{r} + \frac{ds}{s} + \frac{dt}{t} \),

and \( d(vy) = \frac{dv}{v} + \frac{dy}{y} \); therefore, \( \frac{du}{u} = \frac{dr}{r} + \frac{ds}{s} + \frac{dt}{t} - \frac{dv}{v} - \frac{dy}{y}. \)

It also follows, from what has been shown, that if the numerator of a fraction consist of any number of factors \( r, s, t, \) and the denominator of any number \( v, y, \) then

\[ d\left( \frac{rst}{vy} \right) = \frac{rst}{vy} \left( \frac{dr}{r} + \frac{ds}{s} + \frac{dt}{t} - \frac{dv}{v} - \frac{dy}{y} \right). \]

13. We shall next investigate a rule for the differential of any power of a function \( y, \) which may be either itself an independent variable, or else some function of another variable.

First, let \( u = y^n, \) \( n \) being any whole number. The function may be put under this form,

\[ u = y \cdot y \cdot y \cdot \ldots \text{to } n \text{ terms}, \]

and then (11) we have

\[ \frac{du}{u} = \frac{dy}{y} + \frac{dy}{y} + \frac{dy}{y} + \ldots \text{to } n \text{ terms}; \]

that is, \( \frac{du}{u} = ndy/y, \)

and \( du = ndy/y = ny^{n-1} dy. \)

Next, let us suppose that the function has a fractional exponent, and that \( u = y^m, \) then \( u^n = y^m. \)

and \( nu^{n-1} du = my^{m-1} dy, \) by art. 9,

and \( du = \frac{m}{n} y^{m-1} dy; \)

but \( u = y^m \) and \( u^{n-1} = y^{m-n}; \)

therefore \( y^{m-1} = y^{m-n}, \)

and \( du = \frac{m}{n} y^{m-n}. \)

Lastly, let us suppose \( n \) to be a negative whole number or fraction, so that \( u = \frac{1}{y^n}. \) In this case we may apply the rule for a fraction (12); and observing that the numerator here is a constant, of which the differential must be accounted \( = 0, \) we have

\[ du = -ny^{n-1} dy = -ny^{n-1} dy; \]

thus it appears, that whether \( n \) be whole or fractional, positive or negative,

\[ d(y^n) = ny^{n-1} dy. \]

Hence this rule: To differentiate any power of a function, multiply it by the exponent, diminish the exponent by an unit, and multiply by the differential of the variable.

If the function have a co-efficient, the differential must be multiplied by that co-efficient. Thus

\[ d(ax^n) = anx^{n-1} dx. \]

The determination of the differential of a power of a variable might have been shortened by assuming the truth of the binomial theorem, as found in our article ALGEBRA, art. 160, 161. That theorem however may be derived from the differential calculus, and we shall give its investigation as one of its applications.

It frequently happens that the differential of \( \sqrt{y} = y^{1/2}, \) the square root of a function, is to be found; therefore a rule for that case will be convenient. By the general rule for a power,

\[ d(y^{1/2}) = \frac{1}{2} y^{-1/2} dy, \]

that is, \( d(\sqrt{y}) = \frac{dy}{2\sqrt{y}}. \)

Hence it appears that the differential of the square root of a function is the differential of the function divided by twice its square root.

14. Let \( y \) be a function of a variable \( x, \) and let \( u \) be a function of \( y; \) it is proposed to investigate a rule for finding the differential of \( u \) relatively to \( x. \)

Suppose \( x \) to change its value and become \( x + h, \) then \( y \) becomes \( y + ph + qh^2 + rh^3 + \ldots, \) &c.; or, putting \( ph + qh^2 + rh^3 + \ldots, \) &c., \( = k, \) \( y \) becomes \( y + k. \) But \( u \) being a function of \( y, \) when \( y \) becomes \( y + k, \) then \( u \) becomes \( u' = u + p'h + q'h^2 + r'h^3 + \ldots, \) &c.; here \( p', q', r', \ldots \) denote certain functions of \( y, \) which are independent of \( k \) and also of \( h. \) Hence, by substituting for \( k \) in this last series, its value \( ph + qh^2 + \ldots, \) &c., it appears that when \( x \) becomes \( x + h, \) \( u \) becomes

\[ u' = u + p'h + (p'q + q'p')h^2 + \ldots, \] &c.

and hence \( \frac{u' - u}{h} = p'h + (p'q + q'p')h^2 + \ldots, \) &c.

Suppose now \( h \) to decrease continually, we have

\[ \lim_{h \to 0} \frac{u' - u}{h} = p'h. \]

Now, \( y \) being a function of \( x, \) \( p = \frac{dy}{dx}, \) and \( u \) being considered as a function of \( y, \) \( p' = \frac{du}{dy}; \) but if we consider \( u \) as a function of \( x, \) then

\[ \frac{du}{dx} = \lim_{h \to 0} \frac{u' - u}{h} = p'h. \]

Hence it follows that

\[ \frac{du}{dx} = \frac{du}{dy} \times \frac{dy}{dx}, \]

and \( du = \left( \frac{du}{dy} \times \frac{dy}{dx} \right) dx. \)

Hence this rule: To find, relatively to \( x, \) the differential of \( u, \) a function of \( y, \) the quantity \( y \) being a function of \( x, \) Find the differential co-efficient of \( u \) considered as a function of \( y \) only, and the differential co-efficient of \( y \) considered as a function of \( x; \) multiply the product of these co-efficients by the differential of \( x, \) and the result is the differential of \( u. \)

It has been found that

\[ \frac{du}{dy} \times \frac{dy}{dx} = \frac{du}{dx}; \]

Now if we suppose that \( u = x, \) then this expression becomes

\[ \frac{dx}{dy} \times \frac{dy}{dx} = \frac{dx}{dx} = 1; \]

and hence it appears that

\[ \frac{dx}{dy} = \frac{1}{\frac{dy}{dx}}. \]

This shows that the differential co-efficient of \( x \) considered as a function of \( y, \) is the reciprocal of the differential co-efficient of \( y \) considered as a function of \( x; \) a conclusion which may also be deduced from art. 8.

15. Let \( v, y, z \) be functions of a variable \( x, \) and let it be proposed to find the differential of

\[ u = a + bx + cy - ez, \]

where \( a, b, c, e, \) denote constants.

Let us suppose that when \( x \) becomes \( x + h, \) then

\[ v \text{ becomes } v + ph + qh^2 + \ldots, \] &c.

\[ y \text{ becomes } y + p'h + q'h^2 + \ldots, \] &c.

\[ z \text{ becomes } z + p'h + q'h^2 + \ldots, \] &c. Let \( u' \) be the corresponding value of \( u \), so that

\[ u = \frac{a + bx + cy - ez}{h} \]

Hence, putting for the first term of the second member of the equation its value \( u \), and transposing and dividing by \( h \), we get

\[ \frac{u' - u}{h} = bp + cp' - ep' + (bp + cp' - ep')h, \text{ &c.} \]

And passing to the limits, observing that the limit of \( \frac{u' - u}{h} \) is the differential co-efficient of \( u \), also that \( p, p' \), and \( p'' \) are the differential co-efficients of \( v, y, \) and \( z \) respectively, we get

\[ du = bdv + cdy - edz, \]

and \( du = bdv + cdy - edz \).

From this it appears that the differential of a function made up of others simply by addition and subtraction is composed in like manner of the differentials of the several terms, each with the sign of the function from which it was derived; the differential of a constant being reckoned \( = 0 \).

16. These rules are sufficient for the differentiation of any explicit algebraic function, and we shall now give examples.

1. Let \( u = ax^5 \); this is a particular case of the general function \( u = ax^n \); therefore, by the rule (13) \( du = 5ax^4dx \).

2. Let \( u = \frac{a}{x^5} = ax^{-5} \); then \( du = -5ax^{-6}dx = -\frac{5adx}{x^6} \).

3. Let \( u = x^{\frac{3}{2}} = x^{1.5} \); in this case \( du = \frac{3}{2}x^{0.5}dx = \frac{3}{2}dx\sqrt{x} \).

4. Let \( u = ax^2 + bx + c + e \);

then \( du = 3ax^2dx + 2bdx + edx \),

or \( du = (3ax^2 + 2bx + c)dx \).

Here the constant \( e \), which is a term of the function, has disappeared by the differentiation.

5. Let \( u = (a + bx^n)^p \); we may put \( y = a + bx^n \), and then \( u = y^p \); and \( du = py^{p-1}dy \).

Now \( y^{p-1} = (a + bx^n)^{p-1} \), and \( dy = bx^{n-1}dx \);

therefore \( du = p(a + bx^n)^{p-1}x^{n-1}dx \).

We might have dispensed with using the symbol \( y \), and regarded \( a + bx^n \) as a single function, and \( u \) as a power of that function, and found the differential by the rule of art. 13.

As an example of the rule of art. 10, for the product of two functions,

6. Let \( u = x^3(a + x)^2 \);

put \( p = x^3 \), and \( q = (a + x)^2 \); then \( u = pq \),

and \( du = qdp + pdq \).

Now \( dp = 3x^2dx \), and \( dq = 2(a + x)dx \);

therefore \( du = 3x^2(a + x)^2dx + 2x^3(a + x)dx \)

\( = x^2(a + x)(3a + 5x)dx \).

In practice, the introduction of the symbols \( p \) and \( q \) may be omitted.

7. Let \( u = x(1 + x)(1 + x^2) \). In this case, \( u \) is the product of three functions; therefore, by the rule of art. 11,

\( du = (1 + x)(1 + x^2)dx + x(1 + x^2)dx + 2x^2(1 + x)dx \).

This, when abbreviated by multiplication, becomes

\( du = (1 + 2x + 3x^2 + 4x^3)dx \).

We might have proceeded otherwise by the rule

\[ d(xz) = rst \left\{ \frac{dr}{r} + \frac{ds}{s} + \frac{dt}{t} \right\}; \]

accordingly, we would have had

\[ du = x(1 + x)(1 + x^2) \left\{ \frac{dx}{x} + \frac{dx}{1 + x} + \frac{2xdx}{1 + x^2} \right\}, \]

an expression reducible to the former.

8. As an example of a fractional function, let \( u = \frac{x}{1 + x^2} \);

we have, following the rule of art. 12,

\[ du = \frac{(1 + x^2)dx - 2x^2dx}{(1 + x^2)^2} = \frac{(1 - x^2)dx}{(1 + x^2)^2}. \]

9. Again, let \( u = \frac{x^2 + x}{x^4 - x^2 + 1} = \frac{x(x^2 + 1)}{x^4 - x^2 + 1} \); this may exemplify the rule

\[ d\left(\frac{rs}{t}\right) = \frac{rs}{t} \left\{ \frac{dr}{r} + \frac{ds}{s} - \frac{dt}{t} \right\}. \]

Accordingly, making \( x = r, x^2 + 1 = s, x^4 - x^2 + 1 = t \), we have

\[ du = \frac{x^2 + x}{x^4 - x^2 + 1} \left\{ \frac{dx}{x} + \frac{2xdx}{1 + x^2} - \frac{(4x^3 - 2x)dx}{x^4 - x^2 + 1} \right\}. \]

This by reduction becomes

\[ du = \frac{-\{x^6 + 4x^4 - 4x^2 - 1\}dx}{(x^4 - x^2 + 1)^2}. \]

10. As an example of the rule in art. 14, let \( u = 3y^2 \),

and \( y = x^3 + ax \); then

\[ du = 6y, \quad dy = 3x^2 + a; \]

\[ \frac{du}{dy} \cdot \frac{dy}{dx} = 6y(3x^2 + a) = 18x^2y + 6ay, \]

and \( du = \frac{du}{dy} \cdot \frac{dy}{dx} \cdot dx = 18x^2ydx + 6aydx \).

We may in such an example proceed otherwise, thus;

because \( u = 3y^2, \quad du = 6ydy \);

and since \( y = x^3 + ax, \quad dy = 3x^2 + adx \),

substitute this value of \( dy \) in the expression for \( du \), and it becomes

\[ du = 18x^2y + 6aydx, \]

the same as before.

17. The subjects which are to follow require the application of the binomial theorem, we shall therefore give its investigation, as the first application of the calculus.

Supposing \( n \) to be any number whatever, whole or fractional, positive or negative, it is easy to infer, from the consideration of particular cases, that

\[ (1 + x)^n = 1 + Ax + Bx^2 + Cx^3 + Dx^4 + \text{&c.} \]

the co-efficients \( A, B, C, \) &c., in the series being numbers altogether independent of \( x \), and deducible from the index \( n \) alone. The expression \( (1 + x)^n \) and its development being equal for all values of \( x \), they form an identical equation, and must have equal differentials, hence (art. 9) we have, taking the differential of each term,

\[ n(1 + x)^{n-1}dx = Adx + 2Bxdx + 3Cxdx + 4Dxdx + \text{&c.} \]

Leaving now \( dx \) out of each term, and multiplying both sides by \( 1 + x \), the result is

\[ n(1 + x)^n = A + (A + 2B)x + (2B + 3C)x^2 + (3C + 4D)x^3 + \text{&c.} \]

But from the assumed series we have

\[ n(1 + x)^n = n + nAx + nBx^2 + nCx^3 + \text{&c.} \]

Now the terms of these two expansions of \( n(1 + x)^n \) must be identical; therefore

\[ \begin{align*} A &= n, \\ A + 2B &= nA, \\ 2B + 3C &= nB, \\ 3C + 4D &= nD, \\ &\text{&c.} \end{align*} \]

\[ \begin{align*} A &= n; \\ B &= \frac{n-1}{2}A; \\ C &= \frac{n-2}{3}B; \\ D &= \frac{n-3}{4}C; \\ &\text{&c.} \end{align*} \] By substituting for A its value in B, and for B the resulting value in C, and so on, there is found

\[ (1+x)^n = 1 + \frac{n}{1}x + \frac{n(n-1)}{1 \cdot 2}x^2 + \frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3}x^3 + \ldots \]

and again, by putting \( \frac{e}{a} \) instead of \( x \), and multiplying both sides of the equation by \( a^n \), we get

\[ (a + x)^n = a^n + \frac{n}{1}a^{n-1}x + \frac{n(n-1)}{1 \cdot 2}a^{n-2}x^2 + \ldots \]

18. In the investigation of the binomial theorem we have employed a principle of analysis of continual use in the calculus, namely, that in such an equation as this (called an Identical Equation),

\[ a + bx + cx^2 + ex^3 + \ldots = A + Bx + Cx^2 + Ex^3 + \ldots \]

in which \( a, b, c, \ldots, A, B, C, \ldots \) are constants, and \( x \) is variable; in order that the equation may hold universally true, it is necessary that \( a = A, b = B, c = C, e = E, \ldots \). For since \( x \) is indeterminate, we may suppose it to decrease until it vanish; then all the terms into which it enters also vanish, and we have \( a = A \); therefore, leaving these equal terms out of both sides of (1), and dividing all the remaining terms by \( x \), we get

\[ b + cx + ex^2 + \ldots = B + cx + Ex^2 + \ldots \]

By proceeding with this equation as with the former, we obtain \( b = B, c = C, e = E, \ldots \), and so on.

19. We come now to the investigation of the differentials of transcendental functions of a variable, and begin with the exponential function \( u = a^x \), the exponent \( x \) being variable, and the quantity \( a \) constant. Assuming that when \( x \) becomes \( x + h \), \( u \) becomes \( u' \), we have

\[ u = a^x, \quad u' = a^{x+h} = a^x a^h; \]

therefore,

\[ u' - u = a^x a^h - a^x = a^x (a^h - 1); \]

and

\[ \frac{u' - u}{h} = a^x \frac{a^h - 1}{h}. \]

The determination of the limit of the ratio of the increments requires that we determine the limit to which the value of the fraction \( \frac{a^h - 1}{h} \) tends when \( h \) is supposed to decrease continually. Let us put \( a = 1 + e \); by the binomial theorem (art. 17),

\[ a^h = (1 + e)^h = 1 + he + \frac{h(h-1)}{1 \cdot 2}e^2 + \frac{h(h-1)(h-2)}{1 \cdot 2 \cdot 3}e^3 + \ldots \]

therefore,

\[ \frac{a^h - 1}{h} = e + \frac{h-1}{2}e^2 + \frac{(h-1)(h-2)}{2 \cdot 3}e^3 + \ldots \]

Now \( h \) being supposed to decrease continually, the second side of this formula manifestly approaches to

\[ e - \frac{1}{2}e^2 + \frac{1}{2 \cdot 3}e^3 + \frac{1}{2 \cdot 3 \cdot 4}e^4 + \ldots \]

that is, to \( e - \frac{1}{2}e^2 + \frac{1}{2 \cdot 3}e^3 - \frac{1}{2 \cdot 3 \cdot 4}e^4 + \ldots \).

Now it has been shown in Algebra (art. 173), and will be proved in the sequel, that this series expresses Napier's logarithm of the number \( a \). Therefore,

\[ \text{limit of } \frac{a^h - 1}{h} = \text{Nap. log } a, \]

and hence,

\[ \frac{du}{dx} = \text{limit } \frac{u' - u}{h} = ax (\text{Nap. log } a), \]

and \( du = (\text{Nap. log } a) a^x dx \).

We have now this rule for differentiating a function which is a variable power of a constant quantity.

Multiply the function by the differential of the variable index, and by Napier's log. of the constant quantity.

Note.—The logarithm of a number \( a \), in any system whatever, will in what follows be expressed by the abbreviation \( \log_a \); but when the system is that of Napier, we shall denote it thus, \( l_a \).

20. To find the differential of the transcendental \( u = \log_a x \), the base of the system being \( a \). By the definition of a logarithm (Algebra, art. 165),

\[ x = a^u, \]

and, supposing that \( x \) becomes \( x + h \), and that \( u \) becomes \( u' \),

\[ x + h = a^{u'}; \]

therefore,

\[ h = a^{u'} - a^u = a^u (a^{u'-u} - 1) = x (a^{u'-u} - 1); \]

and hence, making \( k = u' - u \),

\[ \frac{h}{u' - u} = \frac{x}{k}; \]

and

\[ \frac{u' - u}{h} = \frac{1}{x} \frac{k}{a^u - 1}. \]

Passing now to the limits of the two sides of the equation, and observing (as was proved in last article) that \( k \) being supposed to decrease continually,

\[ \text{limit of } \frac{k}{a^u - 1} = \frac{1}{l_a}, \]

we have

\[ \frac{du}{dx} = \frac{1}{l_a} \frac{1}{x}. \]

Let \( M \) denote the constant factor, viz. the reciprocal of Napier's logarithm of the base of the system of logarithms, which is the modulus of the system (Algebra, art. 172), and we have

\[ du = \frac{M dx}{x}; \]

That is, The differential of the logarithm of a number is found by multiplying the differential of the number by the modulus of the system and dividing by the number.

Note.—In Napier's system \( M = 1 \).

21. The differentials of the trigonometrical functions \( \sin x \) and \( \cos x \) are next to be investigated.

Let \( u = \sin x \), and \( u' = \sin (x + h) \); then (Algebra, 240),

\[ u' - u = \sin (x + h) - \sin x = 2 \cos (x + \frac{1}{2}h) \sin \frac{1}{2}h; \]

and

\[ \frac{u' - u}{h} = \cos (x + \frac{1}{2}h) \sin \frac{1}{2}h. \]

Observing now that \( h \) being understood to decrease continually until it vanish, the limit of \( \cos (x + \frac{1}{2}h) \) is \( \cos x \), and the limit of \( \frac{\sin \frac{1}{2}h}{\frac{1}{2}h} = 1 \) (Algebra, 266), we have

\[ \frac{du}{dx} = \text{limit } \frac{u' - u}{h} = \cos x; \]

and \( du = dx \cos x \).

Next let \( u = \cos x \), and \( u' = \cos (x + h) \); we have now

\[ u' - u = \cos (x + h) - \cos x = -2 \sin (x + \frac{1}{2}h) \sin \frac{1}{2}h; \]

and

\[ \frac{u' - u}{h} = -\sin (x + \frac{1}{2}h) \sin \frac{1}{2}h. \]

Passing now to the limits by supposing \( h \) to decrease continually,

\[ \frac{du}{dx} = \text{limit } \frac{u' - u}{h} = -\sin x; \]

and \( du = -dx \sin x \).

Hence it appears that The differential of the sine of an arc is the product of the differential of the arc and its cosine: and that the differential of the cosine is the product of the differential of the arc and its sine with the sign prefixed.

Note.—The negative sign indicates, that while the arc increases the cosine decreases when it is positive.

The differential of the cosine might have been found otherwise, by regarding it as a function of the sine. Thus, putting \( v \) for the sine, and \( y \) for the cosine; because \( y^2 = 1 - v^2 \), by differentiating, \( ydy = -v dv \); now \( x \) being the arc, \( dv = ydx \); therefore \( ydy = -vydx \), and \( dy = -vdx \).

22. The functions \( u = \tan x \), \( u = \cot x \), &c. may be considered as formed from the elementary functions \( \sin x \) and \( \cos x \).

(1.) Let \( u = \tan x = \frac{\sin x}{\cos x} \), then (art. 12 and 20),

\[ du = \frac{d(\sin x)}{\cos x} - d(\cos x) \cdot \frac{\sin x}{\cos^2 x} \]

\[ = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} dx = \frac{dx}{\cos^2 x}. \]

Hence, again, \( du = dx \sec^2 x = dx (1 + \tan^2 x) \).

(2.) In the same way, if \( u = \cot x \), we find

\[ du = \frac{-dx}{\sin x} = -dx \csc^2 x = -dx (1 + \cot^2 x). \]

(3.) Let \( u = \sec x = \frac{1}{\cos x} \); then

\[ du = \frac{dx \sin x}{\cos^2 x} = dx \tan x \sec x. \]

(4.) Let \( u = \csc x = \frac{1}{\sin x} \);

\[ du = \frac{-dx \cos x}{\sin^2 x} = dx \cot x \csc x. \]

23. Corresponding to the direct functions \( u = \sin x \), \( u = \cos x \), &c., there are the reverse functions,

\( u = \text{arc} (\sin x) \), \( u = \text{arc} (\cos x) \), &c.

The first, viz. the arc whose sine is \( x \), is sometimes by British writers expressed thus; \( u = \sin^{-1} x \); and the second, the arc whose cosine is \( x \), thus; \( u = \cos^{-1} x \).

(1.) Let \( u = \text{arc} (\sin x) = \sin^{-1} x \);

then \( x = \sin u \); and \( dx = du \cos u = du \sqrt{1 - x^2} \);

and \( du = \frac{dx}{\sqrt{1 - x^2}} \).

(2.) Let \( u = \text{arc} (\cos x) = \cos^{-1} x \);

then \( x = \cos u \); and \( dx = -du \sin u = -du \sqrt{1 - x^2} \);

and \( du = \frac{-dx}{\sqrt{1 - x^2}} \).

(3.) Let \( u = \text{arc} (\tan x) = \tan^{-1} x \);

then \( x = \tan u \) and \( dx = du (1 + x^2) \);

and \( du = \frac{dx}{1 + x^2} \).

(4.) Let \( u = \text{arc} (\cot x) = \cot^{-1} x \);

then \( x = \cot u \) and \( dx = -du (1 + u^2) \);

and \( du = \frac{-dx}{1 + x^2} \).

(5.) Let \( u = \text{arc} (\sec x) = \sec^{-1} x \);

then \( x = \sec u \); and \( dx = du \sec u \tan u = du x \sqrt{x^2 - 1} \),

and \( du = \frac{dx}{x \sqrt{x^2 - 1}} \).

(6.) Let \( u = \text{arc} (\csc x) = \csc^{-1} x \),

then, proceeding as in the foregoing examples, we find

\[ du = \frac{-dx}{x \sqrt{x^2 - 1}}. \]

24. Having now found rules for the differentiation of the elementary transcendental and trigonometrical functions, we proceed to exemplify their application to the differentiation of complex functions.

(1.) Let the function be \( u = x^y \), that is a variable \( x \) raised to a power \( y \), a function of the variable.

By the theory of logarithms \( l.u = y l.x \).

Put \( v = l.u \) and \( z = l.x \); then \( v = yz \);

and \( dv = ydz + zdv \) (art. 10).

Now \( dv = d(l.u) = \frac{du}{u} \); and \( dz = d(l.x) = \frac{dx}{x} \) (20);

therefore, \( \frac{du}{u} = \frac{ydx}{x} + l.x \cdot dy \),

and \( du = \left\{ \begin{array}{l} \frac{y}{x} dx + l.x \cdot dy \\ x^y \left\{ \frac{y}{x} dx + l.x \cdot dy \right\} \\ x^{y-1} ydx + x^y l.x \cdot dy. \end{array} \right. \)

(2.) If \( u = x^x \), then, by last example,

\[ du = x^x \left\{ 1 + l.x \right\} dx. \]

(3.) Let \( u = a^x \); put \( b^x = y \), then \( u = a^y \),

and \( du = l.a^y dy \), also \( dy = l.b \cdot b^x dx \) (19);

therefore \( du = l.a \cdot l.b \cdot a^b dx \).

In the following examples we shall suppose the logarithms to be those of Napier's system.

(4.) Let \( u = \frac{x}{\sqrt{a^2 + x^2}} \). Making \( \frac{x}{\sqrt{a^2 + x^2}} = z \),

we have \( du = \frac{dz}{z} \); but

\[ dz = \frac{dx \sqrt{a^2 + x^2} - \frac{x^2 dx}{\sqrt{a^2 + x^2}}}{a^2 + x^2} = \frac{a^2 dx}{(a^2 + x^2)^{3/2}}; \]

therefore \( du = \frac{a^2 dx}{x(a^2 + x^2)} \).

(5.) Let \( u = 1. \left\{ \frac{\sqrt{1 + x} + \sqrt{1 - x}}{\sqrt{1 + x} - \sqrt{1 - x}} \right\} \);

put \( y = \sqrt{1 + x} + \sqrt{1 - x} \); \( z = \sqrt{1 + x} - \sqrt{1 - x} \);

then \( u = l. \left( \frac{y}{z} \right) = l.(y) - l.(z) \);

and \( du = \frac{dy}{y} - \frac{dz}{z} \).

Now \( dy = \frac{dx}{2\sqrt{1 + x}} - \frac{dx}{2\sqrt{1 - x}} \),

\[ = \frac{-dx}{2\sqrt{1 - x^2}} \left\{ \sqrt{1 + x} - \sqrt{1 - x} \right\} = \frac{-xdx}{2\sqrt{1 - x^2}}; \]

and \( dz = \frac{dx}{2\sqrt{1 + x}} + \frac{x}{2\sqrt{1 - x}} \),

\[ = \frac{dx}{2\sqrt{1 - x^2}} \left\{ \sqrt{1 + x} + \sqrt{1 - x} \right\} = \frac{xdx}{2\sqrt{1 - x^2}}; \]

therefore \( \frac{dy}{y} - \frac{dz}{z} = \frac{-2dx}{2\sqrt{1 - x^2}} - \frac{xdx}{2\sqrt{1 - x^2}} \). For example, from the function \( u = x^n \), we deduce

\[ \frac{du}{dx} = nx^{n-1} \]

putting now \( p = nx^{n-1} \), we hence derive

the differential co-efficient \( \frac{dp}{dx} = n(n-1)x^{n-2} \); and putting \( q = n(n-1)x^{n-2} \), we find \( \frac{dq}{dx} = n(n-1)(n-2)x^{n-3} \).

We may proceed in this way, until the result of a differentiation be a constant, and then the process will stop; but in some cases it may be continued indefinitely.

There is an appropriate notation to express the relation in which the successive differentials, \( p, q, \) &c. stand to the original function \( u \), from which they have been derived: since

\[ p = \frac{du}{dx}, \quad q = \frac{dp}{dx}, \quad \text{therefore} \quad q = \frac{d\left(\frac{du}{dx}\right)}{dx}. \]

The symbol \( d\left(\frac{du}{dx}\right) \) will be more simply expressed by \( d^3u \cdot dx \); and thus we have

\[ p = \frac{du}{dx}, \quad q = \frac{dp}{dx} = \frac{d^2u}{dx^2}; \]

so that by the expression \( q = \frac{d^2u}{dx^2} \) it is to be understood that \( q \) is the result of two differentiations performed on the function \( u \), the differential of the variable, viz. \( dx \), being considered as constant.

From the function \( q \) we may now deduce

\[ r = \frac{dq}{dx} = \frac{d^3u}{dx^3}, \]

and so on continually; but the different import of the characters \( d^3u \) and \( d^3x \) must be carefully attended to; the former implies that the operation of differentiation has been performed three times on the function \( u \), and the latter denotes the third power of \( dx \). The combination of the two, \( d^3u \cdot dx^3 \), expresses the function that results from the three differentiations, in each of which the factor \( dx \) has been left out, and the final result divided by its third power, or, if retained, that it has been regarded as constant.

(1.) As a particular case of the function \( u = ax^n \), let \( u = ax^n \);

then \( \frac{du}{dx} = 5ax^4, \quad \frac{d^2u}{dx^2} = 4 \cdot 5ax^3, \)

\( \frac{d^3u}{dx^3} = 3 \cdot 4 \cdot 5ax^2, \quad \frac{d^4u}{dx^4} = 2 \cdot 3 \cdot 4 \cdot 5ax, \)

\( \frac{d^5u}{dx^5} = 2 \cdot 3 \cdot 4 \cdot 5a. \)

In this example we have come to a differential co-efficient which is a constant, and the series stops; but if the index of the power \( x^n \) be a fraction or negative, the series may be continued indefinitely.

The expression \( \frac{du}{dx} \) is called the first differential co-efficient of the function \( u \), or the differential co-efficient of the first order; those which follow are called the second, third, &c. or the differential co-efficients of the second, third, &c. orders.

In the case of the function \( u = x^n \), writers on the calculus of fluxions call \( du = 5x^4dx \) the first fluxion of the function; \( d^2u = 20x^3dx^2 \) its second fluxion; \( d^3u = 60x^2dx^3 \) its third fluxion; and so on, the terms of the series constituting the different orders of fluxions. Other examples of successive differentiation:

(2.) Let \( u = a^x \); then (article 9), \( \frac{du}{dx} = \ln a \cdot a^x \),

\[ \frac{d^2u}{dx^2} = (\ln a)^2 \cdot a^x, \quad \frac{d^3u}{dx^3} = (\ln a)^3 \cdot a^x, \quad \frac{d^4u}{dx^4} = (\ln a)^4 \cdot a^x, \quad \text{etc.} \]

(3.) Let \( u = \log x \); then (art. 20) \( \frac{du}{dx} = \frac{M}{x} \),

\[ \frac{d^2u}{dx^2} = -\frac{M}{x^2}, \quad \frac{d^3u}{dx^3} = -\frac{1.2M}{x^3}, \quad \frac{d^4u}{dx^4} = \frac{1.23M}{x^4}, \quad \text{etc.} \]

(4.) Let \( u = \sin x \); then \( \frac{du}{dx} = \cos x \),

\[ \frac{d^2u}{dx^2} = -\sin x, \quad \frac{d^3u}{dx^3} = -\cos x, \quad \frac{d^4u}{dx^4} = \sin x, \quad \text{etc.} \]

(5.) Let \( u = \cos x \); then \( \frac{du}{dx} = -\sin x \),

\[ \frac{d^2u}{dx^2} = -\cos x, \quad \frac{d^3u}{dx^3} = \sin x, \quad \frac{d^4u}{dx^4} = \cos x, \quad \text{etc.} \]

(6.) Let \( u = \frac{a^x}{a^x + x^x} \); then \( \frac{du}{dx} = \frac{-2a^x}{(a^x + x^x)^2} \),

\[ \frac{d^2u}{dx^2} = \frac{-2a^x + 6a^x x^x}{(a^x + x^x)^3}, \quad \frac{d^3u}{dx^3} = \frac{24a^x x - 24a^x x^x}{(a^x + x^x)^4}, \quad \text{etc.} \]

Taylor's Theorem.

28. We have assumed for the foundation of the differential calculus an important analytical principle, first particularly recognised by Euler, viz. Let \( f(x) \) denote any function of a variable quantity \( x \); if instead of \( x \) there be substituted \( x + h \), \( h \) being any indeterminate quantity, so that the function becomes \( f(x + h) \), this new value may always be expanded into a series of this form,

\[ f(x) + ph + qh^2 + rh^3 + \text{etc.} \]

in which the quantities \( p, q, r, \text{etc.} \) are new functions of \( x \) derived from the primitive function, and independent of the indeterminate quantity \( h \).

29. The truth of this principle has been exemplified by induction from particular cases. Lagrange has, however, given a demonstration of it in his *Théorie des Fonctions*, which is to the following effect:

In the first place, the development of the function \( (x + h) \) cannot contain any fractional power of \( h \), so long as \( x \) is entirely indeterminate. For the radicals of \( h \) can only come from radicals in the primitive function; and it is manifest that the substitution of \( x + h \) instead of \( x \) can neither increase nor diminish their number, nor change their nature, while \( x \) and \( h \) are indeterminate. On the other hand, by the theory of equations, every radical expression has as many different values as there are units in its exponent; from which it follows, that every irrational function has as many distinct values as there are combinations of the different values of the radicals which it contains. Therefore, if the development of the function \( f(x + h) \) could contain a term of the form \( uh^n \), the function \( f(x) \) must necessarily be irrational, and contain a certain number of different values, which must be the same for its development; but this development being represented by the series \( f(x) + ph + qh^2 + \text{etc.} + uh^n + \text{etc.} \), every value of the function \( f(x) \) (the first term) might be combined with each of the \( n \) values of the radical \( \sqrt[n]{x} \), so that the function \( f(x + h) \) expanded would have more values than the same function unexpanded, which is impossible.

This demonstration is general and rigorous so long as \( x \) and \( h \) continue indeterminate; but it fails if determinate values be given to \( x \), for it may happen that these values may destroy some radicals in the function \( f(x) \), which may yet exist in the function \( f(x + h) \).

30. It being proved that the development of the function cannot have any fractional powers of \( h \), it is easy to be assured that it cannot contain negative powers. For if among the terms one had the form \( \frac{r}{h^m} \), \( m \) being a positive integer number, then, in making \( h = 0 \), that term would become infinite, and the function \( (x + h) \), and of course \( f(x) \), would in this case be infinite, which cannot be, unless a particular value be given to \( x \). Thus it is clearly established that the development of the function can contain neither fractional nor negative exponents.

31. The general form of the development of the function, viz.

\[ f(x + h) = f(x) + ph + qh^2 + rh^3 + \text{etc.} \]

being thus ascertained, the next question is, what is the law of relation between the original function \( f(x) \) and the functions \( p, q, r, \text{etc.} \) which are derived from it? We owe the discovery of this relation to a celebrated English mathematician, Brooke Taylor, who gave it in his *Methodus Incrementorum*, published in the year 1715 in the form of an analytical theorem, which is now called by his name: we shall now give its investigation. The following is that of Lagrange.

Let \( f(x) = u \) be any function of \( x \); then, supposing \( x \) to become \( x + h \), \( u \) becomes

\[ f(x + h) = u + ph + qh^2 + rh^3 + \text{etc.} \]

Suppose now that \( x \) changes again its value, and becomes \( x + k \), and here \( k \), like \( h \), is independent of \( x \). Then

\[ f(x + h) \text{ becomes } f(x + h + k). \]

There are two ways of finding what the series

\[ f(x + h) = u + ph + qh^2 + rh^3 + sh^4 + \text{etc.} \]

becomes, when \( x + h \) becomes \( x + h + k \). 1st. We may have the value of the series by substituting \( h + k \) in every term instead of \( h \). 2dly, We may also have its value by substituting \( x + k \) in the functions \( p, q, r, \text{etc.} \) for \( x \); that is, instead of \( p, q, r, \text{etc.} \) we must put the values they have when \( x + k \) is supposed to be substituted in them instead of \( x \).

By the first-mentioned process we find \( f(x + h + k) = u + ph + qh^2 + rh^3 + sh^4 + \text{etc.} \)

\[ + pk + 2qh + 3rhk + 4shk^2 + \text{etc.} \]

\[ + qk^2 + 3rhk^2 + 6shk^2 + \text{etc.} \]

\[ + rk^3 + 4shk^3 + \text{etc.} \]

\[ + sk^4 + \text{etc.} \]

And, in employing the second, it must be considered that, when \( x \) becomes \( x + k \), then \( u \) becomes \( u + pk + qk^2 + rk^3 + sk^4 + \text{etc.} \), because \( k \) here takes the place of \( h \) in the function \( f(x + h) \), and its development \( u + ph + qh^2 + rh^3 + \text{etc.} \).

The supposition that \( x \) changes its value to \( x + k \), leads to corresponding changes in the functions \( p, q, r, \text{etc.} \); so that

\[ p \text{ becomes } p + pk + pk^2 + pk^3 + \text{etc.} \]

\[ q \text{ becomes } q + qk + qk^2 + qk^3 + \text{etc.} \]

\[ r \text{ becomes } r + rk + rk^2 + rk^3 + \text{etc.} \]

\[ s \text{ becomes } s + sk + sk^2 + sk^3 + \text{etc.} \]

\[ \text{etc.} \] Here \( p', p'', \ldots \) denote functions of \( x \) derived from \( p \), just as \( p, q, r, \ldots \) are derived from the function \( u \); and a like remark applies to \( q', q'', \ldots \) also to \( r', r'', \ldots \) and so on. Substituting now these new values of \( u, p, q, r, \ldots \) in the series \( u + ph + qh^2 + rh^3 + sh^4 + \ldots \), we have

\[ f(x + h + k) = u + ph + qh^2 + rh^3 + sh^4 + \ldots \]

By comparing the co-efficients of like powers, or products of powers, of \( h \) and \( k \) in the two expressions for \( f(x + h + k) \), it appears that, to make them identical, we must have

\[ 2q = p', 3r = q', 4s = r', \text{ and so on.} \]

Therefore \( q = \frac{p'}{2}, r = \frac{q'}{3}, s = \frac{r'}{4}, \ldots \)

Now, by the definition of a differential, \( pdx \) is the differential of \( u \), where \( p \) is the co-efficient of \( h \) in the series \( u + ph + qh^2 + \ldots \). Therefore \( p = \frac{du}{dx} \). Similarly,

\( qdx \) is the differential of \( q \), where \( q' \) is the co-efficient of \( k \) in the development \( q - q'k + q'k^2 + \ldots \). Therefore,

\[ q = \frac{dq}{dx} \quad \text{and in like manner, } r = \frac{dr}{dx} \quad \text{and } s = \frac{ds}{dx}, \]

and so on; hence

\[ p = \frac{dy}{dx} \]

\[ q = \frac{1}{2}p' = \frac{1}{2}\frac{dp}{dx} = \frac{1}{1.2}\frac{d^2u}{dx^2}, \]

\[ r = \frac{1}{3}q' = \frac{1}{3}\frac{dq}{dx} = \frac{1}{1.2.3}\frac{d^3u}{dx^3}, \]

\[ s = \frac{1}{4}r' = \frac{1}{4}\frac{dr}{dx} = \frac{1}{1.2.3.4}\frac{d^4u}{dx^4}, \]

\[ \text{etc.} \]

These values of \( p, q, r, s, \ldots \) being substituted in the series \( u + ph + qh^2 + rh^3 + sh^4 + \ldots \), we obtain

\[ f(x + h) = u + \frac{du}{dx}h + \frac{d^2u}{dx^2}h^2 + \frac{d^3u}{dx^3}h^3 + \ldots \]

where \( u = f(x) \); this is Taylor's theorem.

32. We shall now give another investigation of this important formula, which is remarkable for its brevity and simplicity. It rests on the following analytic principle.

If \( u \) be any function whatever of \( v + z \), which relation may be expressed thus, \( u = f(v + z) \); the differential co-efficient of \( u \), found on the supposition that \( v \) is variable and \( z \) constant, will be the very same as if it were found on the supposition that \( z \) is variable and \( v \) constant.

To exemplify this in particular cases, suppose \( u = (v + z)^n \), then, making \( v \) variable and \( z \) constant, \( \frac{du}{dv} = n(v + z)^{n-1} \); and making \( z \) variable and \( v \) constant, \( \frac{du}{dz} = n(v + z)^{n-1} \), the same as before. Again, let \( u = a^{v+z} = a^v a^z \), then \( \frac{du}{dv} \) and \( \frac{du}{dz} \) are expressed by the same quantity \( l.a.a^v a^z = l.a.a^{v+z} \).

The truth of the principle is almost self-evident; for if we make \( v + z = x \), so that \( f(v + z) = f(x) \); then, when we suppose \( v \) to vary and to become \( v + h \), while \( z \) remains the same, or \( z \) to vary and become \( z + h \), while \( v \) remains constant, the result is the very same, viz.

\[ f(v + z + h) = f(x + h); \]

therefore, in the development of \( f(v + z + h) = f(x + h) \), viz.

\[ f(x) + ph + qh^2 + rh^3 + sh^4 + \ldots, \]

in which \( p, q, r, \ldots \) are functions of \( x \); that is, of \( v + z \), these functions will be the very same, whichever of the two, \( v \) and \( z \), has been considered as the variable, the other remaining constant. Now, by the nature of a differential (art. 7), we have

\[ p = \frac{d\{f(x)\}}{dx} = \frac{d\{f(v + z)\}}{d(v + z)}; \]

and upon the supposition that \( v \) is variable and \( z \) constant, then

\[ p = \frac{d\{f(v + z)\}}{dv}; \]

but supposing \( z \) variable and \( v \) constant,

\[ p = \frac{d\{f(v + z)\}}{dz}; \]

therefore \( \frac{d\{f(v + z)\}}{dv} = \frac{d\{f(v + z)\}}{dz} \);

the first side of this equation being the result of the differentiation, supposing \( v \) variable, and the second the result, on the hypothesis that \( z \) is variable. Since the two expressions \( \frac{d\{f(v + z)\}}{dv} \) and \( \frac{d\{f(v + z)\}}{dz} \) are identical and the same function of \( v + z \), we may represent that function by the symbol \( F(v + z) \); then, by reasoning as before, we shall have

\[ \frac{d\{F(v + z)\}}{dv} = \frac{d\{F(v + z)\}}{dz}; \]

that is, \( \frac{d\{f(v + z)\}}{dv} = \frac{d\{f(v + z)\}}{dz} \).

Thus it appears that the second differential co-efficient of the function \( f(v + z) \) is the same expression, whether \( v \) or \( z \) be regarded as the variable; and the reasoning may be extended to the coefficient of any order.

33. Supposing now \( f(z) = u \) to be any function of \( x \), and \( f(x + h) = u' \) to be its value when \( x \) becomes \( x + h \), it has been proved (art. 29, 30), that the expansion of \( u \) is

\[ u' = u + ph + qh^2 + rh^3 + sh^4; \]

and here \( p, q, r, \ldots \) are functions of \( x \), into which \( x \) does not at all enter; while, on the other hand, \( h \) is entirely independent of \( x \). From the identity of the function \( u' \) and its expansion, their differentials, taken on the same hypothesis, must be equal, whichever of the quantities \( x \) and \( h \) be regarded as the variable. Supposing, first, \( x \) variable and \( h \) constant, we have

\[ \frac{du'}{dx} = \frac{du}{dx} + \frac{dp}{dx}h + \frac{dq}{dx}h^2 + \frac{dr}{dx}h^3 + \frac{ds}{dx}h^4 + \ldots, \]

and next, supposing \( h \) to be variable and \( x \) constant,

\[ \frac{du'}{dh} = ph + 2qh + 3rh^2 + 4sh^3 + \ldots. \]

Now it has been shown in last article that \( \frac{du'}{dx} = \frac{du'}{dh} \); therefore the series which are their expansions must be identical; and hence the terms independent of \( h \) must be equal, also the co-efficients of the same power of \( h \), so that

\[ p = \frac{du}{dx}, \quad 2q = \frac{dp}{dx}, \quad 3r = \frac{dq}{dx}, \quad 4s = \frac{dr}{dx}, \ldots \] Direct Method, and since \( p = \frac{du}{dx} \), therefore \( q = \frac{1}{2} \frac{dp}{dx} = \frac{1}{2} \frac{d^2u}{dx^2} \) and \( r = \frac{1}{3} \frac{dq}{dx} = \frac{1}{2 \cdot 3} \frac{d^3u}{dx^3} \); also \( s = \frac{1}{4} \frac{dr}{dx} = \frac{1}{2 \cdot 3 \cdot 4} \frac{d^4u}{dx^4} \), &c.; and,

on the whole, \( f(x + h) = u + \frac{du}{dx} h + \frac{d^2u}{dx^2} \frac{h^2}{2!} + \frac{d^3u}{dx^3} \frac{h^3}{3!} + \frac{d^4u}{dx^4} \frac{h^4}{4!} + \ldots \).

From this formula it appears, that if \( h \) be the increment of the variable \( x \); the whole increment of \( u \), any function of \( x \), is

\[ u' - u = \frac{du}{dx} h + \frac{d^2u}{dx^2} \frac{h^2}{2!} + \frac{d^3u}{dx^3} \frac{h^3}{3!} + \ldots \]

a series composed of the successive differential co-efficients of the function, each multiplied by an integer power of \( h \), and divided by the products \( 1, 1 \cdot 2, 1 \cdot 2 \cdot 3, \ldots \).

Application of Taylor's Theorem to the Development of Functions.

We shall now give some applications of this most important analytic formula to the development of functions.

34. Let the function \( f(x) \) be \( u = x^n \), so that \( f(x + h) = (x + h)^n \). By successive differentiation (art. 13 and 27),

\[ \begin{align*} \frac{du}{dx} &= nx^{n-1}, \\ \frac{d^2u}{dx^2} &= n(n-1)x^{n-2}, \\ \frac{d^3u}{dx^3} &= n(n-1)(n-2)x^{n-3}, \\ &\vdots \\ \frac{d^nu}{dx^n} &= n(n-1)(n-2)\cdots(n-n)x^{n-n}. \end{align*} \]

These differential co-efficients, when substituted in the formula, give

\[ (x + h)^n = x^n + nx^{n-1}h + \frac{n(n-1)}{1 \cdot 2} x^{n-2}h^2 + \frac{n(n-1)(n-2)}{1 \cdot 2 \cdot 3} x^{n-3}h^3 + \ldots \]

This is the well-known binomial theorem (Algebra, art. 160).

35. Next, let \( f(x) \) be \( u = a^x \), a variable power of the constant quantity \( a \). Then, putting

\[ A = (a-1) - \frac{1}{2}(a-1)^2 + \frac{1}{3}(a-1)^3 - \frac{1}{4}(a-1)^4 + \ldots \]

By article 19,

\[ \begin{align*} \frac{du}{dx} &= Aa^x, \\ \frac{d^2u}{dx^2} &= A^2a^x, \\ \frac{d^3u}{dx^3} &= A^3a^x, \\ &\vdots \\ \frac{d^nu}{dx^n} &= A^na^x. \end{align*} \]

Remarking now that \( f(x + h) = a^{x+h} = a^x a^h \), we have

\[ a^x a^h = e^x \left\{ 1 + Ah + \frac{A^2h^2}{1 \cdot 2} + \frac{A^3h^3}{1 \cdot 2 \cdot 3} + \frac{A^4h^4}{1 \cdot 2 \cdot 3 \cdot 4} + \ldots \right\}; \]

and dividing both sides by \( a^x \), and changing \( h \) into \( x \),

\[ a^x = 1 + Ax + \frac{A^2x^2}{1 \cdot 2} + \frac{A^3x^3}{1 \cdot 2 \cdot 3} + \frac{A^4x^4}{1 \cdot 2 \cdot 3 \cdot 4} + \ldots \]

If we suppose \( x = 1 \), then,

\[ a = 1 + A + \frac{A^2}{1 \cdot 2} + \frac{A^3}{1 \cdot 2 \cdot 3} + \frac{A^4}{1 \cdot 2 \cdot 3 \cdot 4} + \ldots \]

and if we make \( x = \frac{1}{A} \), we have

\[ a^{\frac{1}{A}} = 1 + 1 + \frac{1}{1 \cdot 2} + \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{1 \cdot 2 \cdot 3 \cdot 4} + \ldots \]

Hence it appears that \( a^{\frac{1}{A}} \) is a constant number, which is the sum of this series. By taking the sum of a sufficient number of its terms, we find

\[ a^{\frac{1}{A}} = 2.718281828459. \]

Let this number, which will frequently recur, be denoted by the letter \( e \), and we have

\[ \frac{1}{e} = e^{-1}, \text{ and } a = e^{\frac{1}{A}} \]

\[ e^{Ax} = e^x = 1 + Ax + \frac{A^2x^2}{1 \cdot 2} + \frac{A^3x^3}{1 \cdot 2 \cdot 3} + \frac{A^4x^4}{1 \cdot 2 \cdot 3 \cdot 4} + \ldots \]

and making \( y = Ax \), so that \( y \) may be any number,

\[ e^y = 1 + y + \frac{y^2}{1 \cdot 2} + \frac{y^3}{1 \cdot 2 \cdot 3} + \frac{y^4}{1 \cdot 2 \cdot 3 \cdot 4} + \ldots \]

This formula expresses a remarkable property of the number \( e \).

36. Let the function \( f(x) \) be \( u = \log_a x \) to base \( a \), then, by article 20,

\[ \begin{align*} \frac{du}{dx} &= \frac{1}{Ax}, \\ \frac{d^2u}{dx^2} &= \frac{-1}{Ax^2}, \\ \frac{d^3u}{dx^3} &= \frac{1}{Ax^3}, \\ &\vdots \\ \frac{d^nu}{dx^n} &= \frac{(-1)^{n-1}}{Ax^n}. \end{align*} \]

Hence, by the theorem, \( f(x + h) = \log_a (x + h) = \log_a x + \frac{1}{A} \left\{ \frac{h}{x} - \frac{h^2}{2x^2} + \frac{h^3}{3x^3} - \frac{h^4}{4x^4} + \ldots \right\}. \)

Make \( x = 1 \), and observe that \( \log_1 1 = 0 \), and change \( h \) into \( x \), and we have

\[ \log_a (1 + x) = \frac{1}{A} \left\{ x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \right\}. \]

This is the logarithm of \( 1 + x \) in the system whose base is \( a \).

In the preceding article we found that \( a = e^{\frac{1}{A}} \), hence, from the theory of logarithms (Algebra, sect. xix.), \( A \) is the logarithm of \( a \) to the base \( e \). Now

\[ A = a - 1 - \frac{1}{2}(a-1)^2 + \frac{1}{3}(a-1)^3 - \frac{1}{4}(a-1)^4 + \ldots \]

Therefore, in the system whose base is \( e \),

\[ \log_e a = a - 1 - \frac{1}{2}(a-1)^2 + \frac{1}{3}(a-1)^3 - \frac{1}{4}(a-1)^4 + \ldots \]

and hence, putting \( 1+x \) for \( a \), and \( x \) for \( a-1 \), in that system,

\[ \log_e (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \]

Now we find that in the system whose base is \( a \),

\[ \log_a (1+x) = \frac{1}{A} \left\{ x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \right\}. \]

By comparing these two expressions for \( \log_e (1+x) \), it appears that in the system whose base is \( e \), the constant quantity \( A \), which is

\[ e - 1 - \frac{1}{2}(e-1)^2 + \frac{1}{3}(e-1)^3 - \frac{1}{4}(e-1)^4 + \ldots \]

is equal to unity. Logarithms computed according to this system are the same as those first given to the world by Napier, the celebrated inventor of logarithms, who was, however, led to them by a very different path from that here followed. It so happens that the same logarithms express hyperbolic areas, and hence they were called hyperbolic logarithms, a distinction quite improper, because hyperbolic areas may be expressed by logarithms of any system. Accordingly, it is now common to denominate them, in honour of the inventor, Napierian logarithms. In speaking of these, then, it must be understood that their base is \( e = 2.718281828459 \), &c.; and remembering that the letter \( l \) prefixed to any expression of a function means Napier's logarithm of that function (art. 19).

\[ l.(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \]

and since \( A = \text{Nap. log } a \) (see last article), therefore in the system whose base is \( a \),

\[ \log_a (1+x) = \frac{1}{A} \left\{ x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \right\}. \] The constant multiplier \( \frac{1}{a} \) called the *modulus* of the system, is, for Briggs's logarithms,

\[ \frac{1}{l(10)} = 4342944819033 \quad \text{and its reciprocal,} \]

\[ \frac{1}{l(10)} = 2.302585092994. \]

37. Let the function be \( f(x) = u = \sin x \), then

\[ \frac{du}{dx} = +\cos x, \quad \frac{d^2u}{dx^2} = -\sin x, \quad \frac{d^3u}{dx^3} = -\cos x, \]

\[ \frac{d^4u}{dx^4} = +\sin x, \quad \frac{d^5u}{dx^5} = +\cos x, \quad \text{&c.} \]

These values of the differential co-efficients being substituted in the theorem, it gives \( f(x+h) = \sin (x+h) \)

\[ = \sin x + \cos x \frac{h}{1!} - \sin x \frac{h^2}{2!} - \cos x \frac{h^3}{3!} + \sin x \frac{h^4}{4!} + \text{&c.} \]

Let the function be \( f(x) = u = \cos x \), then

\[ \frac{du}{dx} = -\sin x, \quad \frac{d^2u}{dx^2} = -\cos x, \quad \frac{d^3u}{dx^3} = \sin x, \]

\[ \frac{d^4u}{dx^4} = \cos x, \quad \frac{d^5u}{dx^5} = -\sin x, \quad \text{&c.} \]

and by substituting in the formula \( f(x+h) = \cos (x+h) \)

\[ = \cos x - \sin x \frac{h}{1!} - \cos x \frac{h^2}{2!} + \sin x \frac{h^3}{3!} + \cos x \frac{h^4}{4!} + \text{&c.} \]

Put \( P = 1 - \frac{h^2}{2!} + \frac{h^4}{4!} - \frac{h^6}{6!} + \text{&c.} \)

\( Q = h - \frac{h^3}{3!} + \frac{h^5}{5!} - \frac{h^7}{7!} + \text{&c.} \)

Then \( \sin (x+h) = P \sin x + Q \cos x, \)

\[ \cos (x+h) = P \cos x - Q \sin x. \]

Hence, by eliminating \( Q \) and \( P \) in succession,

\[ \sin (x+h) \sin x + \cos (x+h) \cos x = P, \]

\[ \sin (x+h) \cos x - \cos (x+h) \sin x = Q. \]

Therefore (Algebra, 239), \( \cos h = P, \sin h = Q; \) and exchanging \( h \) for \( x, \)

\[ \cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \text{&c.} \]

\[ \sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \text{&c.} \]

The same results might have been obtained easier by making \( x = 0 \) in the series for \( \sin (x+h) \) and \( \cos (x+h). \)

38. Let the function be the arc whose tangent is \( x, \)

that is, \( u = \tan^{-1} x; \)

then \( du = \frac{dx}{1+x^2} = dx \cos^2 u \) (art. 23),

and \( \frac{du}{dx} = \cos u \cos u. \)

The succeeding differential co-efficients will be found by formulae (1) and (2) of article 26.

By (1) \( \frac{du}{dx} = -\sin 2u \cos^2 u, \)

By (2) \( \frac{du}{dx^2} = -2 \cos 3u \cos^3 u, \)

By (1) \( \frac{du}{dx^3} = 2.3 \sin 4u \cos^4 u, \)

By (2) \( \frac{du}{dx^4} = 2.3.4 \cos 5u \cos^5 u, \)

By (1) \( \frac{du}{dx^5} = 2.3.4.5 \sin 6u \cos^6 u, \)

By (2) \( \frac{du}{dx^6} = 2.3.4.5.6 \cos 7u \cos^7 u, \text{&c.} \)

Hence, if we put \( \tan u = x, \) and \( \tan u' = x + h, \) we deduce from Taylor's theorem this very general and remarkable formula:

\[ u = u + \cos u \cos u \frac{h}{1!} - \sin 2u \cos^2 u \frac{h^2}{2!} - \cos 3u \cos^3 u \frac{h^3}{3!} \]

\[ + \sin 4u \cos^4 u \frac{h^4}{4!} + \cos 5u \cos^5 u \frac{h^5}{5!} \]

\[ - \sin 6u \cos^6 u \frac{h^6}{6!} + \text{&c.} \]

If the arc \( u \) be supposed \( = 0, \) then \( \sin u, \sin 2u, \text{&c.} \) are each \( 0, \) and the powers of \( \cos u \) each \( = 1, \) and the formula becomes

\[ u = \tan u - \frac{1}{3} \tan^3 u + \frac{1}{5} \tan^5 u - \frac{1}{7} \tan^7 u + \text{&c.} \]

a formula due to James Gregory. We have found this series in a very different way (ALGEBRA, art. 270), and have there applied it to the determination of the ratio of the diameter to the circumference. It will be therefore sufficient to state the result here. Let \( \sigma \) denote half the circumference of a circle, of which the rad. \( = 1, \) then

\[ \sigma = 3.141592653590. \]

Maclaurin's Theorem.

39. There is a general formula nearly related to that of Taylor, and indeed an easy deduction from it, commonly called Maclaurin's Theorem. It is given at page 610 of the second volume of his fluxions (printed in 1742); but it is proper to say, that the same formula, in substance, was given in 1717 by Stirling, in his Lineae Tertii Ordines Newtonianae, p. 32.

We have found that \( f(x) = u \) being any function of a variable \( x, \)

\[ f(x+h) = u + \frac{du}{dx} h + \frac{d^2u}{dx^2} \frac{h^2}{2!} + \frac{d^3u}{dx^3} \frac{h^3}{3!} + \frac{d^4u}{dx^4} \frac{h^4}{4!} + \text{&c.} \]

Suppose now that \( x = 0, \) and that by making this assumption

\[ f(x) = u \text{ becomes } U, \]

also that \( \frac{du}{dx} \) becomes \( U', \)

\[ \frac{d^2u}{dx^2} \text{ becomes } U'', \quad \frac{d^3u}{dx^3} \text{ becomes } U''', \quad \text{&c.} \]

Then Taylor's theorem becomes

\[ f(h) = U + U'h + U'' \frac{h^2}{2!} + U''' \frac{h^3}{3!} + \text{&c.} \]

or, changing \( h \) into \( x, \)

\[ f(x) = U + U'x + U'' \frac{x^2}{2!} + U''' \frac{x^3}{3!} + \text{&c.} \]

From this it appears, that provided \( u \) be such a function of \( x \) as admits of being expanded into a series of the form

\[ u = A + Bx + Cx^2 + Dx^3 + \text{&c.} \]

where \( A, B, C, D, \text{&c.} \) are constant quantities, then, \( U \) being the value of \( u \) when \( x = 0, \) and \( U', U'', U''', \text{&c.} \) be- ing the values of the differential co-efficients \( \frac{du}{dx} \), \( \frac{d^2u}{dx^2} \), \( \frac{d^3u}{dx^3} \), &c., found on that hypothesis, we have

\[ A = U, \quad B = U', \quad C = U'', \quad D = U''', \quad \text{&c.} \]

and consequently,

\[ u = U + Ux + U'x^2 + \frac{U''x^3}{2!} + \frac{U'''x^4}{3!} + \text{&c.} \]

This is Maclaurin's theorem, of which we shall now give some applications.

40. Let \( u = (a + x)^n \). The expansion of this expression has the requisite form; then \( \frac{du}{dx} = n(a + x)^{n-1} \),

\[ \frac{d^2u}{dx^2} = n(n - 1)(a + x)^{n-2}, \] \[ \frac{d^3u}{dx^3} = n(n - 1)(n - 2)(a + x)^{n-3}, \quad \text{&c.} \]

The supposition that \( x = 0 \) makes

\[ u = a^n = U, \quad \frac{du}{dx} = na^{n-1} = U', \quad \frac{d^2u}{dx^2} = n(n - 1)a^{n-2} = U'', \] \[ \frac{d^3u}{dx^3} = n(n - 1)(n - 2)a^{n-3} = U''', \quad \text{&c.} \]

The proper substitutions being made in the general theorem, we have

\[ (a + x)^n = a^n + na^{n-1}x + \frac{n(n - 1)}{1 \cdot 2}a^{n-2}x^2 + \frac{n(n - 1)(n - 2)}{1 \cdot 2 \cdot 3}a^{n-3}x^3 + \text{&c.} \]

41. Let \( u = a^x \). This function is also of the kind that admits of expansion by the theorem; the differential co-efficients are (art. 19), putting \( A = 1, a \),

\[ \frac{du}{dx} = Aa^x, \quad \frac{d^2u}{dx^2} = A^2a^x, \quad \frac{d^3u}{dx^3} = A^3a^x, \quad \text{&c.} \]

The supposition that \( x = 0 \) gives \( u = 1 \), and

\[ \frac{du}{dx} = A, \quad \frac{d^2u}{dx^2} = A^2, \quad \frac{d^3u}{dx^3} = A^3, \quad \text{&c.} \]

Hence \( a^x = 1 + Ax + \frac{A^2x^2}{1 \cdot 2} + \frac{A^3x^3}{1 \cdot 2 \cdot 3} + \text{&c.} \)

as was found in art. 35.

42. The formula \( u = \log x \) does not admit of being expanded into a series of the prescribed form, for in this case \( \frac{du}{dx} = \frac{1}{Ax} \); the supposition that \( x = 0 \) makes \( u \) and \( \frac{du}{dx} \) as well as all the following differential co-efficients, infinite.

The function \( u = \log (a + x) \), however, admits of the application of the formula. In this case (art. 20),

\[ \frac{du}{dx} = \frac{1}{A(n + x)}, \quad \frac{d^2u}{dx^2} = \frac{-1}{A(n + x)^2}, \quad \frac{d^3u}{dx^3} = \frac{1}{A(n + x)^3}, \quad \text{&c.} \]

The assumption that \( x = 0 \) makes \( u = \log a = U \), and

\[ \frac{du}{dx} = \frac{1}{An} = U', \quad \frac{d^2u}{dx^2} = \frac{-1}{An^2} = U'', \quad \frac{d^3u}{dx^3} = \frac{1}{An^3} = U''', \quad \text{&c.} \]

We have now, by the general formula,

\[ \log (a + x) = \log a + \frac{1}{A}\left\{\frac{x}{n} - \frac{x^2}{2n} + \frac{x^3}{3n} - \text{&c.}\right\}. \]

By making \( n = 1 \), and therefore \( \log n = 0 \), we have the expression for \( \log (1 + x) \), as already found.

43. The theorem may be applied to the functions \( u = \sin x \), and \( u = \cos x \), without the least difficulty. For the first we have

\[ U = 0, \quad U' = 1, \quad U'' = 0, \quad U''' = -1, \quad \text{&c.} \]

and \( \sin x = x - \frac{x^3}{1 \cdot 2 \cdot 3} + \frac{x^5}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} + \text{&c.} \);

for the second function

\[ U = 1, \quad U' = 0, \quad U'' = -1, \quad U''' = 0, \quad \text{&c.} \]

44. Let \( u \) denote the arc whose sine is \( x \); that is, let \( u = \sin^{-1} x \). In this case (23),

\[ \frac{du}{dx} = (1 - x^2)^{-\frac{1}{2}}, \] \[ \frac{d^2u}{dx^2} = x(1 - x^2)^{-\frac{3}{2}}, \] \[ \frac{d^3u}{dx^3} = (1 - x^2)^{-\frac{5}{2}} + 3x^2(1 - x^2)^{-\frac{3}{2}}, \] \[ \frac{d^4u}{dx^4} = 3 \cdot 3x(1 - x^2)^{-\frac{5}{2}} + 3 \cdot 5x^2(1 - x^2)^{-\frac{7}{2}}, \quad \text{&c.} \]

Making \( x = 0 \) we get \( U = 0, U' = 1, U'' = 0, U''' = 1, U^{IV} = 0 \); and, by continuing the process of differentiation, \( U^V = 3 \cdot 3, \quad \text{&c.} \) therefore

\[ u = x + \frac{x^3}{1 \cdot 2 \cdot 3} + \frac{3 \cdot 3x^5}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5} + \frac{3 \cdot 5x^7}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6 \cdot 7} + \text{&c.} \]

Cases in which Taylor's Theorem fails.

45. We have seen that Taylor's theorem, viz. \( u = f(x) \) being any function of a variable \( x \),

\[ f(x + h) = u + \frac{du}{dx}h + \frac{d^2u}{dx^2} \frac{h^2}{1 \cdot 2} + \frac{d^3u}{dx^3} \frac{h^3}{1 \cdot 2 \cdot 3} + \text{&c.} \]

will always hold true, provided that the variable \( x \) be regarded as indeterminate. If however particular values be given to \( x \), there are cases in which the development is of no use, because in these cases it is not applicable to the function.

For example, let \( u = b + (x - a)^{\frac{1}{2}} \); in this case,

\[ \frac{du}{dx} = \frac{1}{2(x - a)^{\frac{1}{2}}} \frac{d^2u}{dx^2} = \frac{-1}{4(x - a)^{\frac{3}{2}}} \quad \text{&c.} \]

and by the theorem,

\[ f(x + h) = b + (x - a + h)^{\frac{1}{2}} = b + (x - a)^{\frac{1}{2}} + \frac{1}{2(x - a)^{\frac{1}{2}}}h - \frac{1}{8(x - a)^{\frac{3}{2}}}h^2 + \text{&c.} \]

Now if we give to \( x \) any particular value greater than \( a \), the development is perfectly significant, and expresses truly the value of \( b + (x - a + h)^{\frac{1}{2}} \). If however we give to \( x \) the particular value \( a \), then \( x - a = 0 \), and the development becomes \( b + 0 + \frac{h}{2(0)^{\frac{1}{2}}} - \frac{h^2}{8(0)^{\frac{3}{2}}} + \text{&c.} \)

from which no conclusion whatever can be drawn, because all the terms having zero in the denominator are infinite.

If in the expression \( b + (x - a + h)^{\frac{1}{2}} \) we make \( x = a \), it becomes \( b + h^{\frac{1}{2}} \). This expression has two values, viz. \( b + \sqrt{h}, b - \sqrt{h} \). The development given by the theorem contains, however, only positive integer powers of \( h \); therefore it can have only a single value; and hence it is impossible that it should express the function in the particular case of \( x = a \). In such cases the theorem has been said to fail; this however happens not from any imperfection in the theorem, but merely because it is unsuitable to the purpose to which it has been applied; and this want of applicability is indicated by the co-efficients of the powers of \( h \) failing to express anything definite.

46. Again, let the function be \( u = \frac{b}{(x-a)^{\frac{1}{2}}} \); then This is an identical equation, involving \( x \) only, which must hold true whatever value be given to \( x \). Let the expression \( f(x, X) = f(x, y) \) be denoted briefly by \( u \). Suppose now \( x \) to change its value and become \( x + h \); by Taylor's theorem, its new values will be

\[ u' = u + \frac{du}{dx} h + \frac{d^2u}{dx^2} h^2 + \frac{d^3u}{dx^3} h^3 + \ldots \]

The equation \( f(x, y) = u = 0 \) must hold true whatever value be given to \( x \); therefore we must have \( u' = 0 \), and this requires that the co-efficients of the different powers of the indeterminate quantity \( h \) be separately \( = 0 \) (art. 18), so that from the equation \( u = f(x, y) = 0 \) there may be derived a series of equations,

\[ \frac{du}{dx} = 0, \quad \frac{d^2u}{dx^2} = 0, \quad \frac{d^3u}{dx^3} = 0, \ldots \]

which must all hold true at the same time as the original equation.

50. Let us take for example the equation

\[ y^2 + x^2 = a^2, \quad \text{or} \quad y^2 + x^2 - a^2 = 0 \quad \ldots \quad (1) \]

in which \( a \) is a constant quantity; in this case,

\[ u = y^2 + x^2 - a^2; \]

hence we have by differentiation

\[ \frac{du}{dx} = 2y \frac{dy}{dx} + 2x = 0, \]

and

\[ \frac{d^2u}{dx^2} = -\frac{x}{y} \quad \ldots \quad (2). \]

To determine the differential co-efficient of the second order, put \( \frac{dy}{dx} = p \), and we have

\[ \frac{du}{dx} = 2yp + 2x; \]

as \( p \) is a function of \( x \) and \( y \), and \( y \) is a function of \( x \), therefore \( p \) is a function of \( x \). Taking now the differentials, and dividing by \( dx \), the result is

\[ \frac{d^2u}{dx^2} = 2 \left( \frac{dp}{dx} + \frac{dy}{dx} + 1 \right) = 0; \]

hence, because \( p = \frac{dy}{dx} \), and \( \frac{dp}{dx} = \frac{d^2y}{dx^2} \),

\[ \frac{d^2u}{dx^2} = 2 \left( \frac{d^2y}{dx^2} + \frac{dy}{dx} + 1 \right) = 0 \quad \ldots \quad (3). \]

From this equation we get

\[ \frac{d^2y}{dx^2} = -\frac{1}{y} \left( \frac{dy}{dx} + 1 \right), \]

or, since \( \frac{dy}{dx} = -\frac{x}{y} \),

\[ \frac{d^2y}{dx^2} = -\frac{x^2}{y^3} - \frac{1}{y} \quad \ldots \quad (4). \]

To find the differential co-efficient of the third order, we put \( \frac{dy}{dx} = p \); and substituting \( p \) for \( \frac{dy}{dx} \) in equation (3), it becomes

\[ \frac{d^3u}{dx^3} = 2 \left( yp + p^2 + 1 \right) = 0; \]

observing that \( p \) and \( q \) are functions of \( x \) and \( y \), and consequently of \( x \), we may find by differentiation an expression which will involve, besides \( x \) and \( y \), these quantities,

\[ \frac{dq}{dx}, \quad \frac{dp}{dx} = \frac{d^2y}{dx^2}, \quad q = \frac{d^2y}{dx^2}, \quad p = \frac{dy}{dx}; \]

and from this equation \( \frac{dq}{dx} = \frac{d^2y}{dx^2} \) may be determined, observing that \( \frac{d^2y}{dx^2} \) and \( \frac{dy}{dx} \) are expressed by \( x \) and \( y \) in equations (4) and (2).

51. The preceding analysis gives the following rule for determining the differential co-efficients of \( y \), a function of \( x \), when the relation between \( x \) and \( y \) is expressed by an equation.

Take the differentials of the terms of the equation, considering \( y \) as a function of \( x \), and dividing by \( dx \), the result will be an equation which gives the value of \( \frac{dy}{dx} \).

Again, take the differentials of the terms of this equation, considering \( y \) and \( \frac{dy}{dx} \) as functions of \( x \), and the result will be an equation involving \( \frac{d^2y}{dx^2} \) and \( \frac{dy}{dx} \), which, combined with the former, serves to determine \( \frac{d^2y}{dx^2} \). A third equation may be found from this, by taking the differentials, and considering \( y \), \( \frac{dy}{dx} \), and \( \frac{d^2y}{dx^2} \) as functions of \( x \); and this, combined with the other two, gives \( \frac{d^3y}{dx^3} \), and so on to any number of equations.

Let \( y \) be such a function of \( x \) that

\[ y^2 - 2mxy + x^2 = a^2 = 0 \quad \text{(1)} \]

It is proposed to determine its differential co-efficients.

By a first differentiation

\[ (y - mx) \frac{dy}{dx} - (my - x) dx = 0, \]

and

\[ \frac{dy}{dx} = \frac{my - x}{y - mx}. \quad \text{(2)} \]

Again, by differentiating,

\[ \frac{d^2y}{dx^2} = \frac{(1 - m^2)x}{(y - mx)^2} \cdot \frac{dy}{dx} - \frac{(1 - m^2)y}{(y - mx)^2}. \quad \text{(3)} \]

and, substituting for \( \frac{dy}{dx} \) its value,

\[ \frac{d^2y}{dx^2} = -(1 - m^2) \frac{y^2 - 2mxy + x^2}{(y - mx)^3} = -(1 - m^2)a^2. \quad \text{(4)} \]

If we take the differentials of both sides of equation (3), and consider \( \frac{dy}{dx}, \frac{d^2y}{dx^2} \) as functions of \( x \), we shall obtain

\[ \frac{d^3y}{dx^3} = P \frac{d^2y}{dx^2} + Q \frac{dy}{dx} + R. \quad \text{(5)} \]

Here \( P, Q, R \) denote expressions composed of \( y \) and \( x \).

By substituting for \( \frac{d^2y}{dx^2} \) and \( \frac{dy}{dx} \) their values as given by equations (4) and (2), there will be obtained the value of \( \frac{d^3y}{dx^3} \), in terms of \( x \) and \( y \).

52. The equations which may be deduced by differentiation from a proposed equation are called fluxional or differential equations, and the equation from which they have been deduced is called the primitive equation. The equation which gives the value of \( \frac{dy}{dx} \) is said to be of the first order, and that which involves \( \frac{d^2y}{dx^2} \) is of the second order, and so on for the higher orders.

Thus, the primitive equation being

\[ y^2 - 2mxy + x^2 = a^2 = 0, \]

we have found, for the differential equation of the first order,

\[ \frac{dy}{dx} = \frac{my - x}{y - mx} = 0; \]

and that of the second order,

\[ \frac{d^2y}{dx^2} = \frac{(1 - m^2)x}{(y - mx)^2} \cdot \frac{dy}{dx} - \frac{(1 - m^2)y}{(y - mx)^2} = 0, \]

equivalent to

\[ \frac{d^2y}{dx^2} + (1 - m^2) \frac{y^2 - 2mxy + x^2}{(y - mx)^3} = 0. \]

53. Differential equations of all orders hold true simultaneously with the primitive from which they have been derived, therefore any combination of them will also hold true; and whatever number of values the function \( y \) has (in the preceding example it has two), the differential co-efficients into which \( y \) enters will have the same number.

It has appeared (art 15) that in functions of a certain form the constants they contain vanish from their differentials. A like remark applies to differential equations. Thus, if \( y^2 = ax + b \) be a primitive equation, in which \( a \) and \( b \) are constants, the differential equation is \( 2ydy = adx \); and this belongs to every particular equation which can be formed from the primitive, by giving all possible values to \( b \). The constant \( a \) may also be made to disappear by putting the equation under this form,

\[ \frac{y^2}{x} = a + \frac{b}{x}; \]

for this by differentiation gives

\[ \frac{2ydy}{x^2} - \frac{y^2dx}{x^2} = -\frac{bdx}{x^2}, \]

and \( 2ydy - (y^2 - b)dx = 0 \); or \( \frac{dy}{dx} = \frac{y^2 - b}{2xy} \).

This equation expresses a relation which subsists between the quantities \( x, y, \frac{dy}{dx} \), independently of any particular value of the constant \( a \).

The very same equation will be obtained by eliminating, by the ordinary process (Algebra, sect. 7), \( a \) from the two equations

\[ y^2 = ax + b, \quad 2ydy = adx. \]

If the constant quantity which is eliminated is not of the first degree in the proposed equation, the result will contain powers of the co-efficient \( \frac{dy}{dx} \) higher than the first.

For example, let the equation be

\[ y^2 - 2ay + x^2 = a^2; \]

then, by differentiating,

\[ 2ydy - ady + xdx = 0, \]

hence \( a = \frac{ydy + xdx}{dy} \).

This value of \( a \) being substituted in the proposed equation, we obtain, after due reduction,

\[ (x^2 - 2y^2) \frac{dy^2}{dx^2} - 4xy \frac{dy}{dx} - x^2 = 0; \]

an equation that expresses a relation between \( x, y, \frac{dy}{dx} \), which is independent of any particular value of \( a \).

By resolving the equation in respect of \( a \), we find

\[ a = -y \pm \sqrt{2y^2 + x^2}; \]

and hence, the constant \( a \) forming a term by itself, it disappears by differentiation, and we have

\[ \frac{dy}{dx} = \frac{2ydy + xdx}{\sqrt{2y^2 + x^2}} = 0, \]

an equation which, being from the radical sign, will be identical with the former.

54. Any number of constants whatever may be made to disappear by repeating the process of differentiation as often as there are quantities to be eliminated; for example, let the equation be \( y^2 = m(a^2 - x^2) \); by a first differentiation \( y \frac{dy}{dx} = -mx \); taking the differentials a se- cond time, we get \( y \frac{d^2y}{dx^2} + \frac{dy}{dx} = -m \). This value of \( m \) being substituted in the former equation, it becomes

\[ y \frac{dy}{dx} - x \frac{d^2y}{dx^2} - xy \frac{dy}{dx} = 0, \]

a result which is independent of both \( a \) and \( m \).

**Of Vanishing Fractions.**

55. When the numerator and denominator of a fraction are such functions that, by giving a particular value to their variable, they both become \( = 0 \) at the same time, the fraction is then called a *vanishing* fraction. The fraction \( \frac{x^2 - a^2}{x - a} \) is of this kind; for when \( x = a \), the numerator and denominator both vanish, and the fraction takes the form \( \frac{0}{0} \), from which we can draw no conclusion as to its true value, although it be evident that the fraction has then a determinate value; for

\[ \frac{x^2 - a^2}{x - a} = (x + a)(x - a) = x + a; \]

and when \( x = a \), its true value is as \( a + a = 2a \).

It appears that the above fraction has taken the peculiar form \( \frac{0}{0} \) when \( x = a \), from the circumstance of its having the factor \( x - a \) in the numerator and denominator. The same is true of this other fraction,

\[ \frac{x^3 - x^2a - xa^2 + a^3}{ax - a^2}, \]

which takes form \( \frac{0}{0} \) when \( x = a \). The numerator and denominator have a common divisor, \( x - a \); when freed from this, it becomes \( \frac{x^2 - a^2}{a} \); when \( x = a \), this fraction becomes \( \frac{0}{a} \), which is truly \( = 0 \). Again, this fraction

\[ \frac{ax - a^2}{x^2 - 2ax + a^2}, \]

which has the same property when \( x = a \), by reason of the common factor \( x - a \), when freed from the factor is \( \frac{a}{x - a} \). Thus when \( x = a \) becomes \( \frac{a}{0} \), that is, the value is infinite. In these cases the common factor is obvious; but it is not so always. Take this as an example, \( \frac{1 - \sin x + \cos x}{\sin x + \cos x - 1} \); when \( x \) is a quadrant, or \( = \frac{\pi}{2} \), this fraction becomes \( \frac{0}{0} \). For its true value in this case, see ex. 4 of art. 57.

56. When the terms of the fraction are algebraic functions, their greatest common measure may be found by an elementary operation in *Algebra* (art. 29), and the fraction disengaged from it by division. A more simple and general solution may, however, be obtained from the differential calculus.

Let \( \frac{P}{Q} \) denote a fraction, the terms of which are functions of \( x \) that vanish when \( x = a \), some given quantity. Suppose now that \( x \) becomes \( x + h \), then, by Taylor's theorem, the fraction will become

\[ \frac{P}{Q} + \frac{dP}{dx} \frac{h}{1!} + \frac{d^2P}{dx^2} \frac{h^2}{2!} + \frac{d^3P}{dx^3} \frac{h^3}{3!} + \ldots, \]

and

\[ \frac{Q}{Q} + \frac{dQ}{dx} \frac{h}{1!} + \frac{d^2Q}{dx^2} \frac{h^2}{2!} + \frac{d^3Q}{dx^3} \frac{h^3}{3!} + \ldots, \]

By hypothesis, when \( x = a \), then \( P \) and \( Q \) both vanish; therefore, leaving them out of the expression, and putting \( P' \), \( P'' \), \( P''' \), &c. to denote the differential co-efficients of \( P \) in the numerator, and \( Q' \), \( Q'' \), \( Q''' \), &c. those of \( Q \) in the denominator, and dividing the terms by \( h \), the fraction will be expressed by

\[ \frac{P'}{Q'} + \frac{P''}{Q''} \frac{h}{1!} + \frac{P'''}{Q'''} \frac{h^2}{2!} + \ldots, \]

When \( h = 0 \), this expression becomes \( \frac{P'}{Q'} \); and when \( a \) is put in this fraction instead of \( x \), the result will be the true value of \( \frac{P}{Q} \); for it is manifestly the same thing to suppose, first, that \( x \) becomes \( x + h \), and then that \( x = a \) and \( h = 0 \), as to suppose at once that \( x = a \).

If it happen that one of the two quantities, \( P' \), \( Q' \), becomes \( = 0 \) when \( a \) is substituted instead of \( x \), then the fraction \( \frac{P}{Q} \) will be \( = 0 \), or infinite, according as the vanishing quantity is the numerator or denominator; if both become \( = 0 \) at once, then, leaving them out, the expression becomes, by reduction,

\[ \frac{P''}{Q''} + \frac{P'''}{Q'''} \frac{h}{1!} + \ldots, \]

and making \( h = 0 \), it becomes \( \frac{P''}{Q''} \), which, putting \( a \) instead of \( x \), will be the value of the fraction \( \frac{P}{Q} \), and so on.

57. Hence we have this rule: To find the fraction \( \frac{P}{Q} \) in the particular case of \( x = a \), supposing that \( P \) and \( Q \) are both reduced to \( 0 \) by this supposition. Divide the differential co-efficient of the numerator by that of the denominator; let the result be \( \frac{P'}{Q'} \), in which make \( x = a \); then, if this expression does not become \( \frac{0}{0} \), it is the value sought; but if it takes the form \( \frac{0}{0} \), treat the fraction \( \frac{P'}{Q'} \) in all respects like the original fraction, and deduce from it a new fraction \( \frac{P''}{Q''} \), and proceed in this way until an expression be found which does not become \( \frac{0}{0} \) when \( a \) is put for \( x \); and the first expression that is found having this property is the value sought.

Ex. 1. The sum of the geometrical series

\[ 1 + x + x^2 + x^3 + \ldots + x^{n-1} \]

is \( \frac{x^n - 1}{x - 1} \). Find its value when \( x = 1 \).

In this case, \( P = x^n - 1 \), \( Q = x - 1 \); hence we deduce

\[ \frac{dP}{dx} = nx^{n-1} = P'; \quad \frac{dQ}{dx} = 1 = Q; \quad \frac{P'}{Q'} = nx^{n-1}; \]

when \( x = 1 \), the fraction \( \frac{P'}{Q'} = n \), the value of the sum of the series, as is manifest.

Ex. 2. Let the fraction be \( \frac{ax^2 + ac^2 - 2acx}{bx^2 - 2bcx + bc^2} = \frac{P}{Q} \),

which becomes \( \frac{0}{0} \) when \( x = c \), to find its value. In this case \( \frac{dP}{dx} = 2ax - 2ac = P'; \quad \frac{dQ}{dx} = 2bx - 2bc = Q' \); Making \( x = c \), this fraction becomes \( \frac{0}{0} \) therefore, proceeding as before,

\[ \frac{d^2P}{dx^2} = 2a = P', \quad \frac{d^2Q}{dx^2} = 2b = Q', \]

hence it appears that when \( x = c \), the value of the fraction is \( \frac{a}{b} \).

Ex. 3. The fraction \( \frac{a^x - b^x}{x} \) becomes \( \frac{0}{0} \) when \( x = 0 \). Find its value then. In this case,

\[ \frac{dP}{dx} = 1, \quad (a) \quad a^x - 1, \quad (b) \quad b^x = P'; \quad \frac{dQ}{dx} = 1 = Q'. \]

When \( x = 0 \), \( \frac{P'}{Q'} \) becomes \( 1, (a) - 1, (b) = 1 \), the value required.

Ex. 4. Let the fraction be \( \frac{1 - \sin x + \cos x}{\sin x + \cos x - 1} \), which becomes \( \frac{0}{0} \) when \( x = \frac{1}{2}\pi \). Here

\[ \frac{dP}{dx} = -\cos x - \sin x = P', \quad \frac{dQ}{dx} = \cos x - \sin x = Q'; \]

\[ \frac{P'}{Q'} = \frac{-\cos x - \sin x}{\cos x - \sin x}. \]

When \( x = \frac{1}{2}\pi \), a quadrant, then \( \cos x = 0 \), and \( \sin x = 1 \); the value of the proposed fraction is therefore \( \frac{-1}{1} = 1 \).

Ex. 5. Let the fraction be \( \frac{a^n - x^n}{a - x} \); to find its value when \( x = a \). In this case \( P = a^n - x^n \), \( Q = a - x \).

\[ \frac{dP}{dx} = -x^n - n \left( 1 + \frac{x^n}{x} \right); \]

\[ \frac{dQ}{dx} = -1, \]

\[ \frac{P'}{Q'} = x^n - n \left( 1 + \frac{x^n}{x} \right); \]

making \( x = a \), we have \( a^n - n \left( 1 + \frac{a^n}{a} \right) \) for the value required.

58. The rule of last article will not apply when Taylor's theorem fails to give the developments of the functions \( P, Q \), in the case when \( x = a \). When this happens, we may substitute \( a + h \) instead of \( x \) in the fraction \( \frac{P}{Q} \) and expand the numerator and denominator into ascending series, which proceed by the powers of \( h \); we shall then have

\[ \frac{Ah^a + Bh^b + \ldots}{A'h'^a + B'h'^b + \ldots}, \]

instead of the proposed fraction. If in this result we suppose \( h = 0 \), we shall have the value of the fraction \( \frac{P}{Q} \) when \( x = a \). Now there are here three cases; viz. \( a > a' \), \( a = a' \), and \( a < a' \). In the first case, we may write the expression thus:

\[ \frac{Ah^a + Bh^b + \ldots}{A'h'^a + B'h'^b + \ldots}; \]

Under this form, it is manifest that as long as \( a \) is greater than \( a' \), the supposition of \( h = 0 \) will make the fraction equal to nothing, and that it will become \( \frac{A}{A'} \) when \( a = a' \); but when \( a < a' \), we write the expression thus:

\[ \frac{A + Bh^b - a + B'h'^b - a + \ldots}{A'h'^a - a + B'h'^b - a + \ldots}; \]

and by the supposition of \( h = 0 \), this result becomes infinite. In all cases the true value depends on the first term alone of each series.

59. The following rule extends to all fractions which can occur under the indeterminate form \( \frac{0}{0} \):

Take the first term of each of the series which express the development of the numerator and denominator when \( x = a + h \); reduce the resulting fraction to its most simple form, and then make \( h = 0 \).

The fraction \( \frac{(x^a - a)^2}{(x - a)^2} \), whose value cannot be found by differentiation, when \( x = a \) becomes by this method

\[ \frac{(2ah + h^2)^{\frac{3}{2}}}{h^2} = (2a + h)^{\frac{3}{2}}, \]

when \( x \) is changed into \( x + h \); making now \( h = 0 \), we have its true value \( (2a)^{\frac{3}{2}} \).

This last rule will sometimes be preferable to the former, by differentiation, even when that process is applicable. For example, four successive differentiations must be performed on the numerator and denominator of this fraction,

\[ \frac{x^2 + 4ax^2 + 7a^2x - 2a^3 - 2a^2 \sqrt{2ax - a^2}}{x^2 - 2ax - a^2 + 2a \sqrt{2ax - a^2}}, \]

in order to find its true value when \( x = a \); by writing \( a + h \) instead of \( x \), it becomes

\[ \frac{2a^3 - 2a^2h - ah^2 + h^2 - 2a^2 \sqrt{a^2 + 2al}}{-2a^2 + h^2 + 2a \sqrt{a^2 - h^2}}; \]

and reducing the radical quantities into series, we have

\[ \sqrt{a^2 + 2ah} = a + h - \frac{h^2}{2a} + \frac{h^3}{2a^2} - \frac{5h^4}{8a^3}, \ldots; \]

\[ \sqrt{a^2 - h^2} = a - \frac{h^2}{2a} - \frac{h^4}{8a^3}, \ldots; \]

By substituting these series in the fraction, and putting \( h = 0 \), we obtain \( -5a \) for the value sought.

Of the greatest and least Values of a Function.

60. In supposing that a function changes its value, we may assume that the variable increases continually from 0 to infinity, either positively or negatively. To get distinct notions, it may be convenient to suppose that the variable increases or decreases by successive equal differences. The function however may not increase or decrease continually along with its variable. It may first increase to a certain magnitude, and afterwards decrease; or it may decrease and afterwards increase; or it may alternately increase and decrease several times. We may represent the variable \( x \) by the abscissa of a curve, and the function \( y \) by the corresponding ordinate. The curve, from its nature, may recede continually from its axis, like the parabola and hyperbola; or it may first recede from and afterwards approach to the axis, like the circle and ellipse; or it may be sinuous, alternately approaching and receding.

When a variable function, by a continual change in its magnitude, first increases and afterwards decreases, there will be one value of the function which will be greater than those which immediately preceded it, and also greater than those which immediately follow it. In this state the value of the function is the greatest possible, at least within certain limits on each side of that extreme value; and then it is said to be a maximum. On the other hand, if a function, by a continual change in its value, first decreases, until it be reduced to a certain magnitude, and afterwards increases, it is said to be a minimum, when it is just changing from the state of a decreasing to that of an increasing quantity.

A variable function may have several values, each of which will be a maximum or minimum, according to the definition. The essential character of a maximum consists in its being greater than the values which preceded and which follow it; and that of a minimum in being less than the preceding and following values. So that as many changes as there may be from increase to decrease, or the contrary, so many maxima and minima there will be.

61. Before proceeding farther, it will be convenient to establish this principle. In a series of the form

\[ ph + qh^2 + rh^3 + sh^4 + \ldots \]

in which \( h \) is indeterminate, and independent of \( p, q, r, \ldots \), such a value may be given to \( h \), that any term shall exceed the amount of all that follow it. For, putting the series under this form,

\[ h(p + qh + rh^2 + sh^3 + \ldots) \]

by supposing \( h \) to decrease, \( qh + rh^2 + sh^3 + \ldots \) may become less than any assignable quantity, and therefore less than \( p \); consequently, \( rh^2 + sh^3 + \ldots \) may be less than \( ph \), in any ratio of inequality. In the same way it will appear that \( rh^2 + sh^3 + \ldots \) may be less than \( qh + rh^2 + \ldots \), and so on. Hence it follows, that in the series \( ph + qh^2 + rh^3 + sh^4 + \ldots \), \( h \) may have such a value that the amount of all the terms beginning with any assigned term shall have the sign of that term, \( + \) if it be \( + \), and \( - \) if it be \( - \).

62. Let \( y \) denote any function which may become a maximum or minimum when its variable has attained some particular value; then, if \( x = h \) and \( x = h \) be substituted successively in the function instead of \( x \), the two results must be both less than the maximum value, and both greater than the minimum value. Denoting now the value of the function which corresponds to \( x = h \) by \( y_1 \), and the value which corresponds to \( x = h \) by \( y_2 \), we have, by Taylor's theorem,

\[ y = y - \frac{dy}{dx} h + \frac{d^2y}{dx^2} \frac{h^2}{2!} - \frac{d^3y}{dx^3} \frac{h^3}{3!} + \ldots \]

\[ y_1 = y + \frac{dy}{dx} h + \frac{d^2y}{dx^2} \frac{h^2}{2!} + \frac{d^3y}{dx^3} \frac{h^3}{3!} + \ldots \]

Now, it is impossible that \( y \) and \( y_1 \) can at the same time be both less or both greater than \( y \), if the coefficient \( \frac{dy}{dx} \) be not \( = 0 \); for if it have some value different from zero, then such a value may be given to \( h \) that \( \frac{dy}{dx} h \) shall exceed in any given ratio of inequality all the terms which follow it, and one of the two series beginning with \( \pm \frac{dy}{dx} h \) will be a positive quantity and the other negative; and if this were true, \( y \) would be of an intermediate magnitude between \( y \) and \( y_1 \), which cannot be, when it is a maximum or minimum. Therefore, in order that \( y \) may be a maximum or minimum, we must have \( \frac{dy}{dx} = 0 \), and then, in the expressions

\[ y = y + \frac{d^2y}{dx^2} \frac{h^2}{2!} - \frac{d^3y}{dx^3} \frac{h^3}{3!} + \ldots \]

\[ y_1 = y + \frac{d^2y}{dx^2} \frac{h^2}{2!} + \frac{d^3y}{dx^3} \frac{h^3}{3!} + \ldots \]

such a value may be given to \( h \) as shall make \( y \) and \( y_1 \) both less than \( y \) in the case of a maximum, or both greater in that of a minimum.

It may, however, happen that the value of \( x \) which makes \( \frac{dy}{dx} = 0 \) shall also make \( \frac{d^2y}{dx^2} = 0 \), so that we have

\[ y = y - \frac{d^3y}{dx^3} \frac{h^3}{3!} + \frac{d^4y}{dx^4} \frac{h^4}{4!} + \ldots \]

\[ y_1 = y + \frac{d^3y}{dx^3} \frac{h^3}{3!} + \frac{d^4y}{dx^4} \frac{h^4}{4!} + \ldots \]

Then the function cannot be a maximum or minimum, unless the same value of \( x \) make \( \frac{d^3y}{dx^3} = 0 \), and give a finite value for \( \frac{d^4y}{dx^4} \).

63. On the whole, we have this rule, to determine the maximum or minimum value of \( y \), a function of \( x \).

Make the differential co-efficient \( \frac{dy}{dx} = 0 \): Find the value of \( x \), and substitute it in \( \frac{d^2y}{dx^2} \); and if the result be negative, the function is a maximum; but if it be positive, the function is a minimum. And if it be \( = 0 \), then put \( \frac{d^3y}{dx^3} = 0 \), and find \( x \), and substitute it in \( \frac{d^4y}{dx^4} \); and draw the same conclusions from the signs as before, and so on.

The rule for determining a maximum or minimum of a function was first given correctly by Maclaurin, in his Fluxions, chap. ix.

64. We shall now apply the theory to the resolution of problems, of which there may be an endless variety, and many highly interesting.

Ex. 1. To determine whether the function \( u = 2ax - x^2 \) has a maximum or minimum value.

In this case the differential co-efficient \( \frac{du}{dx} = 2(a-x) \), which being put \( = 0 \), we have \( x = a \). The second differential co-efficient \( \frac{d^2u}{dx^2} = 2 \), a negative quantity, therefore the function has a maximum value (art. 63), viz. when \( x = a \); and that value is \( u = a^2 \), as is indeed evident from the nature of the function.

Ex. 2. Let \( u = x^3 + 3x + 2 \); then \( \frac{du}{dx} = 2x + 3 \). This put \( = 0 \), gives \( x = -\frac{3}{2} \). Again, \( \frac{d^2u}{dx^2} = 2 \), a positive quantity, therefore the function is a minimum when \( x = -\frac{3}{2} \). In that case \( u = -\frac{5}{8} \).

Ex. 3. Let \( u = x^3 - 15x^2 + 56x - 60 \). Here \( \frac{du}{dx} = 3x^2 - 30x + 56 \), \( \frac{d^2u}{dx^2} = 6x - 30 \). The equation \( \frac{du}{dx} = 3x^2 - 30x + 56 = 0 \) being resolved, we find \( x = \frac{15 \pm \sqrt{57}}{3} \). Making \( x = \frac{15 \pm \sqrt{57}}{3} \), we find

\[ \frac{d^2u}{dx^2} = 6x - 30 = \pm 2\sqrt{57}. \]

Hence it appears that the function has a minimum, and also a maximum value; the former corresponding to \( x = \frac{15 + \sqrt{57}}{3} \), and the latter to \( x = \frac{15 - \sqrt{57}}{3} \).

If \( x = \frac{15 + \sqrt{57}}{3} \), \( u = -30 - \frac{38}{9}\sqrt{57} \), a minimum.

If \( x = \frac{15 - \sqrt{57}}{3} \), \( u = -30 + \frac{38}{9}\sqrt{57} \), a maximum.

Ex. 4. Let \( u = x^5 - 5x^4 + 5x^3 + 1 \);

then \( \frac{du}{dx} = 5x^4 - 20x^3 + 15x^2 = 5x(x^2 - 4x + 3) \);

\[ \frac{d^2u}{dx^2} = 20x^3 - 60x^2 + 30x = 10x(2x^2 - 6x + 3). \]

The equation \( \frac{du}{dx} = 5x(x^2 - 4x + 3) = 0 \),

gives four values of \( x \), viz. \( x = 3, x = 1 \), the other two are \( = 0 \).

These being substituted in \( \frac{d^2u}{dx^2} \), show that the function has a maximum and minimum:

when \( x = 3 \), \( u = -26 \), a minimum;

\( x = 1 \), \( u = 2 \), a maximum;

when \( x = 0 \), then \( \frac{d^2u}{dx^2} \) vanishes, therefore there is neither maximum or minimum in this case.

Ex. 5. To determine the maxima and minima values of the function \( y = x^x \); we have found (art. 24, ex. 2), that

\[ \frac{dy}{dx} = x^x(1 + 1.x); \]

and hence \( \frac{d^2y}{dx^2} = x^x \left\{ \frac{1}{x} + (1 + 1.x)^2 \right\} \);

The function \( x^x \) cannot be \( = 0 \); we must therefore put \( 1 + 1.x = 0 \);

hence \( 1.x = -1 \), and \( x = e^{-1} = \frac{1}{e} \).

This value of \( x \), substituted in the second differential coefficient, makes

\[ \frac{d^2u}{dx^2} = \left( \frac{1}{e} \right)^{\frac{1}{e}}, \text{a positive quantity}; \]

therefore, when \( x = \frac{1}{e} \), \( x^x = \left( \frac{1}{e} \right)^{\frac{1}{e}} \), a minimum.

Ex. 6. To find the greatest space that can be enclosed by four given straight lines:

Let the given lines be \( AB = a \), \( BC = b \), \( CD = c \), \( DA = f \). Let \( x = \text{angle } B \), \( v = \text{angle } D \), \( y = \text{the area } ABCD \); Draw the diagonal \( AC \).

By trigonometry, \( a^2 + b^2 - 2ab \cos x = AC^2 \);

\[ c^2 + f^2 - 2cf \cos v = AC^2; \]

Hence, \( 2ab \cos x - 2cf \cos v = a^2 + b^2 - c^2 - f^2 \)...........(1)

And because trian. \( ABC = \frac{1}{2}ab \sin x \); trian. \( ADC = \frac{1}{2}cf \sin v \),

therefore, \( y = \frac{1}{2}ab \sin x + \frac{1}{2}cf \sin v \).........................(2)

This last equation by differentiation gives, by the nature of a maximum,

\[ \frac{dy}{dx} = \frac{1}{2}ab \cos x + \frac{1}{2}cf \cos v = 0 \].................(3)

By differentiating equation (1), we find

\[ ab \sin x - ef \sin v = 0; \]

and hence \( ef \frac{dv}{dx} = ab \frac{\sin x}{\sin v} \);

Therefore, from eq. (3), \( ab \left\{ \cos x + \frac{\sin x \cos v}{\sin v} \right\} = 0 \);

and hence \( \sin v \cos x + \cos v \sin x = 0 \);

that is (Algebra, art. 239), \( \sin(x + v) = 0 \), and \( x + v = \pi \).

Thus it appears that the opposite angles \( B, D \) must together make two right angles; therefore the figure will be a maximum when it is inscribed in a circle; and since in this case

\[ \cos x = -\cos v, \sin x = \sin v; \]

from equations (1) and (2),

\[ 2(ab + ef) \cos x = a^2 + b^2 - c^2 - f^2, \]

\[ 2(ab + ef) \sin x = 4y. \]

By squaring both sides of these equations, and adding, and observing that \( \cos^2 x + \sin^2 x = 1 \), we have

\[ 4(ab + ef)^2 = (a^2 + b^2 - c^2 - f^2)^2 + 16y^2; \]

and \( 16y^2 = 4(ab + ef)^2 - (a^2 + b^2 - c^2 - f^2)^2 \).

The right-hand side of this formula admits of being resolved into these two factors, viz.

\[ 2ab + 2ef + a^2 + b^2 - c^2 - f^2 = (a + b)^2 - (e - f)^2; \]

\[ 2ab + 2ef - a^2 - b^2 + c^2 + f^2 = (e + f)^2 - (a - b)^2; \]

and these again admit of this further resolution,

\[ (a + b + e - f)(a + b - e + f), \]

\[ (e + f + a - b)(e + f - a + b); \]

If we now put \( s = \frac{1}{2}(a + b + e + f) \),

we have \( a + b + e - f = 2(s - f) \),

\( a + b - e + f = 2(s - e) \),

\( e + f + a - b = 2(s - b) \),

\( e + f - a + b = 2(s - a) \).

Hence \( 16y^2 = 16(s - a)(s - b)(s - e)(s - f) \);

and \( y = \sqrt{(s - a)(s - b)(s - e)(s - f)} \).

We have now found by the calculus an elegant geometrical theorem.

Ex. 7. To determine the dimensions of a cyindric measure, open at top, which shall contain a given quantity of grain, liquor, &c. under the least internal superficies. Let \( x = AB \) be the diameter of the base, \( v = AC \) its depth, \( \sigma \) the number \( 3\cdot1416 \) (viz. half the circumference of a circle whose rad. = 1), and \( c \) the content of the cylinder. By Geometry, the area of the base \( = \frac{\pi r^2}{4} \), and its circumference \( = \pi x \), and the internal surface of the cylinder \( = \pi x r \). Now, the content of the cylinder \( c = \frac{\pi x^2}{4} \); and therefore \( v = \frac{4c}{\pi x^2} \); therefore the internal surface \( = \frac{4c}{x} \); and putting \( y \) for the whole inside surface of the measure,

\[ y = \frac{\pi x^2}{4} + \frac{4c}{x}, \]

therefore,

\[ \frac{dy}{dx} = \frac{\pi x}{2} - \frac{4c}{x^2} = 0; \]

\[ \frac{d^2y}{dx^2} = \frac{\pi}{2} + \frac{8c}{x^3}. \]

The first of these equations gives

\[ \pi x^3 = 8c = 2\pi x^2 e, \quad \text{and } x = 2v; \]

hence the diameter of the base must be twice the depth:

The second differential co-efficient being a positive quantity, shows the surface to be a minimum.

Ex. 8. A candle stands on a horizontal table, directly over a point at a given distance from a small object on the table; what ought to be the height of the flame when the object is illuminated the most possible.

Fig. 3.

Let \( A \) be the object on the table, \( B \) the point under the candle, and \( C \) the flame, considered as condensed at a point. The intensity of the illumination on the object \( A \) depends on its distance from \( C \), and on the angle which the rays make with the surface (supposed to be horizontal). By the principles of optics, the intensity at different distances, the angle of obliquity being the same, will be inversely as the square of the distance; and with different degrees of obliquity, the distance being the same, as the sine of the angle which the rays make with the surface. Therefore the intensity, as depending on both obliquity and distance, will be expressed by

\[ \frac{1}{AC^2} \sin CAB = \frac{BC}{AC^3}. \]

Put \( a = AB \), \( x = \sin CAB \), and let \( y \) denote the illuminating power on the surface at \( A \); then

\[ y = \frac{BC}{AC} \times \frac{AB^2}{AC^4} \times \frac{1}{AB^5} = \frac{\sin x \cos^2 x}{a^2}, \]

\[ \frac{dy}{dx} = \left\{ \cos^3 x - 2 \sin^2 x \cos x \right\} \frac{1}{a^3} = 0; \]

hence, \( 2 \sin^2 x \cos x = \cos^3 x \),

and \( \sin^2 x = \frac{1}{1 + \cos^2 x} = \frac{1}{2} \), and \( \cos^2 x = \frac{1}{2} \).

By the trigonometrical tables, \( 2x = 70^\circ 32' \), and \( z = 35^\circ 16' \); this gives \( BC = AB \times .71 \) nearly: so that the height of the flame must be about \( \frac{1}{10} \) of the distance \( AB \).

Ex. 9. To find the position of the planet Venus in respect of the earth and sun when her light is the greatest.

The planet does not appear brightest when her disc is perfectly round; she is then too remote to produce that effect; and besides, she is seen in the direction of the sun. In her inferior conjunctions her crescent is too narrow, almost the whole illuminated part being turned towards the sun. It is therefore in some intermediate position, which is to be determined, that she is brightest.

Let \( S \) be the sun, \( E \) the earth, and \( ACBD \) Venus, \( ADB \) its illuminated hemisphere which is turned towards the sun, and \( CBD \) its hemisphere towards the earth: produce \( SV \) to \( F \).

Fig. 4.

The portion of the illuminated surface turned towards the earth is contained between two planes \( DV, BV \), perpendicular to the plane \( ESV \); and this surface will manifestly be projected into a crescent, the breadth of which is the versed sine of the angle \( BVD \), which is equal to \( EVF \), because if \( BVE \) be added to both, each is a right angle. Now the area of the crescent is always as its breadth; therefore, the whole disk being taken as an unit, the illuminated part of it will be expressed by the versed sine of the angle \( EVF \), or by \( 1 + \cos ESV \).

Again, the brightness of the planet at different distances is inversely as the square of the distance; therefore the brightness depending on its position in respect of the sun and its distance from the earth jointly, will be proportional to \( \frac{1 + \cos ESV}{ESV^2} \).

Let \( a = ES \) the distance of the earth from the sun,

\( b = VS \) the distance of Venus from the sun,

\( x = VE \) the distance of Venus from the earth;

Then, \( \cos ESV = \frac{x^2 + b^2 - a^2}{2bx} \),

and \( 1 + \cos ESV = \frac{x^2 + 2bx + b^2 - a^2}{2bx} \)

\[ = \frac{(x + b + a)(x + b - a)}{2bx}; \]

therefore, putting \( y \) to denote the brightness of the planet,

\[ y = \frac{(x + b + a)(x + b - a)}{2bx} \quad \text{a maximum}; \]

and this, by differentiation, gives

\[ \frac{dy}{dx} = \frac{(x + b + a)(x + b - a)}{2bx^2} \left\{ \frac{1}{x + b + a} + \frac{1}{x + b - a} - \frac{3}{x} \right\} = 0; \]

and hence, \( \frac{3(a^2 - b^2) - 4bx - x^2}{2bx^2} = 0 \);

\[ x^2 + 4bx = 3(a^2 - b^2), \]

\( x = \pm \sqrt{3a^2 + b^2 - 2b} \).

The negative value of \( x \) is not applicable, and we have

\[ x = \sqrt{3a^2 + b^2 - 2b}. \]

In numbers, \( a = 10000, b = 7233 \), therefore \( x = 4304 \); the angles \( E = 39^\circ 43' 30'', V = 117^\circ 55' 20'', S = 22^\circ 21' 10'' \).

Ex. 10. Let \( ADBBA' \) be a kirb roof supported by Direct four rafters \(AB\), \(BD\), \(DB'\), \(B'A'\), two and two of which are equal, viz. \(AB = A'B'\) and \(BD = B'D\), and all perfectly free to turn about the points \(A\), \(B\), \(D\), \(B'\), \(A'\) as joints. Suppose also that \(AA'\) the span, and \(DC\) the height of the roof, are given, and that \(AB\) has to \(BD\) a given ratio. Determine the position of the rafters when by their weight they form an equilibrium.

Fig. 5.

Suppose the roof to be inverted so as to form a polygon hanging suspended from the points \(A\), \(A'\); by a known principle in statics, when it forms an equilibrium, the centre of gravity of the mass of matter in the polygon will be the lowest possible. If now the polygon were reversed and placed in its position as a roof, without changing the angles at the joints, the equilibrium would not be disturbed, and in this position the centre of gravity would be the highest possible.

Draw a line from \(B\) to \(B'\), which will be parallel to \(AA'\); and draw the perpendiculars \(BE\), \(B'E'\). Put the angle \(BAC = B'A'C = x\), \(DBF = DB'F = v\); let the given ratio of \(AB\) to \(BD\) be that of a given number \(m\) to a given number \(n\); these numbers may represent the length, also the weight, of the rafters, which may be supposed placed in their centres of gravity, or middle of the beams.

By trigonometry, \(BE = m \sin x\), \(DF = n \sin v\), therefore \(DC = m \sin x + n \sin v\); and, by statics, the height of the centre of gravity of \(AB\), also of \(A'B'\), above \(AA\) will be \(\frac{1}{2} BE = \frac{1}{2} m \sin x\), and the height of the centre of gravity of \(BD\), also of \(B'D\), will be \(\frac{1}{2} (BE + DC) = m \sin x + \frac{1}{2} n \sin v\); and since the distance of the common centre of gravity of all the beams from \(AA'\) will be found by multiplying the distance of each centre by the mass, supposed placed at that centre, and adding the results into one sum, and dividing by the sum of all the masses; the height of the centre of gravity of the whole system of beams above \(AA'\) will be

\[ \frac{m^2 \sin x + n (2m \sin x + n \sin v)}{2(m + n)} \]

Let this height be denoted by \(y\); then, rejecting the constant denominator \(2(m + n)\),

\[ y = \frac{(m^2 + 2mn) \sin x + n^2 \sin v}{m + n}, \text{a maximum}; \]

and since \(AC\), half the span of the roof, is a given quantity, we have

\[ m \cos x + n \cos v = AC, \text{a constant}. \]

From the first equation,

\[ \frac{dy}{dx} = (m^2 + 2mn) \cos x + n^2 \cos v \frac{dv}{dx} = 0; \]

and from the second,

\[ m \sin x + n \sin v \frac{dv}{dx} = 0; \text{ hence } \frac{dv}{dx} = -\frac{m \sin x}{n \sin v}. \]

This value of \(\frac{dv}{dx}\) being substituted in the value of \(\frac{dy}{dx}\), we obtain

\[ (m^2 + 2mn) \cos x - mn \cos v \frac{\sin x}{\sin v} = 0; \]

and from this again,

\[ \tan x = \frac{m}{n} + 2\ldots\ldots\ldots\ldots\ldots\ldots\ldots(1.) \]

This formula expresses a property of all roofs made of four beams, which are equal two and two, when by their position they form an equilibrium.

Join \(AD\), and put \(\gamma = \text{angle } DAC\) which is known, because \(\tan \gamma = \frac{DC}{AC}\). The angle \(BAD = x - \gamma\), and \(BDA = \gamma - v\). Now, in the triangle \(ABD\), \(\sin A : \sin D = DB : BA\), that is, \(\sin (x - \gamma) : \sin (\gamma - v) = n : m\); therefore,

\[ \frac{\sin (x - \gamma)}{\sin (\gamma - v)} = \frac{n}{m}\ldots\ldots\ldots\ldots(2.) \]

These equations (1), (2), determine the angles \(x, v\); when the ratio of \(AB\) to \(BD\) is given; from their form, the angles may be easily found by trials in the trigonometrical tables. The elimination of one of the angles would produce an equation of the fourth degree; for this reason, the method of trials is the best way of proceeding. When the angles \(x\) and \(v\) are known, every thing else may be readily found.

If we suppose the beams \(AB\), \(BD\), to be of equal length, then \(m = n\), and our two equations become

\[ \tan x = 3, \quad \sin (x - \gamma) = \sin (\gamma - v). \]

In this case, \(x + v = 2\gamma\); then, from the equation

\[ \tan x = \frac{\sin x \cos v}{\cos x \sin v} = 3, \]

we get

\[ \sin x \cos v - \cos x \sin v = \frac{\sin (x - v)}{\sin (x + v)} = \frac{1}{2}; \]

and hence \(\sin (x - v) = \frac{1}{2} \sin (x + v) = \frac{1}{2} \sin 2\gamma\);

and putting for \(v\) its value \(2\gamma - x\),

\[ \sin 2(x - \gamma) = \frac{1}{2} \sin 2\gamma. \]

This determines the angle \(x\) and \(v = 2\gamma - x\).

The theory of equilibriums opens a wide field for the application of the calculus, as must be evident from the preceding solution; but our limits will not permit us to enter on it farther.

**Determination of Tangents to Curves.**

65. Let \(HPK\) be any curve referred to an axis \(AC\); and let \(AB = x\) and \(PB = y\) be the co-ordinates of any point \(P\) in the curve; the point \(A\) being the origin of the abscissas. Let \(PT\), a tangent to the curve at \(P\), meet the axis in \(T\); and let \(SPQ\), any straight line drawn through \(P\), meet the curve in \(Q\) and the axis in \(S\). Draw \(QC\) perpendicular to the axis, and \(PR\) perpendicular to \(QC\); and let \(PR = BC = h\), and \(QC = v'\), the ordinate corresponding to the absciss \(x + h\).

Fig. 6.

By similar triangles, \(QR : RP = PB : BS\), that is,

\[ y' - y : h = y : BS. \]

Now, by Taylor's theorem (art. 31),

\[ y' - y = \frac{dy}{dx} h + \frac{d^2y}{dx^2} h^2 + \&c. \]

hence, dividing the first antecedent and consequent by \(h\), This proposition holds true whatever be the magnitude of \( h \), the increment of \( x \). But now let us suppose \( h \) to decrease continually; the point \( Q \) will approach to \( P \), and the line \( QPS \) approaching to coincidence with \( PT \), the point \( S \) will approach to \( T \); and when \( h \) vanishes, \( Q \) will be at \( P \), and \( S \) at \( T \), and \( BS \) will have become \( BT \); our proportion will now be

\[ \frac{dy}{dx} : 1 = y : BT. \]

The line \( BT \), between the ordinate \( PB \) and tangent \( PT \), is called the subtangent; from this proportion it appears, that \( x \) and \( h \) being co-ordinates at any point in the curve,

\[ BT, \text{ the subtangent, } = y - \frac{dy}{dx} = y \frac{dx}{dy}; \]

that is, the subtangent is equal to the ordinate divided by its differential co-efficient; or it is a third proportional to the differential of the ordinate, the differential of the abscissa and the ordinate. The subtangent being known, the position of the tangent is determined.

Let \( PD \), a perpendicular to the tangent at \( P \), and consequently to the curve, meet the axis in \( D \); the line \( PD \) is called a normal to the curve; and \( BD \), the segment of the axis between the ordinate and normal, is called the subnormal.

The triangles \( TBP \), \( PBD \) are similar, therefore

\[ TB : BP = BP : BD, \text{ or } dx : dy = y : BD, \]

hence the subnormal \( BD = \frac{dy}{dx} \).

Because the tangent of the angle \( PTB = \frac{PB}{TB} \), and the tangent of the angle \( PDB = \frac{PB}{BD} \); therefore,

\[ \tan \text{ angle } T = \frac{dy}{dx}, \text{ and } \tan D = \frac{dx}{dy}. \]

In the right-angled triangles \( TBP \), \( PBD \), we have \( PT^2 = PB^2 + BT^2 \), and \( PD^2 = PB^2 + BD^2 \); therefore,

\[ \tan \text{ angle } PT = y \sqrt{1 + \frac{dx^2}{dy^2}}; \]

\[ \text{normal } PD = y \sqrt{1 + \frac{dy^2}{dx^2}}. \]

66. We shall now apply these general formulæ to some particular curves.

Ex. 1. Let the curve be the parabola, and supposing \( AC \) to be the axis, and \( A \) its vertex; by the nature of the curve, putting \( a \) for the parameter, \( y^2 = ax \),

\[ \text{hence } 2ydy = adx; \quad \frac{dx}{dy} = \frac{2y}{a}, \]

subtan. \( BT = \frac{dx}{dy} y = \frac{2y^2}{a} = \frac{2ax}{a} = 2x \);

that is, the subtangent is double the absciss \( AB \).

Again, for the subnormal, we have

\[ \text{subnor. } BD = \frac{dy}{dx} y = \frac{adx}{2dx} = \frac{a}{2}. \]

Thus it appears that the subnormal is constant, and equal to half the parameter.

Ex. 2. Let the curve be an ellipse, of which \( OA \) half the greater axis \( = a \), \( OB \) half the lesser axis \( = b \); and supposing the origin of the co-ordinates to be at \( O \) the centre, let \( OQ = x \), \( PQ = y \). By the nature of the curve,

\[ a^2 : b^2 = (a + x)(a - x) : y^2 \text{ (Conics)}; \]

hence \( y^2 = \frac{b^2}{a^2}(a^2 - x^2) \), and \( ydy = -\frac{b^2}{a^2}xdx \);

and subtan. \( QT = \frac{dx}{dy} y = \frac{a^2y^2}{b^2x} = \frac{x^2 - a^2}{x} = \frac{(a^2 - x^2)}{x} \).

Fig. 7.

The general formula for the subtangent was found on the hypothesis that it was on the same side of the ordinate as the absciss; in this case they are supposed to be on opposite sides, and hence its value has the negative sign; leaving the direction of the subtangent in respect of the absciss out of consideration, we have

\[ QT = \frac{a^2}{x} - x, \text{ and } OT = \frac{a^2}{x}, \]

hence it appears that \( OQ : OA = OA : OT \).

When the axes are equal, the ellipse becomes a circle, but the subtangent is independent of the ratio of the axes.

Fig. 8.

Ex. 3. Let the curve be the hyperbola, of which \( a = OA \) the semitransverse axis, and \( b = \) the semiconjugate; then, putting \( OQ = x \), and \( PQ = y \), we have (by Conics),

\[ y^2 = \frac{b^2}{a^2}(x^2 - a^2), \quad ydy = \frac{b^2}{a^2}xdx; \]

subtan. \( QT = \frac{dx}{dy} y = \frac{a^2y^2}{b^2x} = \frac{x^2 - a^2}{x} = x - \frac{a^2}{x} \);

hence \( OT = OQ - QT = \frac{a^2}{x} \).

This shows that the tangent always passes between the centre and the vertex when \( x \) has a finite value; but when \( x \) becomes infinite, then \( OT = 0 \), and the tangent becomes an asymptote to the hyperbola.

Fig. 9.

Ex. 4. We shall next draw a tangent to the cycloid; but first the nature of the curve must be expressed analytically. Conceive a circle to roll along a straight line, moving always in the same plane, just as the wheel of a carriage rolls along a straight road. By this motion any given point in the circumference of the circle will describe Let HPK be such a curve, generated by the revolution of a variable radius AP about a given point A in AC, a line given in position. Through A draw AT' perpendicular to AP; let a tangent to the curve at P meet this line in T' and the axis in T; draw PQ perpendicular to the axis AC, and draw the normal PD, meeting TA in D. Then AT' is the subtangent and AD the subnormal. Let AQ = x, PQ = y, the revolving radius AP = r, the variable angle PAC = v, and the angle PTA = t. The lines x, y, r are now to be considered as functions of the angle v.

The angle TPA being the supplement of the sum of PTA and PAT; that is, of \(v + t\), we have (Algebra, art. 236 and 242),

\[ \tan \text{angle } TPA = -\tan (v + t) = \frac{\tan v + \tan t}{\tan v \tan t - 1}. \]

Now \(\tan v = \frac{y}{x}\) and \(\tan t = \frac{PQ}{TQ} = \frac{dy}{dx}\);

therefore \(\tan TPA = \frac{y - dy}{xdx} = \frac{zdy - ydx}{xdx + ydx}\).

Again, \(\frac{zdy - ydx}{xdx + ydx} = d\left(\frac{y}{x}\right) = d(\tan v) = \frac{dv}{\cos^2 v}\),

therefore \(zdy - ydx = dv \frac{x^2}{\cos^2 v} = r^2 dv\);

and \(xdx + ydy = \frac{1}{2}d(x^2 + y^2) = \frac{1}{2}d(r^2) = rdr\);

therefore \(\tan TPA = \frac{r^2 dv}{rdr} = \frac{rdv}{dr}\);

and since by trigonometry TA = AP tan TPA, therefore

subtangent TA = \(\frac{dv}{dr} r^2\);

also, since TA : AP = AP : AD; therefore

subnormal AD = \(\frac{dr}{dv}\).

Ex. Let the curve PH be the spiral of Archimedes, the nature of which is defined by the equation \(2\pi r = av\), where \(a = 2\) right angles, and \(a\) is a given line, viz. the value of \(r\) when \(v = 2\pi\). In this case

subtangent \(= \frac{dv}{dr} r^2 = \frac{2\pi r^2}{a}\).

68. In some curves the segment of the axis between the origin of the co-ordinates and the tangent increases with the variable \(x\), and may exceed any given line. In others the tangent cuts the axis always at a finite distance from the origin; it is then an asymptote to the curve. If from the subtangent QT = \(\frac{dx}{dy} y\) (fig. 8) the absciss \(AQ = x\) be subtracted, the remainder \(\frac{dx}{dy} y - x\) is the general expression for AT. Now if when \(x\) is infinite this expression is finite, we may conclude that the curve has asymptotes; but if it be infinite, the curve has no asymptotes.

In the conic hyperbola we found (ex. 3, art. 66)

\[ \frac{dx}{dy} y = x - \frac{a^2}{x^2}; \quad \text{therefore} \quad \frac{dx}{dy} y - x = -\frac{a^2}{x^2}; \]

when \(x\) is infinite, the expression \(-\frac{a^2}{x^2} = 0\), hence it follows that the hyperbola has an asymptote which passes through the centre.

In the parabola, ex. 1, we found \(\frac{dx}{dy} y = \frac{2y^2}{ax}\), therefore Put PB = r, the angle PBO = v, and the curvilinear space PCDB = s; also let BQ = x, PQ = y, then \( ds = d \) (space CDQP) + \( d \) (triangle PQB).

Now \( d \) (space CDQP) = \( -ydx \), which is negative, because \( x \) decreases when the space increases; and \( d \) (triangle PQB) = \( \frac{1}{2}xy \) = \( \frac{1}{2}ydx + \frac{1}{2}xdy \); therefore,

\[ ds = -ydx + \frac{1}{2}ydx + \frac{1}{2}xdy = \frac{1}{2}(xdy - \frac{1}{2}ydx) \]

\( = xd \left( \frac{y}{x} \right) \) (art. 12).

But in the right-angled triangle PBQ,

BQ = \( x = r \cos v \), \( \frac{y}{x} = \tan v \), \( \frac{dy}{dx} = \frac{dv}{\cos^2 v} \) (art. 22);

hence \( ds = \frac{1}{2}r^2 dv \).........................(3).

In this case The differential of the area is half the product of the square of the revolving radius, and the differential of the angle which it makes with the axis.

71. We have already found (art. 23) various expressions for the differential of the arc of one particular curve, viz. the circle. We shall now investigate general expressions for the differential of the arc of any curve. For this purpose it will be convenient to establish the following geometrical proposition.

Let CPMD be a curve, that between the points P and M continually approaches to or continually recedes from an axis AB. Let PQ, MN be ordinates to the axis at the extremities of the arc PM, and PK, MH tangents to the curve at the points P, M, meeting the ordinates in K and H: The length of the arc is always of an intermediate magnitude between the lengths of the tangents PK, MH.

Draw the chord PM, and let I be the intersection of the tangents. It is easy to see that FK, one of the tangents, meets the parallel ordinates in an angle K more acute than M, the angle made by the chord PM and ordinate MN; and the other in an angle H less acute; therefore the chord will be less than the former and greater than the latter. Now the arc is greater than the chord; therefore MH, the least of the tangents, is less than the arc PM.

Again, in the two triangles PIH, MIK, it is evident that the two parts PI, IK, of the longest tangent, are respectively greater than HI, IM, those of the shorter; therefore PK is longer than the sum of PI and IM; but the arc PM is by an axiom of geometry less than the sum of PI and IN, therefore the tangent PK is greater than the arc PM.

72. Let A be a given point in the axis, and C in the curve; draw PE, HF parallel to AB; put AQ = \( x \), AN = \( x' \), PQ = \( y \), CP = \( z \), CM = \( z' \); then PE = \( x' - x \), and arc PM = \( z' - z \). Because PM = \( z' - z \) is less than PK, but greater than MH, therefore Now \( \frac{PK}{PE} \) and \( \frac{HM}{HF} \) are the secants of the angles KPE and MHF; therefore the ratio \( \frac{z' - z}{x' - x} \) is expressed by a quantity less than the secant of one of these angles, and greater than the secant of the other angle. Suppose now \( z' - z = x' - x \), the terms of the ratio, to be continually diminished; the angles at H and P will approach to equality, and at last become equal, therefore the limit of the ratio, viz. \( \frac{dz}{dx} \), is equal to the secant of the angle KPE; but it was shown (art. 65) that the tangent of the angle which a line touching a curve makes with the axis, or any line parallel to the axis, is equal to \( \frac{dy}{dx} \) and consequently its secant to \( \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \); therefore

\[ \frac{dz}{dx} = \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \quad \text{and} \quad dz = \sqrt{dx^2 + dy^2} \tag{1} \]

Hence it appears that The square of the differential of the arc of a curve is equal to the sum of the squares of the differentials of its rectangular co-ordinates.

73. In the case of curves expressed by a polar equation, let A be the given point about which the variable radius AP turns; put \( v \) for the variable angle which the line AP = r makes with AB, a line given in position; let the rectangular co-ordinates be AQ = x, PQ = y; then

\[ x = r \cos v, \quad y = r \sin v; \]

\[ dx = -r \sin v \, dv + \cos v \, dr = -y \, dv + \frac{x}{r} \, dr; \]

\[ dy = r \cos v \, dv + \sin v \, dr = x \, dv + \frac{y}{r} \, dr; \]

\[ dx^2 = y^2 \, dv^2 + \frac{2xy}{r} \, dv \, dr; \]

\[ dy^2 = x^2 \, dv^2 + \frac{2xy}{r} \, dv \, dr; \]

therefore, \( dx^2 + dy^2 = (x^2 + y^2) \left( \frac{dv^2}{r^2} + \frac{dr^2}{r^2} \right) \).

But \( dx^2 + dy^2 = dx^2 \), and \( x^2 + y^2 = r^2 \);

therefore, \( dx^2 = r^2 \, dv^2 + dr^2 \),

and \( dz = \sqrt{r^2 \, dv^2 + dr^2} \). \tag{2}

This formula gives the differential of an arc of a curve in terms of the differential of the variable radius vector, and the angle which it makes with the axis.

74. We shall now give a method of defining a curve by a polar equation, which leads to a simple and elegant expression for the differential of an arc of the curve.

Let CPD be any curve, and A a given point in AB, a line given by position; draw a tangent PF at any point P in the curve, and a perpendicular AF from the given point A on the tangent; if now we know the relation between the perpendicular AF, and the angle FAB, which it makes with the line AB, the length of the perpendicular corresponding to any given value of the angle can be found; thus the position of the tangent will be known; the calculus gives a general formula for PF, the segment of the tangent between the point of contact and its intersection with the perpendicular; thus the point of contact P is known, and in this way may any number of points in the curve be determined.

75. Let D be another point in the curve, DH a tangent, and AH a perpendicular on the tangent, and let us put angle FAB = \( u \), perp. AF = \( p \), tan PF = \( t \), arc CP = \( z \); HAB = \( u' \), AH = \( p' \), DH = \( r \), arc CD = \( z' \).

From the position of the lines in the figure,

\[ HD - FP = HE + ED - (FK + KE - EP) = PE + ED - FK - (KE - HE). \]

By trigonometry, FK = AF, tan FAK = \( p \) tan (\( u' - u \)); and because in the equiangular triangles AFK, EHK, the angles at A and E are equal, therefore HE = KE, \( \cos(u' - u) \); hence the above equation may be expressed thus,

\[ t - t' = PE + ED - p \cdot \tan(u' - u) - \{ 1 - \cos(u' - u) \} KE; \]

and dividing by \( u' - u \),

\[ \frac{t - t'}{u' - u} = \frac{PE + ED}{u' - u} - \frac{p \cdot \tan(u' - u)}{u' - u} - \frac{1 - \cos(u' - u)}{u' - u} KE. \]

Suppose now the point D to approach to coincidence with P, we have these limits to the ratios:

\[ \lim_{u' \to u} \frac{t - t'}{u' - u} = \frac{dt}{du}; \quad \lim_{u' \to u} \frac{PE + ED}{u' - u} = \frac{dz}{du}; \]

\[ \lim_{u' \to u} \frac{\tan(u' - u)}{u' - u} = 1; \quad \lim_{u' \to u} \frac{1 - \cos(u' - u)}{u' - u} = 0; \]

hence, on the whole,

\[ \frac{dt}{du} = \frac{dz}{du} - p, \quad \text{and} \quad dz = dt + pdu, \]

or \( d(z - t) = pdu \). \tag{3}

In the figure from which the formula has been investigated, the arc and tangent lie in the same direction from the point of contact. If they had proceeded in contrary directions, the formula would have been \( d(z + t) = pdu \), so that, in general,

\[ d(z - t) = pdu. \tag{3} \]

The differential of the arc may be expressed independently of \( t \); for

\[ AH - AF = KH + AK - AF; \]

Now KH = HE, tan HEK = HE tan (\( u' - u \)), and AF = AK cos (\( u' - u \)); therefore, dividing by \( u' - u \) we have \[ \frac{p'}{u' - u} = HE \tan(u' - u) + \frac{1 - \cos(u' - u)}{u' - u} AK; \]

and, passing to the limits, observing that \( \lim_{u' \to u} HE = t, \)

\[ \lim_{u' \to u} \tan(u' - u) = 1, \text{ and } \lim_{u' \to u} \frac{1 - \cos(u' - u)}{u' - u} = 0, \]

we find \( \frac{dp}{du} = t; \) if the tangent and arc lie in contrary directions, then \( -\frac{dp}{du} = t, \) so that, including both cases in one formula, \( t = \pm \frac{dp}{du}. \)

The sign \( + \) is to be taken when the arc and tangent lie in the same direction, but the sign \( - \) when they have contrary directions; and because \( dt = \pm \frac{dp}{du}, \) therefore in every curve

\[ dz' = pdu + \frac{dp}{du} du = \left( p + \frac{dp}{du} \right) du. \]

This is the formula which was to be found.

**Generation of Curves by Evolution: Radius of Curvature: Evolutes.**

76. The method of generating curves by evolution is a geometrical theory invented by Huygens, and given in his *Horologium Oscillatorium*. He explained it upon geometrical principles, for his work appeared many years before the calculus was invented.


Let BHE be a model of a plane curve made of a solid material, having some thickness, and let one end of a thread HEC be fastened at a point B on the edge of the model, and applied along its convexity, so as to coincide with it entirely between B and E, the remaining portion EC of the thread forming a straight line touching the curve at E. Suppose now the thread to be gradually unlooped from the curve BE, keeping it always tight; the end C will now describe another curve CPD.

All curves, in which the curvature is neither infinitely great nor infinitely small, may be generated in this way by the evolution of a thread from another curve.

77. The curve EHB, by means of which the curve CPD is generated, is called its **Evolute**; and the curve CPD thus generated by evolution, is called the **Involute** of the other curve.

From the way in which the involute is generated from its evolute, we may infer,

(I.) That PH, the part of the thread disengaged from the arc EH, is a tangent to the evolute at H.

(2.) That the line PH is perpendicular to a tangent to the curve CPD at P, and that at the same time it describes an element of the curve, and an element of a circle whose momentary centre is H and radius HP.

(3.) That to a certain extent the part of the involute adjoining to P, and towards C, is more incurvated than a circle whose radius is PH; because, supposing P to proceed towards C, the variable radius PH is gradually becoming shorter. On the other hand, a portion of the curve from P towards D is less incurvated than a circle whose radius is CP, because the generating radius is continually becoming longer.

(4.) That since the curvature of the involute on one side of P, and adjoining to it, is less than that of a circle whose radius is PH, and the curvature on the other side of P is greater, it follows that at the point P the curvature of the involute is exactly the same as that of a circle described with the radius HP. The point H is therefore called the **centre of curvature** of the curve CPD in the point P; and PH, its **radius of curvature** at that point; and a circle described with the line PH is the **circle of equal curvature** with the curve at the point P. From the close union of the circle and curve they have been said to osculate. It must, however, be kept in mind that the two curves have no common segment; they meet in a geometrical point at P, and separate on both sides of it. We may also infer that the circle of curvature has a more intimate contact with the curve than any other circle that passes through that point.

(5.) That the evolute of a curve is the **locus** of the centre of curvature of every point in the curve.

78. We shall now investigate general formulae for the radius of curvature, and for the evolute of any curve, by which these may be found when the nature of the curve is known.

Let AB be the common axis of the involute CPD, and its evolute EHB, and A the origin of co-ordinates; let H be the centre, and PH the radius of curvature at P; and H' the centre, and PH' the radius of curvature of another point P', near the former. Let the radii produced, if necessary, meet the axis in N and N'; and let O be the point in which they intersect each other.

Put AQ = z, PQ = y, arc CP = z, arc CP' = z', angle PNA = v, angle PN'A = v', rad. of curv. PH = r.

By trigonometry, PG = OG · sin POG; PG = OG · sin POG; therefore, OG = \[ \frac{PG + PG}{\sin POG + \sin P'OG}. \]

Suppose now the points P, P' to approach continually towards coincidence, the line GO will manifestly have for its limit PH, the radius of curvature at P. The limit of the sum of the tangents PG, PG' will be the arc PP' = z' - z. The limit of the sines of the angles POG, POG' will be the arcs which measure the angles, and that of the sum of the sines will be the measure of the angle POP = NON = v' - v. Hence it appears that \( r = \lim_{z' \to z} \frac{dz}{dv} \) and therefore, that

\[ r = \frac{dz}{dv}. \]

This is our first general expression for the radius of curvature. Put \( \frac{dy}{dx} = p, \frac{d^2y}{dx^2} = \frac{dp}{dx} = q. \)

Because (art. 72) \( dz = \sqrt{dx^2 + dy^2} = dx \sqrt{1 + \frac{dy^2}{dx^2}}, \) therefore \( dz = dx \sqrt{1 + p^2}. \)

When treating of tangents to curves, we found (art. 65) that \( \tan v = \frac{dx}{dy}, \) therefore, \( \tan v = \frac{1}{p}, \sec v = \frac{\sqrt{1 + p^2}}{p}, \) and \( d(\tan v) = dx \sec^2 v = -\frac{dp}{p^2} \);

hence, \( \frac{1 + p^2}{p^2} \cdot \frac{dx}{dy} = -\frac{qdx}{p^2} \), and \( de = -\frac{qdx}{1 + p^2} \).

If now the values of \( dz \) and \( dv \) be substituted in our first formula for the radius of curvature, there is obtained

\[ r = -\frac{1}{q}(1 + p^2)^{\frac{3}{2}} = -\frac{(dx^2 + dy^2)^{\frac{3}{2}}}{dxdy^2} \]

This is a second general formula for the radius of curvature.

Let \( s \) denote the normal PN. In the right-angled triangle PQN, by trigonometry, \( \sin N = \frac{PN}{PN} \); that is,

\[ \sin v = \frac{y}{s}; \text{ but since } \]

\[ \tan v = \frac{dx}{dy} = \frac{1}{p}, \text{ therefore } \frac{y}{s} = \sin v = \frac{1}{\sqrt{1 + p^2}}, \]

and \( \sqrt{1 + p^2} = \frac{s}{y} \); and \( (1 + p^2)^{\frac{3}{2}} = \frac{s^3}{y^3} \).

Referring now to our second formula, we have

\[ r = -\frac{s^3}{qy^3} = -\frac{s^3}{y^3} \cdot \frac{dx^2}{dy^2} \]

This is a third formula for the radius of curvature.

If we suppose the nature of the curve (see fig. 14, art. 74) expressed by an equation between \( AF = p \), a perpendicular from a given point in the axis on a tangent \( PF \), and \( FAP = v \), the angle which that perpendicular makes with the axis, and therefore also equal to the angle which the normal PN makes with the axis; then, since

\[ dz = \left( p + \frac{dp}{dt} \right) dt; \text{ from formula (1) we get also } \]

\[ r = p + \frac{dp}{dt} \]

This is a fourth formula for the radius of curvature.

79. To determine the nature of the evolute EHB, from any point H in the curve draw HK perpendicular to AB, and HI perpendicular to PQ; and assuming A, the origin of the co-ordinates of the involute also as the origin of those of the evolute, let \( AK = a \), \( HK = \beta \); in the triangle PHI, \( HI = PH \), \( \cos H = r \cos v \), and \( PI = r \sin v \), therefore

\[ \alpha = x + r \cos v, \quad \beta = y - r \sin v. \]

We have found that \( \tan v = \frac{1}{p} \), and therefore

\[ \cos v = \frac{p}{\sqrt{1 + p^2}}, \quad \sin v = \frac{1}{\sqrt{1 + p^2}}; \]

hence, combining these with the expressions for \( r \), we have

\[ \alpha = x - \frac{p}{q}(1 + p^2), \quad \beta = y + \frac{1}{q}(1 + p^2); \]

or, substituting for \( p \) and \( q \) the quantities they denote;

\[ \alpha = x - \frac{dy}{dx} \cdot \frac{dx^2 + dy^2}{d^2y}, \quad \beta = y + \frac{dx^2 + dy^2}{d^2y}. \]

In these formulae, and in the second and third for the radius of curvature, \( x \) is regarded as the independent variable, and \( y \) as a function of \( x \).

80. From the way in which a curve may be generated by an evolute, we have a method of finding any number of curves which may be rectified, that is, whose length may be assigned in algebraic terms; for \( PH \), the radius of the curvature, is the sum of the straight line CE, and the arc EH of the evolute. Now we can form innumerable curves whose radius of curvature will be known, therefore, corresponding to every curve whose radius of curvature can be found and expressed in algebraic terms, there is a curve which can be exactly rectified.

81. We shall now give examples of the application of the formulas.

Ex. 1. Let the curve be a parabola, of which AC is the axis; let \( c \) denote the parameter, and let \( AQ = x \), and \( PQ = y \), be the co-ordinates of P, any point in the curve. Let CB be the evolute, and let \( AK = a \), \( HK = \beta \), be the co-ordinates of H, the centre of curvature at P, the radius of curvature being \( PH = r \).

By the nature of the curve, \( y^2 = cx \); therefore,

\[ \frac{dy}{dx} = p = \frac{c}{2y}; \quad 1 + p^2 = \frac{c^2 + 4y^2}{4y^2} = \frac{c + 4x}{4x}; \]

\[ \frac{d^2y}{dx^2} = \frac{dp}{dx} = q = -\frac{c}{2} \cdot \frac{dy}{dx} \cdot \frac{1}{y^2} = -\frac{c}{4y^2}; \]

\[ r = -\frac{(1 + p^2)^{\frac{3}{2}}}{q} = \frac{(c^2 + 4y^2)^{\frac{3}{2}}}{2c^2} = \frac{(c^2 + 4cx)^{\frac{3}{2}}}{2c^2} = \frac{1}{2} \sqrt{\frac{(c + 4x)^3}{c}}. \]

This is the expression for the radius of curvature by formula (2). If we put the normal \( PN = s \), we have, by formula (3) (art. 78),

\[ r = -\frac{s^3}{qy^3} = \frac{4s^3}{c^2} \cdot \frac{s^3}{(2c)^3} = \frac{(normal)^3}{(semicub)} \]

Next, to find the evolute, we have

\[ \alpha = x - \frac{p}{q}(1 + p^2) = x + \frac{4x + c}{2} = 3x + \frac{1}{2}c; \]

\[ \beta = y + \frac{1 + p^2}{q} = y - \frac{c^2 + 4y^2}{c^2} = \frac{-4y^3}{c^2}. \]

Hence \( x = \frac{1}{3}(a - \frac{1}{2}c) \), \( y = -\left(\frac{2}{3}\right)^{\frac{3}{2}} \);

and since, by the nature of the parabola, \( cx = y^2 \); therefore the equation of the evolute is

\[ \beta^2 = \frac{16}{27c} \left(x - \frac{1}{2}c\right)^3; \]

Take \( AC = \frac{1}{2}c \), then C is a given point; and \( CK = a - \frac{1}{2}c \). Put \( CK = x' \); \( HK = \beta = -y' \), and the equation of the involute becomes

\[ y'^2 = \frac{16}{27c} x'^3. \]

Hence it appears that the evolute is the semicubical parabola. This is the first curve that was rectified by the infinitesimal calculus.

Ex. 2. Let the curve be an ellipse, of which \( OA = a \) is the semitransverse axis, and \( OB = b \) the semiconjugate. Let \( e = \sqrt{a^2 - b^2} \) be the eccentricity, and, assuming the centre as the origin of co-ordinates, let \( OQ = x \), \( PQ = y \), \( HP \), the radius of curvature at \( P = r \).

The equation of the curve is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \); or \( b^2 x^2 + a^2 y^2 = a^2 b^2 \); Hence By the nature of the cycloid (art. 66, ex. 4), \[ x = v + \sin v, \quad y = 1 - \cos v, \] therefore \( dx = dv (1 + \cos v), dy = dv \sin v, \) \[ \frac{dy}{dx} = p = \frac{\sin v}{1 + \cos v}; \quad 1 + p^2 = \frac{2}{1 + \cos v}, \quad dp = \frac{dv}{1 + \cos v}; \] \[ \frac{d^2y}{dx^2} = \frac{dp}{dx} = q = \frac{1}{(1 + \cos v)^2}. \]

Hence, the radius of cur. \( r = \frac{(1 + p^2)^{3/2}}{q} = \sqrt{8} (1 + \cos v). \)

Draw the chord CD, and because \[ 1 + \cos v = 2 \cos^2 \frac{v}{2} \text{ (ALGEBRA, 249)} = \frac{2}{4} \sin^2 \frac{v}{2} \text{ arc CD} = \frac{1}{2} \text{ chord } CD. \]

Therefore, PH = r = 2 chord CD.

It was shown (art. 66) that a tangent at P is parallel to BD, the chord of the supplement of CD; therefore PH, which is perpendicular to the tangent, is parallel to the chord CD; it is therefore determined in position and magnitude for any given point in the curve.

The evolute AHF may be found by the general formula (art. 79), or otherwise from the expression for PH, the radius of curvature, thus:

Draw AL perpendicular to AC, and equal to BC, but on the opposite side of AC; describe the semicircle AML, and draw HMR perpendicular to AL, cutting the semicircle in M; let PH meet AC in N.

Because CA = semicircle CDB, and CN = DP = arc BD, therefore AN = arc CD. But because PH = 2CD = 2PN; and therefore PN = NH; the parallels PE, HR are equally distant from CA; and hence CE = RA, and arc CD = arc AM and chord CD = chord AM, also angle DCA = angle CAM. Now CD = PN = NH; therefore AM = NH; but AM is parallel to NH; therefore MH = AN = arc CD = arc AM; and HR = arc AM + sin MR; thus it appears that H is a point in a semicycloid having the same base as the cycloid ABA'. Indeed LAF is just the semicycloidal space CBA' transferred to the position AHF, the point B being placed at A, and A' at F. If the other semicycloidal space AB be transferred to the position FAL', the arc APB in the position FA' will form another branch of the evolute, exactly similar to FA, but turned the contrary way.

From what has been found it appears, that if the semicycloids FLA, FL'A' were placed in a vertical plane, with the point F uppermost, and the semi-bases EL, EL' in a horizontal line; and if a weight were suspended from F by a thread equal in length to FB, and made to vibrate in that plane, like a pendulum; the weight would manifestly vibrate in an arc of a cycloid. This elegant property of the curve was discovered by Huygens, who also found that the vibrations would be isochronal, that is, performed Contact of Curves.

82. The ancient geometers paid particular attention to the contact of straight lines, with the few curve lines which they considered. Apollonius even touched on the contact of circles and the conic sections; for he treated of the shortest line that can be drawn from a given point in the axis to the curve. It is only, however, since the invention of the differential calculus that this branch of geometry has been carried to any considerable extent.

83. The curvature of a circle being the same throughout, it serves as the means of comparing the curvature in different points of a curve. It is, however, easy to understand, that in like manner we might consider the nature of the contact of a parabola, or a curve of any kind whatever, with a proposed curve.

Let \( y = f(x) \) and \( v = F(x) \) be the equations of two curves CPD, C'PD', which have a common axis AB, and the same point A for the origin of co-ordinates. Let \( x \)

Fig. 19.

be a common absciss, and \( y \) the corresponding ordinate of any point in the curve CD; also \( v \), the like ordinate in the curve CD'. Suppose now that when \( x \) becomes \( x + h \), \( y \) becomes \( y' \), and \( v \) becomes \( v' \); by Taylor's theorem,

\[ y' = y + \frac{dy}{dx} h + \frac{d^2y}{dx^2} \frac{h^2}{2} + \ldots \]

\[ v' = v + \frac{dv}{dx} h + \frac{d^2v}{dx^2} \frac{h^2}{2} + \ldots \]

Suppose now the curves to have a common point H, at which \( y = v \); then, If the curves be such that \( \frac{dy}{dx} = \frac{dv}{dx} \), their contact at P will be of such a nature, that no third curve can pass between them at that point, unless it has a like property.

Let HK = \( y = v \), KQ = \( h \), PQ = \( y' \), PQ' = \( v' \). Also, let \( u = HK \) be the ordinate of a third curve that passes through H, and \( u' \) its value, corresponding to AQ = \( x + h \), then

\[ u' = u + \frac{du}{dx} h + \frac{d^2u}{dx^2} \frac{h^2}{2} + \ldots \]

We are to prove it to be impossible for the third curve, in leaving the point H, to pass between the arcs HP and HP', unless \( \frac{du}{dx} = \frac{dy}{dx} = \frac{dv}{dx} \). For suppose it possible, then, because in general

\[ PP' = y' - v' = y - v + \left( \frac{dy}{dx} - \frac{dv}{dx} \right) h + \left( \frac{d^2y}{dx^2} - \frac{d^2v}{dx^2} \right) \frac{h^2}{2} + \ldots \]

and, because at the point H, \( y = v \); and, in this particular case, \( \frac{dy}{dx} = \frac{dv}{dx} \), we have

\[ y' - v' = \left( \frac{d^2y}{dx^2} - \frac{d^2v}{dx^2} \right) \frac{h^2}{2} + \left( \frac{d^2y}{dx^2} - \frac{d^2v}{dx^2} \right) \frac{h^3}{6} + \ldots \]

This is the general expression for PP', the difference between the ordinates of the curves CD, CD', whatever be the magnitude of KQ = \( h \).

In like manner, the difference between \( y' \), the ordinate of the first curve, and \( u' \), that of the third, will be generally expressed by

\[ y' - u' = y - u + \left( \frac{dy}{dx} - \frac{du}{dx} \right) h + \left( \frac{d^2y}{dx^2} - \frac{d^2u}{dx^2} \right) \frac{h^2}{2} + \ldots \]

which, on the hypothesis that \( y = u \), becomes

\[ y' - u' = \left( \frac{dy}{dx} - \frac{du}{dx} \right) h + \left( \frac{d^2y}{dx^2} - \frac{d^2u}{dx^2} \right) \frac{h^2}{2} + \ldots \]

Now, if it were possible that the third curve, in leaving the common point H, could pass between the other two, we should, to a certain extent from the point H, have \( y' > v' \) greater than \( y' - u' \), that is, after dividing the expressions for \( y' - v' \) and \( y' - u' \) by \( h \),

\[ \frac{d^2y}{dx^2} - \frac{d^2v}{dx^2} \frac{h^2}{2} + \left( \frac{d^2y}{dx^2} - \frac{d^2u}{dx^2} \right) \frac{h^3}{6} + \ldots \]

And this equation ought to hold true, however small the line KQ = \( h \) be taken. But \( h \) may be taken so small, that all the terms in the expression for \( y' - v' \) shall be less than any assignable quantity, while those of \( y' - u' \) continually approach to \( \frac{dy}{dx} - \frac{du}{dx} \) as a limit, which by hypothesis is not equal to zero. Now these two conclusions are incompatible with each other; therefore it is impossible that the third curve can pass through the point P between the other two, when \( \frac{dy}{dx} \) is not equal to \( \frac{du}{dx} \).

If, however, \( \frac{dy}{dx} = \frac{du}{dx} \), then the third curve may pass between the other two; for it will only be necessary that

\[ \frac{d^2y}{dx^2} - \frac{d^2v}{dx^2} > \frac{d^2y}{dx^2} - \frac{d^2u}{dx^2} \]

that is, \( \frac{d^2u}{dx^2} > \frac{d^2v}{dx^2} \),

which is evidently possible.

84. Again, if there be two curves whose equations are \( y = f(x) \), \( v = F(x) \), and which have a common ordinate \( y = v \), and if they be such, that

\[ \frac{dy}{dx} = \frac{dv}{dx}, \quad \frac{d^2y}{dx^2} = \frac{d^2v}{dx^2}; \]

then no third curve of which the equation is \( u = \varphi(x) \), and which has a common ordinate with the other two, can pass between them, unless at the same time

\[ \frac{du}{dx} = \frac{dv}{dx}, \quad \frac{d^2u}{dx^2} = \frac{d^2v}{dx^2}. \]

This proposition is proved exactly in the same way as that in last article, by means of Taylor's theorem. These are particular cases of a general proposition relating to the contact of curves which may be stated thus: If two curves have a common absciss and ordinate, and if the differential co-efficients of the first order of the ordinates be equal, no third curve passing through their intersection can go between them, unless the differential co-efficient of its ordinate, corresponding to the same absciss, be equal to the differential co-efficients of their ordinates. And if, besides their first differential co-efficients, their second differential co-efficients be equal, then no third curve passing through their intersection can go between them, unless the first and second differential co-efficients of its ordinate be equal to the first and second differential co-efficients of the two curves, and so on to differential co-efficients of all orders. In fact, the two curves meet only in the point in which their ordinates are the same, and the equality of the differential co-efficients merely indicates that no other curve in which the same equality does not take place can pass between them.

From what has been shown, it appears that the contacts of curve lines may be distinguished into different orders. The degree of contact in which the ordinates and also the first differential co-efficients are equal, may be called a Contact of the first order; when, in addition to these, the second differential co-efficients are equal, it is a Contact of the second order; and so on.

85. In illustration of this theory of contacts, let us consider the nature of the contact of any curve HPK (see fig. 6), with a straight line PT. Let A be the common origin of the co-ordinates. In the curve, let AB = x, PB = y. Its nature will be expressed by an equation \( y = f(x) \). In the straight line, let TA = e, AB = x, PB = v, angle PTB = α; the equation of the straight line is \( v = (e + x) \tan \alpha \). The hypothesis that the curve and straight line have a common point, gives \( y = v \); and the equation of the straight line, considering that e and α must be regarded as constants, gives \( \frac{dv}{dx} = \tan \alpha \); now, by our theory, \( \frac{dy}{dx} = \frac{dv}{dx} \), therefore \( \frac{dy}{dx} = \tan \alpha \). This result coincides with what was found in art. 66; and it gives the sub-tangent \( BT = \frac{y}{\tan \alpha} = \frac{dx}{dy} \).

It is impossible that any straight line can pass between the straight line PT and the curve; for, if possible, let the straight line PS pass between them, and meet the axis in S. Put AS = e', AB = x, PB = u, the angle PSB = α'. The equation of the straight line PS is \( u = (e' + x) \tan \alpha' \); and hence \( \frac{du}{dx} = \tan \alpha' \). Now, that the line PS may pass between PT and the curve, we should have \( \frac{du}{dx} = \frac{dy}{dx} \) (art. 83); therefore, \( \tan \alpha' = \tan \alpha \), and \( \alpha' = \alpha \); thus it appears that the lines PS, PT must coincide; therefore the straight line PT is a tangent to the curve, in the strictest sense.

86. Let us next consider the nature of the contact of a circle and any curve. Let CPD be the curve, and C'PD' a circle referred to the same axis AK, in which A is the common origin of the co-ordinates. In the curve CPD let AQ = x, PQ = y. Let H be the centre of the circle; draw the radius PH, and produce it to meet the axis in N, and draw HI, HK perpendicular to PQ and AQ. Put AQ = x, PQ = v, these being co-ordinates of P any point in the circumference of the circle; and put AK = a, HK = β, the co-ordinates of its centre; and PH = r its radius.

The equation of the curve CPD may be expressed generally by \( y = f(x) \). And, since HI = a - x, PI = v - β, and PH = r, the equation of the circle will be

\[ (a - x)^2 + (v - \beta)^2 = r^2 \]

A contact of the first degree requires that \( \frac{dy}{dx} = \frac{dv}{dx} \). Now, observing that \( a, \beta, \) and \( r \) must be considered as constants, we have \( \frac{dy}{dx} = \frac{a - x}{v - \beta} \) hence and because \( y = v \),

\[ \frac{dy}{dx} = \frac{a - x}{y - \beta} \]

The second side of this formula expresses the trigonometrical tangent of the angle PHI or PNQ. Hence (art. 65) the line PN is a normal to the curve at P, and the circle and curve have a common tangent at that point. By a second differentiation

\[ \frac{d^2v}{dx^2} = \frac{1}{v - \beta} - \frac{(a - x)}{(v - \beta)^2} \cdot \frac{dv}{dx} = \frac{(a - x)^2 + (v - \beta)^2}{(v - \beta)^2} \]

Now, that the circle and curve may have a contact of the second degree of closeness, it is necessary that \( \frac{d^2v}{dx^2} = \frac{d^2y}{dx^2} \); therefore, observing that \( v = y \), and that \( (a - x)^2 + (v - \beta)^2 = r^2 \), we have

\[ \frac{d^2y}{dx^2} = \frac{r^2}{(y - \beta)^2} \]

The three equations (1), (2), (3) serve to determine \( r \), the radius of the circle, and \( a, \beta \) the co-ordinates of its centre. From the first and second we get

\[ \alpha - x = \frac{rdy}{(dx^2 + dy^2)^{\frac{1}{2}}} ; \quad y - \beta = \frac{rdx}{(dx^2 + dy^2)^{\frac{1}{2}}} \]

and from these, and equation (3), we get

\[ r = \frac{(dx^2 + dy^2)^{\frac{1}{2}}}{dx^2 + dy^2} \]

These expressions for \( a, \beta, \) and \( r \) determine the position and magnitude of the circle; they are manifestly identical with the radius of the circle of equal curvature, and the co-ordinates of its centre formerly found (art. 78 and 79). Hence we conclude, that the circle of equal curvature with a curve in any point, determined by the theory of evolutes, is identical with a circle, having with the curve a contact of the second order.

And since it appears that only one circle can have with a curve a contact of the second order, and that no circle can pass between that circle and the curve, therefore the same must be true of the circle of equal curvature, found by the theory of evolutes.

87. From what has been shown in regard to the degrees of contact of a straight line and circle with a curve, it appears that these depend on the number of constants which enter into the equation of the touching line. The equation of a straight line contains two constants, and it admits of one order of contact only. The equation of a circle has three constants, and it admits of a contact of the second order. It is easy to see that the number of differential co-efficients that can be made equal in the two curves will always be one less than the number of constants in the touching curve. The equation of a parabola is \( y = ax + bx^2 + cx^3 \). This and the differential equations which may be deduced from it serve to determine the three constants \( a, b, c \), and thence the parabola of equal curvature; but since there are only two differential co-efficients, the parabola can have a contact of the first and also of the second order with any curve, but no higher order of contact. A cubic parabola, whose equation is \( y = a + bx + cx^2 + dx^3 \), may, however, have a contact of the third order, because it involves four constants, and so on.

Development and Differentiation of Functions containing two Variables.

88. Let \( u \) be a function of two independent variables, \( x \) and \( y \); for example, \( u = ax^2 + bxy + cy^2 \), and in general \( u = f(x, y) \). Suppose now that \( x \) changes its value, and becomes \( x + h \), and that \( y \) becomes \( y + k \), so that \( u' \) being the new value of the function, \( u' = f(x + h, y + k) \).

It is proposed to expand \( u' \) into a series of terms proceeding by the integer powers of \( h \) and \( k \). In particular cases the method of proceeding is obvious; a general formula may be found by Taylor's theorem as follows.

It is easy to see that the result will be the same whether we substitute at once \( x + h \) for \( x \), and \( y + k \) for \( y \), in the function \( u \), and then expand it; or, otherwise, first substitute \( x + h \), then expand the function, according to the powers of \( h \), and in this first result put \( y + k \) for \( y \), and afterwards expand it into terms proceeding by the powers of \( k \); or we may reverse this process, beginning with the expansion according to the powers of \( h \), and ending with that proceeding by the powers of \( h \).

Resuming the function \( u = f(x, y) \); when \( x \) becomes \( x + h \), we have by Taylor's theorem \( f(x + h, y) = \)

\[ u + \frac{du}{dx} h + \frac{d^2u}{dx^2} \frac{h^2}{1!2} + \frac{d^3u}{dx^3} \frac{h^3}{1!2!3} + \ldots \]

the differentials being taken upon the supposition that \( x \) is variable, and \( y \) a constant. And in this series \( y \) is contained only in the function \( u \), and the differential coefficients \( \frac{du}{dx}, \frac{d^2u}{dx^2}, \ldots \).

Suppose now that \( y \) becomes \( y + k \), then, in the series, the function \( u \) becomes

\[ u + \frac{du}{dy} k + \frac{d^2u}{dy^2} \frac{k^2}{1!2} + \frac{d^3u}{dy^3} \frac{k^3}{1!2!3} + \ldots \]

the function \( \frac{du}{dx} \) becomes

\[ \frac{du}{dx} + \frac{d^2u}{dx^2} \frac{k}{1!2} + \frac{d^3u}{dx^3} \frac{k^3}{1!2!3} + \ldots \]

the function \( \frac{d^2u}{dx^2} \) becomes

\[ \frac{d^2u}{dx^2} + \frac{d^3u}{dx^3} \frac{k}{1!2} + \frac{d^4u}{dx^4} \frac{k^4}{1!2!3} + \ldots \]

and in the differentiation of the functions \( \frac{du}{dx}, \frac{d^2u}{dx^2}, \ldots \), \( y \) is taken as the variable, and \( x \) as a constant. By making the proper substitutions, we have \( f(x + h, y + k) \) expressed by the series

\[ u + \frac{du}{dy} k + \frac{d^2u}{dy^2} \frac{k^2}{1!2} + \ldots \]

If the substitutions of \( x + h \) for \( x \), and \( y + k \) for \( y \), had been made in an inverse order, we would have had \( f(y + k, x + h) \) equal to

\[ u + \frac{du}{dx} h + \frac{d^2u}{dx^2} \frac{h^2}{1!2} + \ldots \]

The order in which the substitutions are made being arbitrary, the two developments of \( f(x + h, y + k) \) must be identical, consequently the terms containing the same products of \( h \) and \( k \) must be equal. Thus, passing over the terms which are the same, we have

\[ \frac{d^2u}{dx^2} = \frac{d^2u}{dy^2} \]

89. This formula proves an important theorem in the calculus, namely, if a function \( u \) of two independent variables \( x \) and \( y \) be twice differentiated, first, on the hypothesis that \( x \) is variable and \( y \) constant, and then the result on the supposition that \( y \) is variable and \( x \) constant. And again, if the order of the differentiations be reversed, that is, if \( y \) be made variable and \( x \) constant, and then \( x \) variable and \( y \) constant, the final result either way will be the very same.

The symbol \( \frac{d^2u}{dy^2} \), which expresses the result of two differentiations performed on \( u \), a function of two variables \( x \) and \( y \) (the first on the supposition that \( x \) is variable and \( y \) constant, and the second on the reverse supposition), may conveniently be simplified into \( \frac{d^2u}{dxdy} \). By this notation the theorem just enunciated in words will be expressed in symbols thus:

\[ \frac{d^2u}{dxdy} = \frac{d^2u}{dydx} \]

By continuing the two developments a term farther, we would have arrived at the conclusion that (having regard to the notation just premised),

\[ \frac{d^3u}{dxdydz} = \frac{d^3u}{dydzdx} = \frac{d^3u}{dydxz} \]

and that \( \frac{d^3u}{dxdyz} = \frac{d^3u}{dydzx} \), &c.

Suppose, for example, that \( u = x^3 y^5 \), then

\[ \frac{du}{dx} = 3 x^2 y^5, \quad \frac{du}{dy} = 5 x^3 y^4, \]

\[ \frac{d^2u}{dxdy} = 15 x^2 y^4, \quad \frac{du}{dydx} = 15 x^2 y^4, \]

\[ \frac{d^3u}{dx^2dy} = 30 xy^4, \quad \frac{d^3u}{dydx^2} = 30 xy^4. \]

Adopting now the abbreviated notation, when \( u = f(x, y) \), we have \( f(x + h, y + k) = \)

\[ u + \frac{du}{dx} h + \frac{d^2u}{dx^2} \frac{h^2}{1!2} + \frac{d^3u}{dx^3} \frac{h^3}{1!2!3} + \ldots \]

\[ + \frac{du}{dy} k + \frac{d^2u}{dy^2} \frac{k^2}{1!2} + \frac{d^3u}{dy^3} \frac{k^3}{1!2!3} + \ldots \] + \frac{d^2u}{dy^2} h^2 + \frac{d^3u}{dy^2dx} h^2 + &c.

This is the complete development of the function \( u \).

90. Supposing \( u \) to be any function of \( x \), and \( y \) becomes \( x + h \), by which \( u \) becomes \( u + ph + qh^2 + &c. \) or (considering \( h \) as the differential of \( x \)) \( u + pdx + qdx^2 + &c. \). We have defined the differential of \( u \) to be the term \( pdx \); viz., that which contains the simple power of \( h \) or \( dx \), and called the function \( p \), the differential co-efficient, representing it by \( \frac{du}{dx} \).

We may now extend these definitions to \( f(x, y) \), a function of two variables, and changing \( h \) into \( dx \) and \( k \) into \( dy \) in the development of the function, we shall have

\[ df(x, y) = du = \frac{du}{dx} dx + \frac{du}{dy} dy. \]

From this it appears that the complete differential of a function of two variables consists of two parts, viz. \( \frac{du}{dx} dx \), or the differential taken on the supposition that \( x \) alone is variable, and \( \frac{du}{dy} dy \), or the differential taken when \( y \) only is variable.

We may also apply to functions of two variables the rules which have been given (art. 10-33) for those which depend only on one; and for this purpose we must differentiate the function first with respect to one of its variables, and then with respect to another; and take the sum of the results for the complete differential required. By this rule

\[ d(x+y) = dx + dy \\ d(xy) = ydx + xdy. \]

Again, let \( u = x^m y^n \), we have

\[ \frac{du}{dx} = mx^{m-1} y^n dx, \quad \frac{du}{dy} = nx^m y^{n-1} dy, \] \[ du = \frac{du}{dx} dx + \frac{du}{dy} dy = x^{m-1} y^{n-1} (mydx + nxdy). \]

91. The manner in which the differentials of functions which depend on more than one variable are written, gives rise to some important remarks. The expression \( \frac{du}{dx} \) must not be confounded with \( du \), which it might be if \( u \) contained only the variable \( x \); the symbol \( \frac{du}{dx} \) has in this case a particular meaning, and denotes the differential co-efficient taken on the hypothesis of \( x \) only being variable; or it is the quotient of the first term of the development of the difference taken on that supposition divided by the increment \( dx \); and \( \frac{du}{dy} \) signifies the same with respect to \( y \).

The quantities \( \frac{du}{dx}, \frac{du}{dx} \) are commonly called partial differences of the first order of the function \( u \), and generally \( \frac{d^m+n}{dx^m dy^n} \) represents one of those of the order \( m + n \), which arises by differentiating \( m \) times in respect of \( x \), and \( n \) times in respect of \( y \).

A function of a single variable has only one differential co-efficient of any order; but a function of two variables

\[ \frac{d^2u}{dx^2} + \frac{d^2u}{dy^2} + &c. \] On the reverse hypothesis, supposing \( x = F(y) \) a function of \( y \), then, similarly,

\[ h = \frac{dx}{dy} + \frac{d^2x}{dy^2} \cdot \frac{k^2}{2} + \frac{d^3x}{dy^3} \cdot \frac{k^3}{2 \cdot 3} + \text{&c.} \]

Let the value of \( h \), as given by the first equation, be substituted in the second; and, in order to abridge, let us put

\[ \frac{dy}{dx} = y', \quad \frac{d^2y}{dx^2} = y'', \quad \text{&c.} \]

also

\[ \frac{dx}{dy} = x', \quad \frac{d^2x}{dy^2} = x'', \quad \text{&c.} \]

then we have

\[ h = x'(y'h + y'' \cdot \frac{k^2}{2} + y''' \cdot \frac{k^3}{2 \cdot 3} + \text{&c.}) \]

\[ + \frac{x'}{2} \left( y'h + y'' \cdot \frac{k^2}{2} + y''' \cdot \frac{k^3}{2 \cdot 3} + \text{&c.} \right)^2 + \text{&c.} \]

Hence, by actually involving the series to the second, third, powers, &c. and bringing into one term like powers of \( h \), we find

\[ 0 = (x'y' - 1)h + (x'y'' + x''y') \cdot \frac{k^2}{2} + \text{&c.} \]

Now, that this equation may be satisfied, the coefficients of the powers of \( h \) must be each \( = 0 \); thus,

\[ x'y' - 1 = 0, \quad x'y'' + x''y' = 0, \]

\[ x'y'' + 3x''y'' + x''y' = 0, \quad \text{&c.} \]

These equations give us

\[ y' = \frac{1}{x'}, \quad y'' = -\frac{x''}{x'^2}, \quad y''' = \frac{3x''^2 - x'''}{x'^3}, \]

\[ y'' = \frac{15x''^2}{x'^4} + \frac{10x''x'''}{x'^5} - \frac{x'''}{x'^6}, \quad \text{&c.} \]

Reversely, we have

\[ x' = \frac{1}{y'}, \quad x'' = -\frac{y''}{y'^2}, \quad x''' = \frac{3y''^2 - y'''}{y'^3}, \]

\[ x'' = \frac{15y''^2}{y'^4} + \frac{10y''y'''}{y'^5} - \frac{y'''}{y'^6}, \quad \text{&c.} \]

As an application of these formulæ, we may transform the expression for the radius of curvature found (art. 78), on the hypothesis that \( y \) is a function of \( x \), into another in which \( x \) is a function of \( y \). Employing the notation of last article,

\[ r = \frac{(1 + y'^2)^{\frac{3}{2}}}{y''}. \]

Now by the formulæ \( y' = \frac{1}{x'} \) and \( y'' = -\frac{x''}{x'^2} \): this substitution being made, we obtain

\[ r = \frac{(x'^2 + 1)^{\frac{3}{2}}}{x''} = \frac{(dx^2 + dy^2)^{\frac{3}{2}}}{dy^2x}. \]

94. Instead of supposing that one of the two variables \( x, y \), is a function of the other, it is sometimes convenient to regard both as functions of a third quantity. Thus, in geometry, supposing \( x \) and \( y \) to be the co-ordinates of a curve, we may consider the one as a function of the other; or we may consider both as functions of the angle which a tangent makes with the axis, which angle is then the independent variable. Or \( x \) and \( y \) denoting the co-ordinates of the path of a projectile, both may be considered as functions of \( t \), the time of the motion.

Suppose that when \( t \) becomes \( t + i \), \( x \) becomes \( x + h \), and that \( y \) becomes \( y + k \); to abridge, put \( x', x'', \text{&c.} \) for \( \frac{dx}{dt}, \frac{d^2x}{dt^2}, \text{&c.}; \) also \( y', y'', \text{&c.} \) for \( \frac{dy}{dt}, \frac{d^2y}{dt^2}, \text{&c.} \) and \( (y'), (y''), \text{&c.} \) for \( \frac{dy}{dx}, \frac{d^2y}{dx^2}, \text{&c.} \)

By Taylor's theorem,

\[ y = f(x) \text{ gives } k = (y')h + (y'') \cdot \frac{k^2}{2} + \text{&c.} \]

\[ y = F(t) \text{ gives } k = y'i + y'' \cdot \frac{i^2}{2} + \text{&c.} \]

\[ x = \phi(t) \text{ gives } h = xi + x'' \cdot \frac{i^2}{2} + \text{&c.} \]

The three increments \( k, h, i \) are generated together, therefore these equations must all hold true at the same time. Let the value of \( h \), found from the third equation, be substituted in the first, then, putting the two values of \( k \) equal, we have

\[ (y') \left\{ xi + x'' \cdot \frac{i^2}{2} + \text{&c.} \right\} \]

\[ + (y'') \left\{ xi + x'' \cdot \frac{i^2}{2} + \text{&c.} \right\}^2 + \text{&c.} \]

\[ = y'i + y'' \cdot \frac{i^2}{2} + \text{&c.} \]

Hence, by involution, and making the co-efficients of like powers of \( i \) equal to zero, as usual, we find

\[ x'(y') = y', \quad x''(y') + x''(y'') = y'', \quad \text{&c.} \]

therefore \( (y') = \frac{y'}{x'}, \quad (y'') = \frac{y'' - x'(y')}{x'^2}, \quad \text{&c.} \)

As an example, let us take the formula for the radius of curvature, which, by the notation just premised, will be

\[ r = \frac{(1 + y'^2)^{\frac{3}{2}}}{y''}. \]

By substituting for \( (y') \) and \( (y'') \) their values, it becomes

\[ r = \frac{(x'^2 + y'^2)^{\frac{3}{2}}}{y'' - x'(y')}; \]

and hence, putting \( \frac{dx}{dt} \) for \( x' \), \( \frac{dy}{dt} \) for \( y' \), \( \frac{d^2x}{dt^2} \) for \( x'' \), and \( \frac{d^2y}{dt^2} \) for \( y'' \), we get

\[ r = \frac{(dx^2 + dy^2)^{\frac{3}{2}}}{dy^2x - dx^2y}. \]

The independent variable \( t \) does not appear in the formula; nevertheless, in its application, we must keep in mind the hypothesis by which it was found.

Of the Method of Infinitesimals.

95. The ideas which we have of infinity may be embodied in this proposition: A quantity is not infinite so long as it admits of augmentation. If, therefore, in the quantity \( x + a \), we suppose \( x \) to become infinite, then \( a \) must be suppressed, otherwise it would be supposed that \( x \) might be increased by the quantity \( a \), which is contrary to the definition. To show the consistency of the proposition, let there be an equation

\[ \frac{1}{a} + \frac{1}{x} = M. \tag{1} \]

which being multiplied by the product \( ax \), becomes

\[ x + a = Ma. \tag{2} \]

Suppose now that \( x \) increases until it becomes infinite; the fraction \( \frac{1}{x} \), having reached its last degree of diminution, is reduced to 0, and therefore equation (1) becomes

\[ M = \frac{1}{a}, \text{ or } Ma = 1; \] and this value being substituted in equation (2), it becomes \( x + a = x \). This shows that when \( x \) is infinite, \( x + a \) is reduced to \( x \).

The quantity \( a \), in respect of which \( x \) is infinite, is called an infinitesimal in respect of \( x \).

Since we consider here only the ratios of quantities, the preceding demonstration holds good when \( x \) has a finite value, provided that \( a \) be infinitely small in respect of \( x \). Thus, if we compare \( b \), a finite quantity, with the fraction \( \frac{b}{x} \), then \( z \) being supposed to increase continually, \( \frac{b}{z} \) will decrease, and may become smaller than any assignable quantity; and when \( z \) is infinite, the fraction is \( = 0 \), so that, in comparison with \( b \), it may be neglected.

The ratios of quantities are altogether independent of their absolute magnitude. The halves, or any like parts of quantities, have the same ratios as the wholes. The symbols we employ in our reasonings are not the representatives of absolute magnitudes, but of numbers, which have a reference to an unit; they therefore represent ratios only, and the absolute magnitudes of the terms of the ratios are never considered. The differentials of quantities are the ratios to which the ratios of the increments of the variables continually approach; these increments may have any magnitude whatever. Leibnitz supposed them to be infinitely small quantities. Upon this hypothesis a theory has been constructed, which, by following certain ascertained principles, has led to the discovery of the most recondite truths in geometry and physics.

95. The admission of infinitesimals into the mathematical sciences, necessarily leads to a succession of different orders of infinitesimals. In a circle the ratio of the diameter to the chord of an arc is the same as the ratio of the chord to the versed sine; therefore, if the chord be infinitely small in respect of the diameter, or an infinitesimal of the first order, the versed sine will be infinitely small in respect of the chord; and, consequently, in respect of the diameter, will be an infinitesimal of the second order. So also, in a series of continued proportionsals, 1, \( x, x^2, x^3 \), &c. of which the first term is finite, and the second, \( x \), an infinitesimal of the first order, the third, \( x^2 \), will be of the second order, and the fourth, \( x^3 \), of the third, and so on. Upon the same principle, the product \( ab \) of two infinitesimals, \( a, b \) of the first order will be an infinitesimal of the second; for we may consider that \( 1 : a = b : ab \); that is, \( ab \) is a fourth proportional to three quantities 1, \( a \), and \( b \), of which the first is finite. In like manner, the product of three infinitesimals of the first order, also the product of an infinitesimal of the first and of the second order, will be an infinitesimal of the third order.

97. When the infinitely small increments of quantities are once taken as their differentials, the different rules for differentiation are easily found. Thus, if \( u = xy \); then \( u, x, y \) receiving the infinitely small augmentations \( du, dx, dy \), we have

\[ u + du = (x + dx)(y + dy) = xy + xdy + ydx + dxdy; \]

and, taking from these equals the equals \( u \) and \( xy \),

\[ du = xdy + ydx + dxdy. \]

Now \( du, xdy, ydx \) are infinitesimals of the first order, and \( dxdy \) is an infinitesimal of the second, therefore it is incomparably smaller, and in respect of the others ought to be rejected, so that we have simply

\[ du = xdy + ydx. \]

98. In this theory a curve is considered as a polygon of an infinite number of sides, any one of which is the differential of the arc, therefore the differential of the arc will be the hypotenuse of a right-angled triangle, of which the sides are the differentials of the co-ordinates; and these being denoted by \( x \) and \( y \), and the arc by \( z \), we have

\[ dz^2 = dx^2 + dy^2. \]

Proceeding on the same principles, if \( s \) denote the area of a curve, the increment of \( s \) or \( ds \) will be \( ydx + \frac{1}{2}dxdy \); but the second part of this, viz. \( \frac{1}{2}dxdy \), is a differential of the second order, and therefore infinitely less than \( ydx \), which is of the first order, therefore we have simply \( ds = ydx \), agreeing with the formula of art. 69.

In a circle, the infinitely small increments of the cosine, the sine, and the arc form a right-angled triangle similar to that formed by the sine, the cosine, and the radius, so that \( x \) being put for the arc, the radius being unity,

\[ 1 : \cos x = dx : d \sin x, \quad 1 : \sin x = dx : -d \cos x, \]

hence \( d \sin x = dx \cos x \), and \( d \cos x = -xdx \).

It is equally manifest, according to the infinitesimal theory, that the subtangent is a fourth proportional to \( dx \) (the differential of the absciss), \( dy \) (the differential of the ordinate), and \( y \) the ordinate; so that, as has been shown, art. 65,

\[ \tan = \frac{-dy}{dx} y; \]

the positive sign being used if the absciss and ordinate increase together, otherwise the sign \( - \).

The great facility which the theory of infinitesimals gives in the applications of the calculus to the higher geometry, and more especially to the doctrines of physics, is a high recommendation in its favour. Indeed, whatever view be taken of the subject, the mathematicians will hardly be restrained from taking the shortest way he can find to the object he wishes to attain, and that is in general by the infinitesimal calculus. Accordingly we observe that M. Poisson, in the second edition of his Traité de Mécanique, distinctly premises that he will exclusively employ the method of infiniment petits throughout his work.

PART II.

INVERSE METHOD OF FLUXIONS, OR INTEGRAL CALCULUS.

99. The inverse method of fluxions, or integral calculus, treats of the analytic processes by which a function may be found, such, that being differentiated, it shall produce a given differential. This function has been called, by writers on fluxions, the fluent or flowing quantity; and by writers on the differential calculus, the integral of the proposed differential.

To find the integral of a differential is to integrate that differential; and the process by which the integral is found is called integration.

100. When an integral is proposed, its differential may always be found by general rules. There is however no direct rule by which we may return from the differential to the integral, except the obvious one of retracing the steps by which the differential has been deduced from the integral. When these are distinctly indicated, there is no difficulty; but in general the steps of calculation by which the differential might have been deduced from an integral are obliterated or unknown; or the differential may not have been a direct result of differentiation; and then there will be no traces to show how we may pass from it to the integral.

101. It is convenient to have a symbol by which the integral of a differential may be indicated without regard to its particular form: for this purpose the letter \( \int \) (the initial letter of the word sum) is employed. Thus, the integral of the differential \( Xdx \) is expressed by the symbol \( \int Xdx \).

102. We have seen (art. 9) that in differentiating an expression of the form \( X = C \), which is the sum or dif- Inverse Method.

Integration of Rational Functions involving one Variable.

103. The general form of a differential of the first order of a function is $Xdx$, where $X$ denotes any function of a single variable $x$. Now this function may have various forms, and may be rational or irrational.

The rational forms of the function may be

$$Ax^m + Bx^n + Cx^p + \ldots = U,$$

$$Ax^m + Bx^n + Cx^p + \ldots = V,$$

the first of which is integral and the second fractional.

Irrational functions have the form

$$\frac{U}{V},$$

and transcendental functions,

$$f(U, L, U), f(U, \sin v), \text{etc.}$$

104. The simplest case of a rational function is $ax^n$, the differential of which is $nax^{n-1}dx$; therefore, putting $m = n - 1$, so that $n = m + 1$, we have,

$$dy = ax^ndx,$$

then $$y = \int ax^ndx = \frac{a}{m+1} x^{m+1} + C.$$

Hence this rule: To integrate a differential of one term, such as $ax^n dx$, we must increase the exponent of the variable by an unit, and then divide by the new exponent and by $dx$.

We may give the indeterminate constant $C$ the form

$$-\frac{ab^{n+1}}{n+1},$$

and then we shall have

$$\int ax^ndx = \frac{a(x^{n+1} - b^{n+1})}{n+1}.$$

This is the form the integral should have, if it has the property of vanishing in the particular case of $x = b$.

105. There is a particular case to which the general rule of last article will not immediately apply; it is that in which $n = -1$; in that case it gives

$$\int \frac{adx}{x} = \int ax^{-1}dx = \frac{a(x^0 - b^0)}{0} = \frac{1}{0} = 0.$$

From this no conclusion can be drawn, and the integral has been said to fail. This, however, is only in appearance; for, putting $n + 1 = m$, we have (art. 35)

$$x^n = 1 + (l.x)m + (l.x)^2\frac{m^2}{2} + \ldots$$

$$b^n = 1 + (l.b)m + (l.b)^2\frac{m^2}{2} + \ldots$$

$$x^n - b^n = \{l.x - l.b\}m + \{(l.x)^2 - (l.b)^2\}\frac{m^2}{2} + \ldots$$

$$\frac{x^n - b^n}{m} = l.x - l.b + \{(l.x)^2 - (l.b)^2\}\frac{m^2}{2} + \ldots$$

This, when $n + 1 = m = 0$ and $n = -1$, gives

$$\int \frac{adx}{x} = a(l.x - l.b) = l.x + C.$$

This result coincides entirely with the expression for the differential of the logarithm of a number (art. 20).

106. By the rule of art. 104 it is evident that if

$$dy = ax^mdx + bx^n dx + cx^p dx,$$

then $$y = \frac{ax^{m+1}}{m+1} + \frac{bx^{n+1}}{n+1} + \frac{cx^{p+1}}{p+1} + C.$$

This is true, however many terms there may be in the differential: the indeterminate constant $C$ represents the aggregate of all the constants of the integrals of the several terms.

In general, since (art. 15),

$$d(u+v-z) = du + dv - dz,$$

therefore $$u+v-z = \int du + \int dv - \int dz;$$

and in general, $P, Q, R$, being any functions of $x$,

$$\int (Pdx + Qdx - Rdx) = \int Pdx + \int Qdx - \int Rdx.$$

107. It has been found that $u$ and $v$ being any functions of a variable, $d(uv) = udu + vdu$, therefore

$$\int udu = uv - \int vdu.$$

In like manner, from the differential of a fraction we find

$$\int \frac{du}{v^2} = -\frac{u}{v} + \int \frac{du}{v}.$$

Since $d(au) = adu$, we may conclude that

$$\int axdx = a \int Xdx.$$

108. To integrate the differential

$$dy = (ax + b)^mdx,$$

we may expand the given power, and integrate the terms of the result separately; otherwise, make $z = ax + b$, then

$$x = \frac{z-b}{a}, \quad dx = \frac{dz}{a}.$$

By substitution, we have now $dy = \frac{z^mdz}{a}$, and by integration,

$$y = \frac{z^{m+1}}{a(m+1)} + C = \frac{(ax + b)^{m+1}}{a(m+1)} + C.$$

The integral of $dy = (ax^n + b)^m x^n - 1 dx$ may be found in the same way, by putting $ax^n + b = z$; we thus get

$$y = \frac{(ax^n + b)^{m+1}}{na(m+1)} + C.$$

109. We shall next consider fractional functions, and begin with the simple case of $dy = \frac{Ax^n dx}{(ax + b)^n}$; making $z = ax + b$, we find

$$x = \frac{z-b}{a}, \quad dx = \frac{dz}{a},$$

and consequently

$$dy = \frac{A(z-b)^mdz}{a^{m+1}z^n}.$$

By expanding $(z-b)^m$ into a series, then multiplying the terms by $dz$, and dividing by $z^n$, the differential will be transformed into a series of terms of the form $cz^p dz$, which may be integrated severally by the rule (art 104). As a particular example, let \( m = \frac{3}{2}, n = 2 \), then

\[ dy = \frac{A(z-b)^3 dz}{a^3 z} \]

Hence, integrating by the rule,

\[ y = \frac{A}{a^3} \left\{ \frac{z^2}{2} - 3bz + 3b^2 z - b^3 z^{-1} \right\} + C. \]

We now substitute for \( z \) its value, and find

\[ y = \frac{A}{a^3} \left\{ \frac{1}{2}(ax+b)^2 - 3b(ax+b) + b^2 \right\} + C. \]

110. All differentials which are rational fractions may be reduced to this form,

\[ (Ax^n + Bx^{n-1} + Cx^{n-2} + \ldots + T)dx \]

The general method of integrating such a differential consists in decomposing it into others whose denominators are more simple, which are called partial fractions. These are obtained as follows.

Assume the denominator of the proposed fraction equal to zero, thus forming the equation

\[ x^n + A'x^{n-1} + B'x^{n-2} + \ldots + T' = 0. \]

Let the roots of this equation be

\[ x = -a, x = -a', x = -a'', x = -a''', \ldots \]

and supposing them all different, the first side of the equation will be (Algebra, art. 99) the product of \( n \) factors,

\[ x + a, x + a', x + a'', x + a''', \ldots \]

We now assume that the proposed fraction is the sum of the fractions

\[ \frac{Ndx}{x+a} + \frac{N'dx}{x+a'} + \frac{N''dx}{x+a''} + \ldots \]

of which the denominators are the factors of the proposed fraction, and the numerators are undetermined constants. These being reduced to a common denominator, and added, the result will be a fraction identical with the proposed fraction; and by putting the co-efficients of like powers of \( x \) in their numerators equal to one another, a series of equations will be had, by which the undetermined constants \( N, N', N'' \), &c. may be found.

As an example, suppose the differential to be integrated is

\[ (Ax^2 + Bx + C) dx \]

and that by the resolution of an equation we know that

\[ x^2 + Ax^2 + Bx + C = (x+a)(x+a') + (x+a)(x+a''). \]

setting aside \( dx \), we assume that the fraction is equivalent to

\[ \frac{N}{x+a} + \frac{N'}{x+a'} + \frac{N''}{x+a''}. \]

The sum of these is

\[ N(x+a')(x+a'') + N(x+a)(x+a') + N(x+a)(x+a''). \]

The numerator of this fraction is, by multiplying its factors,

\[ (N+N'+N'')x^2 + \{N(a+a') + N(a+a'') + N'(a+a'')\}x \]

\[ + Na'a' + Na'a'' + Na'a'''; \]

and as this must be equal to the numerator of the proposed differential, whatever be the value of \( x \), the co-efficients of like powers of \( x \) will be equal (art. 18); hence we have these three equations,

\[ N + N' + N'' = A. \]

Inversely Method:

\[ N(a+a') + N'(a+a') + N''(a+a') = B; \]

\[ Na'a' + Na'a'' + Na'a''' = C; \]

which are only of the first degree with respect to the undetermined quantities \( N, N', N'' \). Resolving these equations, we find \( N, N', N'' \), and have

\[ \frac{(Ax^2 + Bx + C)dx}{x^2 + Ax^2 + Bx + C} \]

\[ = \frac{Ndx}{x+a} + \frac{N'dx}{x+a'} + \frac{N''dx}{x+a''}; \]

Making \( x + a = z \), we have \( dx = dz \), and

\[ \frac{Ndx}{z} = \text{the integral of which is } N.l.z = N.l.(x+a); \]

similarly,

\[ \int \frac{N'dx}{x+a'} = N'.l.(x+a'); \int \frac{N''dx}{x+a''} = N''.l.(x+a''). \]

So that we have now

\[ \int \frac{(Ax^2 + Bx + C)dx}{x^2 + Ax^2 + Bx + C} \]

\[ = N.l.(x+a) + N.l.(x+a') + N.l.(x+a'') + \text{const.} \]

This process, which may be extended to all rational fractions whose denominators are decomposable into unequal factors of the first degree, has no difficulty, independent of the decomposition of the denominator into its factors, which, however, requires the numerical resolution of equations.

111. If some of the factors of the denominator of the proposed fraction are equal, the process will require to be modified to suit that case. Suppose the denominator \( x^n + A'x^{n-1} + \ldots + T' \) to contain a factor \( (x+a)^p \), then, for every such factor, it will be necessary to assume a partial fraction of the form

\[ (Px^p + Qx^{p-1} + \ldots + Y)dx \]

\[ (x+a)^p \]

This and the other partial fractions being reduced to a common denominator, and added, and the terms of the numerator in the result being put equal to those of the numerator of the proposed differential, as before, as many independent equations will be obtained as there are coefficients to be determined.

112. It is not difficult to see that the partial fraction suited to the case of equal factors may be transformed into an expression of the form

\[ \frac{Ndx}{(x+a)^p} + \frac{N'dx}{(x+a)^{p-1}} + \ldots + \frac{N''dx}{x+a}; \]

therefore we may at once substitute this instead of the equivalent single function. We shall now have to integrate a series of differentials of the form

\[ \frac{Ndx}{(x+a)^p}, \]

which, by making \( x + a = z \), and therefore \( dx = dz \), gives

\[ \int \frac{Ndx}{(x+a)^p} = \int \frac{Ndz}{z^p} = \frac{Nz^{-p+1}}{1-p} \]

\[ = \frac{N}{(1-p)(x+a)^{p-1}}; \]

These will be all algebraical except the last, \( \int \frac{Ndx}{x+a} \),

which will involve a logarithm.

113. If there be imaginary factors, these will occur in pairs of the form \( x + a + \beta \sqrt{-1}, x + a - \beta \sqrt{-1}, \) Inverse Method.

and their product will be \( x^2 + 2ax + a^2 + \beta^2 \). In this case the best way to proceed will be to resolve the denominator of the proposed differential into real factors of the first and of the second degree, which is always possible (see Equations). If there be several pairs of these imaginary factors, each pair the same, the denominator of the proposed differential will have factors of the form \((x^2 + 2ax + a^2 + \beta^2)^q\). Then, for the single factor \(x^2 + 2ax + a^2 + \beta^2\), we must assume, in addition to the partial fractions, for factors of the first degree, one of this form,

\[ \frac{(Kx + L)dx}{x^2 + 2ax + a^2 + \beta^2}; \]

and for factors of the second form, this fraction,

\[ \frac{(Qx^3 - 1 + Rx^2 - 2 \ldots + Y')dx}{(x^2 + 2ax + a^2 + \beta^2)^q}; \]

or, instead of this, a series of fractions,

\[ \frac{(Kx + L)dx}{(x^2 + 2ax + a^2 + \beta^2)^q} + \frac{(K'x + L')dx}{(x^2 + 2ax + a^2 + \beta^2)^{q-1}} + \ldots + \frac{(K''x + L'')dx}{x^2 + 2ax + a^2 + \beta^2}, \]

the co-efficients to be determined, as before explained.

114. To integrate the fraction

\[ \frac{(Kx + L)dx}{x^2 + 2ax + a^2 + \beta^2}; \]

we observe that \(x^2 + 2ax + a^2 + \beta^2 = (x + a)^2 + \beta^2\);

and if we make \(x + a = z\), then

\[ \frac{(Kz + L - Ka)dz}{z^2 + \beta^2} = \frac{Kzdz}{z^2 + \beta^2} + \frac{L'dz}{z^2 + \beta^2}. \]

Here \(L'\) is put for the constant quantity \(L - Ka\). Now, the integral of the first of these differentials may be expressed by a logarithm; for, making \(z^2 = u\), we have \(zdz = \frac{1}{2}du\), which gives

\[ \int \frac{Kzdz}{z^2 + \beta^2} = K \int \frac{du}{u} = K \frac{1}{2} \ln u = K \frac{1}{2} \ln (z^2 + \beta^2). \]

With respect to the second part, we make \(z = \beta v\), then,

\[ \frac{L'dz}{z^2 + \beta^2} = \frac{L'}{\beta} \frac{dv}{v^2 + 1}. \]

But we have found (art. 23) that \(\frac{dv}{v^2 + 1}\) is the differential of the arc whose tangent is \(v\), which is expressed by \(\tan^{-1} v\), therefore

\[ \frac{L'}{\beta} \tan^{-1} \frac{z}{\beta} + \text{const.} \]

Adding now these results, we get

\[ \int \frac{(Kz + L)dz}{z^2 + \beta^2} = K \frac{1}{2} \ln (z^2 + \beta^2) + \frac{L'}{\beta} \tan^{-1} \frac{z}{\beta} + \text{const.} \]

When the value of \(z\) and \(L'\) are replaced, we have

\[ \int \frac{(Kx + L)dx}{x^2 + 2ax + a^2 + \beta^2} = \text{const.} \]

+ \(K \frac{1}{2} \ln (x^2 + 2ax + a^2 + \beta^2) + \frac{L}{\beta} \tan^{-1} \frac{x + a}{\beta}\).

115. To integrate the differential

\[ \frac{(Kx + L)dx}{(x^2 + 2ax + a^2 + \beta^2)^q}, \]

we shall at once make \(x + a = z\), and \(L - Ka = L'\).

By this the differential is transformed to

\[ (Kz + L')dz = \frac{Kzdz}{(z^2 + \beta^2)^q} + \frac{L'dz}{(z^2 + \beta^2)^q}. \]

To transform the first part, we make \(z^2 + \beta^2 = u\), so that \(zdz = \frac{du}{2}\), and consequently,

\[ \int \frac{Kzdz}{(z^2 + \beta^2)^q} = K \int \frac{du}{u^q} = K \frac{1}{2} \frac{u^{q-1}}{1-q}. \]

To integrate the second part, we assume the equation

\[ \int \frac{dz}{(z^2 + \beta^2)^q} = \frac{Gz}{(z^2 + \beta^2)^{q-1}} + H \int \frac{dz}{(z^2 + \beta^2)^{q-1}}; \]

that is, we assume the integral to be equal to an algebraic quantity, together with another integral whose denominator is less by an unit than the first. We now differentiate the terms of the assumed equation, observing that the removal of \(f\), the sign of an integral, is equivalent to the differentiation of that integral; we also reject such factors as are common to all the terms of the result, and thus have

\[ 1 = G(z^2 + \beta^2) - 2(q-1)Gz^2 + H(z^2 + \beta^2); \]

and \((3-2q)G + H(z^2 + (G + H)\beta^2 - 1 = 0.\)

Now, that \(z\) may remain undetermined, we must have

\[ (G + H)\beta^2 - 1 = 0, \quad (3-2q)G + H = 0. \]

From these equations, we find

\[ G = \frac{1}{(2q-2)\beta^2}, \quad H = \frac{2q-3}{(2q-2)\beta^2}. \]

The values of \(G\) and \(H\) being substituted in our assumed equation, it becomes

\[ \int \frac{dz}{(z^2 + \beta^2)^q} = \frac{1}{(2q-2)\beta^2} \frac{z}{(z^2 + \beta^2)^{q-1}} + \frac{2q-3}{(2q-2)\beta^2} \int \frac{dz}{(z^2 + \beta^2)^{q-1}}. \]

This formula gives the means of depressing the index of the denominator of the proposed fraction; for, putting \(q-1\) instead of \(q\), we find

\[ \int \frac{dz}{(z^2 + \beta^2)^{q-1}} = \frac{1}{(2q-4)\beta^2} \frac{z}{(z^2 + \beta^2)^{q-2}} + \frac{2q-5}{(2q-4)\beta^2} \int \frac{dz}{(z^2 + \beta^2)^{q-2}}; \]

and again, putting \(q-2\) for \(q\), we get

\[ \int \frac{dz}{(z^2 + \beta^2)^{q-2}} = \frac{1}{(2q-6)\beta^2} \frac{z}{(z^2 + \beta^2)^{q-3}} + \frac{2q-7}{(2q-6)\beta^2} \int \frac{dz}{(z^2 + \beta^2)^{q-3}}. \]

In this way the integral \(\int \frac{dz}{(z^2 + \beta^2)^q}\) is expressed by an algebraic quantity, and another integral, \(\int \frac{dz}{(z^2 + \beta^2)^{q-1}}\); and this last by another algebraic quantity, and the integral \(\int \frac{dz}{(z^2 + \beta^2)^{q-2}}\); and so on, until we come to the integral \(\int \frac{dz}{z^2 + \beta^2}\), which is known, and expressible by the arc of a circle. Here the process must stop; for the next transformation would involve the integral \(\int \frac{dz}{(z^2 + \beta^2)^q}\) with an infinite co-efficient, from which no conclusion can be drawn. Here we see the application of a method of integration at once fertile and elegant.

From what has been explained, it appears that differentials which are rational fractions may be always integrated either algebraically or by means of logarithms or circular arcs; all that is necessary is to decompose them into partial fractions, whose denominators are either binomial or trinomial quantities.

116. We shall now give some examples of the integration of differentials which are rational fractions.

1. Let the differential be \( \frac{adx}{x^2 - a^2} \).

In this case the denominator \( x^2 - a^2 = (x - a)(x + a) \); we therefore assume

\[ \frac{adx}{(x - a)(x + a)} = \left\{ \frac{A}{x - a} + \frac{B}{x + a} \right\} dx. \]

We now reduce the second side to a single fraction, and, leaving out \( dx \), have

\[ \frac{a}{(x - a)(x + a)} = \frac{(A + B)x + (A - B)a}{(x - a)(x + a)}; \]

therefore \( a = (A + B)x + (A - B)a \).

Now, \( x \) being indeterminate, according to the principle of art. 18, we must have

\[ (A - B)a = a, \quad A + B = 0; \]

these equations give \( A = \frac{1}{2}, B = -\frac{1}{2} \); the proposed differential is now

\[ \frac{adx}{x^2 - a^2} = \frac{1}{2} \frac{dx}{x - a} - \frac{1}{2} \frac{dx}{x + a}; \]

and, by integrating,

\[ \int \frac{adx}{x^2 - a^2} = \frac{1}{2} l(x - a) - \frac{1}{2} l(x + a) + C, \]

\[ = \frac{1}{2} l(x - a) + C = l(x - a) + C. \]

2. Let the differential be \( \frac{a^3 + bx^2}{a^2x - x^3} \).

The factors of the denominator are \( x \) and \( a^2 - x^2 \), which last resolves into \( (a - x)(a + x) \); therefore, leaving out \( dx \), we assume

\[ \frac{a^3 + bx^2}{(a - x)(a + x)} = \frac{A}{x - a} + \frac{B}{x + a} + \frac{C}{x}. \]

The second side of this equation, by reducing its terms to a common denominator, and adding, is

\[ \frac{Aa^2 + a(B + C)x + (B - A - C)x^2}{(a - x)(a + x)}. \]

By equating the co-efficients of the powers of \( x \), we have

\[ Aa^2 = a^3, \quad a(B + C) = 0, \quad B - A - C = b. \]

Here there are three unknown quantities, viz. \( A, B, C \), and three simple equations, therefore they may be found, as taught in Algebra, sect. viii.; and accordingly we have

\[ A = a, \quad B = \frac{1}{2}(a + b), \quad C = -\frac{1}{2}(a + b), \]

\[ \frac{a^3 + bx^2}{a^2x - x^3} dx = \frac{adx}{x} + \frac{a + b}{2(a - x)} dx - \frac{a + b}{2(a + x)} dx; \]

and, integrating,

\[ \int \frac{a^3 + bx^2}{a^2x - x^3} dx = a.l.x - \frac{a + b}{2} l.(a - x) \]

\[ - \frac{a + b}{2} l.(a + x) + C. \]

By the nature of logarithms \( l.P + l.Q = l.(PQ) \) and \( l.P = l.P^n \); hence the integral may be otherwise expressed thus,

\[ \int \frac{a^3 + bx^2}{a^2x - x^3} dx = l.\frac{a^3}{(a - x)^2} + C. \]

Inverse Method.

3. Let the differential be \( \frac{3x - 5}{x^2 - 6x + 8} dx \).

To resolve the denominator into simple factors, we make \( x^2 - 6x + 8 = 0 \), and find the roots of this equation to be \( x = 2 \) and \( x = 4 \); therefore, \( x^2 - 6x + 8 = (x - 2)(x - 4) \). To decompose the fraction, leaving out \( dx \), we now assume

\[ \frac{3x - 5}{(x - 2)(x - 4)} = \frac{A}{x - 2} + \frac{B}{x - 4}. \]

Hence, equating the like terms in the numerators, we have

\[ A + B = 3, \quad -4A - 2B = -5. \]

From these equations we find \( A = \frac{7}{2}, B = \frac{7}{2} \), and

\[ \int \frac{3x - 5}{x^2 - 6x + 8} dx = \frac{7}{2} l.(x - 4) - \frac{7}{2} l.(x - 2) + C. \]

4. As an example of equal simple factors in the denominator, let the differential be

\[ \frac{x^2 dx}{(x - a)^2(x + a)} = \frac{x^2 dx}{(x - a)^2(x + a)}. \]

In this case we assume (art. 112),

\[ \frac{x^2}{(x - a)^2(x + a)} = \frac{A}{(x - a)^2} + \frac{B}{x - a} + \frac{C}{x + a} \]

\[ = \frac{A(x + a) + B(x - a)^2 + C(x - a)^2}{(x - a)^2(x + a)} \]

\[ = \frac{Aa - Ba^2 + Ca^2 + (A - 2aC)x + (B + C)x^2}{(x - a)^2(x + a)} \]

By equating like terms of the numerators, we have

\[ Aa - Ba^2 + Ca^2 = 0, \quad A - 2aC = 0, \quad B + C = 1. \]

Hence, by the ordinary process for resolving simple equations involving three unknown quantities, we find

\[ A = \frac{1}{2}a, \quad B = \frac{1}{2}, \quad C = \frac{1}{2}, \]

and the transformed differential is

\[ \frac{x^2 dx}{(x - a)^2(x + a)} = \frac{a}{2} \frac{dx}{(x - a)^2} + \frac{1}{2} \frac{dx}{x - a} + \frac{1}{2} \frac{dx}{x + a}. \]

By integration we have

\[ \int \frac{dx}{(x - a)^2} = \frac{-1}{x - a} \int \frac{dx}{x - a} = l.(x - a), \quad \int \frac{dx}{x + a} = l.(x + a); \]

therefore, supplying the co-efficients of the partial fractions, and, as usual, an indeterminate constant, we have

\[ \int \frac{x^2 dx}{(x - a)^2(x + a)} = \frac{-a}{2} \frac{1}{x - a} + \frac{1}{2} l.(x - a) \]

\[ + \frac{1}{2} l.(x + a) + const. \]

5. Let the differential be \( \frac{x^3 + x^2 + 2}{x(x + 1)^2(x - 1)^2} dx \).

The denominator contains two pairs of equal factors; we therefore assume (art. 112)

\[ \frac{x^3 + x^2 + 2}{x(x + 1)^2(x - 1)^2} = \frac{A}{x} + \frac{B}{(x + 1)^2} + \frac{C}{(x + 1)} + \frac{D}{(x - 1)^2} + \frac{E}{x - 1}. \]

By proceeding as in the foregoing examples, uniting the partial fractions into one fraction, and equating the co-efficients of like powers of \( x \), we find

\[ A = 2, \quad B = -\frac{1}{2}, \quad C = -\frac{1}{2}, \quad D = 1, \quad E = -\frac{1}{2}. \]

The proposed differential is now decomposed into these, Inverse Method.

\[ \frac{2dx}{x} - \frac{1}{(x+1)^2} - \frac{4}{x+1} + \frac{dx}{(x-1)^2} - \frac{4}{x-1}. \]

Hence, by integration,

\[ \int \frac{x^3 + x^2 + 2}{x(x+1)^2(x-1)} dx = 2l(x+1) + \frac{1}{x+1} - \frac{4}{x-1}(x+1) - \frac{1}{x-1} - \frac{4}{x-1} + \text{const}. \]

6. As an example of a fraction, in which one of the factors in the denominator is a function of the second degree, which does not admit of resolution into real factors of the first degree, let the differential to be integrated be

\[ \frac{a + bx}{x^2 - 1} dx. \]

The denominator \( x^2 - 1 = (x-1)(x+1) \). The expression \( x^2 + x + 1 \) is of that kind which cannot be resolved into real factors, although it may into two imaginary factors. For, making \( x^2 + x + 1 = 0 \), resolving this quadratic equation, we find \( x = \frac{-1 \pm \sqrt{-3}}{2} \).

and \( x^2 + x + 1 = (x + \frac{1}{2} + \frac{1}{2} \sqrt{-3})(x + \frac{1}{2} - \frac{1}{2} \sqrt{-3}) \).

To avoid impossible quantities, we assume (art. 113),

\[ \frac{a + bx}{(x-1)(x^2 + x - 1)} = \frac{A}{x-1} + \frac{Bx + C}{x^2 + x - 1}. \]

Reducing now the partial fractions to a common denominator, and comparing the co-efficients of like powers, we find

\[ A = \frac{1}{2}(a + b), B = -\frac{1}{2}(a + b), C = \frac{1}{2}(2a - b). \]

The proposed differential and its equivalent transformed expression will therefore be

\[ \frac{a + bx}{(x-1)(x^2 + x + 1)} dx = \frac{a + b}{3} \cdot \frac{dx}{x-1} - \frac{1}{2} \frac{(a + b)x + 2a - b}{x^2 + x + 1} dx. \]

The first integral of these differentials is expressible by a logarithm; to transform the second, we put \( x^2 + x + 1 \) under the form \( (x + \frac{1}{2})^2 + \frac{3}{4} \), and make \( x + \frac{1}{2} = z \); then, \( z = x - \frac{1}{2}, dz = dx, (a + b)x + 2a - b = (a + b)z + \frac{1}{2}(a - b) \), and

\[ \frac{1}{2} \frac{(a + b)x + 2a - b}{x^2 + x + 1} dx = \frac{a + b}{3} \cdot \frac{dz}{z^2 + \frac{3}{4}} + \frac{a - b}{2} \cdot \frac{dz}{z^2 + \frac{3}{4}}. \]

The first of these two is also expressible by a logarithm. To simplify the second, we make \( z = \frac{\sqrt{3}}{2} v \); so that \( dz = \frac{\sqrt{3}}{2} dv \), and \( \frac{dz}{z^2 + \frac{3}{4}} = \frac{2}{\sqrt{3}} \cdot \frac{dv}{1 + v^2} \). On the whole, the differential prepared for integration by known rules is

\[ \frac{(a + bx)dx}{x^2 - 1} = \frac{a + b}{3} \cdot \frac{dx}{x-1} - \frac{a + b}{3} \cdot \frac{dz}{z^2 + \frac{3}{4}} - \frac{a - b}{\sqrt{3}} \cdot \frac{dv}{1 + v^2}; \]

and, integrating,

\[ \int \frac{(a + bx)dx}{x^2 - 1} = \frac{a + b}{3} l(x-1) - \frac{a + b}{6} l(z^2 + \frac{3}{4}) - \frac{a - b}{\sqrt{3}} \tan^{-1}v + \text{const}.; \]

this integral, by putting for \( z \) and \( v \) their values in terms of \( x \), will also be

\[ \int \frac{(a + bx)dx}{x^2 - 1} = \frac{a + b}{3} l(x-1) - \frac{a + b}{6} l(z^2 + \frac{3}{4}) - \frac{a - b}{\sqrt{3}} \tan^{-1}\left\{ \frac{2x + 1}{\sqrt{3}} \right\} + \text{const}. \]

117. The method of resolving a rational fraction into its component partial fractions, by the theory of indeterminate co-efficients (art. 110-113), is the most elementary of any; it is however tedious. We shall therefore now show how the labour may be abridged.

Let \( \frac{U}{V} \) be a rational fraction in its lowest terms, the denominator of which has unequal divisors of the first degree; let \( x + a \) be one of them, and \( Q \) the product of all the other factors, whether of the first or second degree; so that

\[ V = (x + a)Q. \]

In this case we assume

\[ \frac{U}{V} = \frac{A}{x+a} + \frac{P}{Q}, \]

where \( A \) is independent of \( x \), and \( P \) and \( Q \) are functions of \( x \), the latter being supposed known. By reducing to a common denominator, and remembering that \( V = (x + a)Q \), we have the identical equation

\[ U = AQ + P(x + a), \]

in which, by hypothesis, neither \( U \) nor \( Q \) are divisible by \( x + a \). Let us make \( x + a = 0 \), that is, \( x = -a \), and upon that hypothesis let \( u \) and \( q \) denote the values of \( U \) and \( Q \) respectively; then the equation becomes \( u = Aq \), and hence \( A = \frac{u}{q} \). This determines the numerator of the partial fraction \( \frac{A}{x+a} \), and in the same way may the numerators of all the partial fractions be found.

As an example, let the fraction to be resolved be

\[ \frac{mx + n}{(x+a)(x-b)} = \frac{A}{x+a} + \frac{B}{x-b}; \]

then, \( \frac{mx + n}{x-b} = A + \frac{B}{x-b}(x+a), \)

and \( \frac{mx + n}{x+a} = B + \frac{A}{x+a}(x-b). \)

Making \( x = -a \), the first of these gives

\[ A = \frac{-ma + n}{-a-b} = \frac{ma - n}{a + b}; \]

and making \( x = +b \), the second gives \( B = \frac{mb + n}{a + b}. \)

Hence we have

\[ \frac{mx + n}{(x+a)(x-b)} = \frac{1}{a+b} \left\{ \frac{ma-n}{x+a} + \frac{mb+n}{x-b} \right\}. \]

118. The differential calculus may be applied to the determination of the constants \( A, B, \) &c. Resuming the equation \( V = (x + a)Q \), we have, by differentiation,

\[ \frac{dV}{dx} = Q + \frac{dQ}{dx}(x+a). \]

Suppose that when \( x = -a \), then \( \frac{dV}{dx} \) becomes \( v' \), and, as before, that \( Q \) becomes \( q \); the differential equation will now become \( v' = q \); and since \( A = \frac{u}{q} \), therefore also \( A = \frac{u}{v'} \).

In the preceding example, in which \( V = (x + a)(x-b) \), we have

\[ \frac{dV}{dx} = 2x + a - b. \]

The supposition that \( x = -a \) makes \( \frac{dV}{dx} = -a - b = v' \), and the supposition that \( x = +b \) makes \( \frac{dV}{dx} = a + b = v' \); these values of \( v' = q \) give the values of \( A \) and \( B \), the same as before. 119. Let us now suppose, that in the fraction \( \frac{U}{V} \) the denominator \( V = (x + a)^n Q \), that is, the product of \( n \) equal factors, and \( Q \), the product of all the remaining factors; so that the assumption is

\[ \frac{U}{V} = \frac{U}{Q(x + a)^n} = \frac{A}{(x + a)^n} + \frac{B}{(x + a)^{n-1}} + \frac{C}{(x + a)^{n-2}} + \cdots + \frac{E}{x + a} + P, \]

where \( A, B, C, \ldots E \) are constant quantities, and \( P, Q \) functions of \( x \). In this case, multiplying all the terms by \( (x + a)^n \), we have

\[ \frac{U}{Q} = A + B(x + a) + C(x + a)^2 + \cdots + E(x + a)^{n-1} + P(x + a)^n. \]

The supposition that \( x = -a \) gives to \( U \) and \( Q \) particular values, which, as before, we denote by \( u \) and \( q \); the terms which contain \( x + a \) all vanish, and we have \( A = \frac{u}{q} \); thus \( A \) is known.

Since \( \frac{U - AQ}{Q} = B(x + a) + C(x + a)^2 + \cdots \),

the numerator, \( U - AQ \), of the first side of the equation must necessarily be divisible by \( x + a \), and may be expressed by \( U_1(x + a) \). We have now

\[ \frac{U}{Q} = B + C(x + a) + \cdots + E(x + a)^{n-2} + \frac{P}{Q}(x + a)^{n-1}. \]

The quantity \( U_1 \), not being divisible by \( x + a \), the assumption that \( x = -a \) will give it a particular value, which we shall denote by \( u_1 \). This assumption will give \( Q \) the same particular value as before; the terms which contain \( (x + a) \) as a factor will all vanish, and we shall have \( B = \frac{u_1}{q} \). In this way may all the remaining numerators \( C, \ldots E \) be found.

As a particular example of this process, let the fraction \( \frac{U}{V} \) be \( \frac{x^2}{(1-x)^3(1+x)} \). We assume

\[ \frac{x^2}{(1-x)^3(1+x)} = \frac{A}{(1-x)^3} + \frac{B}{(1-x)^2} + \frac{C}{1-x} + \frac{P}{1+x}, \]

hence, multiplying by \( (1-x)^3 \), we have

\[ \frac{x^2}{1+x} = A + B(1-x) + C(1-x)^2 + \frac{P}{1+x}(1-x)^3. \]

Making \( 1-x = 0 \), or \( x = 1 \), we have \( A = \frac{1}{2} \), and \( \frac{1}{1+x} = -\frac{1}{2} \).

\[ -A = -\frac{1}{2} + \frac{1}{2(1+x)}(1-x) = B(1-x) + C(1-x)^2 + \frac{P}{1+x}(1-x)^3, \]

and \( -\frac{1}{2(1+x)} = B + C(1-x) + \frac{P}{1+x}(1-x)^3 \);

again, making \( x = 1 \), we find \( B = -\frac{1}{2} \), and

\[ -\frac{1}{2(1+x)} = -\frac{1}{2} + \frac{1}{2(1+x)}(1-x) = C(1-x) + \frac{P}{1+x}(1-x)^3. \]

Lastly, making \( x = 1 \), we have \( C = -\frac{1}{2} \), and the fraction

\[ \frac{x^2}{(1+x)(1-x)^3} = \frac{1}{(1-x)^3} - \frac{1}{(1-x)^2} - \frac{1}{1-x} + \frac{P}{1+x}. \]

120. In this case we may also find the numerators \( A, B, C, \ldots \) by the differential calculus. From the equation of art. 119 we have

\[ U = Q(A + B(x+a) + C(x+a)^2 + \cdots + E(x+a)^{n-1} + P(x+a)^n). \]

If we now differentiate this equation \((n-1)\) times in succession, and then make \( x + a = 0 \), both in this equation and those which we deduce from it, there will arise

\[ \frac{dU}{dx} = A \frac{dQ}{dx} + BQ, \]

\[ \frac{d^2U}{dx^2} = A \frac{d^2Q}{dx^2} + 2B \frac{dQ}{dx} + 2CQ, \]

\[ \frac{d^3U}{dx^3} = A \frac{d^3Q}{dx^3} + 3B \frac{d^2Q}{dx^2} + 6C \frac{dQ}{dx} + 6DQ, \]

\[ \vdots \]

equations which determine each of the unknown quantities \( A, B, C, \ldots \) by means of those which precede it, it being well understood that we substitute after each differentiation \( -a \) in place of \( x \) in the differential co-efficients. In this case the most simple way of determining \( Q \) will be to divide \( V \) by \((x-a)^n \). But it may also be found by differentiation, as in art. 118.

121. Let us now suppose that \( \frac{U}{V} \), the fraction to be decomposed, has a factor of the second degree, viz. \( x^2 + 2ax + a^2 + \beta^2 \), which cannot be resolved into two real factors of the first degree.

In this case, \( V = (x^2 + 2ax + a^2 + \beta^2)Q \), and we assume

\[ \frac{U}{V} = \frac{\Lambda x + B}{x^2 + 2ax + a^2 + \beta^2} + \frac{P}{Q}. \]

Then, \( U = (\Lambda x + B)Q + P(x^2 + 2ax + a^2 + \beta^2) \).

If we substitute for \( x \) one of the imaginary roots of the equation \( x^2 + 2ax + a^2 + \beta^2 = 0 \), the term containing \( P \) will disappear, and the result will contain two kinds of terms; one real, and the other imaginary. We must now put the real quantities equal to each other, and also the imaginary quantities; thus two equations will be obtained by which \( \Lambda \) and \( B \) may be determined.

Let the fraction be

\[ \frac{x}{(x-1)(x^2 + x + 1)} = \frac{\Lambda x + B}{x^2 + x + 1} + \frac{P}{Q}. \]

Then \( x = (\Lambda x + B)(x-1) + P(x^2 + x + 1) \).

Now \( x^2 + x + 1 = 0 \) gives \( x = -\frac{1}{2} \pm \frac{1}{2} \sqrt{-3} \); therefore, by substituting in the last equation, it becomes, after collecting the terms,

\[ -\frac{1}{2} \pm \frac{1}{2} \sqrt{-3} = -\frac{1}{2} B \pm (\frac{1}{2} B - \Lambda) \sqrt{-3}. \]

Hence \( -\frac{1}{2} B = -\frac{1}{2} \) and \( \frac{1}{2} B - \Lambda = \frac{1}{2} \), therefore \( B = \frac{1}{2} \) and \( \Lambda = -\frac{1}{2} \).

122. In the last place, let us consider the case in which the fraction \( \frac{U}{V} \)

\[ \frac{Ax + B}{x^2 + 2ax + a^2 + \beta^2} + \frac{Cx + D}{x^2 + 2ax + a^2 + \beta^2} + \cdots + \frac{P}{Q}, \]

and here \( V = (x^2 + 2ax + a^2 + \beta^2)Q \). We have now

\[ \frac{U}{Q} = (Ax+B) + (Cx+D)(x^2 + 2ax + a^2 + \beta^2) + \cdots + \frac{P}{Q}(x^2 + 2ax + a^2 + \beta^2)V. \]

This case is compounded of the two preceding, and must be treated in the same manner. In the first place, we substitute for \( x \), one of the roots of the equation, \( x^2 + 2ax + a^2 + \beta^2 = 0 \), by which all the terms containing this quantity disappear, and the equation is reduced Inverse Method.

\[ \frac{V}{Q} = Ax + B; \] and here \( V \) and \( Q \) are functions of the particular value of \( x \), which satisfies the above-mentioned equation. As in the last case, we shall have an equation of the form \( r + s\sqrt{-1} = r' + s'\sqrt{-1} \), \( r \) and \( s \) being composed of constants and the indeterminate quantities \( A, B \); and these last will be found from the equations

\[ r = r', \quad s = s'. \]

We now differentiate the preceding equation, and, after omitting the terms in the result that contain \( x^4 + 2ax^3 + ax^2 + bx^2 \), have

\[ \frac{d(V)}{dx} = A + (2x + 2a)(Cx + D). \]

By substituting in this expression the imaginary value of \( x \), which satisfies the equation \( x^2 + 2ax + a^2 + b^2 = 0 \), and putting the real and imaginary terms on each side equal, two equations will be obtained which determine \( C \) and \( D \).

As an example, let the fraction be

\[ \frac{x^3 - 2x^2 + x - 3}{(x^2 - 2x + 2)^2}. \]

We assume it

\[ \frac{Ax + B}{(x^2 - 2x + 2)^2} + \frac{Cx + D}{x^2 - 2x + 2}. \]

By reducing to a common denominator, we find

\[ x^3 - 2x^2 + x - 3 = Ax + B + (Cx + D)(x^2 - 2x + 2), \]

one of the roots of the equation \( x^2 - 2x + 2 = 0 \) is \( x = 1 + \sqrt{-1} \); this being substituted for \( x \), the equation becomes

\[ -4 - \sqrt{-1} = A + B + A\sqrt{-1}. \]

Hence, to determine \( A \) and \( B \), we have \( A + B = -4 \), and \( A = -1 \); therefore \( B = -3 \). Substituting the values of \( A \) and \( B \), and transposing, we have

\[ x^3 - 2x^2 + 2x = (Cx + D)(x^2 - 2x + 2); \]

and, taking the differentials, and dividing by \( dx \),

\[ 3x^2 - 4x + 2 = (Cx + D)(2x - 2) + \text{etc.} \]

By again substituting \( 1 + \sqrt{-1} \) for \( x \), this equation becomes

\[ -2 + 2\sqrt{-1} = -2C + 2(C + D)\sqrt{-1}; \]

hence \( C = 1, D = 0 \), and the proposed fraction is equivalent to

\[ \frac{x + 3}{(x^2 - 2x + 2)^2} + \frac{x}{x^2 - 2x + 2}. \]

There is an extensive class of differentials, rational fractions, in which the denominator has the form \( x^{m} - 2x^{n} \cos a + 1 \); also of this form, \( x^{n} = 1 \). How these functions may be resolved into factors of the first and second degrees has been explained in Algebra, art. 274-278. This resolution being effected, the integration may be performed as has been explained.

Integrals of Irrational Fractions.

123. When a differential involves irrational functions, if by any means these can be transformed into others entirely rational, their integration may be effected by the rules already explained.

For example, let the differential be

\[ \frac{(1 + \sqrt{x} - \sqrt[3]{x^2})}{1 + \sqrt{x}} \]

It is evident, that making \( x = z^2 \), all the extractions indicated by the radical signs may be effected; and since \( dx = 2zdz \), the differential is transformed to

\[ \frac{6z^2dz(1 + z^3 - z^5)}{1 + z^3}, \]

and by actually dividing the numerator by the denominator, this last may be expressed thus,

\[ -6(z^7 - z^6 - z^5 + z^4 - z^3 + 1)dz - \frac{6dz}{1 + z^3}, \]

whose integral is

\[ -6 \left( \frac{z^8}{8} - \frac{z^7}{7} - \frac{z^6}{6} + \frac{z^5}{5} - \frac{z^4}{4} + z - \tan^{-1}z \right) + \text{const.} \]

By replacing the value of \( z \), viz. \( x^{\frac{1}{2}} \), the integral may be expressed in terms of \( x \).

124. We shall now consider those irrational functions which include the radical \( \sqrt{a + bx + cx^2} \) only, and which appear under one or other of the two forms,

\[ Pdx\sqrt{a + bx + cx^2}, \quad \frac{Pdx}{\sqrt{a + bx + cx^2}}, \]

where \( P \) denotes any rational function of \( x \). These may, however, be included in one; for if the first be multiplied by the radical, and the same quantity written under it as a denominator, it becomes

\[ \frac{P(a + bx + cx^2)dx}{\sqrt{a + bx + cx^2}}, \]

an expression of the same kind as the second.

We begin with the simple form, \( \frac{dx}{\sqrt{a + bx + cx^2}} \), which, however, has two cases, according as the sign of \( c \) in the denominator is positive or negative.

125. I. To integrate the differential \( \frac{dx}{\sqrt{a + bx + cx^2}} \)

Assume \( \sqrt{a + bx + cx^2} = py + \frac{q}{y} \)...........(1)

Here \( p \) and \( q \) are indeterminate constant quantities, to which such values are to be given as shall serve to transform the proposed differential into another that may be a rational fraction; we have now

\[ a + bx + cx^2 = p^2y^2 + 2pq + \frac{q^2}{y^2}. \]

To make the left-hand side of this equation a perfect square, we transpose its first term, then multiply both sides by \( c \), and add \( \frac{b^2}{4c} \) to each side of the result; it then becomes

\[ \frac{b^2}{4} + bcx + cx^2 = c(p^2y^2 + 2pq - a + \frac{b^2}{4c} + \frac{q^2}{y^2}). \]

The second side will also become a square if we make

\[ 2pq - a + \frac{b^2}{4c} = -2pq, \]

or \( 4pq = \frac{4ac - b^2}{4c} \),

which is evidently always a possible assumption; by this, the equation becomes

\[ \frac{b^2}{4} + bcx + cx^2 = c(p^2y^2 - 2pq + \frac{q^2}{y^2}); \]

and hence, by taking the square roots,

\[ \frac{b}{2} + cx = \sqrt{c}\left(py - \frac{q}{y}\right) \]...................(2)

From this, by differentiation, we obtain

\[ \sqrt{c}dx = \frac{dy}{y}\left(py + \frac{q}{y}\right) = \frac{dy}{y}\sqrt{a + bx + cx^2}; \]

and hence,

\[ \frac{dx}{\sqrt{a + bx + cx^2}} = \frac{1}{\sqrt{c}}\frac{dy}{y} \]...................(3) Dividing now both sides of equation (2) by \( \sqrt{c} \), and adding the results to equation (1), we get

\[ \frac{b}{2\sqrt{c}} + x\sqrt{c} + \sqrt{a + bx + cx^2} = 2py; \]

and, taking the logarithms,

\[ \ln \left( \frac{b}{2\sqrt{c}} + x\sqrt{c} + \sqrt{a + bx + cx^2} \right) = \ln y + \ln (2p)...(4) \]

Now \( l.y = \int \frac{dy}{y} + C \); here, \( C \) being any constant whatever, we may suppose it \( = C' - \ln (2p) \), where \( C' \) denotes another indeterminate constant; therefore,

\[ l.y + \ln (2p) = \int \frac{dy}{y} + C'. \]

By comparing this last equation with (3) and (4), we find

\[ \int \frac{dx}{\sqrt{a + bx + cx^2}} = \frac{1}{\sqrt{c}} \left( \frac{b}{2\sqrt{c}} + x\sqrt{c} + \sqrt{a + bx + cx^2} \right) + C, \]

\[ = \frac{1}{\sqrt{c}} \left( b + 2cx + 2\sqrt{c}\sqrt{a + bx + cx^2} \right) + C. \]

These logarithmic functions differ from each other only by the constant \( \ln (2\sqrt{c}) \).

126. II. To integrate the differential \( \frac{dx}{\sqrt{a + bx + cx^2}} \)

assume \( \sqrt{a + bx + cx^2} = p \sin \phi \)...........(1)

\( p \) representing here an indeterminate constant, and \( \phi \) a variable angle. Then

\[ \frac{a + bx + cx^2}{p^2} = p^2 \sin^2 \phi = p^2 - p^2 \cos^2 \phi; \]

and hence, proceeding as in last article,

\[ \frac{b}{4} - bcx + c^2x^2 = cp^2 \cos^2 \phi + \frac{b}{4} + ac - p^2c. \]

The left-hand side of the equation is now a complete square, and to make the right also a square, we assume that

\[ cp^2 = \frac{b^2}{4} + ac, \text{ or } p\sqrt{c} = \frac{\sqrt{b^2 + 4ac}}{2}...........(2) \]

and thus obtain

\[ \frac{b}{4} - bcx + c^2x^2 = cp^2 \cos^2 \phi, \]

and, taking the square roots,

\[ \frac{b}{2} - cx = \sqrt{p} \cos \phi...........(3) \]

and hence, differentiating, and dividing by \( \sqrt{c} \),

\[ dx = \frac{dp}{\sqrt{c}} \sin \phi = \frac{dp}{\sqrt{c}} \sqrt{a + bx + cx^2}; \]

and hence,

\[ \frac{dx}{\sqrt{a + bx + cx^2}} = \frac{1}{\sqrt{c}} dp. \]

Taking now the integral, we have

\[ \int \frac{dx}{\sqrt{a + bx + cx^2}} = \frac{1}{\sqrt{c}} \phi + \text{const}. \]

To determine \( \phi \), we have, (3) and (2),

\[ \cos \phi = \frac{b - 2cx}{\sqrt{b^2 + 4ac}}; \]

Therefore,

\[ \int \frac{dx}{\sqrt{a + bx + cx^2}} = \frac{1}{\sqrt{c}} \cos^{-1} \frac{b - 2cx}{\sqrt{b^2 + 4ac}} + C. \]

We have also

\[ \int \frac{dx}{\sqrt{a + bx + cx^2}} = \frac{1}{\sqrt{c}} \sin^{-1} \frac{2cx - b}{\sqrt{b^2 + 4ac}} + C; \]

for the angle whose cosine is any number \( v \) differs from the angle whose sine is \( -v \) by a right angle, which is a constant, and therefore the differentials of these angles will be the same expression.

127. We shall now integrate the formula

\[ \int \frac{x^n dx}{\sqrt{a + bx + cx^2}}, \]

\( n \) being any whole number. We put

\[ X = \sqrt{a + bx + cx^2}, \quad y = x^n \sqrt{a + bx + cx^2} = x^n X, \]

then \( y^2 = ax^{2n} + bx^{2n+1} + cx^{2n+2} \),

and differentiating, we have \( 2ydy = \)

\[ (2ma^{2m-1} + (2m + 1)bx^{2m} + (2m + 2)cx^{2m+1})dx; \]

and hence, dividing by \( y = x^n X \),

\[ 2dy = 2ma \frac{x^{n-1} dx}{X} + (2m + 1)b \frac{x^{n+1} dx}{X} \]

\[ + (2m + 2)c \frac{x^{n+2} dx}{X}. \]

Let us now make \( m + 1 = n \), and therefore \( m = n - 1 \),

\( m - 1 = n - 2 \), \( 2m = 2n - 2 \), \( 2m + 1 = 2n - 1 \), \( 2m + 2 = 2n \); thus we obtain

\[ 2dy = (2n - 2)a \frac{x^{n-2} dx}{X} + (2n - 1)b \frac{x^{n-1} dx}{X} \]

\[ + 2nc \frac{x^n dx}{X}; \]

and hence, by proper arrangement of the terms,

\[ \frac{x^n dx}{X} = \frac{dy}{nc} - \frac{2n - 1}{2a} \cdot \frac{b}{c} \frac{x^{n-1} dx}{X} - \frac{n - 1}{n} \cdot \frac{a}{c} \frac{x^{n-2} dx}{X}; \]

and, by integration, and putting for \( y \) and \( X \) the functions they represent, we have

\[ \int \frac{x^n dx}{\sqrt{a + bx + cx^2}} = \frac{1}{nc} x^n - 1 \sqrt{a + bx + cx^2} \]

\[ - \frac{2n - 1}{2a} \cdot \frac{b}{c} \int \frac{x^{n-1} dx}{\sqrt{a + bx + cx^2}} \]

\[ - \frac{n - 1}{n} \cdot \frac{a}{c} \int \frac{x^{n-2} dx}{\sqrt{a + bx + cx^2}}. \]

The proposed integral is now expressed by an algebraic function, and two other integrals of the same form, but in which the exponents of \( x \) are of lower degrees. We have found its value (art. 125, 126) when \( n = 0 \), a case to which the formula just found does not apply, because then, \( a \) entering into the denominators of the fractions, they become infinite. When \( n = 1 \), the last of the two integrals in the second side of the equation vanishes, because its coefficient \( = 0 \), and we have

\[ \int \frac{x dx}{\sqrt{a + bx + cx^2}} = \frac{1}{c} \sqrt{a + bx + cx^2} \]

\[ - \frac{b}{2c} \int \frac{dx}{\sqrt{a + bx + cx^2}}. \]

Knowing now the integrals in the cases of \( n = 0 \) and \( n = 1 \), the preceding general formula gives it in the case of \( n = 2 \); and in the case of \( n = 3 \), it is found from the cases of \( n = 1 \) and \( n = 2 \); and so on to any extent, supposing \( n \) to be a positive number.

128. When \( n \) is negative, the formula will be better under another form. Writing \( -n \) instead of \( +n \), and arranging the terms of the result anew, we have In this formula, \( n \) is understood to be positive.

129. This formula of transformation will not apply to the case of \( n = 1 \), because then the fractions which have \( n - 1 \) in the denominator become infinite. It is therefore necessary to investigate the integral of \( \frac{dx}{x\sqrt{a + bx + cx^2}} \) separately. Let \( x = \frac{1}{y} \); then \( dx = -\frac{dy}{y} \), and

\[ \frac{dx}{x\sqrt{a + bx + cx^2}} = \frac{-dy}{y}. \]

This transformed differential has two cases, according as \( a \) is positive or negative; when \( a \) is positive, the integral is, by article 125,

\[ \frac{1}{\sqrt{a}} \left( b + 2ay - 2\sqrt{a} \sqrt{c + by + ay^2} \right) + C. \]

We are at liberty to give the radical \( \sqrt{a} \) either the sign \( + \) or the sign \( - \); therefore, giving it the latter, the integral is

\[ \frac{1}{\sqrt{a}} \left( b + 2ay - 2\sqrt{a} \sqrt{c + by + ay^2} \right) + C; \]

and hence, substituting for \( y \) its equivalent \( \frac{1}{x} \), we have

\[ \int \frac{dx}{x\sqrt{a + bx + cx^2}} = \frac{1}{\sqrt{a}} \left( b + 2ax - 2\sqrt{a} \sqrt{c + bx + ax^2} \right) + C; \]

observing, as above stated, that the radical \( \sqrt{a} \) may have the sign \( + \) and also the sign \( - \).

In the second case we have

\[ \frac{dx}{x\sqrt{-a + bx + cx^2}} = \frac{-dy}{y}. \]

Now, by art. 126,

\[ \int \frac{-dy}{\sqrt{c + by - ax^2}} = \frac{1}{\sqrt{a}} \sin^{-1} \frac{2ay - b}{\sqrt{b^2 + 4ac}} + C \]

\[ = \frac{1}{\sqrt{a}} \cos^{-1} \frac{2ay - b}{\sqrt{b^2 + 4ac}} + C. \]

Therefore, putting \( \frac{1}{x} \) instead of \( y \),

\[ \int \frac{dx}{x\sqrt{-a + bx + cx^2}} = \frac{1}{\sqrt{a}} \cos^{-1} \frac{2a - bx}{\sqrt{b^2 + 4ac}} + C. \]

130. The formulæ investigated from articles 125 to 128 serve for the integration of innumerable differentials. Thus, from art. 125 we have

\[ \int \frac{dx}{x\sqrt{a + x^2}} = \frac{1}{\sqrt{a}} (x + \sqrt{a + x^2}) + C \]

\[ \int \frac{dx}{x\sqrt{bx + x^2}} = \frac{1}{\sqrt{b}} (\frac{1}{2}b + x + \sqrt{bx + x^2}) + C; \]

and, from the first formula of art. 129,

\[ \int \frac{adx}{x\sqrt{a^2 + cx^2}} = \frac{1}{\sqrt{a^2 + cx^2}} + C. \]

The second of these integrals is obtained from the first by passing \( 1/(2a) \) into the arbitrary constant \( C \). It may yet have another form; for

\[ \int \frac{2adx}{x\sqrt{a^2 + cx^2}} = \frac{1}{\sqrt{a^2 + cx^2}} + C. \]

Now \( -cx^2 = (a + \sqrt{a^2 + cx^2})(a - \sqrt{a^2 + cx^2}); \)

therefore, substituting and reducing,

\[ \int \frac{adx}{x\sqrt{a^2 + cx^2}} = \frac{1}{\sqrt{a^2 + cx^2}} + C. \]

This example shows that the integral is much modified by the arbitrary constant which enters it.

131. Corresponding to these logarithmic integrals there is a set of similar integrals which are expressed by angles or arcs of a circle. Thus we have, art. 126,

\[ \int \frac{dx}{\sqrt{x^2 - a^2}} = \sin^{-1} \frac{x}{a} + C, \]

\[ \int \frac{dx}{\sqrt{ax - x^2}} = \cos^{-1} \left( 1 - \frac{x}{a} \right) + C, \]

and by formula 2 of art. 129,

\[ \int \frac{adx}{x\sqrt{x^2 - a^2}} = \sec^{-1} \frac{x}{a} + C = \cos^{-1} \frac{a}{x} + C. \]

132. As an example of the way of proceeding in the integration of a differential which involves radical quantities, let it be proposed to integrate the differential

\[ \frac{dx}{x\sqrt{a^2 - x^2}}. \]

By an obvious transformation, we find

\[ \frac{dx}{x\sqrt{a^2 - x^2}} = \frac{a^2 dx}{x\sqrt{(a^2 - x^2)(x^2 - b^2)}}; \]

\[ = \frac{xdx}{\sqrt{(a^2 - x^2)(x^2 - b^2)}}. \]

The first of these differentials may be otherwise expressed thus,

\[ \frac{1}{x} \frac{d}{dx} \left( \frac{1}{\sqrt{\left( \frac{a^2}{x^2} - 1 \right)\left( \frac{1}{x^2} - \frac{1}{b^2} \right)}} \right). \]

Therefore we have

\[ \frac{dx}{x\sqrt{a^2 - x^2}} = \frac{xdx}{\sqrt{(a^2 - x^2)(x^2 - b^2)}}; \]

\[ = \frac{1}{x} \frac{d}{dx} \left( \frac{1}{\sqrt{\left( \frac{a^2}{x^2} - 1 \right)\left( \frac{1}{x^2} - \frac{1}{b^2} \right)}} \right), \]

and it is evident that the two differentials have precisely the same form.

To integrate the first of these, let us assume

\[ x^2 = a^2 \cos^2 \phi + b^2 \sin^2 \phi; \]

then \( xdx = (b^2 - a^2) \sin \phi \cos \phi d\phi, \)

\[ a^2 - x^2 = (b^2 - a^2) \sin^2 \phi, \]

\[ x^2 - b^2 = (b^2 - a^2) \cos^2 \phi. \]

\[ \frac{-xdx}{\sqrt{(a^2 - x^2)(x^2 - b^2)}} = \frac{(a^2 - b^2) \cos \phi \sin \phi d\phi}{(a - b) \cos \phi \sin \phi} = d\phi. \]

Exactly in the same way, if we make

\[ \frac{1}{x^2} = \frac{1}{a^2} \cos^2 \psi + \frac{1}{b^2} \sin^2 \psi, \] we get

\[ \frac{1}{x} \frac{d}{dx} \left( \frac{1}{x^2} - \frac{1}{a^2} \right) = \left( \frac{1}{b^2} - \frac{1}{a^2} \right) \sin \psi \cos \psi d\psi, \]

\[ \frac{1}{x^2} - \frac{1}{a^2} = \left( \frac{1}{b^2} - \frac{1}{a^2} \right) \sin^2 \psi \cdots \cdots \cdots (3) \]

\[ \frac{1}{x^2} - \frac{1}{b^2} = \left( \frac{1}{b^2} - \frac{1}{a^2} \right) \cos^2 \psi \cdots \cdots \cdots (4) \]

\[ \frac{1}{x} \frac{d}{dx} \left( \frac{1}{x^2} - \frac{1}{a^2} \right) = \left( \frac{1}{b^2} - \frac{1}{a^2} \right) \sin \psi \cos \psi d\psi \]

On the whole, if \( \phi \) and \( \psi \) be such angles that

\[ \cos \phi = \sqrt{\frac{x^2 - b^2}{a^2 - b^2}}, \quad \cos \psi = \frac{a}{x} \cos \phi; \]

then

\[ \int \frac{dx}{x} \sqrt{\frac{a^2 - x^2}{x^2 - b^2}} = \phi + \psi + \text{const}. \]

**Integration of Binomial Differentials.**

133. These differentials are represented by the formula

\[ x^{m-1} dx (a + bx^n)^p, \]

whose generality will not be affected by supposing that \( m \) and \( n \) are whole numbers. The object we propose is to find in what case this differential may be made rational.

We assume \( a + bx^n = z^q \), so that \( (a + bx^n)^p = z^p \); we then find

\[ x^n = \frac{z^q - a}{b}, \quad x^m = \left( \frac{z^q - a}{b} \right)^{\frac{m}{n}}, \]

\[ x^{m-1} dx = \frac{q}{nb} z^{q-1} \left( \frac{z^q - a}{b} \right)^{\frac{m}{n}-1}; \]

and the proposed differential is now transformed into

\[ \frac{q}{nb} z^{q+1} - 1 dx \left( \frac{z^q - a}{b} \right)^{\frac{m}{n}-1}, \]

an expression which is evidently rational whenever \( \frac{m}{n} \) is a whole number.

The differential \( x^{m-1} dx (a + bx^n)^p \) satisfies this condition, since \( m = 9, n = 3, \frac{m}{n} = 3 \), and it is transformed into

\[ \frac{q}{3b} z^{p+1} - 1 dx \left( \frac{z^q - a}{b} \right)^{\frac{m}{n}-1}. \]

The differential \( x^{m-1} dx (a + bx^n)^p \) admits of another form, by making the index of \( x \) between the brackets negative, or by dividing the quantity \( a + bx^n \) by \( x^n \); then we have

\[ x^{m-1} dx (a + bx^n)^p = x^{m-1} dx \left( \frac{(ax - b)x^n}{1} \right)^p \]

\[ = x^{m+\frac{np}{q}-1} dx (ax - b)^p; \]

and by the process preceding, the last of these expressions may be made rational, whenever \( \frac{m+\frac{np}{q}}{n} \) is a whole number; or, what is the same thing, whenever \( \frac{m}{n} + \frac{p}{q} \) is an integer.

The differential \( x^p dx (a + bx^n)^p \) is of this description, since \( \frac{m}{n} = \frac{5}{3}, \frac{p}{q} = \frac{1}{3}, \frac{m}{n} + \frac{p}{q} = \frac{6}{3} = 2 \).

In applying to the differential

\[ x^{m+\frac{np}{q}-1} dx (ax - b)^p, \]

the substitution indicated for the first form of this differential, we shall make \( ax - b = z^q \), and thence deduce \( a + bx^n = z^q x^n \); and if we transform immediately the expression \( x^{m-\frac{np}{q}-1} dx (a + bx^n)^p \) by means of the equation preceding, we shall evidently obtain the same result as if we had at first given it the form

\[ x^{m+\frac{np}{q}-1} dx (ax - b)^p. \]

134. Since it is not possible to integrate in every case the formula \( x^{m-1} dx (a + bx^n)^p \), we may try to change it into another more simple, as has been done in integrating a differential (art. 115). We have found that \( \int u dv = uv - \int v du \). If therefore we can decompose the differential into two factors, one of which being integrable, may be represented by \( dv \), and the other by \( u \), the integration of the proposed differential will be made to depend on that of \( vdu \), which in some cases will be more simple than the proposed differential. This method, which is at once extensive and elegant, is called integration by parts.

For the sake of abridging, we shall write simply \( p \) instead of \( \frac{p}{q} \), supposing \( p \) to represent a fractional number.

The formula will then become

\[ x^{m-1} dx (a + bx^n)^p. \]

Among the different ways of resolving this differential into factors, we choose this:

\[ x^{m-n} x^{n-1} dx (a + bx^n)^p. \]

One of the factors, \( x^{m-n} dx (a + bx^n)^p \), is integrable, whatever be the value of \( p \) (art. 108); representing it therefore by \( dv \), we have

\[ v = \frac{(a + bx^n)^{p+1}}{nb(p+1)}, \quad u = x^{m-n}, \]

whence there results

\[ \int x^{m-1} dx (a + bx^n)^p = \frac{x^{m-n}(a + bx^n)^{p+1}}{nb(p+1)} - \frac{m-n}{nb(p+1)} \int x^{m-n-1} dx (a + bx^n)^{p+1}; \]

but

\[ \int x^{m-n-1} dx (a + bx^n)^{p+1} \]

\[ = \int x^{m-n-1} dx (a + bx^n)^p (a + bx^n) = \]

\[ a \int x^{m-n-1} dx (a + bx^n)^p + \]

\[ b \int x^{m-n-1} dx (a + bx^n)^p, \] putting this last in the preceding equation; and collecting the terms containing the integral \( \int x^{m-1} dx (a + bx^n)^p \), we obtain, after reduction,

\[ (A.) \quad \int x^{m-1} dx (a + bx^n)^p = \frac{x^m - n}{b (pn + m)} \left( a + bx^n \right)^{p+1} \]

It is obvious that since we may reduce the integration of \( x^{m-1} dx (a + bx^n)^p \) to that of \( x^{m-n-1} dx (a + bx^n)^p \), we may reduce this last to the integration of \( x^{m-2n-1} dx (a + bx^n)^p \) by writing \( m-n \) in the place of \( m \) in equation (A); then, by changing \( m-n \) into \( m-2n \), we shall be able to determine \( \int x^{m-2n-1} dx (a + bx^n)^p \) by means of \( \int x^{m-3n-1} dx (a + bx^n)^p \), and so on.

In general, if \( r \) denote the number of reductions, we shall at last come to \( \int x^{m-rn-1} dx (a + bx^n)^p \), and the last formula will be

\[ \int x^{m-rn-1} dx (a + bx^n)^p = \frac{x^m - rn}{b (pn + m - (r-1)n)} \left( a + bx^n \right)^{p+1} \]

It appears from this formula, that if \( m \) be a multiple of \( n \), the integration of the proposed formula \( x^{m-1} dx (a + bx^n)^p \) may be effected in algebraic terms, since the last integral of the transformed expression will be multi- plied by \( m-rn = 0 \). This result agrees with that of art. 133.

135. There is another way of reduction by which the exponent of the quantity within the parenthesis may be diminished by an unit; for this purpose it is only neces- sary to observe that

\[ \int x^{m-1} dx (a + bx^n)^p = \int x^{m-n-1} dx (a + bx^n)^p - 1 \]

Formula (A), by changing \( m \) into \( m+n \), and \( p \) into \( p-1 \), gives

\[ \int x^{m+n-1} dx (a + bx^n)^p = \frac{x^m - n}{b (pn + m)} \left( a + bx^n \right)^{p+1} \]

Substituting this value in the preceding equation, we have

\[ (B.) \quad \int x^{m-n-1} dx (a + bx^n)^p = \frac{x^m - n}{b (pn + m)} \left( a + bx^n \right)^{p+1} \]

By this formula we may take away successively from \( p \)

as many units as it contains; and by its application, along with formula (A), we may make the integral

\[ \int x^{m-1} dx (a + bx^n)^p \text{ depend on } \]

\[ \int x^{m-n-1} dx (a + bx^n)^p - s, \quad mn \text{ being the greatest } \]

multiple of \( m \) contained in \( m-1 \), and \( s \) the greatest whole number contained in \( p \).

The integral \( \int x^p dx (a + bx^n)^p \), for example, may be re- duced by formula (A) successively to

\[ \int x^p dx (a + bx^n)^p, \quad \int x^p dx (a + bx^n)^p; \]

and by formula (B), \( \int x^p dx (a + bx^n)^p \) is reduced to

\[ \int x^p dx (a + bx^n)^p, \quad \text{and that again to } \int x^p dx (a + bx^n)^p. \]

136. It is evident that if \( m \) and \( n \) were negative, the formula (A) and (B) would not accomplish the purpose for which they have been investigated; they would then increase the exponent of \( x \) without the parenthesis, as well as that of the parenthesis itself; but by reversing them we shall find others suited to the case in question. From formula (A) we deduce

\[ \int x^{m-n-1} dx (a + bx^n)^p = \frac{x^m - n}{b (pn + m - (r-1)n)} \left( a + bx^n \right)^{p+1} \]

In this expression, substitute \( m+n \) in place of \( n \), and it becomes

\[ (C.) \quad \int x^{m-n-1} dx (a + bx^n)^p = \frac{x^m - n}{b (pn + m - np)} \left( a + bx^n \right)^{p+1} \]

This formula diminishes the exponent without the pa- renthesis, since \( m+n-1 \) becomes \( m+n-1 \) when \( m \) is put instead of \( m+n \).

To reverse the formula (B), we take

\[ \int x^{m-n-1} dx (a + bx^n)^p = \frac{x^m - n}{b (pn + m - np)} \left( a + bx^n \right)^{p+1} \]

then we write \( p+1 \) instead of \( p \), and thus find formula

\[ (D.) \quad \int x^{m-n-1} dx (a + bx^n)^p = \frac{x^m - n}{b (pn + m - np)} \left( a + bx^n \right)^{p+1} \]

This formula answers the proposed purpose, since \( p+1 \) becomes \( p+1 \), when \( p \) is negative.

137. Let the formula be \( \int \frac{x^{m-1} dx}{\sqrt{1-x^2}} \), where \( m \) is a whole positive number. Making \( a = 1, b = -1, n = 2, p = -1 \), we find

\[ \int \frac{x^{m-1} dx}{\sqrt{1-x^2}} = \frac{x^{m-2} \sqrt{1-x^2}}{m-1} + \frac{m-2}{m-1} \int \frac{x^{m-3} dx}{\sqrt{1-x^2}} \] Writing \( m \) in place of \( m - 1 \), we have

\[ \int \frac{x^m dx}{\sqrt{1-x^2}} = \frac{x^{m-1} \sqrt{1-x^2}}{m} + \frac{m-1}{m} \int \frac{x^{m-2} dx}{\sqrt{1-x^2}}. \]

If we give to \( m \) successive different values, beginning with the uneven numbers, we have

\[ \int \frac{x^3 dx}{\sqrt{1-x^2}} = -\sqrt{1-x^2} + \text{const.} \]

\[ \int \frac{x^5 dx}{\sqrt{1-x^2}} = -\frac{1}{2} x^4 \sqrt{1-x^2} + \frac{3}{2} \int \frac{x^3 dx}{\sqrt{1-x^2}}, \]

\[ \int \frac{x^7 dx}{\sqrt{1-x^2}} = -\frac{1}{2} x^6 \sqrt{1-x^2} + \frac{3}{2} \int \frac{x^5 dx}{\sqrt{1-x^2}}, \]

\[ \int \frac{x^9 dx}{\sqrt{1-x^2}} = \frac{1}{2} x^8 \sqrt{1-x^2} + \frac{3}{2} \int \frac{x^7 dx}{\sqrt{1-x^2}}, \]

and hence we deduce

\[ \int \frac{x^3 dx}{\sqrt{1-x^2}} = -\left(\frac{1}{2} x^2 + \frac{1.3}{1.3}\right) \sqrt{1-x^2}, \]

\[ \int \frac{x^5 dx}{\sqrt{1-x^2}} = -\left(\frac{1}{2} x^4 + \frac{1.4}{3.5} x^2 + \frac{1.24}{1.35}\right) \sqrt{1-x^2}, \]

\[ \int \frac{x^7 dx}{\sqrt{1-x^2}} = -\left(\frac{1}{2} x^6 + \frac{1.6}{5.7} x^4 + \frac{1.46}{3.57} x^2 + \frac{1.246}{1.357}\right) \sqrt{1-x^2}, \]

&c.

The law of formation of the integrals is evident; an arbitrary constant is to be annexed to each.

Let us now consider the even values of \( m \). Supposing \( m = 2, 4, 6, &c. \) we have

\[ \int \frac{x^2 dx}{\sqrt{1-x^2}} = -\frac{1}{2} x \sqrt{1-x^2} + \frac{1}{2} \int \frac{dx}{\sqrt{1-x^2}}, \]

\[ \int \frac{x^4 dx}{\sqrt{1-x^2}} = -\frac{1}{2} x^3 \sqrt{1-x^2} + \frac{3}{2} \int \frac{x^2 dx}{\sqrt{1-x^2}}, \]

\[ \int \frac{x^6 dx}{\sqrt{1-x^2}} = -\frac{1}{2} x^5 \sqrt{1-x^2} + \frac{3}{2} \int \frac{x^4 dx}{\sqrt{1-x^2}}, \]

&c.

Here all the integrals depend on

\[ \int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1} x + \text{const. (art. 23)} \]

Let us call \( A \) the arc whose sine is \( x \), and we have

\[ \int \frac{dx}{\sqrt{1-x^2}} = A + \text{const.} \]

\[ \int \frac{x^2 dx}{\sqrt{1-x^2}} = -\frac{1}{2} x \sqrt{1-x^2} + \frac{1}{2} A + \text{const.} \]

\[ \int \frac{x^4 dx}{\sqrt{1-x^2}} = -\left(\frac{1}{2} x^2 + \frac{1.3}{2.4} x\right) \sqrt{1-x^2} + \frac{1.3}{2.4} A + \text{const.} \]

\[ \int \frac{x^6 dx}{\sqrt{1-x^2}} = -\left(\frac{1}{2} x^4 + \frac{1.5}{4.6} x^2 + \frac{1.35}{2.46}\right) \sqrt{1-x^2} + \frac{1.35}{2.46} A + \text{const.} \]

138. Let us now consider the formulas which answer to the case of \( m \) negative; then by formula (C) art. 136,

\[ \int \frac{x^{m-1} dx}{\sqrt{1-x^2}} = -\frac{x^{m-1} \sqrt{1-x^2}}{m} + \frac{m-1}{m} \int \frac{x^{m-2} dx}{\sqrt{1-x^2}}. \]

Writing \( -m \) instead of \( -m - 1 \), this becomes

\[ \int \frac{dx}{x^m \sqrt{1-x^2}} = -\frac{\sqrt{1-x^2}}{(m-1)x^{m-1}} + \frac{m-2}{m-1} \int \frac{dx}{x^{m-2} \sqrt{1-x^2}}. \]

We cannot make \( m = 1 \), because that value would make the denominator zero and the coefficient infinite. We have already found the integral in this case, viz.

\[ \int \frac{dx}{x \sqrt{1-x^2}} = -\frac{1}{x} + \frac{\sqrt{1-x^2}}{x} + \text{const. (art. 130)} \]

Proceeding now, making \( m = 3, m = 5, &c. \) we have

\[ \int \frac{dx}{x^3 \sqrt{1-x^2}} = -\frac{\sqrt{1-x^2}}{2x^2} + \frac{1}{2} \int \frac{dx}{x^2 \sqrt{1-x^2}}, \]

\[ \int \frac{dx}{x^5 \sqrt{1-x^2}} = -\frac{\sqrt{1-x^2}}{4x^4} + \frac{3}{2} \int \frac{dx}{x^4 \sqrt{1-x^2}}, \]

\[ \int \frac{d}{x^7 \sqrt{1-x^2}} = -\frac{\sqrt{1-x^2}}{6x^6} + \frac{3}{2} \int \frac{dx}{x^6 \sqrt{1-x^2}}, \]

&c.

Making \( m = 2, m = 4, m = 6, &c. \) we find

\[ \int \frac{dx}{x^2 \sqrt{1-x^2}} = -\frac{\sqrt{1-x^2}}{x} + \text{const.} \]

\[ \int \frac{dx}{x^4 \sqrt{1-x^2}} = -\frac{\sqrt{1-x^2}}{3x^3} + \frac{3}{2} \int \frac{dx}{x^3 \sqrt{1-x^2}}, \]

\[ \int \frac{dx}{x^6 \sqrt{1-x^2}} = -\frac{\sqrt{1-x^2}}{5x^5} + \frac{3}{2} \int \frac{dx}{x^5 \sqrt{1-x^2}}, \]

&c.

From these two series of equations we deduce, as in the former article, a series of differentials integrated by logarithms, and another by algebraic formulæ.

Integration by Series.

139. The integral \( \int X dx \) is easily found when the function \( X \) is expanded into a series, because then we have only to integrate a series of single terms in succession, to which the rule of art. 104 immediately applies.

Thus, let \( X = Ax^m + Bx^{m+n} + Cx^{m+2n} + Dx^{m+3n} + &c. \)

If both members of this equation be multiplied by \( dx \), and each term be separately integrated, we have

\[ \int X dx = \frac{Ax^{m+1}}{m+1} + \frac{Bx^{m+n+1}}{m+n+1} + \frac{Cx^{m+2n+1}}{m+2n+1} + &c. + c. \]

If in the expansion of \( X \) any term of the form \( \frac{E}{x} \) occur, the integral corresponding to that term will be \( E \ln x \) (art. 20). We shall now give particular examples.

Ex. 1. To integrate \( \frac{dx}{a+x} \) by a series.

We know that the integral comprehends in it Napier's logarithm of \( a + x \) (art. 20). Now, by division,

\[ \frac{1}{a+x} = \frac{1}{a} - \frac{x}{a^2} + \frac{x^2}{a^3} - \frac{x^3}{a^4} + &c. \]

Therefore, multiplying by \( dx \), and integrating,

\[ \int \frac{dx}{a+x} = \frac{x}{a} - \frac{x^2}{2a^2} + \frac{x^3}{3a^3} - \frac{x^4}{4a^4} + &c. + C. \]

This series expresses the general integral, without any regard to its particular application. If, however, we consider it as expressing \( \ln(a+x) \), then we have The constant quantity C is now no longer indeterminate; it must have such a value as satisfies this equation whatever be the value of x. If we make \(x = 0\), then all the terms of the series vanish, and we have \(1 + a = C\).

Thus the value of C is determined, and we have

\[1 + \frac{x}{a} + \frac{x^2}{2a^2} + \frac{x^3}{3a^3} + \ldots + C.\]

This result agrees with what we found (art. 86).

Ex. 2. To integrate \(\int \frac{dx}{1 + x^2}\) by a series. By division,

\[\frac{1}{1 + x^2} = 1 - x^2 + x^4 - x^6 + \ldots + C.\]

Multiplying now both sides of this equation by \(dx\), and integrating, we have

\[\int \frac{dx}{1 + x^2} = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \ldots + C.\]

As a formula purely analytical, the constant C must be regarded as indeterminate. We know, however, that \(\frac{dx}{1 + x^2}\) is the integral of an arc of which \(x\) is the tangent, the radius being unity (art. 22, Ex. 3). Therefore, if we mean to express \(\tan^{-1} x\) by a series, then

\[\tan^{-1} x = x - \frac{x^3}{3} + \frac{x^5}{5} - \frac{x^7}{7} + \ldots + C.\]

The constant C has now a precise value, and it may be found by giving any corresponding known values to \(\tan^{-1} x\) and \(x\). Now we know that when \(\tan^{-1} x = 0\), then \(x = 0\). Therefore, considering that when all the terms in the series vanish, we have \(0 = 0 + C\); thus it appears that \(C = 0\).

Ex. 3. In the last example, the integral of \(\int \frac{dx}{1 + x^2}\) was expressed by a series of terms formed from the ascending powers of \(x\). We may, however, express that integral by a descending series. By division

\[\frac{1}{x^2 + 1} = \frac{1}{x^2} - \frac{1}{x^4} + \frac{1}{x^6} - \ldots + C.\]

And \[\int \frac{dx}{x^2 + 1} = -\frac{1}{x} + \frac{1}{3x^3} - \frac{1}{5x^5} + \ldots + C.\]

If we now regard the integral as the angle or arc whose tangent is \(x\), we have

\[\tan^{-1} x = -\frac{1}{x} + \frac{1}{3x^3} - \frac{1}{5x^5} + \ldots + C.\]

The value of C cannot now be found by making \(x = 0\), because then the arc \(= 0\), and the substitution of \(x = 0\) makes all the terms of the series infinite. If we suppose the arc to be a quadrant \(= \frac{\pi}{2}\), then \(x\) is infinite, and the terms of the series are each \(= 0\); by this assumption, \(\frac{\pi}{2} = C\). We have now

\[\tan^{-1} x = \frac{1}{x} - \frac{1}{3x^3} + \frac{1}{5x^5} - \ldots + C.\]

Ex. 4. The differential of the arc whose sine is \(x\) is \(\frac{dx}{\sqrt{1 - x^2}}\). It is proposed to integrate this differential by a series.

By the binomial theorem, \(\frac{1}{\sqrt{1 - x^2}} = (1 - x^2)^{-\frac{1}{2}} = 1 + \frac{1}{2}x^2 + \frac{1}{2 \cdot 4}x^4 + \frac{1}{2 \cdot 4 \cdot 6}x^6 + \ldots + C.\)

Hence we have \(\sin^{-1} x = \int \frac{dx}{\sqrt{1 - x^2}} = x + \frac{1}{2}x^3 + \frac{1}{2 \cdot 4}x^5 + \frac{1}{2 \cdot 4 \cdot 6}x^7 + \ldots + C.\)

When the arc \(= 0\), then the sine \(= 0\); therefore both sides of this equation vanish together, and no constant is wanted to adjust it to equality.

140. The object of integration by series being to obtain approximate values of the integrals when they cannot be found accurately, it is important to have several series, so that we may be able to choose one convergent for a proposed value of \(x\). Ascending series, or those which proceed by positive and increasing powers of \(x\), do not converge sufficiently unless \(x\) be a small fraction, as happens with the series of Ex. 2; whilst those which proceed by the negative powers of \(x\), or descending series, converge only so as to be of use when \(x\) is a large number.

141. In having recourse to series, it is sufficient to find one, of which the terms are separately integrable, although each may not have the form \(Ax^n dx\); they may have any form, the integral of which can be expressed by an algebraic quantity, or a logarithm, or an arc of a circle; or, in short, any function, the numerical value of which is most readily obtained, provided always that the convergence be such as to admit of the sum of all the terms which are not taken in being rejected, because of its smallness in comparison to the sum of the terms taken in.

In the application of the calculus recourse is only to be had to series as a last resource, when the integral, from its nature, cannot be found in finite terms, or by logarithms or arcs of a circle, which in fact are the sums of infinite series. Indeed it would be very convenient to have various other tables, ready computed, of the value of a function corresponding to given successive values of the variable. The labour of their computation, their necessary great extent, and the infrequency of their use, have hitherto limited this valuable aid in calculation to a table of Napier's Logarithms, in addition to the Trigonometrical Tables.

Integration of Logarithmic Functions.

142. Let it be required to integrate \(Pdx (1.x)^n\), that is, the \(n\)th power of Napier's logarithm of \(x\) multiplied by \(Pdx\), \(P\) being an algebraic function of \(x\). Here we apply the formula \(f u'dx = u - f'udu\); and making \(u = (1.x)^n\), from which we find \(du = n(1.x)^{n-1} \frac{dx}{x}\), also \(dv = Pdx\), we have, making in order to abridge \(f'Pdx = N\),

\[\int Pdx (1.x)^n = N(1.x)^n - n \int \frac{dx}{x} (1.x)^{n-1} N.\]

If now we represent the integral \(\int \frac{dx}{x} N(1.x)^{n-1}\) by \(M\), we shall have

\[\int \frac{dx}{x} N(1.x)^{n-1} = M(1.x)^{n-1} - (n-1) \int \frac{dx}{x} (1.x)^{n-2} M.\]

By this reduction the proposed integral is made to depend on another of the same kind, in which the exponent of the logarithm is an unit less. Thus, if \(P = x^n\), we have

\[\int x^n dx (1.x)^n = \frac{x^{n+1}}{n+1} (1.x)^n - \frac{n}{n+1} \int (1.x)^{n-1} x^n dx.\] By applying the formula to this last integral, and again to the result, we find \( \int x^n dx (l.x)^n = \)

\[ x^{n+1} \left\{ \frac{(l.x)^n}{m + 1} - \frac{n(l.x)^{n-1}}{(m + 1)^2} + \frac{n(n-1)(l.x)^{n-2}}{(m + 1)^3} - \ldots \right\} + C. \]

It is evident that this will always terminate when \( n \) is a whole number.

143. If \( n \) be a whole negative number, we must apply the formula \( \int u dv = uv - \int v du \) so as to increase the exponent of the logarithm. Thus in the differential

\[ \frac{Pdx}{(l.x)^n} = \frac{P}{n+1} \cdot (-n + 1) \cdot (l.x)^{-n} \frac{dx}{x}, \]

we make \( Px = u \), and \( (l.x)^{-n} \frac{dx}{x} = dv \), by which

\[ (l.x)^{-m+1} = v; \text{ then we have } \int \frac{Pdx}{(l.x)^n} = \]

\[ \frac{P}{n+1} (l.x)^{-n+1} + \frac{1}{n-1} \int (l.x)^{-n+1} d(Px). \]

Let us suppose \( P = x^m \), then this formula becomes

\[ \int \frac{x^m dx}{(l.x)^n} = \frac{-x^{m+1}}{(n-1)(l.x)^{n-1}} + \frac{m+1}{n-1} \int \frac{x^m dx}{(l.x)^{n-1}}. \]

By repetitions of this transformation, we at last make the proposed integral depend on \( \int \frac{x^m dx}{l.x} \). Now, put \( x^{m+1} = z \), then \( l.x = \frac{l.z}{m+1} \) and \( x^m dx = \frac{dz}{m+1} \); therefore

\[ \frac{x^m dx}{l.x} = \frac{dz}{l.z}. \]

The integral of this last quantity is a transcendental of a peculiar kind, which cannot be otherwise expressed than by infinite series.

**Integration of Exponential Functions.**

144. It appears from art. 19, that \( d(a^x) = l.(a) a^x dx \), therefore

\[ \int a^x dx = \frac{a^x}{l.a} + C. \]

And because \( dx = \frac{d(a^x)}{a^x l.a} \); therefore, if \( V \) be any algebraic function of \( a^x \), then, putting \( a^x = u \), we have \( Vdu = \frac{Vdu}{u l.a} \), an expression which, in respect of \( u \), has an algebraic form.

For example, let \( V = \frac{a^x}{\sqrt{1+a^x}} \), then,

\[ \int \frac{a^x dx}{\sqrt{1+a^x}} = \frac{du}{\sqrt{1+u^2}}. \]

The integral of this last expression may be found by rules which have been investigated.

Let \( z \) be any function of \( x \), then, \( e \) being the number of which Nap. log \( = 1 \), we have \( d(ze^x) = e^x dz + e^x z dx \), therefore

\[ \int e^x dx \left( z + \frac{dz}{dx} \right) = ze^x + C. \]

For example, let \( z = x^2 - 1 \), then \( \frac{dz}{dx} = 3x^2 \), and

\[ \int e^x dx (3x^2 + x^2 - 1) = e^x (x^3 - 1) + C. \]

145. In other cases we may have recourse to the method of integration by parts (art. 134). Thus, let the differential be \( x^n dx \), then, by the formula \( \int u dv = uv - \int v du \), making \( u = x^n \), \( dv = a^x dx \), we find

\[ \int a^x x^n dx = \frac{a^x x^n}{l.a} - \frac{n}{l.a} \int a^x x^{n-1} dx. \]

By treating the differential \( a^x x^{n-1} dx \) in the same way, and repeating this as often as necessary, we find \( \int a^x x^n dx = \)

\[ \frac{a^x}{l.a} \left\{ \frac{x^n}{l.a} - \frac{n x^{n-1}}{(l.a)^2} + \frac{n(n-1)x^{n-2}}{(l.a)^3} - \ldots \right\} + C. \]

146. If the exponent \( n \) be negative, by following the same method we may increase the exponent of \( x \). Accordingly, from the formula \( \int u dv = uv - \int v du \), making \( u = a^x \), and \( dv = \frac{dx}{x} \), we find

\[ \int \frac{a^x dx}{x^n} = \frac{-a^x}{(n-1)x^{n-1}} + \frac{1}{n-1} \int \frac{a^x dx}{x^{n-1}}. \]

By repeating this transformation, we bring the integral \( \int \frac{a^x dx}{x^n} \) to depend at last on \( \int \frac{a^x dx}{x} \).

If we make \( a^x = z \), then \( x l.a = l.z \), and \( x = \frac{l.z}{l.a} \), and \( dx = \frac{1}{l.a} \frac{dz}{z} \); hence \( \frac{a^x dx}{x} = \frac{1}{l.a} \frac{dz}{z} \), and

\[ \int \frac{a^x dx}{x} = \frac{1}{l.a} \int \frac{dz}{z}. \]

These two integrals involve the same difficulty, and have greatly exercised the ingenuity of analysts. It appears they can only be integrated by infinite series. See on this subject, Lacroix, *Traité du Calcul Différentiel*, vol. iii. No. 1224.

147. If \( n \) be a fraction, either of the preceding methods will apply to reduce the exponent of \( x \) to some fraction between 0 and \( +1 \) or \( -1 \); and then recourse must be had to the method of infinite series.

Whatever has been done in regard to the integral of \( x^n a^x dx \), will apply to \( Pa^x dx \), supposing \( P \) to be any function of \( x \) whatever.

**Integration of Angular or Circular Functions.**

148. There are several ways of integrating expressions which contain circular or trigonometrical functions of a variable.

**Method I.**—Let \( x \) be an arc whose sine or cosine is \( z \); for example, let \( \sin x = z \), then

\[ \cos x = \sqrt{1-z^2}, \quad dx = \frac{dz}{\sqrt{1-z^2}}. \]

\( dx \sin^n x \cos^n x = z^n (1-z^2)^{\frac{n-1}{2}} dx \).

1. If \( n \) be any odd number, then, in the binomial differential, the radical in the transformed expression disappears. 2. If \( m \) is an odd number, the exponent of \( z \) without the parenthesis, when increased by unity, will be a multiple of \( z \) its exponent, within the parenthesis. Thus one of the conditions of art. 132 is satisfied, and the differential may be made rational.

3. If \( m \) and \( n \) are even numbers, then the second condition of art. 132 will be satisfied, and the differential may be integrated.

As an example, let the differential be \( dx \sin^3 x \). This, by making \( z = \sin x \), becomes \( \frac{z^3}{\sqrt{1 - z^2}} \). Now, by art. 137,

\[ \int \frac{z^3 dz}{\sqrt{1 - z^2}} = \frac{1}{3} \cos x (2 + \sin^2 x) + C. \]

149. Method II.—The powers of the sine or cosine may be transformed into series, of which the terms are sines or cosines of the multiples of the arc; the terms to be integrated will then have the form \( dx \cos kx \), \( dx \sin kx \). Now

\[ \int dx \cos kx = \frac{1}{k} \sin kx + C; \] \[ \int dx \sin kx = -\frac{1}{k} \cos kx + C. \]

Ex. 1. Let the differential be \( dx \cos^2 x \). By the calculus of sines (Algebra, 258), \( \cos^2 x = \frac{1}{2} (\cos 2x + 1) \); therefore

\[ \int dx \cos^2 x = \frac{1}{2} \sin 2x + \frac{1}{2} x + C. \]

This method is often used, because the sines and cosines of the multiples of an arc are more easily found than the powers of the sine and cosine.

Ex. 2. Let the differential be \( dx \cos^2 x \sin^3 x \). In the first place, we have (Algebra, art. 260),

\[ \cos^2 x \sin^3 x = \frac{1}{2} (\sin 5x + \sin 3x + 2 \sin x). \]

Hence, multiplying by \( dx \) and integrating,

\[ \int dx \cos^2 x \sin^3 x = \frac{1}{2} \cos 5x - \frac{1}{2} \cos 3x - \frac{1}{2} \cos x + C. \]

150. Method III.—It has been found (Algebra, art. 270), that \( e \) being the base of Napier's logarithms,

\[ \cos x = \frac{e^{x \sqrt{-1}} + e^{-x \sqrt{-1}}}{2}; \] \[ \sin x = \frac{e^{x \sqrt{-1}} - e^{-x \sqrt{-1}}}{2 \sqrt{-1}}. \]

These exponential expressions for the sine and cosine enable us to transform differentials containing circular functions into others involving exponential and logarithmic functions, and to integrate by the methods explained (art. 140-145).

151. Method IV.—We may reduce the integration of such a differential as \( \sin^m x \cos^n x \) \( dx \), to that of another in which the indices of the sine or cosine are smaller numbers, by the formula \( fudv = uv - \int vdu \). For, making \( u = \sin^{m-1} x \) and \( dv = dx \sin x \cos^n x \), we find \( v = \frac{\cos^{n+1} x}{n+1} \), and

\[ \int dx \sin^m x \cos^n x = -\frac{\sin^{m-1} x \cos^{n+1} x}{m+1} + \frac{m-1}{m+1} \int dx \cos^{n+2} x \sin^{m-2} x. \]

From this expression, by putting for \( \cos^{n+1} x \), its value \( \cos^n x (1 - \sin^2 x) \), and transposing, we find

\[ \int dx \sin^m x \cos^n x = -\frac{\sin^{m-1} x \cos^{n+1} x}{m+1} + \frac{m-1}{m+1} \int dx \cos^{n+2} x \sin^{m-2} x. \]

By resolving the proposed differential into the two factors \( dx \cos x \sin^{mx} \) and \( \cos^{n-1} x \), and proceeding in the same way, we find

\[ \int dx \sin^m x \cos^n x = \frac{\sin^{m+1} x \cos^{n-1} x}{m+n} + \frac{n-1}{m+n} \int dx \sin^m x \cos^{n-2} x. \]

One of these formulæ serves to depress the exponent of the sine, and the other that of the cosine, and by their joint application the differential may be found when \( m \) and \( n \) are two positive integer numbers. For example,

\[ \int dx \sin^m x \cos^n x = -\frac{1}{2} \sin^2 x \cos^3 x + \frac{1}{2} \int dx \sin^m x \cos^n x; \] \[ \int dx \sin^m x \cos^n x = \frac{1}{2} \sin^2 x \cos^3 x + \frac{1}{2} \int dx \sin^m x \cos^n x; \]

this last term \( = -\frac{1}{2} \cos x + C \), after collecting the terms, we have \( \int dx \sin^m x \cos^n x = \cos x (-\frac{1}{2} \sin^2 x \cos^3 x + \frac{1}{2} \sin^2 x - \frac{1}{2}) + C \).

152. When \( m \) and \( n \) are negative, these formulæ require some modification. The first, by changing \( m \) into \( -n \), gives

\[ \int dx \sin^m x \cos^n x = \frac{\sin^{m-1} x}{(m-n) \cos^{n-1} x} + \frac{m-1}{m-n} \int dx \sin^{m-2} x \cos^n x. \]

This makes the integral sought depend on that of \( \frac{dx \sin x}{\cos^n x} \) or \( \frac{dx}{\cos^n x} \), according as \( m \) is odd or even. By making \( \cos x = z \), the first of these becomes \( \frac{dz}{2} \) of which the integral is obvious. The integral of the other will be shown presently in art. 153.

The second of the two general formulæ, by making \( n \) negative, and bringing the integral in the second side of the equation to stand alone, and lastly, changing \( n \) into \( n-2 \), gives

\[ \int dx \sin^m x \cos^n x = \frac{\sin^{m+1} x}{(n-1) \cos^{n-1} x} + \frac{m-1}{m-n} \int dx \sin^{m-2} x \cos^n x. \]

By this the integral sought is reduced to that of \( dx \sin^m x \), or to \( \frac{dx \sin^m x}{\cos^n x} \), according as \( n \) is even or odd. The first is found by formula (a) (art. 151), the other is presently to be noticed.

153. If we make \( m \) or \( n = 0 \), we have

\[ \int dx \sin^m x = -\frac{\cos x \sin^{m-1} x}{m} + \frac{m-1}{m} \int dx \sin^{m-1} x, \] \[ \int dx \cos^n x = \frac{\sin x \cos^{n-1} x}{n} + \frac{n-1}{n} \int dx \cos^{n-1} x, \] \[ \int \frac{dx}{\sin^n x} = \frac{-\cos x}{(m-1) \sin^{m-1} x} + \frac{m-2}{m-1} \int \frac{dx}{\sin^{m-1} x}, \] \[ \int \frac{dx}{\cos^n x} = \frac{\sin x}{(n-1) \cos^{n-1} x} + \frac{n-2}{n-1} \int \frac{dx}{\cos^{n-1} x}. \]

154. When the exponents of the sine and cosine are both negative, we multiply the numerator by \( \cos^2 x + \sin^2 x = 1 \), and have Ex. 1. Let the curve be a parabola, of which \( p \) is the parameter. By the nature of the curve, \( y^2 = px \), and the integral by art. 104,

\[ s = \int y dx = \frac{3}{2} p^{\frac{1}{2}} x^{\frac{3}{2}} + c = \frac{3}{2} yx + C \quad \text{(1)} \]

Let \( AD = a \), \( CD = b \); when \( s = 0 \), then \( x = a \), \( y = b \). These values being substituted in the above general expression for the area, it becomes

\[ 0 = \frac{3}{2} ab + C \]

If we now subtract the sides of equation (2) from the corresponding sides of (1), the indeterminate constant \( C \) will disappear, and we shall have

\[ s = \frac{3}{2} (xy - ab) \]

This is a convenient way of eliminating the indeterminate constant. If \( A \), the origin of the abscissa \( x \), be the vertex of the parabola, and if the area \( s \) be supposed to begin at the vertex, then \( a = 0 \), \( b = 0 \); and the area comprehended between the axis, an ordinate, and the curve, is \( \frac{3}{2} xy \), that is, \( \frac{3}{2} \) of the circumscribing parallelogram.

Ex. 2. Let the curve be a circle \( Apa \) (see fig. 7, art. 66), of which \( Ap \) is a diameter. Suppose the origin of co-ordinates to be at \( O \), the centre; let \( Ob \) be the radius perpendicular to \( OA \); put \( OQ = x \), \( PQ = y \). The radius \( = a \), the space \( OQpb = s \); by the nature of the circle \( x^2 + y^2 = a^2 \), and \( y = \sqrt{a^2 - x^2} \); hence

\[ s = \int y dx = \int dx \sqrt{a^2 - x^2} \]

This integral cannot be expressed in finite algebraic terms; therefore recourse must be had to an infinite series. By the binomial theorem, \( \sqrt{a^2 - x^2} = (a^2 - x^2)^{\frac{1}{2}} \)

\[ = a - \frac{1}{2} \cdot \frac{x^2}{a^2} - \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{x^4}{a^4} - \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{x^6}{a^6} - \ldots \]

Each term of this series being multiplied by \( dx \), and the integral taken, we have

\[ s = ax - \frac{1}{2} \cdot \frac{x^3}{3a} - \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{x^5}{5a^5} - \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{x^7}{7a^7} - \ldots + C \]

To determine \( C \), we consider that when \( x = 0 \), then \( s = 0 \); therefore \( C = 0 \); when \( x = a \), then \( s = \frac{1}{2} \) of the area of the circle; therefore the area of a circle whose radius \( = a \) is

\[ 4a^2 \left( \frac{1}{1} - \frac{1}{2} \cdot \frac{1}{3} - \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{5} - \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{2} \cdot \frac{1}{7} - \ldots \right) \]

This series converges slowly, so that a considerable number of terms would be required to give its value near the truth. If we make \( x = \cos \text{arc } 60^\circ \), then, drawing the radius \( Op \), the area \( OQpb \) will manifestly be made up of the sector \( PoB \), which is \( \frac{1}{12} \) of the area of the circle, and the triangle \( OQP \), which is \( \frac{1}{2} OQ \cdot PQ = \frac{\sqrt{3}}{8} a^2 \).

Putting \( \frac{a}{2} \) for \( x \) in the series, it will now converge much faster, and the sum of seven of its terms will be \( 4783057a^2 \), from which subtracting \( \frac{\sqrt{3}}{8} a^2 = 2165063a^2 \), there remains \( 2617994a^2 \) for \( \frac{1}{12} \) of the area, and \( 3 \cdot 1415928a^2 \) for the whole area of the circle. We have found this number by a less laborious calculation, in Algebra, 273. We may make \( A \), one end of the diameter (fig. 7, art. 66), the origin of the co-ordinates, so that \( AQ = x \), \( PQ = y \); segment \( ApQ = s \); then, because \( y = \sqrt{2ax - x^2} \), the area

\[ s = \int dx \sqrt{2ax - x^2} = \int dx \sqrt{2ax} \left( 1 - \frac{x}{2a} \right)^{\frac{1}{2}} \]

The integral of this expression will be had by convert- Inverse Method.

ing \((1 - \frac{x}{2a})^{\frac{1}{2}}\) into a series, then multiplying the terms separately by \(dx\sqrt{2ax}\), and integrating.

Ex. 3. To find the area of the elliptic segment APQ (fig. 7, art. 66). Let OA, the semitransverse, = a, OB, the semiconjugate, = b, AQ = x, PQ = y, the area APQ = s.

By the nature of the ellipse, \(y = \frac{b}{a} \sqrt{2ax - x^2}\); therefore

the elliptic segment

\[s = \int y dx = \frac{b}{a} \int dx \sqrt{2ax - x^2}.\]

Let APA be a semicircle on the same axis, and let \(s'\) denote the segment cut off by the ordinate pPQ, which is common to both curves. We found in last example that \(s' = \int dx \sqrt{2ax - x^2}\); therefore \(s = \frac{b}{a} s'\). Thus it appears that the area of APQ, any segment of an ellipse, has to the area of ApQ the corresponding segment of the circumscribing circle, the constant ratio of the lesser axis to the greater. Neither of the integrals require a constant, because both vanish when \(x = 0\).

Ex. 4. To find the area of the hyperbolic sector PCA, C being the centre of the hyperbola, and CA the semitransverse axis.

Draw the ordinate PQ; let the semitransverse axis CA = a, the semiconjugate axis = b, CQ = x, PQ = y, the sector PAC = s.

Because \(s = \text{triangle CPQ} - \text{space APQ} = \frac{1}{2} xy - \int y dx\); therefore \(ds = \frac{1}{2} xdy + \frac{1}{2} ydx - ydx = \frac{1}{2} (xdy - ydx)\).

Now, by the nature of the curve, \(y = \frac{b}{a} \sqrt{x^2 - a^2}\), therefore \(dy = \frac{b}{a} \frac{x dx}{\sqrt{x^2 - a^2}}\), and

\[ds = \frac{b}{2a} \left\{ \frac{x^2 dx}{\sqrt{x^2 - a^2}} - dx \sqrt{x^2 - a^2} \right\} = \frac{1}{2} ab \frac{dx}{\sqrt{x^2 - a^2}}.\]

Now, by art. 130, \(\int \frac{dx}{\sqrt{x^2 - a^2}} = l(x + \sqrt{x^2 - a^2}) + C\).

Therefore \(s = \frac{ab}{2} l(x + \sqrt{x^2 - a^2}) + C\).

Here C denotes an indefinite constant; To determine its value, we consider that when \(x = a\), then \(s = 0\); therefore in this particular case,

\[0 = \frac{ab}{2} l.a + C.\]

Hence, by subtracting equals from equals, and observing that the difference of two logarithms is the logarithm of their quotient, we eliminate C, and have

\[s = \frac{ab}{2} l \frac{x + \sqrt{x^2 - a^2}}{a}.\]

Because \(\sqrt{x^2 - a^2} = ay/b\), therefore also

\[s = \frac{ab}{2} l \left\{ \frac{x}{a} + \frac{y}{b} \right\}.\]

Ex. 5. To find the area of s, the hyperbolic space APQ, we have \(s = \text{triangle CPQ} - \text{sector CAP}\), or

\[s = \frac{1}{2} xy - \frac{ab}{2} l \left\{ \frac{x}{a} + \frac{y}{b} \right\}.\]

Ex. 6. Let the curve be an equilateral hyperbola, of which AX, AY are the asymptotes: To find s, the area included by the hyperbolic arc CF, the straight lines CE, FQ, which are parallel to one asymptote, and EQ, the segment of the other asymptote between them; suppose CE one of the parallels to be given in position. Put AE = a, EC = b, AQ = x, EQ = y. By the nature of the curve, \(xy = ab\); hence \(y = \frac{ab}{x}\), and

\[s = ydx = ab \frac{dx}{x} = ab l.x + C.\]

Fig. 23.

Now, when \(x = a\), then \(s = 0\); in this case \(0 = ab l.a + C\), or \(C = -ab l.a\); hence, substituting for C its value, and putting \(l \frac{x}{a}\) for \(l.x - l.a\), we have

\[s = ab l \frac{x}{a}.\]

From this formula we find \(l \frac{x}{a} = \frac{s}{ab}\). Thus it appears that Napier's logarithm of any number \(x/a\) may be expressed by a hyperbolic area; hence it has happened that the logarithms of his system have been called hyperbolic.

If M be the modulus of any system, the log of \(x/a\) in that system will be \(M l \frac{x}{a}\); therefore, in any system of which the modulus is M, we have

\[\log_{\frac{x}{a}} = \frac{M}{ab} s.\]

Thus we see that logarithms of any system may be expressed geometrically by hyperbolic areas, and therefore that the name hyperbolic logarithms cannot with propriety be given to one kind of logarithms rather than to another.

Ex. 7. Let the curve be the cycloid (fig. 9, art. 66), of which AC is the axis, and AH a perpendicular to the axis at the vertex A. From any point in the curve draw PG perpendicular to AH; it is proposed to find s, the area of the external space contained by the straight lines AG, GP, and the cycloidal arc AP.

Let O be the centre of the generating circle; draw PDQ perpendicular to AC, meeting the circle in D, and join OD. Put \(x = AG, y = PG, a = AO, v = \text{angle AOD}\); then (art. 66) \(PQ = a(v + \sin v), AQ = a(1 - \cos v)\), therefore,

\[x = a(v + \sin v), y = a(1 - \cos v), dx = ade(1 + \cos v), ds = a^2(1 - \cos v)(1 + \cos v)de = a^2 dv \sin^2 v,\]

and integrating by 153,

\[s = a^2 \int dv \sin^2 v = \frac{a^2}{2} (v - \sin v \cos v) + C.\]

When \(v = 0\), then s ought to vanish; this requires that \(C = 0\). Putting now the arc AD for \(av\), and DQ · OQ for \(a^2 \sin v \cos v\), we have

\[s = \frac{1}{2} AO \cdot \text{arc AD} - \frac{1}{2} DQ \cdot OQ = \text{circ. seg. ADQ}.\] Ex. 8. To find the area contained by AF, a line drawn from the focus of a parabola to the vertex, FP a line drawn to any point of the curve, and the intercepted arc AP.

Let \(a\) denote the parameter = 4AF, \(u\) = the angle AFP, \(r\) = FP, \(s\) = the parabolic sector AFP. Draw PQ perpendicular to the axis AF. By the nature of the curve, PF = AQ + AF. Now, supposing PFA to be an obtuse angle, \(FQ = r \cos v\), \(AQ = \frac{1}{2}a - r \cos v\), \(AF = \frac{1}{2}a\); therefore, \(r = \frac{1}{2}a - r \cos v\), and \(r(1 + \cos v) = \frac{1}{2}a\); hence,

\[1 + \cos v = 2 \cos^2 \frac{1}{2}v, \quad r = \frac{a}{4 \cos^2 \frac{1}{2}v}.\]

Now, by formula 3, article 69, \(ds = \frac{1}{2}r^2dv\), therefore

\[ds = \frac{a^2}{16 \cos^4 \frac{1}{2}v}.\]

To integrate this differential, we consider that

\[\frac{1}{\cos^2 \frac{1}{2}v} = \sec^2 \frac{1}{2}v \cdot \frac{dv}{2} = d(\tan \frac{1}{2}v);\]

and that

\[\frac{1}{\cos^4 \frac{1}{2}v} = \sec^4 \frac{1}{2}v = 1 + \tan^2 \frac{1}{2}v;\]

therefore, multiplying these equations, we get

\[\frac{1}{\cos^4 \frac{1}{2}v} = d(\tan \frac{1}{2}v) + \tan^2 \frac{1}{2}v d(\tan \frac{1}{2}v);\]

and hence, integrating by the rule art. 104, we find

\[\int \frac{1}{\cos^4 \frac{1}{2}v} dv = \tan \frac{1}{2}v + \frac{1}{2} \tan^2 \frac{1}{2}v + C.\]

Observing now that when \(v = 0\), then \(s = 0\); therefore \(C = 0\), and we have

\[s = \frac{a^2}{16} (\tan \frac{1}{2}v + \frac{1}{2} \tan^2 \frac{1}{2}v).\]

This integral might have been found by the formula of art. 153; not so easily, however. The formula here found is wanted in the astronomy of comets.

157. Since the area of any plane curve is expressed by an integral \(f ydx\), in which \(y\) the ordinate is some function of \(x\) the absciss, so, on the other hand, any integral may be represented geometrically by the area of a curve. The geometrical representation of an integral shows distinctly wherein it differs from a common analytic function, such as \(ax + bx^2\), or \(ax\), or \(\sin x\), &c. These last have determinate values corresponding to any given value of \(x\), and each is independent of its preceding and following values; but the magnitude of the space represented by an integral \(f ydx\) is the increment that a certain area receives while the variable \(x\) passes from one state of magnitude to another. Thus the absolute magnitude of the space CDQP (fig. 11), which represents an integral \(f ydx\), and which, when the curve is a parabola, is expressed by \(\frac{3}{2}xy + C\), cannot be known before \(C\) is determined, or else eliminated from the expression.

Whatever be the nature of the curve, the integral \(f ydx =\) space CPQD is correctly indicated, so as to admit of being computed, when we say that it is generated while the variable \(x\) increases from any value = AD, to another value = AQ, the lines AD, AQ being supposed inverse given in magnitude, and the relation between \(x\) and \(y\) Method known. The same thing is briefly indicated by this notation:

\[\text{area CPQD} = \int_a^x y dx,\]

in which \(a\) and \(x\) denote the lines AD and AQ.

From what has been shown it appears that an integral is an indeterminate quantity, the result of a change in the value of its variable; and that to know its absolute value we must know the values of the variable at its commencement and completion.

158. The analogy between a curvilinear area and an integral indicates a general method of finding its numerical value to any degree of nearness. Supposing \(y\) to be any function of \(x\), let it be proposed to find the integral \(f ydx\) between \(x = a\) and \(x = b\).

Let CPD be a curve, of which the co-ordinates are \(x\) and \(y\). In the axis AB, take AQ = \(a\), the least value of \(x\), and AQ' = \(b\), its greatest value, and draw the ordinates PQ, P'Q'. The area PQQ'P' will be the geometrical expression of the integral between the proposed limits; and by whatever means that area can be found, the same will apply to the determination of the integral.

Divide QQ' into any number of equal parts at the points Q', Q", &c., and draw the ordinates PQ', P'Q", &c. These will divide the whole figure into the spaces PP'QQ', PP'Q'Q", &c. Let a series of rectangles PQ, P'Q', P"Q" be formed, each having the least of two adjoining ordinates for a side: these will fill within the curve, supposing it to be entirely concave or convex towards the axis. Let another series P'Q, P"Q', P"Q" be constructed, each having the longest of two adjoining ordinates for a side: the aggregate of these will exceed the area of the curve. The length of the base of each rectangle will be known, and, from the nature of the curve, its height will also be known; hence the areas of all the rectangles may be found. The sum of the areas of the inscribed rectangles will be less than the integral, and the sum of the areas of the circumscribing rectangles will be greater. The difference between these sums is the rectangle PMSR, which, by supposing its base PM = QQ' to be sufficiently small, may be less than any assignable space. Thus an approximate value of the integral may be found, which shall differ from the truth by as small a quantity as we please.

Draw the chords PP', P'P", P"P", thus forming a series of trapeziums, which will differ from the curvilinear area by less than the sum of either the inscribed or circumscribing parallelograms, and will be a mean between them. This will be a nearer approximation than either sum to the integral.

As an example of the application of this method, let it be proposed to approximate to the integral \(\int_0^1 \frac{dx}{1+x^2}\) that is, to the arc whose tangent is \(x\), between the values of \(x = 0\) and \(x = 1\), which arc is \(\frac{1}{4}\) of a semicircle, the radius Inverse Method.

being supposed = 1. In this case the equation of the curve

\[ y = \frac{1}{1+x^2} \]

Let us suppose the base QQ' to be divided into ten equal parts; then, making \( x = 0, x = 1, x = 2, \ldots \) to \( x = 1 \), we obtain eleven equidistant ordinates or values of \( y \) as follows:

| The 1st | 1.00000 | |---------|----------| | 2d | -99010 | | 3d | -96154 | | 4th | -91743 | | 5th | -86207 | | 6th | -80000 |

The 7th | -73529 | |--------|--------| | 8th | -67114 | | 9th | -60975 | | 10th | -55249 | | 11th | -50000 |

By the elements of geometry, the area of the rectilineal figure formed by the trapeziums is found by adding together all the ordinates except the first and last, and half the sum of the first and last, and multiplying the result by the common breadth of the trapeziums, which is 1. This gives 784981 for the area or value of the integral

\[ \int_0^1 \frac{dx}{1+x^2}. \]

The true value expressed by a series is 1 -

\[ \frac{1}{3} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \ldots \]

which converges too slowly to be of any practical use; by other methods the more correct value of the integral is 7854 nearly.

159. Supposing the ordinates PQ, PQ', &c., to go on continually increasing, the inscribed rectangles will be constructed on the 1st, 2d, 3d, &c., ordinates, and the circumscribing rectangles on the 2d, 4th, 6th, &c.

Observing now that the ordinates are the values of \( y \) corresponding to AQ, AQ', &c., values of \( x \), which differ by the common interval QQ', we have manifestly the following rule for approximating to an integral \( \int y dx \) between the limits of \( x = AQ = a \) and \( x = AQ' = b \). Divide the interval QQ' = \( b - a \) into \( n \) equal parts, each equal to \( h \).

Find \( Y, Y', Y'', \ldots, Y^{(n)} \) the particular values of \( Y \) corresponding to \( x = a, x = a + h, x = a + 2h, \ldots, x = a + nh = b \).

Then supposing the ordinates to go on increasing,

\[ \int_a^b y dx > h (Y + Y' + Y'' + \ldots + Y^{(n-1)}); \]

\[ \int_a^b y dx < h (Y + Y' + Y'' + \ldots + Y^{(n)}). \]

The difference of these is \( h(Y^{(n)} - Y) \), which, by taking \( x \) sufficiently small, may be as small as we please. We may reason in the same way when the ordinates form a decreasing series. If they first increase and then decrease, the interval \( b - a \) may be divided into two or more portions, such that \( y \) increases or decreases continually from one end of each to the other.

Whatever be the values of \( y \), provided they be always finite from \( x = a \) to \( x = b \), putting for them \( Y, Y', Y'', \ldots, Y^{(n)} \), we have

\[ \int_a^b y dx = Yh + Y'h + Y''h + \ldots + Y^{(n-1)}h. \]

Leibnitz and his followers considered the integral \( \int y dx \) as the sum of the infinitely little elements \( Yh, Y'h, \ldots \). Hence the origin of the terms integral, to integrate, integration, &c. When \( h \) is infinitely little, and represented by \( dx \), the differential of \( x \), then \( y dx \) is the differential of the integral.

Rectification of Curves.

160. We have found (art. 72) that \( z \) being any arc of a curve referred to an axis by rectangular co-ordinates \( x, y \), the differential of the arc, viz. \( dz = \sqrt{dx^2 + dy^2} \). We are now to apply the calculus to particular curves.

Ex. 1. Let the curve be a parabola (fig. 16) of which the parameter is \( 2a \), the absciss \( AQ = x \), the ordinate \( PQ = y \). The equation of the curve, \( 2ax = y^2 \), gives

\[ adx = ydy \quad \text{and} \quad dz = \sqrt{dx^2 + dy^2} = \frac{1}{a} \sqrt{a^2 + y^2}. \]

If in formula B, art. 135, we make \( m = 1, n = 2, p = \frac{1}{2}, b = 1 \), and put \( y \) instead of \( x \), and \( a^2 \) instead of \( a \), we have

\[ \int dy \sqrt{a^2 + y^2} = \frac{1}{2} y \sqrt{a^2 + y^2} + \frac{1}{2} a^2 \int \frac{dy}{\sqrt{a^2 + y^2}}. \]

The integral in the second side of this equation is, by the formula of art. 130, I. \( (y + \sqrt{a^2 + y^2}) + C \); therefore \( z = \frac{y}{2a} \sqrt{a^2 + y^2} + \frac{a}{2} \ln(y + \sqrt{a^2 + y^2}) + C \).

If we suppose \( z \) to begin at the vertex, then, when \( z = 0, y = 0 \); therefore \( C = -\frac{a}{2} \ln a \), and

\[ z = \frac{y}{2a} \sqrt{a^2 + y^2} + \frac{a}{2} \ln \left( \frac{y + \sqrt{a^2 + y^2}}{a} \right). \]

Ex. 2. Let the curve be the second cubic parabola of which the equation is \( y^2 = ax^3 \). In this case the general formula gives

\[ dz = dy \sqrt{1 + \frac{9y}{4a}} \quad \text{and} \quad z = \frac{y}{3} \sqrt{\left( 1 + \frac{9y}{4a} \right)^3} + C. \]

This curve is perfectly rectifiable. Indeed it is the evolute of the parabola (art. 81, ex. 1).

Ex. 3. Let the curve be a circle (fig. 7), of which the radius \( OA = a \). Reckoning the co-ordinates from the centre, let \( OQ = x, PQ = y, BP = z \). The equation of the curve is \( x^2 + y^2 = a^2 \); hence \( y = \sqrt{a^2 - x^2} \),

\[ dy = -\frac{x dx}{\sqrt{a^2 - x^2}} \quad \text{and} \quad dz = \sqrt{a^2 - x^2}. \]

This integral cannot be otherwise expressed than by an infinite series. We have already found it (art. 139, ex. 4) on the supposition that the radius = 1; when the radius = \( a \), we have only to put \( \frac{x}{a} \) instead of \( x \), thence we get

\[ z = x + \frac{1}{2} \cdot \frac{x^3}{3a^2} + \frac{1.3}{24} \cdot \frac{x^5}{5a^4} + \frac{1.3.5}{24.6} \cdot \frac{x^7}{7a^6} + \ldots \]

There is no constant wanted; because, from the nature of the case, \( z \) and \( x \) begin together.

From this example, also from ex. 2, art. 156, and ex. 2, art. 139, it appears that the quadrature, also the rectification of the circle, is equivalent to the determination of any one of these integrals,

\[ \int dx \sqrt{1 - x^2}, \int \frac{dx}{\sqrt{1 - x^2}}, \int \frac{dx}{1 + x^2}, \]

which is a problem in pure analysis, and which does not seem to admit of a solution in finite terms. The solution of that celebrated geometrical problem is therefore hopeless.

Ex. 4. Let the curve be an ellipse (fig. 7), of which the semitransverse axis \( OA = 1 \), the semiconjugate \( OB = b \), the eccentricity, which is \( \sqrt{1 - b^2} = e \). Let \( O \) the centre be the origin of the co-ordinates \( OQ = x, PQ = y \); and, reckoning the arc from the extremity of the lesser axis, let \( BP = z \). By the nature of the curve, \( y = b \sqrt{1 - x^2} \), therefore \( dy = \frac{-bx dx}{\sqrt{1 - x^2}} \) and

\[ dz = \sqrt{dx^2 + dy^2} = \frac{dx \sqrt{1 - (1-b^2)x^2}}{\sqrt{1 - x^2}} = \frac{dx \sqrt{1 - e^2 x^2}}{\sqrt{1 - x^2}}. \] The integral of this differential is of such a nature as not to admit of being expressed either by arcs of a circle or logarithms. Therefore it can only be expressed by infinite series.

By the binomial theorem \( \sqrt{1 - e^2 x^2} = 1 - \frac{1}{2} e^2 x^2 - \frac{1}{2!} e^4 x^4 - \frac{1}{3!} e^6 x^6 - \frac{1}{5!} e^8 x^8 - \ldots \)

The terms of this series must now be multiplied each by \( \frac{dx}{\sqrt{1 - x^2}} \) and the integrals found, and their sum will express the length of the arc \( z \).

The integrals, leaving out the numeral co-efficients, will all have the form \( \int \frac{x^n dx}{\sqrt{1 - x^2}} \); therefore they will be found by the second formula of article 137. Thus, putting \( \varphi \) for the arc whose sine is \( x \), we have

\[ \int \frac{dx}{\sqrt{1 - x^2}} = \varphi, \]

\[ \int \frac{x dx}{\sqrt{1 - x^2}} = -\frac{1}{2} x \sqrt{1 - x^2} + \frac{1}{2} \varphi, \]

\[ \int \frac{x^3 dx}{\sqrt{1 - x^2}} = -\left( \frac{1}{4} x^3 + \frac{1}{2} x \right) \sqrt{1 - x^2} + \frac{1}{2} \varphi, \]

&c.

These integrals being multiplied by their respective co-efficients, we have the elliptic arc \( z = \varphi \left( 1 - \frac{1}{2} e^2 x^2 - \frac{1}{2!} e^4 x^4 - \frac{1}{3!} e^6 x^6 - \ldots \right) \)

\[ + \frac{1}{2} e^2 \left( \frac{1}{2} x^2 + \frac{1}{2} x \right) \sqrt{1 - x^2} \]

\[ + \frac{1}{2} e^4 \left( \frac{1}{2} x^2 + \frac{1}{2} x \right) \sqrt{1 - x^2} \]

&c.

Here no constant is wanted, because the functions on both sides of \( z \) vanish when \( x = 0 \), as they should.

If we make \( x = 1 \), then all the terms multiplied by \( \sqrt{1 - e^2 x^2} \) vanish; and as in this case \( \varphi = \) a quadrant \( = \frac{1}{2} \pi \), we get the elliptic quadrant \( AB = \)

\[ \frac{1}{2} \pi \left( 1 - \frac{1}{2} e^2 - \frac{1}{2!} e^4 - \frac{1}{3!} e^6 - \ldots \right) \]

This expression converges pretty fast when \( e \) is small, but when \( e \) is nearly \( = 1 \), it is hardly of any use. To have a complete solution, we ought to have another series suited to the case of \( e = 1 \) nearly. Our limits, however, do not admit of our entering on its investigation here.

The integration of the differential \( \frac{dx \sqrt{1 - e^2 x^2}}{\sqrt{1 - x^2}} \), which, putting \( \varphi \) for the arc whose sine \( = x \), may be also expressed thus, \( d\varphi \sqrt{1 - e^2 \sin^2 \varphi} \), has greatly engaged the attention of mathematicians. On this subject consult the works quoted in the conclusion of our History of Fluxions, page 640.

The length of a hyperbolic arc may be found exactly in the same way as that of the ellipse from its equation \( y = b \sqrt{x^2 - 1} \).

161. We shall here explain the remarkable property of the ellipse, first discovered by Fagnano, as has been mentioned, page 640, and which has proved the fertile germ of some of the finest discoveries in the modern analysis.

In the ellipse \( ABA \), let \( DE \) be a tangent at any point \( D \) in the curve, draw \( CE \) from the centre perpendicular to the tangent, and parallel to it draw the semidiameter \( CH \), which will be the conjugate to that drawn from \( C \) through \( D \); and draw \( HK \) perpendicular to the axis. Let \( CA = 1 \), \( CB = b \), \( \sqrt{1 - b^2} \), the eccentricity, \( = e \). Make the perpendicular \( CE = p \), the tangent \( DE = t \), the angle \( ACE = CHK = \varphi \), and the arc \( AD = z \). By the nature of the ellipse, \( CA^2 - HK^2 = CB^2 (CA^2 - CK^2) \), that is, because \( HK = CH \cos \varphi \) and \( CK = CH \sin \varphi \), \( CH^2 \cos^2 \varphi = b^2 (1 - CH^2 \sin^2 \varphi) \), and hence \( b^2 = CH^2 (\cos^2 \varphi + b^2 \sin^2 \varphi) = CH^2 (1 - e^2 \sin^2 \varphi) \), and \( CH = \frac{b}{\sqrt{1 - e^2 \sin^2 \varphi}} \).

Now, by the nature of the ellipse, \( CH = \frac{CA \cdot CB}{CE} = \frac{b}{p} \),

therefore \( p = \sqrt{1 - e^2 \sin^2 \varphi} \). Now it has been shown (art. 75) that in every curve \( d(z + t) = pd\varphi \), and therefore \( z + t = \int pd\varphi \); hence, in the ellipse, \( z + t = \int \varphi \sqrt{1 - e^2 \sin^2 \varphi} \).

In the semi-axis \( CA \) take \( CQ = CA \cdot \sin ECA = \sin \varphi \); draw the ordinate \( QP \); and put \( z = \) elliptic arc \( BP \). By what was shown in last article, \( dz = d\varphi \sqrt{1 - e^2 \sin^2 \varphi} \),

and \( z = \int d\varphi \sqrt{1 - e^2 \sin^2 \varphi} \); thence it appears that the functions \( z + t \) and \( z \) have the same differential, therefore they can only differ by a constant, that is, \( z + t = z + C \).

Now when \( z = 0 \), then \( \varphi = 0 \), and \( z' = 0 \), also \( t = 0 \); therefore the variables \( z \), \( z' \), and \( t \) begin together, and the constant \( C = 0 \); thus we have \( z' + t = z \) and \( z - z' = t \).

Hence we have this very remarkable property of an ellipse. At any point \( D \) in the curve draw a tangent \( DE \), and from the centre \( C \) draw \( CE \) perpendicular to the tangent. In \( CA \) take \( CQ = CA \cdot \sin ECA \); draw the ordinate \( PQ \). The difference of the elliptic arcs \( BP \), \( AD \) (that is, \( BP - AD \)), is equal to the tangent \( DE \). This is in substance Fagnano's theorem. Hence it follows that any arc of an ellipse being given, another may be found by a geometrical construction, such, that their difference shall be an assignable straight line.

We have found (art. 75) that in all curves \( t = \frac{dp}{d\varphi} \). In the ellipse \( p = \sqrt{1 - e^2 \sin^2 \varphi} \) and \( \frac{dp}{d\varphi} = -e^2 \sin \varphi \cos \varphi \).

Therefore, without any reference to the length of the tangent,

\[ z - z' = \frac{e^2 \sin \varphi \cos \varphi}{\sqrt{1 - e^2 \sin^2 \varphi}} \]

There are corresponding properties of the hyperbola, but our limits forbid our entering on them.

**Cubature of Solids.**

162. Let \( AB \) be the axis of any plane curve \( APP' \), and \( PQ, P'Q' \) ordinates perpendicular to the axis; draw \( PH \) perpendicular to the ordinate \( P'Q' \), and \( PK \) perpendicular to \( PQ \). Conceive now the plane figure \( APQ' \) to revolve about the axis \( AB \); the curve \( AP \) will generate a solid \( AP'EP' \), called a solid of revolution; and the rectangles... Put AQ = x, AQ' = x', PQ = y, P'Q' = y'; the solid generated by the space APQ = s, that generated by APQ' = s'. Let π = 3.1416 = the area of a circle whose radius is 1. The content of the cylinder whose base is PEp will be π(x' - x)y^2, and that of the cylinder whose base is P'E'p' will be π(x' - x)y'^2.

The solid generated by the curvilinear space PQQ', being greater than the cylinder whose base is PEp, but less than the cylinder whose base is P'E'p', we have

\[ s' - s > \pi(x' - x)y^2, \quad \text{and} \quad s' - s < \pi(x' - x)y'^2; \]

therefore \( \frac{s' - s}{x' - x} > \pi y^2, \quad \text{and} \quad \frac{s' - s}{x' - x} < \pi y'^2. \)

Conceive now the circle P'E'p' to approach continually to the circle PEp; they will ultimately coincide and be equal; therefore \( \pi y^2 \) is the limit of \( \frac{s' - s}{x' - x} \), and consequently \( \frac{ds}{dx} = \pi y^2 \), and

\[ ds = \pi y^2 dx, \quad \text{and} \quad s = \int \pi y^2 dx. \]

By this formula the content of a solid of revolution generated by any given plane curve may be found.

Ex. 1. Let the solid be a paraboloid generated by the revolution of a parabola AP about its axis AQ. By the nature of the curve, \( y^2 = ax \); therefore, in this case,

\[ s = \int \pi ax dx = \frac{1}{2} \pi ax^2 + C = \frac{1}{2} \pi xy^2 + C. \]

If the integral begin when \( x = 0 \), then also \( s = 0 \) and \( C = 0 \); now \( \pi xy^2 \) is the content of a cylinder having the same base as the paraboloid, therefore, a paraboloid is half a cylinder of the same base and altitude.

From this geometrical theorem we may find the content of the solid between two sections of a paraboloid by planes perpendicular to its axis, which is called a frustum of a paraboloid. Let A be the area of the base of the frustum, a the area of its top, h its height, x the segment of the axis between the vertex of the paraboloid and top of the frustum. The content of the whole paraboloid \( = \frac{1}{3}(a + h)A \), and the content of the part cut off \( = \frac{1}{3}xa \), therefore the content of the frustum \( = \frac{1}{3}(a + h)A - xa \).

Now by the nature of the solid \( \frac{x + h}{x} = \frac{A}{a} \), hence \( xA - xa = ha \), therefore the content of the frustum is \( \frac{1}{3}(A + a)h \). Hence it appears that a frustum of a paraboloid is equal to a cylinder of the same altitude, and whose base is a mean between the top and bottom of the frustum.

Ex. 2. Let the solid be a cone whose vertex is A, and base P'E'p', a plane figure of any kind.

Let AB be the altitude of the cone, and PEp a section parallel to the base, meeting AB in Q. Let b = the area of the base P'E'p', v = area of section PEp, e = AB, x = AQ, s = content of solid A-PEp. In this case, \( ds = edx \), an expression the same as for a solid of revolution, and found in the same way. Now, sections of similar cones are proportional to the squares of their like dimensions; therefore \( c^2 : x^2 = b : v \), hence \( v = \frac{bx^2}{c^2} \) and

\[ ds = \frac{b}{c^2} x^2 dx, \quad \text{and} \quad \int ds = \frac{b}{3c^2} x^3 + C = \frac{\pi r^3}{3} + C. \]

If \( x \) and \( s \) begin together, then \( C = 0 \) and \( s = \frac{1}{3} \pi r^3 \). Thus it appears that a cone of any kind is \( \frac{1}{3} \) of its circumscribing cylinder.

To find the content of a frustum of a cone, or space between the base and a plane parallel to its base; put A = area of base, a = area of top, h = height of frustum, x = height of cone cut off above the frustum. The content of the whole cone \( = \frac{1}{3} (h + x) A \), and the content of the part cut off \( = \frac{1}{3} xa \), therefore the content of the frustum \( = \frac{1}{3} (hA + xA - xa) = \frac{1}{3} hA + \frac{1}{3} xA - \frac{1}{3} xa \).

Now \( \frac{A}{a} = \frac{(x + h)^2}{x^2} \), and \( \sqrt{A} = \frac{x + h}{x} \),

\[ \text{hence} \quad x = \frac{\sqrt{A}}{\sqrt{A} - \sqrt{a}} h, \quad \text{and} \quad x(A - a) = \sqrt{a} (\sqrt{A} + \sqrt{a}) h = (\sqrt{Aa} + a) h. \]

On the whole, the content of the frustum \( = \frac{1}{3} (\sqrt{Aa} + a) h \).

Ex. 3. To find the content of a spheroid generated by the revolution of an ellipse about either of its axes.

Let PQp be a section of the solid perpendicular to the fixed axis AB. Put AB = a, the revolving axis DE = b, also AQ = x, PQ = y. From the nature of the ellipse,

\[ y^2 = \frac{b^2}{a^2} (ax - x^2), \quad \text{hence} \quad s = \frac{\pi b^2}{a^2} \int (ax - x^2) dx = \frac{\pi b^2}{a^2} \left( \frac{1}{2} ax^2 - \frac{1}{3} x^3 \right) + C. \]

If we suppose \( s = 0 \) when \( x = 0 \), then \( C = 0 \). If we make \( x = a \), then \( s \), the whole spheroid, \( = \frac{\pi b^2}{6} a \times \text{area of middle section}. \) Hence it appears that the whole solid is \( \frac{2}{3} \) of its circumscribing cylinder. This proposition was discovered by Archimedes.

If the axes \( a, b \) be supposed equal, the solid is a sphere; hence it appears that the content of a sphere \( = \frac{\pi}{6} a^3 \), \( a \) being the diameter.

Ex. 4. Let the solid be a parabole spindle which is generated by the revolution of APB, an arc of a parabola, about ACB, an ordinate to its axis. From P any point in the revolving curve, draw PQ perpendicular to the ordinate AB, and PR perpendicular to the axes of the parabola. Make DC = p, AC = q, CQ = PR = x, PQ = CR = y, the solid (between the sections DEF, PGp) = s. By the nature of the curve, PR² : AC² = DR : DC; that is, \(x^2 : q^2 = p - y : p\); hence \(p \cdot x^2 = pq^2 - q^2y\), and \(y^2 = \frac{pq}{q^2} (q^4 - 2q^2x^2 + x^4)\), and

\[s = \frac{pq}{q^2} \int (q^4dx - 2q^2x^2dx + x^4dx)\]

\[= \frac{pq}{q^2} (q^4x - \frac{3}{2}q^2x^3 + \frac{1}{3}x^5) + C.\]

If the integral begin when \(x = 0\), then \(C = 0\). If we make \(x = q\), we get \(\frac{pq}{q^2}\) for half the content of the solid generated by the curve ADB.

**Surfaces of Solids.**

164. Let AB be the axis of any plane curve APP', and PQ, P'Q' ordinates to the axis; draw PE, P'E tangents to the curve at P and P', meeting in E; draw EF perpendicular to AB, and draw the chord PP'. Suppose now the curve to revolve about its axis AB; the arc APP' will generate a curve surface, to be found: the portion PP' of the arc will generate an increment of that surface, which, by an axiom in geometry, will be less than the sum of the conical surfaces generated by the tangents PE, P'E, but greater than the conical surface generated by the chord PP'. Put AQ = x, AQ' = x', PQ = y, PQ' = y', EF = y", arc AP = z, arc AP' = z', surface generated by arc AP = v, surface generated by arc AP' = v'.

By the elements of geometry, the surfaces generated by the tangents PE, P'E will be

\[\pi(PQ + EF)PE = \pi(y + y')PE,\]

\[\pi(PQ' + EF)P'E = \pi(y' + y'')PE,\]

and the surface generated by the chord PP' will be

\[\pi(PQ + PQ')PP' = \pi(y + y')PP'.\]

Therefore \(v' - v > \pi(y + y')\) chord PP', and

\[\frac{v' - v}{z' - z} < \pi(y + y')PE + (y' + y'')PE,\]

\[\frac{v' - v}{z' - z} > \pi(y + y')PE + (y' + y'')PE,\]

Now, the point P being supposed to approach to P', these quantities, one of which is greater than \(\frac{v' - v}{z' - z}\), and the other less, become ultimately equal; for the lines \(y, y', y''\) become equal, and \((y + y')PE + (y + y')PE\) becomes \((y + y')(PE + P'E)\), and the fractions \(\frac{PE + P'E}{arc PP'}\), chord PP' are ultimately each = 1; therefore,

\[\lim_{z' \to z} \frac{v' - v}{z' - z} = 2\pi y, \text{ and } \frac{dv}{dz} = 2\pi y;\]

hence, \(dv = 2\pi ydz = 2\pi y \sqrt{dx^2 + dy^2}\).

Now \(2\pi y\) expresses the circumference of the section \(PP'\); hence it appears that the differential of the surface of revolution is equal to the differential of the arc of the generating curve multiplied by a section of the surface made by a plane perpendicular to the axis.

165. Ex. 1. Let the solid be a sphere, of which the radius is \(a\). Then, making \(AQ = x, PQ = y\), the arc AP = z, the surface of the segment cut off by the plane \(PP' = v\); the equation of the curve being \(y = \sqrt{2ax - x^2}\), we have

\[dy = \frac{adx - xdx}{\sqrt{2ax - x^2}}, dz = \sqrt{dx^2 + dy^2} = \frac{adx}{\sqrt{2ax - x^2}} = \frac{adx}{y},\]

\[dv = 2\pi ydz = 2\pi axdx,\]

and the surface \(v = 2\pi ax\); here no constant is wanted, because we suppose that when \(x = 0\), then \(v = 0\).

Since \(2\pi a\) is the circumference of a circle whose diameter is \(a\), it follows that the curve surface of a segment of a sphere is equal to a rectangle contained by a straight line equal to the circumference of the sphere and the height of the segment; and hence it follows, that the whole surface of the sphere is four times the area of one of its great circles. These conclusions were found by Archimedes.

Ex. 2. To find the surface of a paraboloid. Put AQ = x, PQ = y, arc AP = z, curve surface generated by AP = v, the parameter \(= 2a\). We found (article 160)

\[dz = \frac{dy}{a} \sqrt{a^2 + y^2}, \text{ hence,}\]

\[dv = 2\pi ydz = \frac{2\pi}{a} ydy \sqrt{a^2 + y^2},\]

and, integrating, \(v = \frac{2\pi}{a} \int ydy \sqrt{a^2 + y^2} = \frac{2\pi}{3a} (a^2 + y^2)^{\frac{3}{2}} + C.\)

When \(v = 0\), then \(y = 0\); therefore, at the vertex we have

\[0 = \frac{2\pi a^2}{3a} + C \text{ and } C = -\frac{2\pi a^2}{3a};\]

therefore, substituting for C, we have

\[v = \frac{2\pi}{3a} \left((a^2 + y^2)^{\frac{3}{2}} - a^3\right) = \frac{2}{3} \pi \left(\sqrt{a(2x + a)^2} - a^2\right).\]

**Rectification of Curves of Double Curvature.**

166. Let CPD be a line of double curvature referred to three co-ordinate planes YAX, ZAX, ZAY. From every point in the curve let perpendiculars PP', &c. be drawn to one of the planes, YAX; these will all be in the surface of a cylinder that meets the plane in a curve CPD, which will be the projection of the proposed curve. Again, from P' draw PQ, P'R perpendicular to AX, AY. Let AQ = PR = x, AR = PQ = y, PP' = z, the arc CP = v, its projection C'P' = v': Then \(dv' = \sqrt{dx^2 + dy^2}\) (art 72). Suppose now the cylindric surface to be extended into a plane surface; the curve CPD will then be changed into a common plane curve, and its projection C'P'D' into a straight line of the same length as before. We may now consider

\( Z_1 \) being functions of \( z \) only; and since \( dy = zdz + zdz \), the equation will be, after dividing by \( x^n \), \( zdz + Z_1 \), \( (zdz + zdz) = 0 \), a result which may be put under the form

\[ \frac{dx}{x} + \frac{Z_1 dz}{Z + zZ_1} = 0; \]

from which we get

\[ \int \frac{dx}{x} + \int \frac{Z_1 dz}{Z + zZ_1} = C. \]

170. Ex. 1. Let us apply this transformation to the equation \( xdy + ydx = nydx + (x - ny) dx + ydy = 0 \), which, making \( y = zx \), and therefore \( dy = zdz + zdz \), becomes \( x(1 - nz) dx + x^2dz + x^2dz = 0 \), or \( (1 - nz + z^2) dx + zdz = 0 \), and \( \int \frac{dx}{x} + \int \frac{zdz}{1 - nz + z^2} = C \),

or \( 1.x + \int \frac{zdz}{1 - nz + z^2} = C \). This integral may be simplified by observing that

\[ \frac{zdz}{1 - nz + z^2} = \frac{2zdz - ndz}{1 - nz + z^2} + \frac{ndz}{1 - nz + z^2} \]

for it then becomes

\[ 1.x + \frac{1}{2}l.(1 - nz + z^2) + \frac{1}{2}\int \frac{ndz}{1 - nz + z^2} = C. \]

The integral which yet remains to be found will depend on logarithms, if \( \frac{1}{2}n > 1 \); on arcs of circles, if \( \frac{1}{2}n < 1 \); and will be algebraical if \( \frac{1}{2}n = 1 \). In each case it may be had by the methods given for integrating rational fractions.

Ex. 2. Let it be proposed to integrate the equation

\[ xyy - ydx = dx \sqrt{x^2 + x^2}. \]

By making \( y = zx \), bringing all the terms to one side of the equation, and dividing by \( x \), we shall have \( dx \sqrt{1 + z^2} - zdz = 0 \), and hence

\[ \frac{dx}{x} - \frac{dz}{\sqrt{1 + z^2}} = 0; \]

and by the separate integration of each term,

\[ 1.x - l.(z + \sqrt{1 + z^2}) = l.C, \quad \text{or} \quad \frac{x}{z + \sqrt{1 + z^2}} = C. \]

and replacing \( z \) by its value \( x \), there will result

\[ \frac{x^2}{y + \sqrt{x^2 + y^2}} = C, \quad \text{or} \quad y + \sqrt{x^2 + y^2} = C; \]

and hence again, by transposing \( y \), and squaring, \( x^2 = C^2 + 2Cy \), which is the primitive equation.

The equation

\[ (a + mx + ny) dx + (b + px + qy) dy = 0, \]

which is not homogeneous in its present form, may be transformed so as to become homogeneous. Assume \( x = t + a, y = u + b \), then \( dx = dt, dy = du \), and the proposed equation becomes

\[ (a + ma + n\beta + mt + nu)dt + (b + pa + q\beta + pt + qu)du = 0. \]

To make the constant terms disappear, we assume \( a + ma + n\beta = 0, b + pa + q\beta = 0 \), equations which determine \( a \) and \( \beta \), and then there remains the differential equation

\[ (mt + nu) dt + (pt + qu) du = 0, \]

which is homogeneous in respect of the new variables \( t \) and \( u \). This transformation gives no result when \( my - np = 0 \), a case in which \( a \) and \( \beta \) become infinite; but then \( q = \frac{np}{m} \); and consequently \( m(px + qy) = p(mx + ny) \). The proposed equation being changed to

\[ adx + bdy + (mx + ny)(dx + \frac{p}{m}dy) = 0, \] we now make \( mx + ny = z \), so that \( dy = dz - m dx \); by substitution and proper reduction our equation now becomes

\[ dx + \frac{(bm + pz) dz}{amn - bm^2 + m(n-p)z} = 0. \]

The integral of this equation will contain logarithms, except in the case where \( n = p \), when it will be

\[ x + \frac{2bmx + pz^2}{2m(an - bm)} = C. \]

171. The separation of the variables is easily effected in the equation \( dy + Pydx = Qdx \), where P and Q denote any functions whatever of \( x \). Substituting \( Xz \) and \( zdX + Xxdx \) instead of \( y \) and \( dy \), it becomes

\[ zdX + Xxdx + PXzdx = Qdx. \]

Here X denotes an indeterminate function of \( x \), to which we may give such a form as shall conduce to the separation of the variables. For this purpose we make \( Xz + PXxdx = 0 \), which requires that \( zdX = Qdx \). If we divide the first equation by \( X \), it becomes \( dz + Px dx = 0 \), from which we deduce \( \frac{dz}{z} + P dx = 0 \), and \( 1 \cdot z + f Pdx = 0 \); and passing from logarithms to numbers, \( z = e^{-fPdx} \). (Here e is the base of Napier's logarithms,) we neglect the arbitrary constant in the mean time, as it will be added at the end of the operation. Afterwards, deducing the value of \( dx \) from the second equation, and substituting it in the first, we shall have

\[ dX = e^{fPdx} Qdx, \quad X = e^{fPdx} Qdx + C; \]

and consequently,

\[ y = e^{-fPdx} \left( fe^{fPdx} Qdx + C \right). \]

Ex. Let the equation be \( dy + ydx = ax^2 dx \): then \( P = 1, Q = ax^2, fPdx = x^2 fPdx = e^x, -fPdx = e^{-x}, \)

\[ fe^{fPdx} Qdx = \int ae^{f} dx = ae^x(x^3 - 3x^2 + 6x - 6); \]

therefore \( y = e^{-x} \{ ae^x(x^3 - 3x^2 + 6x - 6) + C \} \)

\[ = Ce^{-x} + a(x^3 - 3x^2 + 6x - 6). \]

172. The early writers on the integral calculus classed differential equations by the number of their terms. In equations which had but two terms, and whose form is consequently \( \beta u^k - \gamma u^l dz = au^f dz \), the variables are separated immediately, since we thence deduce \( \beta \frac{dz}{z} = -\gamma \frac{du}{u} \). This however is not the case with equations of three terms, which are expressed by the formula

\[ \gamma u^k dz + \beta u^l du = \omega uz^f dz. \]

This may be simplified by dividing the terms by \( \gamma u^k z^f \),

it then becomes

\[ z^k - f dz + \beta u^f - i z^h - f du = \alpha u^i - i du. \]

Suppose now \( z^k - f dz = \frac{dy}{g-i+1}, \quad u^f - i du = \frac{dx}{g-i+1} \);

then we have \( z^k - f + 1 = y, u^f - i + 1 = x \), and

\[ dy + \frac{(k-f+1)\beta y}{g-i+1} dx = \frac{(k-f+1)\alpha x}{g-i+1} dx. \]

To abridge, let us put

\[ b = \frac{f-k+1}{1}, \quad a = \frac{k-f+1}{a}, \quad n = \frac{h-f}{k-f+1}, \quad m = \frac{e-g}{g-i+1}. \]

and the equation will become

\[ dy + by dx = ay dx. \]

When \( n = 1 \), the equation belongs to the class treated of in last article. When \( n = 2 \), the equation is

\[ dy + by^2 dx = ax dx, \]

which was first considered by an Italian geometer, Riccati, whose name it bears. When \( m = 0 \), the variables are separable, and we have

\[ dx = \frac{dy}{a-by}. \]

Proceeding from this case, analysts have succeeded in separating the variables when \( m \) is any number of the form \( \frac{-4i}{2j-1} \), \( i \) being supposed an integer number. Riccati's equation was first considered by James Bernoulli, who gave an approximate solution of it. (Jac. Bernoulli Opera, p. 1053.) The general problem, however, remains yet unresolved.

173. When a differential equation is the immediate result of the differentiation of a known function, then that function, put equal to \( C \) constant, will be the integral required. The differential equation \( axy + ydx = 0 \) is of this class; it is the differential of the product \( xy \), therefore the primitive equation is \( xy = C \). A differential equation may, however, not be the immediate result of the differentiation of a function of \( x \) and \( y \); it may be formed by the elimination of some constant quantity contained in the function and also in its differential (art. 54); or it may be the result after the differential of the primitive has been divided by some factor common to all its terms.

Thus, if the primitive be \( y = cx \), or \( y = cx = 0 \), the differential equation is \( dy-cdx = 0 \). If however we eliminate \( c \) by the usual algebraic process, we find \( xy - ydx = 0 \),

an expression which is not the immediate result of any differentiation. The same result will be obtained if the primitive \( x = cy \) be put under the form \( \frac{x}{y} = e \); for then,

by differentiating, \( \frac{ydx-xdy}{y^2} = 0 \), and, rejecting the factor \( \frac{1}{y^2} \), \( ydx-xdy = 0 \).

This expression is not an exact differential; however, it becomes so by restoring the factor \( \frac{1}{y^2} \), and then it is the differential of \( \frac{y}{x} \).

174. In general let \( u = f(x,y) \) be the primitive equation, the differential being \( du = Mdx + Ndy \), where \( M \) denotes \( \frac{du}{dx} \), that is, the differential co-efficient found on the supposition that \( x \) is variable and \( y \) constant, and \( N \) denotes \( \frac{du}{dy} \), the co-efficient taken on the reverse hypothesis of \( y \) being variable and \( x \) constant; now we have found that \( \frac{d^2u}{dx dy} = \frac{d^2u}{dy dx} \) (art. 89); therefore, if \( Mdx + Ndy \) be the differential of \( u = f(x,y) \), then

\[ \frac{dM}{dy} = \frac{dN}{dx}. \]

Hence, If \( Mdx + Ndy \) is an exact differential, the condition expressed by formula (1) will always be satisfied. And conversely, if \( M \) and \( N \) are such functions of \( x \) and \( y \) that \( \frac{dM}{dy} = \frac{dN}{dx} \), then \( Mdx + Ndy \) shall be an exact differential, and in every case its integral may be found. This is called the Condition of Integrability.

To prove the second part of the proposition, let us sup- pose that the integral of $Mdx$ is taken on the supposition that in the function $M$, $x$ is variable and $y$ constant; and let the integral be $P + Y$, where $Y$ is any function of $y$, which comes in the place of the indeterminate constant that enters into every integral, and $P$ is a known function of $x$ and $y$, which results from $fMdx$ relatively to $x$ only, so that $M = \frac{dP}{dx}$.

The complete differential of $P + Y$ is (art. 91) $\frac{dP}{dx} dx + \frac{dy}{dy} dy + dY$, or $Mdx + \frac{dP}{dy} dy + dY$. By comparing this with the differential $Mdx + Ndy$, it appears that if such a value be given to $Y$ that $Ndy = \frac{dP}{dy} dy + dY$, or that

$$dY = \left(N - \frac{dP}{dy}\right) dy \ldots \ldots \ldots \ldots (2)$$

then the integral of $Mdx + Ndy$ will be $P + Y$. Now, by differentiating $M = \frac{dP}{dx}$ in respect of $y$, we have $\frac{dM}{dy} = \frac{d^2P}{dydx}$, but by hypothesis $\frac{dM}{dy} = \frac{dN}{dx}$, and $\frac{d^2P}{dydx} = \frac{d^2P}{dxdy}$ (art. 89), therefore $\frac{dN}{dx} = \frac{d^2P}{dxdy}$ and $\frac{dN}{dx} - \frac{d^2P}{dxdy} = 0$, that is,

$$d\left(N - \frac{dP}{dy}\right) = 0,$$

the differential being taken supposing $x$ only to be variable; therefore $N - \frac{dP}{dy}$ is constant in respect of $x$, and is a function of $y$ only; hence the possibility of finding $Y = \int \left(N - \frac{dP}{dy}\right) dy$ is proved, and we have the integral of $Mdx + Ndy$ expressed by $P + \int \left(N - \frac{dP}{dy}\right) dy$, and here $P = \int Mdx$, the integral being taken on the supposition that $y$ is constant.

We may begin with finding the integral of $Ndy$, supposing $x$ to be constant, proceeding in all respects as has been explained. In general we ought to begin with the term which brings out the integral with least calculation.

Ex. 1. It is proposed to find whether the differential

$$\frac{ydx - xdy}{x^2 + y^2}$$

satisfies the condition of integrability, and, if it does, to integrate it.

We put the differential under the form

$$du = \frac{y}{x^2 + y^2} dx - \frac{x}{x^2 + y^2} dy,$$

therefore $M = \frac{y}{x^2 + y^2}$, $N = -\frac{x}{x^2 + y^2}$,

$$\frac{dM}{dy} = \frac{x^2 - y^2}{(x^2 + y^2)^2}, \quad \frac{dN}{dx} = \frac{x^2 - y^2}{(x^2 + y^2)^2}.$$

The condition of integrability is in this case satisfied, and therefore $\frac{ydx - xdy}{x^2 + y^2}$ is an exact differential.

To determine the integral, we have

$$P = \int Mdx = \int \frac{ydx}{x^2 + y^2} = \int \frac{dx}{1 + \frac{x^2}{y^2}} = \tan^{-1} \frac{x}{y},$$

whence $u = \tan^{-1} \frac{x}{y} + Y$.

Differentiating, and considering the whole as variable,

$$du = \frac{ydx - xdy}{x^2 + y^2} + dY.$$

This, compared with the proposed differential, shows that $dY = 0$, and $Y = C$ a constant.

Ex. 2. To find the integral of the differential

$$du = (2yx + 3y^2) dx + (2xy + 9xy^2 + 8y^3) dy.$$

In this example, $M = 2yx + 3y^2$, $N = 2xy + 9xy^2 + 8y^3$,

$$\frac{dM}{dy} = 4yx + 9y^2 = \frac{dN}{dx};$$

the proposed function is therefore a complete differential.

Integrating in respect of $x$, we find

$$\int Mdx = y^2x^2 + 3yx + Y,$$

and $u = y^2x^2 + 3yx + Y$.

Differentiating this expression in respect of $y$,

$$\frac{du}{dy} = 2x^2y + 9xy^2 + \frac{dY}{dy},$$

but we had $\frac{du}{dy} = 2x^2y + 9xy^2 + 8y^3$;

therefore, $\frac{dY}{dy} = 8y^3$, and $Y = \int 8y^3 dy = 2y^4 + C$,

and consequently the integral is

$$u = y^2x^2 + 3yx + 2y^4 + C.$$

175. If in the equation $Mdx + Ndy = 0$, the condition of integrability, viz. $\frac{dM}{dy} = \frac{dN}{dx}$, is not satisfied, we may next inquire whether some factor, a function of $x$ and $y$, can be found by which the expression $Mdx + Ndy$ can be rendered a complete differential.

Let the equation be put under the form $\frac{dy}{dx} + K = 0$, where $K = \frac{M}{N}$. This equation is the result of the elimination of a constant quantity $c$ between a primitive equation, which may be represented by $f(x, y, c) = 0$ and its immediate differential. Or else $c$ may be eliminated directly by bringing the primitive equation to the form $F(x, y) = c$, which will lead to the same result (art. 53).

Put $u$ for $F(x, y)$, also $Pdx$ for the differential of $u$ taken relatively to $x$, and $Qdy$ for its differential relatively to $y$, so that $du = Pdx + Qdy$, and $\frac{du}{dx} = P + \frac{Qdy}{dx} = 0$,

therefore $P + Q\frac{dy}{dx} = 0$. Since this equation does not contain $C$, it ought to be identical with the equation $\frac{dy}{dx} + K = 0$;

therefore $\frac{dy}{dx} + K = \frac{dy}{dx} + \frac{P}{Q} = \frac{Pdx + Qdy}{Qdx}$, that is, $\frac{dy}{dx} + K = \frac{du}{Qdx}$, and hence

$$Q(dy + Kdx) = du.$$

Now the second side of the equation is a complete integral; hence it appears that there is always a factor $Q$, by which, if the differential equation $dy + Kdx = 0$ be multiplied, the result will be a complete integral.

Let both sides of the equation be multiplied by $U$, any function of $U$, and we have

$$UQ(dy + Kdx) = Udu.$$

Now $Udu$ is still a complete differential, and as the factor $UQ$ may have an infinite variety of forms, it follows that there are an infinite number of factors, such that if a differential equation be multiplied by any one of them, it will be a complete differential. Since it appears that when the equation $Mdx + Ndy = 0$ does not satisfy the condition of integrability, it is because differentiation and subsequent elimination of the arbitrary constant have caused a factor to disappear, which, if it were known and restored, would render the expression a complete differential, the discovery of that factor is a most important problem in the calculus. Its solution, however, transcends the present powers of analysis, although it can be resolved in particular cases. When the functions $M$ and $N$ are homogeneous, the factor can be found; but that case can always be integrated as has been explained. When the variables can be separated, a multiplier can be found, but then it is not wanted. It can also be found when the equation has the form $\frac{dy}{dx} + Pydx = Qdx$; but this also can be integrated by other means.

Euler, who has entered deeply into this subject, has reversed the problem; and, in his Calculus Integralis, supposing the integrating factor given, has investigated the nature of the functions which must enter into a differential equation of a given form, in order that it may be integrable. These investigations, however, frequently lead to differential equations, which cannot be integrated by any known method; and the cases in which they are successful are of no great importance or extent.

176. Let $M$ be any function of $x$ and $y$, and let $u = \int Mdx$, the integral being taken on the supposition that $x$ is variable and $y$ constant: it is sometimes required that the differential co-efficient $\frac{du}{dy}$ be found; or, in other words, that the differential of $fMdx$ relatively to $y$ be found without having actually integrated the expression relatively to $x$. Because $u = \int Mdx$, therefore $\frac{du}{dx} = M$, and

$$\frac{d^2u}{dy^2} = \frac{dM}{dy};$$

but $\frac{d^2u}{dy^2} = \frac{d^2u}{dydx}$ (art. 89), therefore

$$\frac{d^2u}{dydx} = \frac{dM}{dy},$$

and $\frac{du}{dy} = \frac{dM}{dy} dx$, and integrating relatively to $x$,

$$\frac{du}{dy} = \int \frac{dM}{dy} dx.$$

Ex. Let $u = \int \frac{ydx}{\sqrt{y^2 - x^2}}$, here $M = \frac{y}{\sqrt{y^2 - x^2}}$, and

$$\frac{dM}{dy} = \frac{-x^2}{(y^2 - x^2)^{3/2}},$$

$$\frac{du}{dy} = \int \frac{-x^2dx}{(y^2 - x^2)^{3/2}}.$$

The theorem investigated in this article was found by Leibnitz, and was reckoned an important discovery in the calculus. (See Bossut, Traité du Calcul Diff. &c. vol. ii. p. 58.)

177. The elimination of a constant contained in a primitive equation may introduce the second or higher powers of the differentials into the resulting differential equation (art. 53): in such cases we must find the values of $\frac{dy}{dx}$ by the resolution of an equation.

For example, if $dy^2 - ax^2dx = 0$, or $\frac{dy^2}{dx^2} = ax^2$, we have

$$\frac{dy}{dx} = \pm a$$

and $dy + adx = 0$, also $dy - adx = 0$; therefore $y + ax + c = 0$ and $y - ax + c = 0$ are two primitive equations, from each of which the differential equation may be derived; also from their product $(y + ax + c)(y - ax + c) = 0$.

178. When the equation contains only one of the variables, $x$ for example, we deduce from it $\frac{dy}{dx} = X$, and $y = \int Xdx$; but if the equation be more easily resolvable with respect to $x$ than with respect to the co-efficient $\frac{dy}{dx}$, which we will represent by $p$, and if we have also $x = P$, a function of $p$, and thence $dx = dP$; then, since $dy = pdx$, therefore $dy = pdP$, and $y = pP - \int Pdp$. The relation between $x$ and $y$ is now to be found by eliminating $p$ by means of the equations

$$x = P, \quad y = pP - \int Pdp.$$

Let us take as an example $xdx + ady = b\sqrt{x^2 + dy^2}$, or $x + ap = b\sqrt{1 + p^2}$, by writing $p$ in the place of $\frac{dy}{dx}$. This last equation gives immediately $x = -ap + b\sqrt{1 + p^2}$, and consequently

$$y = bp\sqrt{1 + p^2} - \frac{1}{2}ap^2 - b\int dp\sqrt{1 + p^2}.$$

179. When the primitive equation cannot be deduced from the differential equation by any of the known artifices of analysis, then, as a last resource, recourse must be had to approximation by infinite series.

Ex. Let the differential equation be $dy + ydx = mx^n dx$, and suppose it to be known that when $x = a$, then $y = b$: we now make $x = a + t$, $y = b + u$, the equation by this transformation becomes $du + (b + u)dt = m(a + t)dt$. We now assume that

$$u = Ax^n + Bx^{n+1} + Cx^{n+2} + \ldots,$$

and hence, putting instead of $u$ and $du$ their values, and bringing the terms to one side, we have

$$\begin{align*} &Ax^n + (a + 1)Bx^{n+1} + (a + 2)Cx^{n+2} + \ldots \\ &+ b + At^n + Bt^{n+1} + Ct^{n+2} + \ldots \\ &- ma^n - \frac{m}{1}a^{n-1}t - \frac{m}{2}a^{n-2}t^2 - \ldots = 0. \end{align*}$$

In this equation we must suppose $a - 1 = 0$, and we shall then find

$$A = ma^{n-1} - b, \quad B = \frac{ma^{n-1} - ma^n + b}{1.2},$$

$$C = \frac{ma(n-1)a^{n-2} - ma^{n-1} + ma^n - b}{1.2.3}, \ldots$$

These values being substituted in the series, we have $u$ expressed by $t$ and known quantities; we may then put $x = a$ for $t$, and $y = b$ for $u$, and the result will express the relation between $x$ and $y$.

In the series assumed for the value of $u$, the exponents of $t$ form an arithmetical progression, of which the common difference is 1. In many cases, however, the common difference will be a fraction, as in this example, $(dx + dy)y = dx$. Here we assume

$$y = Ax^n + Bx^{n+1} + Cx^{n+2} + \ldots$$

By proceeding as before, there is obtained

$$\begin{align*} &Ax^{2a-1} + ABx^{2a-1} + ACx^{2a-1} + \ldots \\ &+ ABx^{2a-1} + Bx^{2a-1} + \ldots \\ &+ ACx^{2a-1} + \ldots \\ &- 1 + Ax^{2a-1} + Bx^{2a-1} + \ldots \end{align*}$$

Hence $2a - 1 = 0$, $a + b - 1 = a$, $a + c - 1 = b$, &c.; therefore $a = \frac{1}{2}$, $b = 1$, $c = \frac{3}{2}$, &c.

Again, $Aa = 1$, $AB(a + b) + A = 0$, &c.

Hence $A = \sqrt{2}$, $B = -\frac{1}{2}$, $C = \frac{1}{18}\sqrt{2}$, &c.

and $y = x^{\frac{1}{2}}\sqrt{2} - \frac{1}{2}x^{\frac{3}{2}} + \frac{1}{18}x^{\frac{5}{2}}\sqrt{2} - \ldots$

On the integration of differential equations by series, consult Euler, Instit. Col. Integ. vol. i. sect. 2; Lacroix, du Calcul Diff. part ii. chap. 6. Differential Equations of the Second and Higher Orders.

180. The difficulty of the integrations becomes so much the greater the higher the order of the differential coefficients which they involve, and we only succeed in effecting it in a very small number of very limited equations.

Let the expression

\[ f(x, y, c, c') = 0 \]

represent any primitive equation composed of the variables \( x, y \), and two constant quantities \( c, c' \), besides any other constants. By differentiation (as explained art. 48–54), there is formed its differential equation of the first order, which will contain \( \frac{dy}{dx} \). Let this differential co-efficient be represented by \( p \), and let \( q = \frac{dp}{dx} = \frac{d^2y}{dx^2} \). The differential of the first order will contain \( x, y, p, c, c' \), and may be represented by

\[ f'(x, y, p, c, c') = 0 \]

By differentiating this equation we obtain another which contains also \( \frac{dy}{dx} = q \), and may be expressed by

\[ f''(x, y, p, q, c, c') = 0 \]

Since these three equations must hold true at once, we may eliminate the constants \( c, c' \), and the result will be a single equation,

\[ F(x, y, p, q) = 0 \]

in which \( c, c' \) are not found. This will be the differential equation of the second order, deducible from the primitive (1), and which is independent of the constants \( c, c' \).

We may arrive at the very same equation (4) in two other ways. We may give the primitive (1) these forms:

\[ \phi(x, y, c) = c', \quad \psi(x, y, c') = c, \]

from which, by differentiation, two results will be obtained, one entirely free from \( c' \), and the other from \( c \); these again may have the forms

\[ \phi'(x, y, p) = c, \quad \psi'(x, y, p) = c'. \]

By taking now the differentials of these equations, we shall obtain the very same equation from each, which will be identical with equation (4).

For example, let the primitive equation be

\[ x^2 - 2ax + 2by = 0, \]

in which \( a \) and \( b \) represent the constants \( c, c' \); by differentiation, we find

\[ x - a + b \frac{dy}{dx} = 0, \quad 1 + b \frac{d^2y}{dx^2} = 0. \]

After eliminating \( a \) and \( b \) by the three equations, we have this differential equation,

\[ 2y - 2x \frac{dy}{dx} + x^2 \frac{d^2y}{dx^2} = 0. \]

We may otherwise put the primitive equation under these forms:

\[ \frac{x^2 + 2by}{x} = 2a, \quad \frac{x^2 - 2ax}{y} = -2b. \]

Taking now the differentials, and arranging the results so that the constants may stand alone, and putting \( p \) instead of \( \frac{dy}{dx} \) we have

\[ \frac{x^2}{y - xp} = 2b, \quad \frac{2xy - x^2p}{y - xp} = 2a. \]

By differentiating either of these, the constants disappear, and in each case the result is the same, viz.

\[ 2y - 2x \frac{dy}{dx} + x^2 \frac{d^2y}{dx^2} = 0. \]

181. Since two of the constants in a primitive equation may not be found in the differential equation of the second order derived from it, so, reversely, in returning from a differential equation of the second order to its absolute primitive, the latter cannot be complete unless it contain two arbitrary constants, which are not in the former; and as the same differential equation of the second order may be obtained from two distinct differential equations of the first order, it follows that every differential equation of the second order has two distinct differential equations of the first order, each having its own arbitrary constant; and these again have one and the same absolute primitive equation.

The properties of differential equations of the third and higher orders are perfectly analogous to these.

182. The most simple differential equation of the second order is \( \frac{dy}{dx} = X \), a function of \( x \); then \( \frac{dy}{dx} = \int X dx \).

Let \( P \) be the variable part of the integral \( \int X dx \), and \( e \) the constant, so that \( \frac{dy}{dx} = P + e \); hence again \( dy = Pdx + edx \), and by a second integration, \( y = \int Pdx + ex + c \).

Since \( \int Pdx = Px - \int xdP = xfXdx - \int xdx \), therefore,

\[ y = xfXdx - \int xdx + ex + c. \]

The primitive of the differential equation \( \frac{dy}{dx} = X \) \( = 0 \) may be found in the same way. We first put it under the form \( \frac{dy}{dx} = Xdx \); then we have by a first integration

\[ \frac{dy}{dx} = \int Xdx = P + c; \text{ and hence } \frac{dy}{dx} = Pdx + edx; \text{ by a second integration, } \frac{dy}{dx} = \int Pdx + cx + c' = P + cx + c; \text{ here } P' = \int Pdx. \]

We have now \( dy = Pdx + edx + c'dx \); and lastly, \( y = \int Pdx + \frac{1}{2}cx^2 + c'x + c. \)

The primitive thus containing three arbitrary constants: a like differential equation of the fourth order would contain four, and so on.

183. When a differential equation of the second order involves \( \frac{dy}{dx} \) and \( \frac{d^2y}{dx^2} \) and constants; if we put \( \frac{dy}{dx} = p \), then, regarding \( dx \) as constant, \( \frac{d^2y}{dx^2} = \frac{dp}{dx} \); the equation will now involve \( p, dp, dx \) and constants; and it will be of the first order, in respect of \( p \) and \( x \). We may thence find \( dx = Pdp \), \( P \) being here put for some function of \( p \); and since \( dy = pdx = Ppdp \), we have

\[ x = \int Pdp, \quad y = \int Ppdp. \]

These integrals being found, and a constant added to each, by eliminating \( p \), we find an equation expressing the relation between \( x \) and \( y \).

Ex. To integrate the equation \( \frac{(dx^2 + dy^2)^{\frac{3}{2}}}{dx^2y} = a \). By putting \( pdx \) for \( dy \), and \( dpdx \) for \( d^2y \), the equation is changed to \( \frac{(1 + p^2)^{\frac{3}{2}}}{dp} = a \), from which we deduce

\[ \frac{dp}{\sqrt{1 + p^2}} = \frac{apdp}{\sqrt{1 + p^2}}. \]

The integration gives

\[ x = c + \frac{ap}{\sqrt{1 + p^2}}, \quad y = c' - \frac{a}{\sqrt{1 + p^2}}. \]

Hence \( (c - x)^2 + (y - c')^2 = a^2 \).

The proposed differential equation is nothing more than the general expression of the radius of curvature (art. 78) made equal to a constant quantity \( a \). We have resolved the geometrical problem, to find a curve whose radius of

184. We may remark, that the equation \( x = C + \int \frac{dp}{P} \), and \( dy = C' + \int \frac{pdq}{P} \), severally satisfy the differential equation \( \frac{dp}{dx} = P \); and that, by supposing their second members integrated, they will be only of the first order. This agrees with what was observed, art. 181, viz. that there are two equations of the first order which satisfy a differential equation of the second order.

185. Let us now consider differentials of the form \( \frac{dy}{dx} = Y \), representing any function of \( y \). If we make \( dy = pdx \), we thence deduce \( dx = \frac{dy}{p} \), which gives \( \frac{dy}{dx} = \frac{dp}{dx} = \frac{pdq}{dy} \).

Substituting this in the proposed equation, there results from it \( pdq = Ydy \); by integrating, we find \( p^2 = 2fYdy \), \( C \), and hence

\[ p = \frac{dy}{dx} = \sqrt{C + 2fYdy}, \text{ and } x = \int \frac{dy}{\sqrt{C + 2fYdy}} + C. \]

It is proper to observe that the above integration may be effected by multiplying the proposed equation by \( dy \), for thence we find \( \frac{dy}{dx} \cdot \frac{dy}{dx} = Ydy \); and since \( \frac{dy}{dx} = \frac{dy}{dx} \),

we have \( \frac{dy^2}{dx^2} = \int Ydy + C, \text{ or } \frac{dy}{dx} = \sqrt{C + 2fYdy}. \)

Ex. Let the equation be \( \frac{dy}{dx} \cdot \frac{dy}{dx} = dx^2 \); we shall have

\[ \frac{dy}{dx} = \frac{1}{\sqrt{ay}}, \quad \frac{dy}{dx} \cdot \frac{dy}{dx} = \frac{dy}{\sqrt{ay}}, \]

and, by integrating, \( \frac{dy}{dx} = \frac{2}{a} \sqrt{ay} + C; \) changing \( C \) into \( \frac{2c}{\sqrt{a}} \), we shall deduce from thence

\[ \frac{dy^2}{dx^2} = \frac{4}{\sqrt{a}}(\sqrt{y} + c), \quad \frac{2dx}{\sqrt{a}} = \frac{dy}{\sqrt{c + \sqrt{y}}}. \]

Now, making \( c + \sqrt{y} = z \), there will result

\[ \frac{dz}{\sqrt{a}} = \frac{(z - c)}{\sqrt{z}} dz = (z^{1/2} - cz^{-1/2})dz; \]

and finally,

\[ \frac{x}{\sqrt{a}} = \frac{3}{2} z^{3/2} - 2cz^{1/2} + c = \frac{3}{2}(\sqrt{y} - 2c)\sqrt{y} + c. \]

186. We have seen in the differential calculus, art. 93, that beyond the first order the form of the differential equations would be changed, according as \( x \) or \( y \), or even a function of these quantities, was assumed as the independent variable; which comes to the same thing as assuming that \( dx \), or \( dy \), or a given function of these variables, was constant. It is therefore necessary, in integrating equations which exceed the first degree, to know upon which of these hypotheses it has been calculated. The preceding examples all correspond to the case of \( y \) being a function of \( x \), and consequently of \( dx \) being constant; but it will be easy to discover, among relations deduced relatively to other hypotheses, to which of them they may be referred.

It is immediately evident that, if we represent by \( Q \) any function whatever of \( \frac{dx}{dy} \), every equation of the form

\[ \frac{d^2x}{dy^2} = Q, \text{ and in which } dy \text{ is constant, may be treated in the same way as that in art. 183, by making } \frac{dx}{dy} = q; \text{ and } \frac{d^2x}{dy^2} = \frac{dq}{dy}. \]

It may also be reduced immediately to the form \( \frac{d^2y}{dx^2} = P \), by passing by the formula of art. 93, to the hypothesis of \( dx \) being constant, which will be effected by the substitution of \( -\frac{dxdy}{dy^2} \) in place of \( \frac{d^2x}{dy^2} \).

If the proposed equation had been taken on the supposition of \( \sqrt{dx^2 + dy^2} \) being constant, and if it involved only \( dx, dy, d^2y, \) or \( dy, dx, dx^2 \), it might still be treated in the same manner as that of art. 183, after transforming it into one in which \( dx \) was constant.

187. When the equation contains \( \frac{d^2y}{dx^2} \) and the variable \( x \), it may, as before, be transformed into a differential equation of the first order, by substituting \( pdx \) for \( dy \), and \( dpdx \) for \( d^2y \); then, if the primitive of that differential equation, and thence the value of \( p \) in terms of \( x \), can be found, we may have the value of \( y \) from the formula \( y = fpx \); or else, if we have the value of \( x \) in terms of \( p \), then, because \( fpx = px - fxdp \), we shall have \( y = px - fxdp \).

Ex. Let the equation be

\[ \frac{(dx^2 + dy^2)^{3/2}}{dx^2} = X \text{ or } \frac{(1 + p^2)^{3/2}dx}{dp} = X, \]

where \( x \) denotes a given function of \( x \); then

\[ \frac{dx}{X} = \frac{-dp}{(1 + p^2)^{3/2}} \text{ and } \int \frac{dx}{X} + C = \frac{-p}{\sqrt{1 + p^2}}. \]

Let \( V \) represent \( \int \frac{dx}{X} + C \), then \( p = \frac{V}{\sqrt{1 - V^2}} \), and

\[ y = \int pdx + C = \int \frac{Vdx}{\sqrt{1 - V^2}} + C; \] \( y \) is now expressed by means of \( x \) only. This is the analytical solution of the geometrical problem to find the nature of a curve whose radius of curvature shall be a given function of \( x \), the abscissa.

If we suppose \( X = \frac{a^2}{2x} \), then \( \frac{-p}{\sqrt{1 + p^2}} = \int \frac{2xdx}{a^2} = \frac{x}{a} + C, \)

\[ p^2 = \frac{dy^2}{dx^2} = \frac{a^2 + c}{a^2 - (x^2 + c^2)^2} \text{ and } y = \int \frac{(x^2 + c)dx}{\sqrt{a^2 - (x^2 + c^2)^2}}; \]

here \( c \) is put generally for an indeterminate constant. This is the equation of the elastic curve.

188. If the proposed differential equation is composed of \( \frac{d^2y}{dx^2} \) and \( y \), we may, as before, put \( p = \frac{dy}{dx} \), from which we get \( \frac{d^2y}{dx^2} = \frac{dp}{dx} = \frac{pdq}{dy} \); the equation will now involve \( dp, dq, p \) and \( y \) only. When the primitive equation can be found, and thence the value of \( p \) in terms of \( y \), we may find \( x \) by the formula \( x = \int \frac{dy}{p} \); but when \( y \) is expressed by \( p \), we may then employ the formula \( x = \frac{y}{p} + \int \frac{dp}{p^2} \).

189. Differentials of the second order, which have this form,

\[ \frac{d^2y}{dx^2} + P \frac{dy}{dx} + Qy = R, \] in which P, Q, R are any functions of x, are called linear, also equations of the first degree, because y is only of one dimension. It will easily be understood that the difficul- ty of finding the primitive equation will be greater than in the like equation of the first order; and, indeed, except in particular cases, there is no known method of reducing the integration to the finding of the differential of a sin- gle variable; that is, to the quadrature of a curve. If R = 0, in which case the equation is

\[ \frac{dy}{dx} + P \frac{dy}{dx} + Qy = 0, \]

it may be reduced to a differential equation of the first degree, by a very simple transformation. Let e, as usual, denote the base of Napier's logarithms; assume \( y = e^{f(x)} \), then, regarding x as constant, we have

\[ dy = u dx e^{f(x)}, \quad d^2y = e^{f(x)} (du dx + u^2 dx^2); \]

the value of dy and d^2y being substituted in the equation, and the common factors rejected, it becomes

\[ du + (u^2 + Pu + Q) dx = 0. \]

When P and Q are constants, which may be represented by A and B, the equation becomes

\[ du + (u^2 + Au + B) dx = 0; \]

in which the variables are separated, if we give it the form

\[ \frac{du}{u^2 + Au + B} + dx = 0. \]

As it is merely necessary to satisfy this equation, we may make u = m, m being any constant; then du = 0, and m^2 + Am + B = 0. This last equation gives, in general, two values of m; if we represent them by a and b, we shall have two values of f(u)dx; viz. ax + c and bx + c', and hence we have two values of y, viz.

\[ y = e^{ax+c}, \quad y = e^{bx+c'}; \]

or, putting C for e^c, and C' for e^c',

\[ y = Ce^{ax}, \quad y = C'e^{bx}. \]

These, however, are only particular values of the func- tion y, because each contains only a single arbitrary con- stant; but by adding them, we get

\[ y = Ce^{ax} + C'e^{bx} \]

as the complete primitive equation.

To prove this, by differentiating, we get

\[ \frac{dy}{dx} = aCe^{ax} + bC'e^{bx}, \quad \frac{d^2y}{dx^2} = a^2Ce^{ax} + b^2C'e^{bx}. \]

From these and the above primitive equation, we have, after eliminating C and C',

\[ \frac{d^2y}{dx^2} + (a + b) \frac{dy}{dx} - aby = 0. \]

This will agree with the proposed equation, if we give a and b such values that a + b = P, ab = -Q.

If a and b come out impossible quantities, the exponents of a and b in the value of y will have the form

\[ a = a + \beta \sqrt{-1}, \quad b = a - \beta \sqrt{-1}; \]

we have then

\[ y = Ce^{ax + \beta x \sqrt{-1}} + C'e^{ax - \beta x \sqrt{-1}} \] \[ = e^{ax} (C \beta x \sqrt{-1} + C'e^{-\beta x \sqrt{-1}}); \]

This result is rendered real by eliminating the imagi- nary quantities, by means of sines and cosines; thus we have (ALGEBRA, art. 269),

\[ e^{\beta x \sqrt{-1}} = \cos \beta x + \sqrt{-1} \sin \beta x, \] \[ e^{-\beta x \sqrt{-1}} = \cos \beta x - \sqrt{-1} \sin \beta x, \] \[ y = e^{ax} ((C + C') \cos \beta x + (C - C') \sqrt{-1} \sin \beta x); \]

and making C + C' = c, (c - c') \sqrt{-1} = c,

\[ y = e^{ax} (c \cos \beta x + c' \sin \beta x); \]

or, making c = p \sin q, c' = p \cos q,

\[ y = pe^{ax} \sin (\beta x + x). \]

When the roots a and b are equal, the value of y being reduced to

\[ Ce^{ax} + C'e^{ax} = (C + C')e^{ax}, \]

it becomes incomplete. In this case we may proceed, as in art. 58, by supposing that a and b differ by a very small quantity. Let us suppose that b = a + k, there thence results

\[ y = Ce^{ax} + C'e^{ax+kx} = e^{ax} (C + C'e^{kx}); \]

developing e^{kx} according to the powers of k, we have

\[ y = e^{ax} (C + C' + C'kx + C'k^2x^2 + &c.) \]

or, putting C + C' = c and C'k = c'

\[ y = e^{ax} (c + c'x + c'kx^2 + &c.). \]

This last expression, which satisfies the proposed equa- tion for all values of k, agrees with it likewise if k = 0, or b = a; and, in that case, it becomes

\[ y = e^{ax} (c + c'x). \]

190. When P and Q are variable quantities, functions of x, then if v and v' are two values of y which each sa- tisfy the equation

\[ \frac{dy}{dx} + P \frac{dy}{dx} + Qdy = 0, \]

we may take

\[ y = cv + c'v \]

for the complete primitive equation. For then dy = cdv + cdv', d^2y = cdv' + cdv', and the differential equation becomes

\[ c \left( \frac{d^2v}{dx^2} + P \frac{dv}{dx} + Qv \right) + c' \left( \frac{d^2v'}{dx^2} + P \frac{dv'}{dx} + Qv' \right) = 0. \]

Now, by hypothesis,

\[ \frac{dv}{dx} + P \frac{dv}{dx} + Qv = 0; \quad \frac{dv'}{dx} + P \frac{dv'}{dx} + Qv' = 0. \]

Therefore the equation is identical, and so the value of y is truly determined.

As an example of the integration of a differential equa- tion of the second degree, see the article ANCHI (art. 57).

191. We shall now give an example of the integration of a differential equation of the first degree and the se- cond order.

Let the equation be

\[ \frac{d^2y}{dx^2} + ax^n y dx^n = 0. \]

If we suppose

\[ y = Ax^n + Bx^{n-1} + Cx^{n-2} + &c. \]

and that the series of exponents is an increasing one, or that δ is positive, we may, when x is supposed very small, conceive y to reduce itself to its first term, since the others are too small to be compared with this first. On this supposition, we may confine ourselves to as- suming

\[ y = Ax^n; \text{ then } \frac{d^2y}{dx^2} = a(a - 1) Ax^{n-2} dx^2, \]

and the proposed equation will become

\[ a(a - 1) Ax^{n-2} + ax^n = 0. \]

It is not possible to determine a so as to make the two exponents a - 2 and a + n equal, except in the parti- cular case of n = -2; but the exponent of x being greater in the second term than the first, we may neglect one of these terms in comparison with the other; and the equation may then be verified in two ways (by approxi- mation), viz. by making \(a = 0\), and \(a = 1\), since upon either hypothesis the term \(a(a - 1)Ax^2 - 2\) (the greatest in the equation) vanishes; \(A\), therefore, remains indeterminate, and we have two series, one beginning with \(A\), and the other with \(Ax\).

If we take successively

\[y = A + Bx^3 + cx^5 + \ldots\]

\[y = Ax^4 + Bx^3 + cx^2 + \ldots\]

and substitute for \(y\) these values, and for \(dy\) its corresponding values, we shall then find, by properly arranging the terms, that \(b\) must be \(= 2\); and, in either case, determining the values of the co-efficients \(A, B, C, \ldots\) we arrive at these two series,

\[\frac{aAx^n + 2}{(n+1)(n+2)} + \frac{a^2Ax^{2n} + 4}{(n+1)(n+2)(2n+3)(2n+4)} + \ldots\]

\[\frac{a^3Ax^{3n} + 6}{(n+1)(n+2)(2n+3)(2n+4)(3n+5)(3n+6)} + \ldots\]

These developments are only particular cases, since they contain each but one arbitrary constant \(A\); but, on account of the particular form of the proposed example (189), we shall obtain a general expression for \(y\), by writing in the latter of them \(A_1\) for \(A\), and taking their sum.

Having now given as full a view of the principles of the fluxional calculus as we conceive to be compatible with the nature of our work, we shall conclude with a few more examples of its application to geometry.

**Problem 1.**

To find the length of the enlarged meridian in Mercator's, or rather Wright's projection of the sphere.

In this projection, the meridians and parallels of latitude are straight lines, which intersect each other at right angles; and the projection of any small arc of a meridian (as one minute), reckoned from any parallel, is to the projection of a like arc of longitude in that parallel, as radius to the cosine of the latitude of the parallel, that is, in the same ratio as the arcs themselves in the sphere. Supposing the radius of the sphere to be unity, let \(v\) be any arc of latitude reckoned from the equator, \(z\) its projection on the chart, or the enlarged meridian. Let \(v'\) and \(z'\) be any small increments of \(v\) and \(z\); then, by the principles of the projection,

\[z' = \frac{\text{rad}}{\cos v};\] therefore, putting the ratio of the differentials for the limit of the ratio of the increments, \(dz = \frac{1}{\cos v}\) and \(dz = \frac{dv}{\cos v}\). Therefore, taking the integral (art. 155),

\[z = L \left\{ \tan (45^\circ + \frac{1}{2}v) \right\}.\] Here no correction is wanted, because when \(v = 0\), then \(z = L (\tan 45^\circ) = L (1) = 0\), as it should be.

Henry Bond, in the year 1650, discovered, by chance, that the enlarged meridian might be expressed by the logarithmic tangents of half the complements of the latitudes, a rule easily found from the preceding solution; but the difficulty of proving this was then considered so great, that Mercator offered to wager a sum of money against any person that should undertake to prove it either true or false. James Gregory, however, proved it in his *Exercitationes Geometricae*, published in 1668; also Barrow, in his Geometrical Lectures: their demonstrations, however, were intricate. Afterwards Dr Wallis and Dr Halley gave demonstrations which were sufficiently simple and elegant.

**Problem 2.**

If any number of straight lines are drawn according to some determinate law, it is required to find the nature of a curve to which these are tangents.

For example, let \(AE\) be a straight line given by position, and \(K\) a given point without it; let any number of lines \(KD, KD', \ldots\) be drawn to meet \(AE\) in \(D, D', \ldots\); and let perpendiculars \(DC, DC', \ldots\) be drawn to these lines; it is required to find the nature of the curve \(ACC'\), to which these perpendiculars are tangents.

Without attending to the particular case, we shall resolve the general problem, and suppose \(AE\) to be the axis of the curve, \(A\) being the origin of the co-ordinates, and \(CD\) any position of the tangent, which meets the axis in \(D\). From the point of contact \(C\) draw the perpendicular \(CB\); and, considering \(C\) as a point in the curve, put \(AB = x\), \(BC = y\); but again considering \(C\) as any point whatever in the tangent, put \(AB = x'\), \(BC = y'\). Then, whatever be the conditions that determine the position of the tangent, the relation of \(x'\) and \(y'\), the co-ordinates of any point in it, may be expressed by the equation \(y' = Px' + Q\), where \(P\) and \(Q\) are put to denote generally certain functions of constant quantities, and some quantity \(p\), which has the same value for any given position of the tangent, but which changes its value if the tangent changes its position. For example, \(p\) may express the angle which the tangent makes with the axis, or it may represent the subtangent \(BD, \ldots\)

Let us now suppose that the variable quantity \(p\) changes its value, and becomes \(p + h\), and that \(C'D'\) is the new position of the tangent corresponding to \(p + h\); then, considering \(P\) and \(Q\) as functions of \(p\), by Taylor's theorem (art. 31),

\[P \text{ becomes } P + \frac{dP}{dp}h + \frac{d^2P}{dp^2}h^2 + \ldots\]

\[Q \text{ becomes } Q + \frac{dQ}{dp}h + \frac{d^2Q}{dp^2}h^2 + \ldots\]

The relation of \(x'\) to \(y'\) in the new position of the tangent will now be expressed by the equation

\[y' = Px' + Q + \left( \frac{dP}{dp}x' + \frac{dQ}{dp} \right)h + Kh^2 + \ldots\]

Where \(Kh^2 + \ldots\) is put for all the remaining terms of the series.

Now, as this equation holds true of every point in the tangent \( C'D \), and the equation \( y = Px + Q \) holds true of every point in the tangent \( CD \), it follows that at \( c \), the intersection of the two tangents, both equations must be true at the same time; therefore at \( c \) we have

\[ \left( \frac{dP}{dp} x' + \frac{dQ}{dp} h + Kh^2 + &c. = 0; \right) \]

and, dividing by \( h \),

\[ \frac{dP}{dp} x' + \frac{dQ}{dp} + Kh + &c. = 0. \]

Conceive now the two tangents to approach to coincidence; when \( C' \) comes to \( C \), then \( c \) will also fall at \( C \); and \( h \), and all the terms into which it enters, vanish; also \( x' \) and \( y' \) become \( x \) and \( y \); and, to determine the nature of the curve, we have these two equations:

\[ y = Px + Q \quad \text{(1)} \]

\[ 0 = \frac{dP}{dp} x + \frac{dQ}{dp} \quad \text{(2)} \]

By eliminating \( p \) from these, the resulting equation will express the nature of the curve.

Ex. 1. Let us now recur to the particular case of \( AE \), a straight line given by position, \( K \) a given point, and \( KDC \) a right angle: Draw \( KA \) perpendicular to \( AE \); put \( AB = x \), \( BC = y \), \( KA = a \), and let \( AD \) be the variable quantity \( p \). The triangles \( KAD \), \( DBC \) are manifestly similar; therefore \( KA : AD = DB : BC \), that is, \( a : p = x - p : y \); hence \( ay = px - p^2 \), and \( y = \frac{p}{a} x - \frac{p^2}{a} \).

Compare this with equation (1), and it will appear that

\[ P = \frac{p}{a}, \quad Q = -\frac{p^2}{a}; \quad \text{therefore } \frac{dP}{dp} = \frac{1}{a}, \quad \frac{dQ}{dp} = -\frac{2p}{a}; \]

hence the nature of the curve is expressed by the two equations

\[ y = \frac{p}{a} x - \frac{p^2}{a}; \quad 0 = \frac{1}{a} x - \frac{2p}{a}. \]

The second of these equations gives \( p = \frac{1}{2}x \); and hence the first becomes \( y = \frac{x^2}{2a} - \frac{x^4}{4a^2} \); therefore \( 4ay = x^2 \) is the equation of the curve, which is evidently a parabola, of which \( AK \) is the axis, \( K \) the focus, and \( A \) the vertex.

Ex. 2. Suppose a ray of light \( RD \), coming from the sun, to fall upon \( FEG \), the concave surface of a sphere, at \( D \), and to be thence reflected in the direction \( DH \); it is proposed to find the nature of the curve to which this, and all rays reflected in the same manner, are tangents.

Fig. 34.

Draw \( AD \) the radius of the sphere, and \( AE \) parallel to the incident ray \( RD \); let \( C \) be the point in which the reflected ray touches the curve; let \( DC \) meet \( AE \) in \( H \), and draw \( CB \) perpendicular to \( AE \). Put \( AD = a \), \( AB = x \), \( BC = y \), and let \( p \) be the variable angle \( DAE \).

By the principles of optics, \( AD \) bisects the angle \( RDH \), which is equal to \( DHE \), that is, to the sum of the angles \( DAH \), \( ADH \); therefore the angles \( ADH \), \( DAH \) are equal, and angle \( DHE = 2p \). Now, by trigonometry,

\[ AH = \frac{\sin ADH}{\sin AHD} \times AD = \frac{\sin p}{\sin 2p} a, \]

\[ BH = \frac{\cos CHB}{\sin CHB} \times CB = \frac{\cos 2p}{\sin 2p} y; \]

Hence \( x = (AH + BH) = \frac{\sin p}{\sin 2p} a + \frac{\cos 2p}{\sin 2p} y \),

and \( y = \frac{\sin 2p}{\cos 2p} x - \frac{\sin p}{\cos 2p} \).

By comparing this with the general formula (1), it appears that

\[ P = \frac{\sin 2p}{\cos 2p}, \quad Q = -\frac{\sin p}{\cos 2p} a; \quad \text{hence,} \]

\[ \frac{dP}{dp} = \frac{2}{\cos^2 2p}, \quad \frac{dQ}{dp} = -\frac{\cos p}{\cos^2 2p} a - \frac{2 \sin 2p \sin p}{\cos^2 2p} a. \]

Hence, by formula (2),

\[ \frac{2x}{\cos^2 2p} = \left\{ \frac{\cos p}{\cos 2p} + \frac{2 \sin 2p \sin p}{\cos^2 2p} \right\} a. \quad \text{(3)} \]

From equations (a) and (3) we readily find

\[ x = \left( \frac{1}{2} \cos 2p \cos p + \sin 2p \sin p \right) a, \]

\[ y = \left( \frac{1}{2} \sin 2p \cos p - \cos 2p \sin p \right) a; \]

and hence, by observing that \( \sin 2p = 2 \sin p \cos p \), and that \( \cos 2p = 2 \cos^2 p - 1 = 1 - 2 \sin^2 p \) (Arithmetic of Sines, Algebra), we have also

\[ x = \frac{1}{2} \cos p (1 + 2 \sin^2 p) a; \quad y = \sin^2 pa. \]

From these equations it is easy to eliminate the trigonometrical quantities \( \cos p \) and \( \sin p \), and the result will be the equation of the curve, which is an epicycloid. The curve in question is the catacaustic curve to a circle.

It is easy to see that the general problem, of which we have now given two examples, is very comprehensive. We may evidently find, by the formulæ (1) and (2), catacaustic and diacaustic curves in all cases whatever. These, and an infinite number of other geometrical problems, are contained in the following more general problem.

Problem 3.

Determine the nature of a curve which touches an infinite number of lines of a given kind, described upon a plane according to some determinate law. For example, suppose the lines to be parabolas described by a projectile thrown from an engine with a given velocity, at every possible elevation in a vertical plane.

Let \( HCD \) be any one of the lines of a given kind (a parabola, for example), and suppose it referred to an axis \( AB \), by the rectangular co-ordinates \( AB = x' \) and \( BC = y' \). Let \( p \) denote some quantity belonging to the line or curve \( HCD \), which has always the same value in the same curve, but which has different values in different curves. Thus, if the curve be a circle, \( p \) may be its radius; or if the curve be a parabola, \( p \) may be its parameter, &c. This quantity \( p \) may also by analogy be called the parameter of the curve \( HCD \). Let us suppose the nature of this curve to be expressed by the equation

\[ f(x', y', p) = 0. \quad \text{(1)} \]

that is, let some function of \( x' \), \( y' \), and \( p \), be supposed \( = 0 \). If \( p \) now be supposed to change its value, and become \( p + h \), then the curve \( HCD \) will change its figure, and have some other position \( H'C'D' \). Let \( AB' = x'' \), and \( BC' = y'' \). Let \( y \) be any co-ordinates of this other curve, and as, by hypothesis, the two curves are expressed by equations of the same form, we must have

\[ f(x', y', p + h) = 0; \]

and this expression, by Taylor's theorem, is equivalent to

\[ (2) \quad f(x', y', p) + \frac{d}{dp} \left\{ f(x', y', p) \right\} h + Kh^2 + &c., \]

the differential being taken upon the hypothesis that \( p \) alone is variable, and \( Kh^2 + &c. \) being put for all terms of the series following the second, each of which is multiplied by a power of \( h \).

Let the two curves intersect each other in \( c \), and let \( Ab = x \), and \( cb = y \), be the common co-ordinates; then, as equation (1) holds true of every point in the curve HCD, and equation (2) holds true of every point in HCD', both must hold true at once, if we substitute in them \( x \) and \( y \), the co-ordinates belonging to their common point \( c \); that is, we must have

\[ f(x, y, p) = 0, \]

\[ f(x, y, p) + \frac{d}{dp} \left\{ f(x, y, p) \right\} h + Kh^2 + &c. = 0; \]

and hence we must also have

\[ \frac{d}{dp} \left\{ f(x, y, p) \right\} + Kh^2 + &c. = 0. \]

Let C and C' be now supposed the points in which the curves HcD, H'eD' touch the curve PCCQ, whose nature is required; then, if we suppose \( h \) to decrease continually, and at last to vanish, the points C and C will approach to C, and at last will coincide with it, so that \( x \) and \( y \), which are co-ordinates of \( c \), the intersection of the two curves HcD, H'eD', will then become the co-ordinates of the curve PCQ. As all the terms which contain \( h \) will then vanish, we have evidently this rule.

Let the equation of the given curves be

\[ f(x, y, p) = 0, \]

\( x \) and \( y \) being the co-ordinates, and \( p \) a variable parameter. From this equation, by taking the differential, supposing \( p \) to be variable, and all the other quantities constant, we deduce this other equation,

\[ \frac{d}{dp} \left\{ f(x, y, p) \right\} = 0. \]

By these eliminate \( p \), and the result will be an equation, which expresses the nature of the curve that touches all the given curves.

Ex. Let ACD, ACD', &c. be parabolas described by a projectile thrown from a given point \( A \), with a given velocity, in a given vertical plane. It is proposed to find the curve PCQ which touches them all. Let EF be the axis of any one of the curves, AD an ordinate to the axis, \( AP = a \) the height due to the velocity of projection, \( AB = x \), \( BC = y \), the co-ordinates of \( C \), any point in the curve. Put the parameter of the axis \( = p \), and considering AD as a function of \( p \), which is to be regarded as variable, put \( AD = q \).

By the theory of projectiles, \( EF = a - \frac{1}{2} p \), and by the nature of the parabola, \( AF^2 = p \times EF \), and \( AB \times BD = p \times BC \); hence we have these two equations,

\[ q^2 = 4ap - p^2, \]

\[ q^2 = 4ap - p^2, \]

From the first of these,

\[ f(x, y, p) = x^2 + py - qx = 0; \]

and, taking the differentials, considering \( x \) and \( y \) as constant, and \( q \) as a function of the variable quantity \( p \),

\[ \frac{d}{dp} \left\{ f(x, y, p) \right\} = y - \frac{dq}{dp} x = 0, \]

therefore \( \frac{y}{x} = \frac{dq}{dp} \); but from equation (2), taking the differentials, \( dq = 2adp - pdp \), and hence \( \frac{dq}{dp} = \frac{2a - p}{q} \),

and by the first equation \( \frac{y}{x} = \frac{q - x}{p} \), therefore \( \frac{q - x}{p} = \frac{2a - p}{q} \), and hence \( q^2 - qx = 2ap - p^2 \); and substituting for \( q^2 \) its value given by equation (2), we get \( qx = 2ap \); hence, and by equation (1),

\[ p = \frac{x^2}{2a - y}, \quad q = \frac{2ax}{2a - y}. \]

These values of \( p \) and \( q \) being substituted in the second equation, and the common denominator rejected, it becomes

\[ 4a^2x^2 = (8a^2 - 4ay - x^2)x^2. \]

Hence \( 4ay = 4a^2 - x^2 \), and this is the equation of the curve PCQ, which is evidently a parabola having its focus at \( A \), the common intersection of all the parabolas, its axis perpendicular to the horizon, and its parameter \( = 4a \).

The geometrical theory comprehended in the second and third problems has a corresponding analytical theory relating to the integrals of certain differential equations. This is the theory of singular primitive equations, which are not included in the complete primitive equation. Thus, the differential equation

\[ dy \sqrt{(x^2 + y^2 - b)} - ydx - xdx = 0, \]

has for its complete primitive equation

\[ x^2 - 2ay - a^2 - b = 0, \]

where \( a \) is the arbitrary constant quantity; but, besides this, it has a singular primitive equation,

\[ x^2 + y^2 - b = 0, \]

which does not admit of an arbitrary constant, although it equally satisfies the differential equation, as is easily proved by differentiating. The bounds within which it was proper to confine this treatise have not allowed us to enter into this branch of the subject, which, although interesting, is yet not elementary.

Investigation of the Properties of the Catenary.

Suppose a chain or thread of uniform thickness, and perfectly flexible, but inextensible, to be fastened by its extremities at two points \( A, B \) in a vertical plane; by the action of gravity it will take the form of a curve ACPPB, called the catenary.

The curve will evidently be all in one plane, and a horizontal line CD, in that plane, at C its lowest point, will manifestly touch the curve, which is retained at rest by a mutual balancing of the forces produced by the weight of the particles of the chain.

If we suppose its lowest point to be fixed at C, it is easy to understand that this part BC will not, on that account change in the least its figure, because the tension of the chain at C, when it was free, is now replaced by the reaction of the force exerted in the horizontal direction CD. The forces exerted on the points B, C will evidently be the same in quantity and in direction, whether the chain be considered as flexible, or rigid like a solid wire; hence it may be regarded as kept at rest by three forces, viz. its gravity acting vertically, and the re-actions of the tensions at B and C; the former exerted in the direction of a tangent to the curve at B, and the latter in the direction CD, the tangent at the lowest point C.

In like manner any portion PC between P and its lowest point may be considered as kept at rest by the joint action of the gravity of the mass PC, the tension of the chain at C, which is constant, and the tension at P, which is variable; and by the nature of an equilibrium (Mechanics), these forces will be to each other as the sides of any triangle which are parallel to their directions. Draw a horizontal line OX in the plane of the curve as an axis, meeting the vertical through C in O, which may be taken as an origin of co-ordinates; draw PQ perpendicular to OX, and PT touching the curve and meeting the axis in T. From what has been explained, the tension of the curve at C will be to the weight of the matter in PC as TQ to PQ, that is, as radius to the tangent of the angle which PT makes with any horizontal line.

Put OQ = x, PQ = y, arc CP = z, angle PTQ = ϕ; and let a be the tension of the curve at the vertex C: we have then \(a : z = 1 : \tan \phi\); hence

\[ \tan \phi = \frac{z}{a} \tag{1} \]

This is the distinguishing property of the curve, from which we are to deduce its other properties.

From equation (1), by differentiation,

\[ \frac{d\phi}{\cos^2 \phi} = \frac{dz}{a}. \]

Now in all curves,

\[ dz = \frac{dx}{\cos \phi} = \frac{dy}{\sin \phi}; \]

hence we obtain

\[ dx = a \cdot \frac{d\phi}{\cos \phi}, \quad dy = a \cdot \frac{d\phi \sin \phi}{\cos \phi}. \]

From these, by integration,

\[ x = a \cdot \tan \left(45^\circ + \frac{1}{2} \phi\right) + \text{const.} \quad \text{(art. 155)} \]

\[ y = \frac{a}{\cos \phi} + \text{const.} \]

If we assume that OC measures the tension at C, which is a constant quantity, then because when \(ϕ = 0, x = 0,\) and \(y = a,\) we have the relation of the quantities \(x, y, ϕ,\) expressed by these equations,

\[ x = a \cdot \tan \left(45^\circ + \frac{1}{2} \phi\right), \quad y = \frac{a}{\cos \phi} = a \sec \phi. \tag{2} \]

The constant line \(a\) is called the parameter of the curve, also its modulus. From the first of these equations, putting \(ϕ\) for \(45^\circ + \frac{1}{2} \phi,\) we have

\[ \tan ϕ = e^{\frac{x}{a}}, \quad \cot ϕ = e^{-\frac{x}{a}}, \]

\[ \tan ϕ + \cot ϕ = e^{\frac{x}{a}} + e^{-\frac{x}{a}}. \]

Now, \(\tan ϕ + \cot ϕ = \frac{\sin ϕ}{\cos ϕ} + \frac{\cos ϕ}{\sin ϕ} = \frac{2}{\sin 2ϕ} = \frac{2}{\cos ϕ} = \frac{2y}{a};\)

therefore \(y = \frac{a}{2} \left\{ e^{\frac{x}{a}} + e^{-\frac{x}{a}} \right\}. \tag{3}\)

This exponential equation expresses the relation between \(x\) and \(y\) by means of \(e,\) the base of Napier's logarithms. From this last equation, observing that \(y = a \sec ϕ,\) we get

\[ \sec^2 ϕ = \frac{1}{2} \left\{ e^{\frac{x}{a}} + e^{-\frac{x}{a}} + 2 \right\}, \]

\[ \sec^2 ϕ - 1 = \frac{1}{2} \left\{ e^{\frac{x}{a}} + e^{-\frac{x}{a}} - 2 \right\}, \]

\[ \tan ϕ = \sqrt{\sec^2 ϕ - 1} = \frac{1}{2} \left\{ e^{\frac{x}{a}} - e^{-\frac{x}{a}} \right\}. \]

But it was found that \(z = a \tan ϕ,\)

therefore \(z = \frac{a}{2} \left\{ e^{\frac{x}{a}} - e^{-\frac{x}{a}} \right\}. \tag{4}\)

From formulae (3) and (4) we have

\[ y + z = ae^{\frac{x}{a}}, \quad y - z = ae^{-\frac{x}{a}}. \tag{5} \]

therefore \(y^2 - z^2 = a^2. \tag{6}\)

hence again,

\[ y = \sqrt{a^2 + z^2}, \quad z = \sqrt{y^2 - a^2}. \tag{7} \]

and since from formula (5)

\[ \frac{x}{a} = 1, \quad \frac{y + z}{a} = 1, \quad \frac{y - z}{a} = 1. \tag{8} \]

therefore also, \(\frac{x}{a} = 1, \quad \frac{y + \sqrt{y^2 - a^2}}{a} = 1, \quad \frac{y - \sqrt{y^2 - a^2}}{a} = 1. \tag{9}\)

Let \(s\) denote the area bounded by the arc PC and the straight lines CO, OQ, PQ; then, since \(dx = \frac{adϕ}{\cos ϕ},\) and \(y = \frac{a}{\cos ϕ},\) therefore

\[ ds = ydx = \frac{a^2 dϕ}{\cos^2 ϕ} = adz. \]

and \(s = az. \tag{10}\)

Thus it appears that the arc CP is always proportional to the conterminous area CPQA. It is evident that the catenary, if inverted, would form an equilibrated arch. From this property it appears that if a wall, whose height above the crown of the arch is equal to the line CO, be horizontal at the top, the structure thus formed will constitute an equilibrated arch with a straight roadway. (See on this subject our article Arch.)