FLUXIONS

F L U X I O N S.

INTRODUCTION.

THE branch of mathematical analysis which is called in this country the Method of Fluxions, but on the continent the Differential and Integral Calculi, was invented near the end of the 17th century; and Sir Isaac Newton, and Mr Leibnitz, two of the greatest philosophers of that age, have both claimed the discovery.

It will appear very possible that two such men should both fall upon this method of calculation nearly about the same time, if it be considered, that from the beginning of the 17th century its principles were gradually coming into view, in consequence of the united labours and discoveries of a number of mathematicians, such as Napier, Cavallertus, Roberval, Fermat, Barrow, Wallis, and others. And considering the number of men of the first abilities engaged at that time in the study of mathematics, we may reasonably suppose, that the fluxional, or differential calculus would very soon have been found according to the ordinary progress of human knowledge, even although a Newton, or a Leibnitz, had not by the force of superior genius anticipated perhaps by a few years that event. The first intimation that was given of the discovery of the calculus was in the year 1669, when, through the intervention of Dr Barrow, a correspondence was begun between Sir Isaac Newton (then Mr Newton), and Mr Collins, one of the secretaries to the Royal Society. Dr Barrow communicated to the latter a paper by Newton, which had for its title, De analysi per aequationes numero terminorum infinitas. In this paper, besides shewing how to resolve equations by approximation, Newton teaches how to square curves, not only when the expression for the ordinate in terms of the abscissa is a rational quantity,

but also when it involves radical quantities, by first resolving these into an infinite series of rational terms by means of the binomial theorem, a thing which had never before been done. Newton in this paper gives some rather obscure indications of the nature of his calculus, which however serve to shew, beyond all doubt, that he was then in possession of it; and indeed there is good reason to believe that he knew it as early as the year 1665, or even sooner.

These analytical discoveries of Newton were immediately circulated among mathematicians both in this country and abroad, by Dr Barrow, and by Collins and Oldenburg, the two secretaries to the Royal Society.

About the end of the year 1672, Newton communicated to Collins, by letter, a method of drawing tangents to curve lines, illustrated by an example, from which it again plainly appears, that he now possessed his method of fluxions.

In the course of the following year, Leibnitz came to London, and communicated to several members of the Royal Society, some researches relating to the theory of differences. It was however shewn to him, that this subject had been previously treated by Mouton, an astronomer of Lyons; upon this Leibnitz directed his attention to the doctrine of series, which was now considerably advanced, in consequence of the discoveries of the English mathematicians.

The first direct communication that passed between Newton and Leibnitz, was by a letter, which the former addressed to Oldenburg, about the middle of the year 1676. In the beginning of this letter, which was intended to be shewn to Leibnitz, Newton speaks of him with much respect. The letter itself chiefly refers to

the theory of infinite series. In a second letter, written also with a view to its being communicated to Leibnitz, Newton, after bestowing deserved commendation on him, proceeds to explain the steps by which he was led to the discovery of the binomial theorem. He afterwards, among other things, delivers several theorems which have the method of fluxions for their basis; but he does not give their demonstrations, and only observes, that they depend on the solution of a general problem, the enunciation of which he conceals under an anagram of transposed letters, but the meaning of it is this: An equation being given containing any number of flowing quantities, to find their fluxions; and the contrary. This letter affords another proof that Newton was now in full possession of his calculus.

In the end of June 1677, Leibnitz sent to Oldenburg, for the purpose of being communicated to Newton, a letter containing the first essays of his Differential Calculus. The death of Oldenburg, which happened soon after, put an end to the correspondence, and in the year 1684, Leibnitz published his method, in the Leipic Acts for the month of October 1684. The title of the memoir which contained it was, Nova methodus pro maximis et minimis, itemque tangentibus, que nec fractas, nec irrationales quantitates moratur, et singulare pro illis calculi genus. Thus, in whatever way Leibnitz came by his calculus, whether he discovered it solely by the force of his own genius, or founded it on the method of fluxions, previously invented by Newton, both of which hypotheses are possible, his method was certainly published before Newton's, which, except what transpired in consequence of the circulation of his letters and manuscripts, became only known to the world in general for the first time, by the publication of the Principia in the end of the year 1686.

It seems at first to have been allowed, that Leibnitz had invented his calculus, without having any previous knowledge of what had been done by Newton; for in the first edition of the Principia, Newton says, "In the course of a correspondence which ten years ago I carried on with the very learned geometrician Mr Leibnitz, having intimated to him that I possessed a method of determining maxima and minima, of drawing tangents, and resolving such problems, not only when the equations were rational, but also when they were irrational; and having concealed this method, by transposing the letters of the following sentence—An equation being given, containing any number of flowing quantities, to find their fluxions; and the contrary; this celebrated man answered that he had found a similar method, which he communicated to me, and which differs from mine, only in the enunciation, and in the notation." To this, in the edition of 1714 is added, "and in the manner of conceiving the quantities to be

* Principia, generated."

There is reason to suppose that Leibnitz might have continued to enjoy undisturbed the honour of being considered as one of the inventors of the fluxional, or differential calculus, if he had not manifested a disposition to attribute the invention too exclusively to himself. This called forth some remarks respecting the priority of Newton's claim to the discovery. In particular M. Facio asserted, in a treatise on the Line of swiftest descent, published in 1699, "that he was obliged to own Newton as the first inventor of the diffe-

rential calculus, and the first by many years; and that he left the world to judge whether Leibnitz, the second inventor had taken any thing from him."

On the other hand, when Newton's treatise on the quadrature of curves, and on the enumeration of lines of the third order was published, which was in 1704, the Leipic journalists insinuated, in a very illiberal account which they gave of the work, that Leibnitz was the first inventor, and that Newton had taken his method from Leibnitz's, substituting fluxions for differences.

In consequence of this attack on Newton, Dr John Keill asserted, in the Philosophical Transactions for 1708, that Newton was beyond a doubt the first inventor of the arithmetic of fluxions, and that the same arithmetic, having its name and notation changed, was afterwards published by Mr Leibnitz in the Leipic Acts. In answer to this, Leibnitz replied, in a letter to Hans Sloane, secretary to the Royal Society, that no one knew better than Newton himself, that this charge against him implied in Keill's assertion was false; and he required Keill to retract what he had said. To this request however Keill would by no means accede; but on the contrary, he wrote a long letter to the secretary of the Royal Society, in which he endeavoured to prove, not only that Newton had preceded Leibnitz in the invention, but that he had given to the latter such indications of the nature of his calculus, as made it easy for him to fall upon the same. This letter was sent to Leibnitz, who replied, that Keill, although learned, was too young a man to be fit to judge of what had passed between him and Newton, and he requested the Royal Society to put a stop to Keill's clamours.

The Royal Society being thus appealed to as a judge, appointed a committee to examine all the old letters, papers, and documents, which had passed among the several mathematicians, relating to the question. The judgment of the committee was to the following effect. "That Mr Leibnitz was in London in 1673, and went thence to Paris, where he kept a correspondence with Mr Collins, by means of Mr Oldenburg, till about September 1676, and then returned by London and Amsterdam, to Hanover; and that Mr Collins was very free in communicating to able mathematicians what he had received from Newton. That it did not appear, that Mr Leibnitz knew any thing of the differential calculus before his letter of the 21st of June 1677, which was a year after a copy of Newton's letter of the 10th of December 1672 had been sent to Paris, to be communicated to him, and above four years after Mr Collins began to communicate that letter to his correspondents; in which letter, the method of fluxions was sufficiently described to any intelligent person. That Newton was in possession of his calculus before the year 1669, and that those who had reputed Leibnitz the first inventor, knew little or nothing of his correspondence with Mr Collins, and Mr Oldenburg, long before, nor of Newton's having that method above 15 years before Mr Leibnitz began to publish it in the Leipic Acts. That for these reasons, they reckoned Newton the first inventor, and were of opinion, that Mr Keill in asserting the same had been in no ways injurious to Mr Leibnitz."

It is deserving of remark, that the committee delivered no opinion upon the advantage which Leibnitz was accused of having taken of the hints furnished to him,

in

in the course of his correspondence with Newton; they left the decision of this point to the world in general; and to enable every one to judge for himself, the Royal Society ordered the opinion of the committee to be printed, together with all the documents upon which it was founded. These appeared in 1712 under the title of, Commercium Epistolicum de Analyti promoti. This work was carefully circulated over Europe, to vindicate the title of the English nation to the discovery.

The Commercium Epistolicum having appeared, Leibnitz expressed great dissatisfaction, and threatened to reply in such a manner as to confound his adversaries. There seems no reason however to suppose, that any thing he could have said, would have affected Newton's claim to the honour of being the first inventor; for on this point there cannot be any doubt. With respect, however, to the other question, whether Leibnitz took his calculus from Newton, or found it himself, it is impossible to decide with such certainty. Mr Montucla, in his History of Mathematics, says, "There are only three places of the Commercium Epistolicum, which treat of the principles of fluxions in so clear a way, as to prove that Newton had found it before Leibnitz, but too obscurely, it seems, to take from the latter the merit of the discovery. One of these is in a letter from Newton to Oldenburg, who had signified to him, that Slufus and Gregory had each found a very simple method of drawing tangents. Newton replied, that he conjectured what the nature of that method was; and he gave an example of it, which shews it to be in effect the same thing as those geometricians had found. He adds, that it is only a particular case, or rather a corollary to a method much more general, which, without a laborious calculation, applies to the finding of tangents to all sorts of curves, geometrical or mechanical, and that without being obliged to free the equation from radicals. He repeats the same thing without explaining himself farther, in another letter, and he conceals the principle of the method under transposed letters. The only place where Newton has allowed any thing of his method to transpire, is in his Analysis per aequationes numero terminis infinitas. He here discloses, in a very concise and obscure manner, his principle of fluxions, but there is no certainty of Leibnitz's having seen this essay. His opponents have never asserted that it was communicated to him by letter, and they have gone no farther than to suspect, that he had obtained a knowledge of it in the interview which he had with Collins, upon his second journey to London. Indeed, this suspicion is not entirely destitute of probability, for Leibnitz admitted, that in this interview, he saw a part of the Epistolary Correspondence of Collins. However I think it would be rash to pronounce upon this circumstance. If Leibnitz had confined himself to a few essays of his new calculus, there might have been some foundation for that suspicion; but the numerous pieces he inserted in the Leipzic Acta, prove the calculus to have received such improvements from him, that probably he owed the invention of it to his genius, and to the efforts he made to discover a method, which put Newton in possession of so many beautiful truths. This is so much the more likely, as, from the method of tangents discovered by Dr Barrow, the transition to the differential calculus was easy, nor was the step too great for such a genius as that with which Leibnitz appears to have been endowed." Such is the opinion of Mon-

tucula, who being a foreigner, cannot be supposed have been too partial towards Newton, an Englishman. The British mathematicians have hitherto, with few exceptions, entertained an opinion still more decidedly in favour of the claims of their celebrated countryman.

It has been said that Newton took no share in the controversy; this however seems not to have been exactly the case, for besides suppressing in the third edition of his Principia (printed in 1726) the passage we have already quoted, which seems to admit that Leibnitz invented his calculus for himself, he is known to have written the notes which accompany the edition of the Commercium Epistolicum, printed in 1722. Leibnitz had also begun to prepare a Commercium Epistolicum, but he died before it was completed.

Besides the disputes that have happened respecting the inventor of the method of fluxions, the accuracy of the method itself has been the subject of controversy, both in Britain and on the continent. The differential calculus was attacked abroad by Nieuwentijt, a writer of little or no reputation as a mathematician, and by Rolle, who was an expert algebraist, and an indefatigable calculator, but rash, and too confident of the justness of his own opinions, and jealous of the inventions of others. To the first of these writers Leibnitz himself replied, and afterwards Bernoulli and Herman; the attack from Rolle was successfully repelled by Varignon, who was as zealous and intelligent, as his adversary was warm and impetuous.

The very concise manner in which the great inventor of the method of fluxions thought proper to explain its principles, gave occasion to the celebrated Dr Berkley bishop of Cloyne to call in question, not only the logical accuracy of the reasoning employed to establish the theory of fluxions, but also the faith of mathematicians in general, in regard to the truths of religion. The bishop commenced the controversy first in a small work entitled The Minute Philosopher; but his principal attack made its appearance in 1734, under the title of "The Analyst, or A Discourse addressed to an Infidel Mathematician," (understood to be Dr Halley) "wherein it is examined whether the object, principles, and inferences of the modern Analysis are more distinctly conceived than religious mysteries and points of faith." One of the best answers which was made to this work came from the pen of Benjamin Robins, and is entitled, "A discourse concerning the nature and certainty of Sir Isaac Newton's methods of fluxions, and of prime and ultimate ratios." Other mathematicians likewise attempted to defend Newton, and the method of fluxions, against the very cogent and well-directed arguments of the bishop; but the most satisfactory way of removing all objections to the method, was to abandon those obscure and inaccurate modes of expression, of which Berkley had, not without some reason, complained, and to substitute in their place, others more intelligible, and more consonant to the common methods of mathematical reasoning. This was accordingly done by the celebrated MacLaurin, who, in the year 1742, published his Treatise of Fluxions, a work which, although in some respects rather diffuse, placed the principles of the method beyond controversy, by establishing them on the firm basis of geometrical demonstration.

It would extend this introduction to too great a length were we to enter into a detailed account of the various improvements which the calculus has received from

from its first invention to the present time. We shall just briefly observe, that among those who contributed the first and the most effectually to its improvement, we may reckon Newton and Leibnitz themselves, the two illustrious rivals for the honour of its discovery; these were followed by the two brothers James and John Bernoulli, by the Marquis de L'Hospital, and many other foreign mathematicians; and in this country we may reckon Craig, Cheyne, Cotes, Taylor, and De Moivre, as among the earliest of its improvers. It is to Cotes in particular that we are indebted for the discovery of the method of finding the fluents of certain rational fractions, a discovery which was extended by De Moivre, so as to form one of the most beautiful and complete branches of the theory of fluxions.

Besides innumerable memoirs on particular branches of the fluxional calculus, which are to be found in academical collections, many distinct treatises have been written on the subject. Some of the most valuable of these are as follow. The Method of Fluxions and Infinite Series, by Sir Isaac Newton. This work was written in Latin, but was not published till the year 1736, when it was translated into English, and given to the world, along with a comment, by Mr Colson. Harmonia Mensurarum, by Cotes, a most valuable and original work, published in 1716. A Treatise on Fluxions, in two books, by Maclaurin, published in 1742. Many parts of the writings of the celebrated Euler have a reference to the theory of fluxions, or the differential and integral calculus. He has, however, three works in particular that relate to that subject; the first is his Introductio in Analysin Infinitorum, the second his Institutiones Calculi Differentialis, and the third his Institutiones Calculi Integralis.

There is a work on this subject which deserves to be particularly mentioned, both on account of its excellence, and the singular circumstance of its being composed by a lady. Its title is, Analytical Institutions, in four books, originally written in Italian, by Donna Maria Gaetana Agnelli. This lady was Professor of Mathematics and Philosophy in the University of Bologna; her work was

originally published in 1748, and has been styled by her countryman Frisi, Opus nitidissimum, ingeniosissimum, et certe maximum quod adhuc ex femina aliquis calamo prodierit. A part of this work has been published in the French language by Bossut. An English translation was prepared for the press many years ago by the late Professor Colson; it remained, however, unpublished, and might still have continued so, but for the liberality of Baron Maseres, who, after satisfying some pecuniary claims upon the manuscript, caused it, in 1801, to be published (we believe at his own expence), in two volumes quarto. The Doctrine and Application of Fluxions, by Thomas Simpson, is a work deservedly in high estimation. The Doctrine of Fluxions, by Emerson, is also very generally read by the British mathematicians. We are sorry, however, to observe, that there is no work in the English language that exhibits a complete view of the theory of fluxions, with all the improvements that have been made upon it to the present time. We cannot at present acquire any tolerable acquaintance with the subject, without consulting the writings of the foreign mathematicians. There are several excellent works in the French language; we may mention in particular a Traité de Calcul Differential et de Calcul Integral, by Cousin, in 2 vols. 4to.; another by Bossut, in 2 vols. 8vo.; and another by La Croix, in 3 vols. 4to. This last deserves particular notice, as the author intended it to comprehend the substance of the various valuable treatises by Euler, as well as of the most important academical memoirs that relate to this subject. The author has also published an abridgment of his work, in one volume octavo. Principiorum Calculi Differentialis et Integralis, by L'Huillier, published in 1795, contains a very clear exposition of the principles of the calculus. The writings of our countrymen Landen and Waring, and of these foreign mathematicians La Grange, Le Gendre, La Place, and many others, abound with improvements in the calculus. Having given this sketch of the history of this very important branch of mathematical science, we proceed to explain its principles.

PART I. THE DIRECT METHOD OF FLUXIONS.

SECT. I. Principles and Definitions.

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

2. A constant quantity is that which retains always

the same magnitude, however other quantities with which it is connected may be supposed to change; and a variable quantity is that which is indefinite in respect of magnitude, or which may be supposed to change its value. Thus, in the arithmetic of lines, the radius is a constant quantity, while the co-sine, sine, tangent, &c. of an arch, also the arch itself, are variable quantities; and in the conic sections the axes and the parameters of the axes are constant quantities, and any abscissa and its corresponding ordinate are variable quantities.

Constant quantities are usually denoted by the first letters of the alphabet a, b, c, &c. and variable quantities by the last letters x, y, z, &c.

3. Any expression of calculation, containing a variable quantity, along with other constant quantities, is called a Function of that variable quantity. Thus, supposing x to be variable, and the other quantities constant, any

any one of these expressions $a\; x^n$, $\backslash frac\{a\; +\; b\; x^n\}\{c\; x^n\; +\; d\; x^m\}$, $a^x$, $\backslash log.\; x$,

$\backslash cos.\; x$, $\backslash sin.\; x$, &c. is a function of $x$; and in any such equation as $y\; =\; a\; x\; +\; b\; x^2\; +\; c\; x^3\; +\; d\; x^4$, the quantity $y$ is called a function of $x$. Even although the variable quantities $x$ and $y$, should not be separated as in the last example, but should be related to each other as in the following

as, setting aside the consideration of the constant quantities, the value of $y$ depends on that of $x$, and on the contrary the value of $x$ depends upon that of $y$, the quantity $y$ is said to be a function of $x$, and on the other hand $x$ is said to be a function of $y$.

4. If a variable quantity be supposed to change its value, then a corresponding change will take place in the value of any function of that quantity. Let us examine the nature of this change in the magnitude of a function.

First, let us suppose that, $x$ denoting any variable quantity, the function to be considered is any integer power of that quantity, as $x^2$, or $x^3$, or $x^4$, &c.; then, $x$ being supposed to be increased by an indefinite quantity $h$, and thus to become $x+h$, the function will change its value; if it be $x^2$ it will become $(x+h)^2$, or

and if it be $x^3$ it will become $(x+h)^3$ or

and if it be $x^4$ it will become $(x+h)^4$, or

and so on, for other integer powers.

If we compare the new value of the function in each of these cases with its former value, it will immediately appear, that the new value may be resolved into two parts, one of which is the original value of the function, and, therefore, the other is the increment which the function has received, in consequence of the change in the value of the variable quantity $x$. Thus, the function being $x^2$, we have found its new value to be $x^2\; +\; 2\; x\; h\; +\; h^2$, of which expression, the first term $x^2$ is the original value of the function; therefore the other part of the expression viz. $2\; x\; h\; +\; h^2$ is its increment. In like manner the expression $x^3\; +\; 3\; x^2\; h\; +\; 3\; x\; h^2\; +\; h^3$, which is the new value of the function $x^3$, may be resolved into $x^3$, the original value of the function, and $3\; x^2\; h\; +\; 3\; x\; h^2\; +\; h^3$ its increment; and $x^4\; +\; 4\; x^3\; h\; +\; 6\; x^2\; h^2\; +\; 4\; x\; h^3\; +\; h^4$, the new value of the function $x^4$, may be resolved into $x^4$, the function itself, and $4\; x^3\; h\; +\; 6\; x^2\; h^2\; +\; 4\; x\; h^3\; +\; h^4$ its increment.

5. Having seen that, by conceiving the variable quantity $x$ to receive the indefinite increment $h$, the functions $x^2$, $x^3$, $x^4$ receive the increments

respectively, we next observe that each increment is expressed by a series, the first term of which is the first

power of the indefinite quantity $h$ multiplied by some function of the variable quantity $x$ as a co-efficient. The second term of the series consists of the second power of $h$, multiplied also by a function of $x$ as a co-efficient; and, in like manner, the third and following terms are composed of the third and higher powers of $h$ (the exponents forming the arithmetical series 1, 2, 3, 4, &c.) each multiplied by a function of $x$, as a co-efficient; and it appears, that the particular form of the function which constitutes the coefficient of any assigned term of the series depends entirely upon the particular form of the original function. Thus, when the original function is $x^2$, the function which is the coefficient of the first term is $2\; x$; when the original function is $x^3$, the co-efficient of the first term is $3\; x^2$; when the original function is $x^4$, the coefficient of the first term is $4\; x^3$, and so on. It also appears that the functions of $x$, which are the coefficients of the powers of $h$, are composed only of the variable quantity $x$ and given quantities, so that they are entirely independent of the indefinite increment $h$.

6. These observations may be extended to a function that is any power whatever of a variable quantity, by the application of the binomial theorem. Let $x$ be supposed to become $x+h$, then $x^n$ will become $(x+h)^n$; but by the binomial theorem, (see ALGEBRA, Sect. xvii.) $(x+h)^n$ when expanded into a series, is

where it appears, that the first term of the series is the original value of the function, and the following terms are the first, second, and following powers of the increment $h$, each multiplied by a new function of $x$, that is independent of the increment. Let us denote the functions $n\; x^\{n-1\}$, $\backslash frac\{n(n-1)\}\{1\; \backslash cdot\; 2\}\; x^\{n-2\}$, and $\backslash frac\{n(n-1)(n-2)\}\{1\; \backslash cdot\; 2\; \backslash cdot\; 3\}$

$x^\{n-3\}$, &c. by $p$, $q$, $r$, &c. respectively, and it is to be observed that, in the present case, as well as in the case of any other function of $x$ we may hereafter consider, by the letters $p$, $q$, $r$, &c. or the same letters with accents over them, or lastly the capital letters $P$, $Q$, $R$, &c. we do not mean to denote functions of $x$ of any particular form, but functions of $x$ in general, that consist only of $x$ and given quantities. This being kept in view, it appears that the variable quantity $x$ being supposed to change its value, and to become $x+h$, the function $x^n$ changes its value, so as to become

a series, the terms of which have the properties already explained in the two preceding sections.

7. Every rational and integer function of a variable quantity $x$ is necessarily of this form

where $A$, $B$, $C$, &c. and $\backslash alpha$, $\backslash beta$, $\backslash gamma$, are supposed to denote constant quantities.

Let us examine what is the form which the function assumes when the variable quantity $x$ changes its value to

to $x+h$; and to avoid complicated expressions, let us suppose the function to consist of these two terms $Ax^n\; +\; Bx^\backslash beta$. We have already found in last section that $x$ being supposed to become $x+h$, $x^n$ will become

where denote functions of $x$ independent of $h$, as explained in the last section, and consequently $Ax^n$ will become

In like manner $Bx^\backslash beta$ will become

denoting also functions of $x$ independent of $h$; therefore, taking the sum of the two series, it appears that, supposing $x$ to change its value, and to become $x+h$, the function $Ax^n\; +\; Bx^\backslash beta$ becomes

now $p$ and $p\text{'}$ being functions of $x$, $Ap\; +\; Bp\text{'}$ will also be a function of $x$, and may be denoted more simply by $P$, and for the same reason $Aq\; +\; Bq\text{'}$, $Ar\; +\; Br\text{'}$, &c. which are functions of $x$, may be denoted by

thus the expression for the new value of $Ax^n\; +\; Bx^\backslash beta$ is

a series, the form and properties of which are in all respects analogous to those of the series that expresses the new value of the function $x^n$; and although we have supposed the function to consist of but two terms, yet it is obvious, that whatever be the number of terms, still the form of the series and its properties will be the same; that is, it will consist of two parts, one of which is independent of $h$, and is the original value of the function, and the other is a series, the terms of which are the successive powers of the increment $h$, each multiplied by a function of the variable quantity $x$ as a co-efficient. This conclusion may be expressed in symbols concisely thus. Let $u$ denote any rational and integer function of a variable quantity $x$, let $x$ be conceived to change its magnitude, and to become $x+h$, and let $u\text{'}$ denote the new value which the function acquires in consequence of the change in the value of $x$, then

where denote functions of $x$ as already stated.

8. Suppose next the function of $x$ to be of this form:

that is, suppose it to be the $n$th power of a polynomial, consisting of any number of terms whatever. Let the expression between the parenthesis be denoted by $v$, then we are to consider the function $v^n$. Now when $x$ becomes $x+h$, we have already found that $v$ becomes

therefore $v^n$ will become

or, putting ,

and this expression when expanded into a series by the binomial theorem is

where express numbers.

Now from the form of the series denoted by $M$, it is manifest that its square, cube, or any power of it whatever, will be a series proceeding by the powers of $h$, and having for the coefficients of its terms certain combinations of the quantities which being functions of $x$, any combinations of them will also be functions of $x$. Therefore, each of the terms of the above series, expressing the development of $(v+M)^n$, excepting the first term $v^n$, will itself be a series proceeding by the powers of $h$, and having its terms multiplied by functions of $x$, and consequently their sum will be a series of the same nature. Let us as before denote the functions of $x$, which are the coefficients of the successive powers of $h$ by and we shall have upon the whole

expressed by a series of this form

therefore, putting the single letter $u$ for the function $u^n$, or for

and $u\text{'}$ for the new value which $u$ acquires by $x$ changing its value to $x+h$,

a series of the same nature as before.

9. Let us now consider a fractional function of $x$, and let us suppose it to be

Where also denote constant quantities. Let $v$ denote the numerator of the fraction, and $w$ its denominator, then the function is

now when $x$ becomes $x+h$, $v$ becomes

and $w^\{-1\}$ becomes

and consequently $v\; w^\{-1\}$ becomes

and

Direct Method. and the product of these two factors, by actual multiplication is

Now, here as before, it appears that the coefficients of the powers of $h$ are functions of $x$, therefore, denoting these functions by $P$, $Q$, $R$, &c. and observing that $v\; w^\{-1\}$ is $\backslash frac\{v\}\{w\}$, we have the new value of $\backslash frac\{v\}\{w\}$ expressed by the series

or, substituting the single letter $u$ for $\backslash frac\{v\}\{w\}$, that is, for

and putting $u\text{'}$ for the value that $u$ acquires when $x$ becomes $x+h$,

a series in all respects analogous to those already found for the other functions of $x$.

10. In the functions which we have hitherto considered, the exponents of the powers of $x$ were constant quantities. Let us now consider a function in which the exponent is the variable quantity $x$ itself.

Suppose then the function to be $a^x$, where $a$ denotes a given number; then, by supposing $x$ to become $x+h$, the function will become

Now it has been shown in the article ALGEBRA, § 295, that if $A$ be put to denote the quotient arising from the division of a logarithm of $a$ by the logarithm of $2.7182818\backslash dots$ the exponential quantity $a^h$ is expressed by the series

therefore, $a^\{x+h\}$, the new value of the function is

this series, by multiplying all its terms by $a^x$, and putting $p$, $q$, $r$, &c. for that part of each term which is independent of $h$, becomes

so that denoting the function $a^x$ by $u$, and its new value by $u\text{'}$,

a series of the same form as the others.

11. From a due consideration of what has been shown relating to the change that takes place in the magnitude of a variable function, corresponding to the

change that takes place in the magnitude of the variable quantity from which the function is formed, we may conclude the truth of the following general proposition to be sufficiently established.

Let $x$ denote a variable quantity, and $u$ any function whatever of that quantity, let $x$ be supposed to receive any increment $h$, and thus to become $x+h$, and let $u\text{'}$ be the new value which the function acquires by the change in the value of $x$, then, the new value of $u$ may in every case be expressed thus:

where $p$, $q$, $r$, &c. denote quantities that are quite independent of $h$, and consequently can only involve the variable quantity $x$, and given quantities.

12. Having examined what is the general form that any function of a variable quantity acquires by a change in the value of that quantity, and found it to be a series, the first term of which is always the function itself, it is evident that the remaining terms will express the increment that the function receives, in consequence of the change in the magnitude of the variable quantity from which the function is formed. Let us now compare the simultaneous increments of a variable quantity and its function with each other, and that we may at first avoid general reasoning, and fix the mind more completely, let us suppose the functions to have a determinate form, as $x^2$, $x^3$, $x^4$, &c.

Putting $u$ and $u\text{'}$ as before to denote the two succeeding values of the function, first let it be supposed that $u\; =\; x^2$, then $x$ being supposed to receive the indefinite increment $h$, and thus to become $x+h$, and $u$ to change its value to $u\text{'}\; =\; (x+h)^2$, we have

and consequently

Thus it appears, that the simultaneous increments of $x$ and $x^2$ (or $u$) are $h$ and $2\; x\; h\; +\; h^2$, respectively. Let us now compare these increments, not in respect of their absolute magnitudes, but in respect of their ratio to each other, thus we shall have the increment of $u$ to the increment of $x$, as $2\; x\; h\; +\; h^2$ to $h$, that is, (dividing the two last terms of the proportion by $h$) as $2\; x\; +\; h$ to 1. Or, instead of employing an analogy, let us, for the sake of brevity, and in conformity to the algebraic notation, rather express each of these ratios by the quotient arising from the division of the antecedents of the ratio by its consequents, and put the results equal to each other. Then, observing that the symbol $u\text{'}\; -\; u$, which expresses the difference between the succeeding values of the function, may be employed to denote its increment, we have

Hence it appears that the expression for the ratio of the increment of the function $u$ to the increment of the variable quantity $x$ is made up of two parts, one of these, viz. $2\; x$, is quite independent of $h$, the increment of $x$, and the other is in the present case that increment itself. In consequence of this peculiarity in the form

form of the expression for the ratio, it is evident that if the increment $h$ be conceived to be continually diminished, the part of the expression which consists of $h$ will continually diminish, so that the whole expression, viz. $2x+h$, may become more nearly equal to its first term $2x$ than by any assignable difference; therefore $2x$ may be considered as the limit of the ratio $\backslash frac\{u-u\text{'}\}\{h\}$; that is, a quantity to which the ratio may approach nearer than by any assignable difference, but to which it cannot be considered as becoming absolutely equal.

Let us next suppose that $u=x^3$, then $x$ being supposed to become $x+h$, we have $u\text{'}=(x+h)^3\; =\; x^3\; +\; 3x^2h\; +\; 3xh^2\; +\; h^3$, or

and consequently

Here it is evident, as in the former case, that the expression for the ratio $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ is composed of two parts, one of these, viz. the first term $3x^2$, is a function of $x$ that is independent of the increment $h$, but the other, viz. $3xh\; +\; h^2$, or $h(3x+h)$, is the product of two factors, one of which is the increment itself. From the particular form of this latter part of the expression for the ratio, it is plain, that $h$ being supposed to be continually diminished, that part will also diminish, and may become less than any assignable quantity. Therefore in this case, as well as in the former, the ratio $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ has a limit, and that limit is the first term of the general expression for the ratio, namely the quantity $3x^2$.

Suppose, next, that $u=x^4$, and consequently

or,

Here, as in the two former cases, we have only to inspect the general expression for the ratio $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ to discover, that by supposing $h$ to be continually diminished, the latter part of the expression, viz. $h(6x^2\; +\; 4xh\; +\; h^3)$, and which comprises all the terms except the first, will become smaller than any assignable quantity; and consequently, that the first term $4x^3$ is the limit of the ratio.

13. It is easy to see that the property which we have found to belong to the ratio of the simultaneous increments of a variable quantity, and its function in these three particular cases, is an immediate consequence of the form of the expression for the increment of the function, so that it is not peculiar to the functions $x^3$, $x^4$, and $x^4$, but belongs equally to all functions whatever.

For we have found, § 11, that $u$ being supposed to

denote any function of a variable quantity $x$, as for example, $ax^n$, or , or

and $u\text{'}$ being put for the new value which the function acquires when $x$ becomes $x+h$,

where $p$, $q$, $r$, &c. denote functions of $x$ that are independent of $h$, therefore,

and

or,

Thus it appears, that whatever be the form of the function, the ratio $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ is always expressed by a quantity which may be resolved into two parts; one of these, viz. $p$, is independent of the increment $h$, and the other, viz. , is the product of $h$ by a series, the first term of which is a function of $x$, and the remaining terms also functions of $x$ multiplied by the first, second, third, and higher powers of $h$. Now from the particular form of this last part of the general expression for the ratio, it is manifest, that $h$ being conceived to be continually diminished, the quantity will also be continually diminished, and may become less than any assignable quantity; therefore, the limit of the ratio $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ is simply $p$, that is, the function of $x$, which is the coefficient of the first or simple power of $h$ in the general expression for the increment.

14. From what has been just shewn, we may infer the truth of the following general proposition relative to the simultaneous changes that take place in a variable quantity and its function.

Let $x$ denote a variable quantity and $u$ any function of that quantity, let $x$ be conceived to change its value and become $x+h$, where $h$ denotes an arbitrary increment, and let $u\text{'}$ denote the new value that the function acquires, in consequence of the change in the magnitude of $x$. Then, observing that $h$ and $u\text{'}\; -\; u$ are the simultaneous increments of the variable quantity and its function, if $h$ be conceived to be continually diminished, the ratio $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ will continually approach to a certain limit, which will be different for different functions, but always the same for the same function, and in every case quite independent of the magnitude of the increments.

The ratio which is the limit of the ratio of the increments, when these increments are conceived to be continually diminished, may be called the limiting ratio of the increments.

15. The analytical fact contained in the preceding proposition affords the foundation for a mathematical theory of great extent, and which may be divided into

into two distinct branches; one having for its object the resolution of the following problem. Having given the relation of any number of variable quantities to each other, to determine the limiting ratios of their increments; and the other the converse of that problem, namely, Having given the limiting ratios, to determine the relations of the quantities themselves.

The theory to which we have alluded constitutes the METHOD OF FLUXIONS, and in explaining the foundation of the method, we have endeavoured to show, that it rests upon a principle purely analytical, namely the theory of limiting ratios; and this being the case, the subject may be treated as a branch of pure mathematics, without having occasion to introduce any ideas foreign to geometry.

16. Sir Isaac Newton, however, in first delivering the principles of the method, thought proper to employ considerations drawn from the theory of motion. But he appears to have done this chiefly for the purpose of illustration, for he immediately has recourse to the theory of limiting ratios, and it has been the opinion of several mathematicians of great eminence (A) that the consideration of motion was introduced into the method of fluxions at first without necessity, and that succeeding writers on the subject ought to have established the theory upon principles purely mathematical, independent of the ideas of time, and velocity, both of which seem foreign to investigations relating to abstract quantity.

17. That we may conform to the usual method of treating this subject, we proceed to show how the theory of motion is commonly applied to the illustration of the nature of variable quantities, and of the relations that result from their being conceived to change their value.

As quantities of every kind, if we abstract from their position, figure, and such affections, and consider their magnitude only, may be represented by lines, we may consider a variable quantity $x$, and $u$ any function of that quantity, to be represented by two lines $AP$, $BQ$, which have $A$, $B$, one extremity of each given, and which vary by the points $P$, $Q$, their other extremities, moving in the directions $AC$, $BD$, while the equation expressing the relation between $x$ and $u$, or their representative lines $AP$, $BQ$, remains always the same.

18. The lines $AP$, $BQ$ being thus conceived to vary, the relation that is supposed always to subsist between them, in respect of their magnitudes, will necessarily give rise to another relation, namely, that which will constantly subsist between the velocities of the moving points $P$ and $Q$, by which the lines are generated.

With a reference to this particular mode of conceiving variable quantities to exist, the quantities themselves have been called flowing quantities or fluents, and the

measures of the velocities with which the variable quantities increase have been called their fluxions.

19. To simplify the hypothesis, we may suppose that the point which generates the line $AP$, or $x$, moves uniformly; thus the measure of its velocity, or the fluxion of $x$, will be a given quantity, with which the measure of the velocity of the point $Q$, or the fluxion of $u$, may be continually compared. To determine then the fluxions, or rather the ratio of the fluxions of $x$ a variable quantity, and $u$, any function of that quantity, is in effect to resolve the following problem.

Having given an equation expressing the relation at every instant between the spaces passed over by two points, one of which moves with an uniform velocity. It is required to find an expression for the ratio that the measures of the velocities have at every instant to each other.

20. Now it is a fundamental principle in the theory of motion, resulting indeed from the very nature of a variable velocity, that when two velocities are compared together, whether they be both variable, or one of them uniform, and the other variable, the measures of their velocities are any quantities having to each other the ratio that is the limit of the ratio of the spaces described in the same time, when those spaces are conceived to be continually diminished. And hence it follows that the ratio of the fluxions of two variable quantities is no other than the limiting ratio of their simultaneous increments.

That the theory of motion may be applied to the generation of variable algebraic quantities, we have supposed them to be represented by lines; this however is not necessary, if the variable quantities are themselves geometrical magnitudes; for like as a line is conceived to be generated by the motion of a point, so a surface may be considered as generated by the motion of a line, a solid by the motion of a surface, and an angle by the rotation of one of the lines which contain it; and the fluxions of those quantities at any instant, or position, will be the measures of the velocities, or degrees of swiftness, according to which they increase at that instant or position.

But in every case the ratio of the rates of increase, or fluxions of two homogeneous magnitudes, will be the limiting ratio of their simultaneous increments.

21. Having thus found that by conceiving variable quantities as generated by motion, and taking their velocities, or rates of increase, as an object for the mind to contemplate and reason on, we are in the end led to the consideration of the limiting ratios of their increments, a subject which is purely mathematical, and independent of the ideas of time or velocity, we shall exchange the definition of a fluxion given in § 18, which involves those ideas, for another that rests entirely upon the existence of limiting ratios.

By the fluxions then of two variable quantities having any assigned relation to each other, we are in the following treatise always to be understood to mean any

(A) Such as Lagrange, Cousin, La Croix, &c. abroad, and Landen in this country.

In conformity to this definition of fluxions it is evident that we are to consider them, not as absolute, but as relative quantities, which derive their origin from the comparison of variable quantities with each other in respect of their simultaneous variations of magnitude.

22. Sir Isaac Newton employed different symbols at different times to denote the fluxions of variable quantities. It is now however common in Britain to denote them by the same letters employed to express the quantities themselves, and each having a dot over it. Thus $\backslash dot\{x\}$ denotes the fluxion of the variable quantity expressed by $x$, and in the like manner $\backslash dot\{u\}$, $\backslash dot\{v\}$, $\backslash dot\{z\}$, denote the fluxions of the variable quantities $u$, $v$, $z$, respectively.

23. Suppose now that $u$ is any function of a variable quantity $x$, and that the limiting ratio of the simultaneous increments of $u$ and $x$ is the ratio of $p$ to 1, where $p$ denotes some other function of $x$, then, from the definition just given of fluxions, we have

Hence it follows as a consequence of the preceding definition, that the fluxion of $u$, any function of a variable quantity $x$, is the product arising from the multiplication of that function of $x$ which is the expression for the limiting ratio of the increments by the fluxion of the variable quantity $x$ itself.

SECT. II. Investigation of the Rules of the Direct Method of Fluxions.

24. The method of fluxions naturally resolves itself into two parts, as we have already observed, § 15. We proceed to explain the first of these, which is called the Direct Method, and which treats of finding the ratios of the fluxions of variable quantities, having given the relations of those quantities to each other.

25. We shall begin with investigating the ratio of the fluxions of two variable quantities in that particular case, when one of them is any power of the other.

Let us suppose then, that $u$ is such a function of a variable quantity $x$, that $u\; =\; x^n$, where $n$ denotes any number whatever, it is required to determine the ratio of the fluxions of $u$ and $x$.

If we recur to the definition which has been given of the fluxions of variable quantities in § 21, it will appear that we have in effect resolved the problem just proposed in three particular cases, when treating of the limiting ratios of the increments of variable quantities. For it has been shown in § 12, that when $u\; =\; x^1$, and $x$ is supposed to change its value, so as to become $x+h$, where

$h$ denotes an indefinite increment, and in consequence of this change in the magnitude of $x$, $u$ also changes its value, and becomes $u\text{'}$, then, observing that $u\text{'}\; -\; u$ and $h$ are the simultaneous increments of $u$ and $x$, the limiting ratio of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ is 2 $x$. Let this expression for the limiting ratio be put equal to the ratio of the fluxions of $u$ and $x$, that is to $\backslash frac\{\backslash dot\{u\}\}\{\backslash dot\{x\}\}$; thus we have $\backslash frac\{\backslash dot\{u\}\}\{\backslash dot\{x\}\}\; =\; 2x$, and $\backslash dot\{u\}\; =\; 2x\; \backslash dot\{x\}$.

Hence it appears that whatever be the magnitude of the quantity that expresses the fluxion of $x$, the fluxion of $u$ or $x^1$ will be expressed by the fluxion of $x$ multiplied by 2 $x$.

Again, when $u\; =\; x^3$, and $u\text{'}$, $u\text{'}\; -\; u$ and $h$ denote the same as before, it has been shown, § 12, that the limiting ratio of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ is 3 $x^2$, therefore, § 21, $\backslash frac\{\backslash dot\{u\}\}\{\backslash dot\{x\}\}\; =\; 3x^2$, and

$\backslash dot\{u\}\; =\; 3x^2\; \backslash dot\{x\}$; that is, the fluxion of $x^3$ is expressed by the fluxion of $x$ multiplied by 3 $x^2$.

And when $u\; =\; x^4$ it has been shown, § 12, that the limiting ratio of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ is 4 $x^3$. Therefore, § 21,

To resolve the problem generally, or when $u\; =\; x^n$, let us suppose $x$ to become $x+h$, and $u$ to become $u\text{'}$, then $u\text{'}\; =\; (x+h)^n$. But this last quantity, when expanded into a series by the binomial theorem (ALGEBRA, SECT. XVII.) is

where $p$, $q$, $r$, &c. denote functions of $x$, independent of $h$. Therefore,

Therefore, supposing $h$ to be continually diminished, the limit of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ is $p$; but, whatever be the nature of

the exponent $n$, $p$ is always $n\; x^\{n-1\}$, (ALGEBRA, § 267.), therefore, the limit of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ is $n\; x^\{n-1\}$, and consequent-

26. As we shall have frequently occasion to employ the result of this investigation, it will be proper to express it in the form of a practical rule, thus.

To find the fluxion of any power of a variable quantity. Multiply the fluxion of the variable quantity itself by the exponent of the power, and by a power of the quantity whose exponent is less by unity than the given exponent, and the product will be the fluxion required.

(a) We are here to be understood to mean the ratio of the numerical values of the increments, which may always be compared with each other, whether the variable quantities be of the same kind, as both lines, or both surfaces, &c. or of different kinds, as the one a line, and the other a surface.

27. In determining the limit of the ratio of the simultaneous increments of $x$ and $x^n$, we have referred to the binomial theorem, but the only application we have had occasion to make of that theorem was to determine the numeral coefficient of the second term of the development of $(x+h)^n$, when $n$ is supposed to be any number whatever, which is an inquiry of a more simple nature than the general investigation of the theorem. We shall now show how that coefficient may be deduced from the first principles of algebra. Thus the investigation of the fluxion of $x^n$ will be rendered independent of the general demonstration of the binomial theorem; and we shall hereafter show that the theorem itself is easily investigated by the direct method of fluxions.

28. Since $x+h$ is equal to $x(1+\backslash frac\{h\}\{x\})$ it follows that $(x+h)^n\; =\; x^n\; (1+\backslash frac\{h\}\{x\})^n$, thus the development of $(x+h)^n$ is reduced to that of $(1+\backslash frac\{h\}\{x\})^n$, or putting $\backslash frac\{h\}\{x\}=v$, to $(1+v)^n$. Now if we give particular values to $n$, and suppose it to be 1, 2, 3, &c. or -1, -2, &c. or lastly $\backslash frac\{1\}\{2\}$, $\backslash frac\{1\}\{3\}$, &c. we can find the series that expresses the powers of $1+v$, whose exponents are those numbers, by the operations of multiplication, division, and evolution, and in every particular case we shall find, that the powers of $1+v$ are expressed by a series of this form,

Where $A,\; B,\; C,\; D,\; \backslash dots$ denote numeral coefficients which depend for their value only on $n$ the exponent of the power, and not on the quantity $v$; and as the form of the series will be found to be the same whatever particular values we may give to the exponent, we may conclude that it is the same, whether the exponent be positive or negative, whole or fractional.

29. First let us suppose that the exponent is a whole positive number, then, because

if we multiply both sides of this equation by $1+v$, and collect together the like powers of $v$, it will appear that

Hence it appears that the coefficient of the second term of any power of $1+v$ exceeds that of the next less power by unity. Now in the case of the first power of $1+v$, the coefficient of the second term is obviously 1, therefore, in the second power it is 2, in the third power 3, and universally, in the $n$th power it is $n$; so that $n$ being a whole positive number,

30. Let us next suppose the exponent to be a fraction denoted by $\backslash frac\{m\}\{n\}$, so that

Let both sides of this equation be raised to the power $n$, having first substituted $Mv$ for $Av\; +\; Bv^2\; +\; \backslash dots$ then

Now as we have just found that $m$ and $n$ being integers,

Here we stop at the second term, that being the only one whole coefficient is required. Substitute now for $Mv$ its value $Av\; +\; \backslash dots$ then, stopping again at the second term, we get

therefore,

and making the coefficients of $v$ in each series equal to each other,

31. In the last place let us suppose that the exponent is a negative quantity either whole or fractional, so that

then, multiplying both sides by $(1+v)^n$ we get

or, substituting $1\; +\; Av\; +\; Bv^2\; +\; \backslash dots$ for $(1+v)^n$ and actually multiplying the two series,

Now that this equation may subsist, whatever be the value of $v$, it is necessary that

and by these equations we may determine $A\text{'},\; B\text{'},\; C\text{'},\; \backslash dots$. It is however only required at present to find the first of these, viz $A\text{'}$, now we have $A\text{'}\; =\; -A$; but $A$ being the coefficient of the second term of the series expressing $(1+v)^n$, the exponent of which is positive, we have already found it to be $\backslash frac\{m\}\{n\}$, therefore, $A\text{'}\; =\; -\backslash frac\{m\}\{n\}$.

32. As we have found that the coefficients of the second term of the developments of $(1+v)^n$, $(1+v)^\{\backslash frac\{m\}\{n\}\}$ and $(1+v)^\{-\backslash frac\{m\}\{n\}\}$ are $m,\; \backslash frac\{m\}\{n\}$, and $-\backslash frac\{m\}\{n\}$ respectively, it appears that whatever be the number denoted by $n$, the two first terms of the series expressing $(1+v)^n$ are, $1\; +\; nv$, and therefore, substituting for $v$ its value $\backslash frac\{h\}\{x\}$, and multiplying by $x^n$, the two first terms of the series expressing $(x+h)^n$ are $x^n\; +\; nx^\{n-1\}h$, agreeing with what we have

33. The mode of reasoning employed to determine the ratio of the fluxions of $u$ and $x$, when the former is a function of the latter of the form $x^n$, will apply equally when the function has any other assigned form. But instead of investigating in this manner the fluxion of every particular function, it is better to consider a complex function as the sum, or difference, or product or quotient, &c. of other simple functions, and to investigate rules for each of these cases, supposing that the fluxions of the simple functions are previously known.

Therefore, to find the fluxion of the sum of any number of functions, each multiplied by a constant quantity. Multiply the fluxion of each function by its constant coefficient, and the sum of the products is the fluxion required.

34. Let us first suppose that $u$, a function of a variable quantity $x$, is equal to the sum of $v$ and $w$, two other functions of $x$. It is required to find the fluxion of $u$, having given the fluxions of $v$ and $w$.

36. If $c$ denote a constant quantity, and $u,\; v$, be functions of $x$, such, that $u\; =\; c\; +\; v$; then $x$ being supposed to become $x\; +\; h$, and consequently $v$ to become $v\text{'}$, or and $c\; +\; v$ to become we have

Let $x$ be conceived to change its value, and to become $x\; +\; h$, then, as $v$ and $w$ will also change their values, § 11, the one to

and the other to

if $u\text{'}$ as usual denote the corresponding new value of $u$, we have

If we now conceive $h$ to be continually diminished, we shall have the limit of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ expressed by $p\; +\; p\text{'}$. But

and these ratios being always equal, their limits must also be equal; therefore, substituting for the limiting ratios those of the fluxions, we have $\backslash frac\{u\}\{x\}\; =\; \backslash frac\{v\}\{x\}$ and $\backslash dot\{u\}\; =\; \backslash dot\{v\}$; that

is to say, the fluxion of $c\; +\; v$ is $\backslash dot\{v\}$, from which it appears, That the fluxion of any variable function is the very same as the fluxion of the same function, increased or diminished by any constant quantity. This is a remark of great importance in the theory of fluxions, as will appear hereafter.

$p$ is the limit of $\backslash frac\{v\text{'}\; -\; v\}\{h\}$, § 14, and in like manner $p\text{'}$ is

By supposing that $x,\; u,\; v$ and $w$ change their values as usual, we have

the limit of $\backslash frac\{w\text{'}\; -\; w\}\{h\}$, therefore,

limit of $\backslash frac\{u\text{'}\; -\; u\}\{h\}\; =\; \backslash text\{limit\; of\; \}\; \backslash frac\{v\text{'}\; -\; v\}\{h\}\; +\; \backslash text\{limit\; of\; \}\; \backslash frac\{w\text{'}\; -\; w\}\{h\}.$

and this last expression by multiplication becomes

Substitute now the ratio of the fluxions instead of the limiting ratios, and we have

therefore

therefore, $\backslash dot\{u\}\; =\; \backslash dot\{v\}\; +\; \backslash dot\{w\}.$

dividing now by $h$, and taking the limit of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$, we have that limit expressed by $v\; p\text{'}\; +\; w\; p$; but $p\text{'}$ is the limit of $\backslash frac\{v\text{'}\; -\; v\}\{h\}$ § 14, and in like manner $p$ is the limit of $\backslash frac\{w\text{'}\; -\; w\}\{h\}$; therefore

35. If we suppose $s$ to be a function of $x$, and other functions of $x$, such that

the

(c) In this investigation we have supposed $n$ to be a rational number. If, however, it were irrational, still the result would be the same; for, corresponding to every such number, two rational numbers, one greater, and the other less than it, may be found, which shall differ from each other by less than any assignable quantity. Therefore, the general properties of these numbers must belong also to the irrational number which is their limit.

Hence, by substituting for the limiting ratios the ratios of the fluxions, we have

Therefore, to find the fluxion of the product of any two functions, multiply the fluxion of each function by the other function, and the sum of these products is the fluxion required.

38. We have just now seen that when

Let each side of the latter equation be divided by the corresponding side of the former, thus we get

suppose now that the function $w$ is the product of two other functions $s,\; t$, so that

then, because $w\; =\; s\; t$, from what has been shewn it follows that $\backslash frac\{\backslash dot\{w\}\}\{w\}\; =\; \backslash frac\{\backslash dot\{s\}\}\{s\}\; +\; \backslash frac\{\backslash dot\{t\}\}\{t\}$; therefore, substituting this value

of $\backslash frac\{\backslash dot\{w\}\}\{w\}$ in the equation $\backslash frac\{\backslash dot\{u\}\}\{u\}\; =\; \backslash frac\{\backslash dot\{v\}\}\{v\}\; +\; \backslash frac\{\backslash dot\{w\}\}\{w\}$, it becomes

In general, if we suppose that

by reasoning as above it will be found that

whatever be the number of factors.

Suppose the number of factors to be three, so that

then substituting $v\; s\; t$ for $u$ in this last equation, and taking away the denominators, we find

And as a similar result will be found, whatever be the number of factors, we may conclude that The fluxion of the product of any number of functions is equal to the sum of the products of the fluxion of each function by all the other functions.

39. Let us in the last place suppose that $u\; =\; \backslash frac\{v\}\{w\}$, and

that it is required to find the fluxion of $u$, having given the fluxions of $v$ and $w$.

From the given equation we have $v\; =\; u\; w$, and therefore (§ 37.) $\backslash dot\{v\}\; =\; w\; \backslash dot\{u\}\; +\; u\; \backslash dot\{w\}$, let $\backslash frac\{v\}\{w\}$ be substituted for $u$ in this equation, and it becomes $\backslash dot\{v\}\; =\; w\; \backslash dot\{u\}\; +\; \backslash frac\{v\; \backslash dot\{w\}\}\{w\}$, from which we easily obtain

Hence we have the following rule for finding the fluxion of a fraction.

Multiply the fluxion of the numerator by the denominator, and from the product subtract the fluxion of the denominator multiplied by the numerator, and divide the remainder by the square of the denominator; the result is the fluxion required.

40. It will now be proper to shew the application of these general rules for determining the fluxions of variable functions to some particular examples.

Example 1. Suppose $u\; =\; a\; +\; b\; \backslash sqrt\{x\}\; -\; \backslash frac\{c\}\{x\}$. Required the fluxion of $u$.

Here $a$ being a constant quantity, the fluxion of $a\; +\; b\; \backslash sqrt\{x\}\; -\; \backslash frac\{c\}\{x\}$ is the same as the fluxion of $b\; \backslash sqrt\{x\}\; -\; \backslash frac\{c\}\{x\}$, § 36, or $b\; x^\{\backslash frac\{1\}\{2\}\}\; -\; c\; x^\{-1\}$. Now, by § 26 the fluxion of $x^\{\backslash frac\{1\}\{2\}\}$ is $\backslash frac\{1\}\{2\}\; x^\{-\backslash frac\{1\}\{2\}\}\; \backslash dot\{x\}$, which expression is equivalent to $\backslash frac\{1\}\{2\}\; x^\{-\backslash frac\{1\}\{2\}\}\; \backslash dot\{x\}$, or to $\backslash frac\{\backslash dot\{x\}\}\{2\; \backslash sqrt\{x\}\}$, and in like manner the fluxion of $x^\{-1\}$ is $-x^\{-2\}\; \backslash dot\{x\}$ or $-x^\{-2\}\; \backslash dot\{x\}$, or $\backslash frac\{-\backslash dot\{x\}\}\{x^2\}$, therefore, multiplying the fluxion of $x^\{\backslash frac\{1\}\{2\}\}$ by $b$, and the fluxion of $x^\{-1\}$ by $c$, and taking the sum of the products, agreeably to the rule in § 35, we have

Ex. 2. Suppose $u\; =\; a\; +\; \backslash frac\{b\}\{\backslash sqrt\{x^2\}\}\; -\; \backslash frac\{c\}\{x^3\; \backslash sqrt\{x\}\}\; +\; \backslash frac\{d\}\{x^2\}$.

By writing the function thus

the application of the same rules employed in the last example gives us

or, exchanging the fractional indices for the radical sign, and otherwise reducing,

In order to find the fluxion of this function by the rules already laid down, it will be necessary to consider it first as a function of a variable quantity that is itself a function of $x$. Let us then put $a\; +\; bx^m$ equal to $v$, and thus the proposed equation becomes $u\; =\; v^n$; then, $u$ being considered as a function of $v$, we have by § 26. $\backslash dot\{u\}\; =\; n\; v^\{n-1\}\; \backslash dot\{v\}$. Again, $v$ being considered as a function of $x$, from the equation $v\; =\; a\; +\; bx^m$, we find by § 36, and § 26, $\backslash dot\{v\}\; =\; m\; b\; x^\{m-1\}\; \backslash dot\{x\}$; Let this value of $\backslash dot\{v\}$ be substituted in the expression for $\backslash dot\{u\}$, and it becomes

which, by substituting for $v$ its value $a\; +\; bx^m$, is also

Here we are to proceed as in the last example, and first put $a^2\; -\; x^2\; =\; v$, then $u\; =\; \backslash sqrt\{v\}\; =\; v^\{\backslash frac\{1\}\{2\}\}$; and therefore (§ 26) $\backslash dot\{u\}\; =\; \backslash frac\{1\}\{2\}\; v^\{-\backslash frac\{1\}\{2\}\}\; \backslash dot\{v\}\; =\; \backslash frac\{\backslash dot\{v\}\}\{2\backslash sqrt\{v\}\}$. Again, since $v\; =\; a^2\; -\; x^2$, by § 26, we find $\backslash dot\{v\}\; =\; -2x\; \backslash dot\{x\}$. Substitute this value of $\backslash dot\{v\}$ in the expression for $\backslash dot\{u\}$, and we have

which, by restoring $\backslash sqrt\{a^2\; -\; x^2\}$ for $v$, and leaving out the number common to the numerator and denominator, becomes

By proceeding in the same manner as in last example we find

Here the proposed function is the product of these three functions, viz $x$, $a^2\; +\; x^2$, and $\backslash sqrt\{a^2\; -\; x^2\}$. Therefore, its fluxion will be found by proceeding according to the rule in § 38.

Now the fluxion of $x$ is $\backslash dot\{x\}$, and the fluxion of $a^2\; +\; x^2$ is $2x\; \backslash dot\{x\}$, and the fluxion of $\backslash sqrt\{a^2\; -\; x^2\}$ has been found in last example to be $\backslash frac\{-x\; \backslash dot\{x\}\}\{\backslash sqrt\{a^2\; -\; x^2\}\}$. Therefore, multiplying the fluxion of each function by the product of the other two functions, and taking the sum of all these products, we find

and this equation, by reducing all the terms on the latter side of it to a common denominator, is more simply expressed thus,

Here we employ the rule given in § 39, for finding the fluxion of a fractional function; thus we find

which when reduced is

To simplify this expression we put

Thus we have

Now as the fluxion of $a\; -\; y\; +\; z$, is $-\backslash dot\{y\}\; +\; \backslash dot\{z\}$ (§ 35), we find from § 26, that by considering $a\; -\; y\; +\; z$ as a single variable quantity,

but, since $y\; =\; \backslash frac\{b\}\{\backslash sqrt\{x\}\}\; =\; b\; x^\{-\backslash frac\{1\}\{2\}\}$, we have, § 26,

and since $z\; =\; \backslash sqrt\{c^2\; -\; x^2\}\; =\; (c^2\; -\; x^2)^\{\backslash frac\{1\}\{2\}\}$, by considering $c^2\; -\; x^2$ as a single variable quantity, and observing that its fluxion is $-2x\; \backslash dot\{x\}$, we find by § 26, that

Instead of $y$, $z$, $\backslash dot\{y\}$, $\backslash dot\{z\}$, substitute now their values in the expression for the fluxion of $u$, thus it becomes

Ex. 9. Suppose $u\; =\; a\; v^n\; y^n$, where $v$, and $y$ denote any functions of a variable quantity.

Then, § 37,

But

Direct Method. But fluxion of $v^m\; =\; m\; v^\{m-1\}\; \backslash dot\{v\}$, § 26,
and fluxion of $y^n\; =\; n\; y^\{n-1\}\; \backslash dot\{y\}$;

Ex. 10. Suppose $u\; =\; \backslash frac\{v+\backslash dot\{v\}\}\{y^3\}$, where $v,\; \backslash dot\{v\}$ and $y$ denote any functions of a variable quantity. Then, because fluxion $(v+\backslash dot\{v\})\; =\; \backslash dot\{v\}\; +\; \backslash dot\{v\}$, § 34, and fluxion $y^3\; =\; 3\; y^2\; \backslash dot\{y\}$, § 26, we have, § 39,

41. As when $u$ denotes that particular function of $x$ which is $x^n$, we have (§ 25.)

so in general, whatever be the form of the function denoted by $u$, we have always

where $\backslash rho$ denotes a new function of $x$, resulting from the analytical process employed to find the fluxion of the function $u$, and depending for its form upon the particular form of that function: just as in involution, or any of the other operations of algebra, a result is obtained depending upon the particular nature of the operation, and the quantities operated upon.

Let us put $\backslash rho$ to denote the particular function $n\; x^\{n-1\}$, or the expression for $\backslash frac\{\backslash dot\{u\}\}\{x\}$ the ratio of the fluxion of $u$ to the fluxion of $x$ when $u\; =\; x^n$, then, supposing that $n-1$ is not equal to 0, (for in that case $n\; x^\{n-1\}$ would be simply $n$, a given number,) we may reason concerning the ratio of the fluxions of the variable quantities $\backslash rho$ and $x$, in all respects as concerning the ratio of the fluxions of $u$ and $x$; and accordingly, from the equation

we get, by taking the fluxions,

or, considering $\backslash rho$ as denoting generally the function of $x$ that results from the operation of finding the fluxion of the original function $u$, whatever be the form of that function, we have

where $q$ denotes a new function of $x$, derived from $\backslash rho$, the former function, by the same kind of operation as that by which $\backslash rho$ was deduced from $u$. Direct Method.

Suppose now $q$ to denote the particular function $n(n-1)x^\{n-2\}$, then,

where $r$ denotes a function of $x$ derived from $q$, as $q$ was derived from $\backslash rho$, or $\backslash rho$ from the original function $u$. And it is evident that we may proceed in this manner as far as we please, unless it happen that in finding the series of functions $\backslash rho,\; q,\; r$, &c. we at last arrive at a result that is a constant quantity, and then the series of operations will terminate. Thus if the function was $x^4$, we should have

Here the expression for $\backslash frac\{\backslash dot\{r\}\}\{x\}$ is a constant quantity, which has no fluxion.

Hence it appears, that relatively to any function of a variable quantity, there exists a series of limiting ratios, deducible from that function, and from each other, by a repetition of the operation of finding the fluxion of a variable function.

42. In treating of the fluxion of a function, we have hitherto regarded the fluxion of the variable quantity $x$, from which the function is formed, merely as one of the terms of a ratio, without considering whether it was a constant or a variable quantity.

Now as we may assume any hypothesis respecting the nature of the fluxion of $x$, that is not inconsistent with what has been already delivered, we shall suppose it to be constant. This assumption, if we consider the fluxions of variable quantities as the measures of their respective velocities, or rates of increase, is in effect the same thing as to suppose that the variable quantity $x$ increases uniformly. Then, as in the expressions

or these others, which follow from them,

the symbol $\backslash dot\{x\}$ is to be understood as denoting a constant quantity, it follows that if $\backslash rho$ be variable, then $\backslash rho\; x$, or $\backslash dot\{u\}$ will be variable; and if $q$ be variable, then $q\; x$, or $\backslash dot\{\backslash rho\}$, will be

Direct Method. $\backslash dot\{p\}$, will be variable; and if $r$ be variable, then $r\backslash dot\{x\}$, or $\backslash dot\{q\}$ will be variable and so on.

43. Let us now recur to the relation in which the preceding functions $p,\; q,\; r$, &c. stand to the original function $u$.

By performing that particular analytical operation upon the function $u$, which consists in finding its fluxion, we obtain $\backslash dot\{u\}\; =\; \backslash dot\{p\}x$ as the expression for its fluxion, that is, we get $\backslash dot\{u\}\; =\; \backslash dot\{p\}x$; and by repeating the operation on the function $\backslash dot\{u\}$, we get $\backslash ddot\{u\}\; =\; \backslash dot\{q\}x^2$; and therefore $\backslash ddot\{u\}\; =\; \backslash dot\{q\}x^2$; but, $x$ being regarded as a constant quantity, $\backslash ddot\{u\}$ is deduced from $\backslash dot\{u\}$, considered as a function of $x$, just in the same manner as $\backslash dot\{u\}$ is derived from the original function $u$; therefore the expression $\backslash dot\{q\}x^2$ is deduced from the function $u$ by performing the operation of taking the fluxion twice; that is, first upon the function $u$ itself, and then upon $\backslash dot\{u\}$, or $\backslash dot\{p\}x$, the expression for its fluxion; and in this second operation $x$ (or the fluxion of the quantity from which the function is formed) is considered as a constant quantity.

The expression $\backslash dot\{q\}x^2$, obtained in this manner from the function $u$, is called the second fluxion of the function; and to express its relation to the function $u$, it is denoted by $\backslash ddot\{u\}$, that is, by the letter denoting the function itself with two dots over it. Thus, like as $\backslash dot\{u\}\; =\; \backslash dot\{p\}x$, we have

Again, since $\backslash dot\{q\}\; =\; r\backslash dot\{x\}$, it follows that $\backslash dot\{q\}x^2\; =\; r\backslash dot\{x\}^3$; but, as $x$ is constant, $\backslash dot\{q\}x^2$ is derived from $\backslash dot\{q\}x^2$, by the operation of finding its fluxion, considering it as a function of $x$, just in the same manner as $\backslash dot\{u\}$ is derived from $\backslash dot\{p\}x$, or $\backslash dot\{u\}$, and in the same manner as $\backslash dot\{u\}$ is derived from the original function $u$; therefore, like as $\backslash dot\{p\}x$ or $\backslash dot\{u\}$ is the first fluxion of the function, and $\backslash dot\{q\}x^2$ or $\backslash ddot\{u\}$ is its second fluxion, so $r\backslash dot\{x\}^3$ is called its third fluxion, and is denoted by $\backslash ddot\{\backslash dot\{u\}\}$, that is by the letter expressing the function itself, having three dots placed over it, so that

The fourth fluxion of a variable function $u$ is denoted by $\backslash ddot\{\backslash dot\{\backslash dot\{u\}\}\}$, that is by the letter $u$ with four dots over it, and is derived from the third fluxion, in the same manner as the third is derived from the second, or the second from the first, or the first fluxion from the variable function itself; observing, that in repeating the operation of taking the fluxions, the symbol $x$ (or the fluxion of the variable quantity from which the function is formed) is considered as a constant quantity. And the same mode of notation and deduction is to be understood as applying to a fluxion of any order whatever of a variable function.

44. To illustrate what has been said respecting the

second and higher orders of fluxions of a function, let us suppose $u$ to denote the particular function $ax^n$; then, proceeding agreeably to what has been laid down in last section, we obtain, by the rule for finding the fluxion of any power of a variable quantity (§ 26)

Here we have exhibited the first, second, third, and fourth fluxions of the function $ax^n$; the law of continuation is obvious, and it appears that when $n$ is any positive integer, the function $ax^n$ will have as many orders of fluxions, as there are units in $n$, and no more; for if $n$ were supposed $=\; 3$, then, as the fourth fluxion, and all the subsequent ones, are multiplied by $n-3$, or in that case by $3-3=0$, they consequently would vanish, and a similar observation may be made when $n$ is any other whole positive number.

45. That we might be able to apply the rules of § 26, § 34, &c. to the determination of the fluxion of a complex function of a variable quantity, we have found it convenient in some cases to consider such a function as composed of other more simple functions of the same quantity, and we have expressed its fluxion by means of the fluxions of those other functions. In finding the fluxion of any higher order than the first of such a complex function by those rules, we must keep in mind, that it is only the fluxion of $x$, the variable quantity from which the functions are all formed, that is to be considered as constant, and that the fluxions of the functions themselves are in general variable quantities; so that each of them may have a second, third, &c. fluxion, as well as the function which is composed of them.

Let us suppose for example, that

then, considering $a^2\; +\; x^2$ as a function of $x$, and putting $v$ to denote it, we have $a\; =\; \backslash sqrt\{v\}\; =\; v^\{\backslash frac\{1\}\{2\}\}$, and $\backslash dot\{u\}\; =\; \backslash frac\{1\}\{2\}v^\{-\backslash frac\{1\}\{2\}\}\backslash dot\{v\}$ $=\; \backslash frac\{1\}\{2\backslash sqrt\{v\}\}\backslash dot\{v\}$; but since $v\; =\; a^2\; +\; x^2$, it follows that $\backslash dot\{v\}\; =\; 2x\backslash dot\{x\}$;

therefore, substituting for $v$ and $\backslash dot\{v\}$ their respective values, we have

Now, to find the second fluxion of $u$, we may either take the fluxion of this last expression, viz. $\backslash frac\{x\backslash dot\{x\}\}\{\backslash sqrt\{(a^2\; +\; x^2)\}\}$, and consider the symbol $\backslash dot\{x\}$, which is found in it, as denoting a constant quantity; or we may recur to the equation $\backslash dot\{u\}\; =\; \backslash frac\{1\}\{2\backslash sqrt\{v\}\}\backslash dot\{v\}$, and take the fluxion of this other expression for $u$; and in this case, we must consider that both $v$ and $\backslash dot\{v\}$ denote variable functions of $x$, and therefore that the fluxion of $\backslash frac\{1\}\{2\backslash sqrt\{v\}\}\backslash dot\{v\}$ may be found by the rule

for

for finding the fluxion of a function; observing that $\backslash dot\{v\}$ is to be substituted as the fluxion of $v$. Accordingly, proceeding by this last method, and considering that the fluxion of $\backslash sqrt\{v\}$ the denominator of the fraction is $\backslash frac\{1\}\{2\}\; \backslash frac\{\backslash dot\{v\}\}\{\backslash sqrt\{v\}\}$, we find

Now from the equation $v\; =\; a^2\; +\; x^2$, we have $\backslash dot\{v\}\; =\; 2x\; \backslash dot\{x\}$, and (observing that $\backslash dot\{x\}$ is constant) $\backslash ddot\{v\}\; =\; 2\backslash dot\{x\}^2$. Let these values of $v$, $\backslash dot\{v\}$, and $\backslash ddot\{v\}$, be now substituted in the expression for $\backslash dot\{u\}$, and it becomes

The very same expression for $\backslash dot\{u\}$ would have been found if we had employed the other method.

By proceeding as in this last example, the rules already delivered for finding the first fluxion of any function of a variable quantity will apply to the finding of the fluxion of any higher order.

Thus if we had $u\; =\; v\; t$, where $v$ and $t$ denote each a function of another variable quantity $x$, and it were required to find the different orders of fluxions of $u$, considered also as a function of $x$; then, by the rule of § 37, we have

but $v$ and $t$ being variable functions of $x$, we may consider $\backslash dot\{v\}$ and $\backslash dot\{t\}$ as denoting also variable functions of $x$, the fluxions of which are to be denoted by $\backslash ddot\{v\}$ and $\backslash ddot\{t\}$ respectively; now by the rule in § 37, we have

By considering $v$, $\backslash dot\{v\}$, $\backslash ddot\{v\}$, also $t$, $\backslash dot\{t\}$, $\backslash ddot\{t\}$, as denoting each a distinct function of $x$, we may find the third fluxion of $u$ from the second, in the same manner as the second has been found from the first, and so on for the other orders of fluxions of $u$. If it be now required to express the successive orders of fluxions of $u$ in terms of $x$ and its fluxion, we must find the values of $v$, $\backslash dot\{v\}$, $\backslash ddot\{v\}$, also of $t$, $\backslash dot\{t\}$, $\backslash ddot\{t\}$, in terms of $x$ and its fluxion, and these values, also the particular functions of $x$ denoted by $v$ and $t$, being substituted in the expressions found for $u$, $\backslash dot\{u\}$, $\backslash ddot\{u\}$, &c. will give to these expressions the form required.

If for example we suppose that

VOL. VIII. Part II.

and, considering $\backslash dot\{x\}$ as constant,

these values of $v$, $t$, $\backslash dot\{v\}$, $\backslash dot\{t\}$, &c. being substituted in the expressions of $\backslash dot\{u\}$, $\backslash ddot\{u\}$, &c. will give the successive fluxions of $u$ in terms of $x$ and $\backslash dot\{x\}$ only.

46. If the fluxion of a variable quantity be considered as the measure of its rate of increase, if that rate be uniform, then its measure will be a constant quantity; but if it be variable, then its measure will be a variable quantity, which will also have a certain rate of increase or decrease; and the measure of this rate will be its fluxion, or will be the fluxion of the fluxion of the original variable quantity; that is, it will be the second fluxion of the original variable quantity. And if this second fluxion is not a constant quantity, then the measure of its rate of variation will be its fluxion, or will be the third fluxion of the original variable quantity, and so on. Thus a quantity will have a successive order of fluxions till some one fluxion become constant, and then it will have no more.

47. We have hitherto supposed the equation expressing the relation between a variable quantity, and a function of that quantity, to be of such a form, that the function was found alone, and of the first degree on one side of the equation, and some power, or combination of powers, of the variable quantity on the other; as in these examples,

In such cases as these, $u$ is said to be an explicit function of $x$. We are now to consider how the ratio of the fluxions is to be found when the relation between the variable quantity and its function is expressed by an equation, the terms of which involve different powers, both of the function, and the variable quantity; as in the following example,

where we are to consider $y$ as a function of $x$; but from the particular manner in which its relation to $x$ is expressed, it is said to be an implicit function of that quantity.

Now in this example, by the resolution of a quadratic equation, we find

and as $y$ is here an explicit function of $x$, its fluxion, or the ratio of its fluxion to that of $x$, might be determined by the rules already laid down. But it is to be observed that it is only in the particular case of the proposed equation being of the second degree that we can effect the solution generally in this manner. If it

were of a higher order, this particular mode of solution would be often impracticable, for want of a general method of resolving equations.

48. We may however in all cases resolve the problem, without a previous resolution of the equation, by reasoning as follows.

Whatever be the degree of the equation, by giving particular values to $x$, we can, by the theory of equations, obtain corresponding particular values of $y$; therefore, we may be assured that in every case $y$ is expressible by means of $x$ in some way or other, if not in finite terms, at least in the form of a series, the terms of which shall involve powers of $x$. Hence we may infer, as in the case of explicit functions, that when $x$ changes its value, and becomes $x+h$, $y$ will also change its value, and become

where denote functions of $x$, that are independent of the arbitrary quantity $h$. Let us denote the increment of $y$, by the single letter $k$; then $y+k$ is the new value of $y$, corresponding to $x+h$, the new value of $x$. Let these new values be substituted instead of $x$ and $y$ in the proposed equation

and as the result must still be $=0$, we have

which equation, by actually involving its terms, substituting for $k$ its value and arranging the result in the form of a series proceeding by the powers of $h$, becomes

Here denote quantities independent of $h$, and involving that is to say, $x$, and functions of $x$, and therefore are also functions of $x$. Now as this equation must subsist whatever $h$ may be, which is a quantity quite arbitrary and independent of the coefficients by which its powers are multiplied; it follows (as has been observed when treating of the method of indeterminate coefficients, ALGEBRA, § 261.) that the coefficients of the different powers of $h$ must be each equal to 0.

Therefore,

From the first of these equations we can infer nothing, as it is no other than the proposed equation itself; but from the second we find

Now $h$, and being the simultaneous increments of $x$ and $y$, we have therefore, supposing $h$ to be continually diminished, and

putting $\backslash frac\{\backslash dot\{y\}\}\{x\}$ equal to the limit of $\backslash frac\{k\}\{h\}$, we have $\backslash frac\{\backslash dot\{y\}\}\{x\}=p$, therefore

thus we have obtained an expression for the ratio of the fluxions of $y$ and $x$, from which we find

and this is precisely the expression we should have obtained, had we taken the fluxion of each term of

the proposed equation, and put the result equal to 0.

49. But to see that this will always be the case, whatever be the degree of the equation, we have only to observe, that, by the very same process employed to deduce from the original equation

these two others

if we suppose the equation to be generally expressed thus,

where the exponents denote constant quantities, we shall obtain

and hence, by substituting for $p$ its value $\backslash frac\{\backslash dot\{y\}\}\{x\}$, and bringing $\backslash dot\{x\}$ from the denominator,

From which it appears that, when the relation between $x$, a variable quantity, and $y$, a function of that quantity, is expressed by an equation, the terms of which are brought all to one side, so as to produce an expression $=0$; the relation of the fluxions will be found, by taking the fluxion of each term of the equation (considering $y$ as a function of $x$), and putting the sum of these fluxions equal to 0.

50. Having from the equation

found that

if it be required to find the second fluxion of $y$, we have only to take the fluxion of the latter side of this equation,

Direct Method. equation, considering $x$ as constant, and $y$ as a function of $x$; thus we have

an equation which abbreviates to

and from which we may exterminate $\backslash dot\{y\}$ by means of the equation

By the same mode of proceeding we may determine the third, or any higher fluxion of the function $y$.

51. As far as we have yet gone in explaining the principles of fluxions, we have had continually occasion to employ the rule for finding the fluxion of the particular function $x^n$, where $x$ denotes a variable quantity, and $n$ any constant number; and we may therefore, in respect of other functions consider $x^n$ as a simple function. Besides the function $x^n$, writers on Analysis have considered each of the following as also constituting a simple analytic function of a variable quantity; viz.

$a^x$, where $a$ is constant, and $x$ is variable.

Log. $x$, that is the logarithm of $x$, a variable number.

Sin. $x$, that is the sine of $x$, a variable arch of a circle, radius being unity.

Cos. $x$, that is the cosine of $x$, a variable arch of a circle, radius being as before unity.

52. We have already found the fluxion of $x^n$, and we proceed to find the fluxions of the other simple functions of $x$; and, as in the investigation of these we shall have occasion to employ the binomial theorem, it will be proper to show how that theorem may be deduced from the principles already explained. We are then to find

the series that expresses $(a+x)^n$, when $n$ is any number whatever. Or, since $(a+x)^n$ is equal to $a^n(1+v)^n$, where $v$ denotes the fraction $\backslash frac\{x\}\{a\}$, we may leave the quantity $a^n$ out of consideration, as has been formerly ob-

served, § 28, and seek the series that expresses $(1+v)^n$. As we have already pointed out (§ 28) the process of induction by which we may find the general form of the series, we shall not here repeat it, but assume

where denote numbers that are independent of $v$.

Now, as the fluxion of a variable function must be the same, whether that function be expressed by one term, or developed into a series of terms; by performing the operation of taking the fluxion on each side of the above equation, the results must be equal, that is, § 26.

or, leaving out the quantity $\backslash dot\{v\}$, common to each term,

Let both sides of this equation be multiplied by $1+v$, and divided by $n$, thus we shall have

Thus, by performing on the quantities the analytical process of taking their fluxions, we have obtained a new expression for $(1+v)^n$. Let the quantities that are independent of $v$ in each expression be put equal to each other, and also the co-efficients of like powers of $v$; thus we obtain

Or, substituting successively the expression for each coefficient in that which follows it,

Hence it appears that

and therefore, substituting $\backslash frac\{x\}\{a\}$ for $v$, and multiplying by $a^n$,

where the law of continuation is evident.

53. We now proceed to investigate the fluxion of the function $u\; =\; a^x$, $a$ being supposed constant, and $x$ the variable quantity, to which the function is referred.

Let

Let $x$ be supposed, as formerly, to change its value, and to become $x+h$, and put $u\text{'}$ for the new value that the function acquires by this change in the magnitude of $x$, then we have

and, taking the difference between the two succeeding values,

We must now develop the expression $a^h\; -\; 1$ into a series, the terms of which are arranged according to the successive powers of the increment $h$. To effect this, let us put $b\; =\; a\; -\; 1$, so that $a\; =\; 1\; +\; b$, and $a^h\; =\; (1\; +\; b)^h$; but by the binomial theorem, this last expression may be expanded into the following series:

Therefore,

As the terms of this series are not arranged according to the powers of $h$, but according to the powers of $b$, it is necessary that we transform it into another having the required form; now this may be effected by actually multiplying together all the factors that constitute each term, and arranging the series anew in such a manner, that each of its terms may be a power of $h$, multiplied by a coefficient composed only of the powers of $b$, and given numbers;

Accordingly we have

Therefore, by taking the sum of all the quantities on each side of these equations, we get the series,

otherwise expressed thus,

where $A$ is equal to the infinite series

that is, to

and are also quantities composed of the powers of $b$, and consequently are independent of $h$; but as these are all to disappear in the course of the investigation, it is not necessary to express them otherwise.

than by a general symbol. Therefore, we have now got

and consequently,

and

Hence, when $h$ is conceived to be continually diminished, we have the limit of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$, expressed by $Aa^x$, and therefore § 21,

54. In the preceding investigation, we have had occasion to develop the exponential expression $a^h$ into a series of this form,

that is, a series the terms of which are the successive powers of the exponent, each multiplied by a coefficient, which is independent of the exponent.

We have however only determined the coefficients of the first two terms of the series, these being the only ones we had occasion to employ.

The result of the investigation however may be applied to determine all the coefficients by the very same kind of process as that which we have employed in § 53, to determine the coefficients of the terms of the series which constitutes the other expansion of $a^h$.

Instead of denoting the exponent by $h$, let us consider it as a variable quantity, and express it by $x$, then, from what has been shown it appears that

where express constant quantities. Let the operation of taking the fluxions be now performed on both sides of this equation, (observing that the fluxion of $a^x$ is $Aa^x\; \backslash dot\{x\}$) and let all the terms be divided by $\backslash dot\{x\}$, which is common to each, thus we obtain

and, dividing by $A$,

Let the coefficients of the same powers of $x$ in each of the two series expressing $a^x$ be put equal to each other, then,

Therefore,

Therefore, substituting these values of B, C, D, &c. in the original series, we have

the same result as has been found in the article ALGEBRA, § 293. by proceeding in a different manner.

55. If we suppose $x=1$, then the preceding equation becomes

and if we suppose $x\; =\; \backslash frac\{1\}\{A\}$, it becomes

thus the quantity $a^\{\backslash frac\{1\}\{A\}\}$ is equal to a constant number which is the value of $a$ when $A=1$, and which, by taking the sum of the first ten terms of the series, is found to be 2.7182818, or, by taking the sum of a greater number of terms, more accurately

We shall, in the remainder of this treatise, denote this

number always by $e$, then $a^\{\backslash frac\{1\}\{A\}\}\; =\; e$, and $a\; =\; e^A$, and taking the logarithms, $\backslash log.\; a\; =\; A\; \backslash times\; \backslash log.\; e$, hence

56. If we now substitute this value of $A$ in the expression for the fluxion of $u\; =\; a^x$, found in § 53, it becomes

Hence it appears, that the fluxion of the function $a^x$ is equal to the fluxion of $x$ multiplied by the function itself, and by the quotient arising from the division of the logarithm of $a$ by the logarithm of $e$, where $e$ denotes $1\; +\; \backslash frac\{1\}\{1\; \backslash cdot\; 2\}\; +\; \backslash frac\{1\}\{1\; \backslash cdot\; 2\; \backslash cdot\; 3\}\; +\; \backslash frac\{1\}\{1\; \backslash cdot\; 2\; \backslash cdot\; 3\; \backslash cdot\; 4\}\; +\; \backslash dots$ a series whose sum is 2.7182818 nearly.

57. Let us now consider the third simple function of $x$, namely $u\; =\; \backslash log.\; x$. Let $a$ be the radical number of the particular system, in which $u$ is a logarithm, and $x$ the corresponding number; then from the nature of logarithms (see ALGEBRA, § 277.) we have $a^x\; =\; x$. Now, whether we consider $u$ as a function of $x$, or $x$ as a function of $u$, the limiting ratio of their simultaneous increments, and consequently the ratio of their fluxions will be the very same. But by considering $x$ as a function of $u$, we have immediately, from what has been shown in § 53, and § 55,

and therefore, $\backslash dot\{u\}\; =\; \backslash frac\{1\}\{A\}\; \backslash frac\{\backslash dot\{x\}\}\{x\}\; =\; \backslash frac\{\backslash log.\; e\}\{\backslash log.\; a\}\; \backslash frac\{\backslash dot\{x\}\}\{x\}$, but as $a$ is the radical number of the system, $\backslash log.\; a\; =\; 1$, therefore

The number which we have denoted by $e$ occurs very often in analytical investigations; it is the radical number of the system of logarithms first invented by Baron Napier, and called by some writers Hyperbolic logarithms, but by others, with more propriety, Napierian logarithms.

The expression $\backslash frac\{\backslash log.\; e\}\{\backslash log.\; a\}$ is called the modulus of the system of logarithms whose radical number is $a$. In the Napierian system $\backslash frac\{\backslash log.\; e\}\{\backslash log.\; a\}\; =\; \backslash frac\{\backslash log.\; e\}\{\backslash log.\; e\}\; =\; 1$, that is, the modulus is unity; but in the common system, or that in which $a=10$, the modulus $\backslash frac\{\backslash log.\; e\}\{\backslash log.\; a\}\; =\; 0.434294482$. The rule for finding the fluxion of the logarithm of a variable quantity may now be expressed thus:

Multiply the fluxion of the variable quantity by the modulus of the system, and divide the product by the variable quantity itself, the result is the fluxion required.

58. By the application of the rule for finding the fluxion of the logarithm of a variable quantity we may readily find the fluxions of exponential functions in general. Thus, for example, if $u\; =\; x^y$, $x$ and $y$ being both functions of any variable quantity $z$, then $\backslash log.\; u\; =\; y\; \backslash times\; \backslash log.\; x$; and taking the fluxions (considering $y\; \backslash times\; \backslash log.\; x$ as the product of two functions $y$ and $\backslash log.\; x$, and proceeding by the rules of § 37 and last),

and hence

59. We are next to consider the functions $u\; =\; \backslash sin.\; x$; and $u\; =\; \backslash cos.\; x$.

Suppose $x$ to change its value, and to become $x+h$, and $u$ to become $u\text{'}$, then, since

but by the arithmetic of sines (see ALGEBRA, § 353), $\backslash sin.\; (x+h)\; =\; \backslash sin.\; x\; \backslash cos.\; h\; +\; \backslash cos.\; x\; \backslash sin.\; h$, therefore,

In this case, as when treating formerly of other functions, we might consider the above expression for $u\text{'}\; -\; u$, as resolvable into a series $p\; h\; +\; q\; h^2\; +\; \backslash dots$ proceeding by the powers of the increment, and thence we might

might find the limit of $\backslash frac\{u\text{'}-u\}\{h\}$ as before. But we may discover the limit otherwise, by proceeding as follows; Because

Let this value of $1\; -\; \backslash cos\; h$ be substituted in the expression for $u\text{'}\; -\; u$, and it becomes

And hence, dividing by $h$, and arranging the terms so as to exhibit the ratio $\backslash frac\{\backslash sin\; h\}\{h\}$, we get

Conceive now $h$ to be continually diminished, and we shall have the limit of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ equal to the limit of $\backslash frac\{\backslash sin\; h\}\{h\}$ multiplied by the limit of the following expression

Now, the sine of an arch being less than the arch itself, we have . Again, the arch being less than its tangent, $\backslash frac\{\backslash sin\; h\}\{h\}\; >\; \backslash frac\{\backslash sin\; h\}\{\backslash tan\; h\}$; but $\backslash tan\; h\; =\; \backslash frac\{\backslash sin\; h\}\{\backslash cos\; h\}$, and therefore $\backslash frac\{\backslash sin\; h\}\{\backslash tan\; h\}\; =\; \backslash cos\; h$; consequently $\backslash frac\{\backslash sin\; h\}\{h\}\; >\; \backslash cos\; h$. Hence it appears, that the expression for the ratio $\backslash frac\{\backslash sin\; h\}\{h\}$ is less than 1, or radius, but greater than $\backslash cos\; h$. But $h$ being conceived to be continually diminished, $\backslash cos\; h$ continually approaches to 1, and may come nearer to it than by any assignable difference; therefore, the limit of $\backslash frac\{\backslash sin\; h\}\{h\}$ is 1. As to the other expression, $\backslash cos\; x\; \backslash frac\{\backslash sin\; h\; \backslash sin\; x\}\{1\; +\; \backslash cos\; h\}$; when $h$ is supposed to be continually diminished, its second term, to wit, $\backslash frac\{\backslash sin\; h\; \backslash sin\; x\}\{1\; +\; \backslash cos\; h\}$ may become less than any assignable quantity; therefore the limit of the expression is simply $\backslash cos\; x$: thus, upon the whole, we have found that the limit of $\backslash frac\{u\text{'}\; -\; u\}\{h\}$ is $\backslash cos\; x$, and therefore

The fluxion of the other function, $u\; =\; \backslash cos\; x$, is easily deduced from that which we have just found, by proceeding thus:

Put $c$ to denote a quadrant, then $\backslash cos\; x\; =\; \backslash sin(c\; -\; x)$, and therefore

Now, it has been just shewn that

but $\backslash cos(c\; -\; x)\; =\; \backslash sin\; x$, and the fluxion of $c\; -\; x$ is $-\backslash dot\{x\}$, therefore

Thus it appears, that the fluxion of the sine of a variable arch is equal to the fluxion of the arch multiplied by its cosine; and that the fluxion of the cosine is equal to the fluxion of the arch (taken with a negative sign) multiplied by the sine.

60. We can now very readily find the fluxion of any other function of an arch of a circle. Thus, suppose $u\; =\; \backslash tan\; x$; then, because $\backslash tan\; x\; =\; \backslash frac\{\backslash sin\; x\}\{\backslash cos\; x\}$, we have $u\; =\; \backslash frac\{\backslash sin\; x\}\{\backslash cos\; x\}$. This expression being considered as a fractional function of $x$, we have, by § 39, and what has been just now shewn,

or, since $\backslash cos^2\; x\; +\; \backslash sin^2\; x\; =\; 1$, and $\backslash frac\{1\}\{\backslash cos^2\; x\}\; =\; \backslash sec^2\; x$,

Hence also we have $\backslash dot\{x\}\; =\; \backslash frac\{\backslash dot\{u\}\}\{\backslash sec^2\; x\}\; =\; \backslash frac\{\backslash dot\{u\}\}\{1\; +\; u^2\}$.

In like manner, if we suppose $u\; =\; \backslash sec\; x$, then, because $\backslash sec\; x\; =\; \backslash frac\{1\}\{\backslash cos\; x\}$, we have $u\; =\; \backslash frac\{1\}\{\backslash cos\; x\}$, and

or, since $\backslash frac\{\backslash sin\; x\}\{\backslash cos\; x\}\; =\; \backslash tan\; x$, and $\backslash frac\{1\}\{\backslash cos^2\; x\}\; =\; \backslash sec^2\; x$,

Proceeding in this manner, we find that when $u\; =\; \backslash cotan\; x$, then

And when $u\; =\; \backslash text\{cosec\; \}\; x$, then

61. Let us now consider the fluxions of geometrical plane magnitudes: And first let it be required to find the COXIX expression for the fluxion of BDPC the area bounded fig. 1. by CP, a curve line, and by CB, PD, the ordinates at its extremities, and BD, the portion of AE, the line of the abscissas, which lies between those ordinates. Let the numerical measures of AD and PD, the co-ordinates at the point P, be denoted by $x$ and $y$, and the numerical measure of the area BDPC by $s$; then

Direct Method. then $y$ and $s$ may both be considered as functions of the abscissa $x$.

Let $x$, or $AD$, be supposed to change its value, and to become $AD\text{'}$, and let $DP\text{'}$, and $BD\text{'}P\text{'}C$ be the corresponding new values of $y$ and $s$; then $DD\text{'}$, and $DD\text{'}P\text{'}P$ will be the geometrical expressions for the simultaneous increments of the abscissa and area. But, as one of these quantities is a line, and the other a space, they cannot be compared in respect of their ratio. Therefore, let us consider $a$ as denoting a line whose numerical value is unity, and then the numerical values of the increments of the abscissa and area may be considered as analogous to the geometrical quantities $DD\text{'}\; \backslash times\; a$, and the area $DD\text{'}P\text{'}P$ respectively, which quantities being homogeneous may now be compared with each other. We are now to investigate the limit

of $\backslash frac\{\backslash text\{area\; \}\; DD\text{'}P\text{'}P\}\{a\; \backslash times\; DD\text{'}\}$, the general expression for the ratio of the increments of $s$ and $x$. Draw $PM$ and $PN$ parallel to $AE$, meeting the ordinates in $M$ and $N$. The curvilinear area $DD\text{'}P\text{'}P$ is greater than the rectangle $DD\text{'}MP$, that is greater than $PD\; \backslash times\; DD\text{'}$; but less than the rectangle $DD\text{'}PN$, that is, less than $PD\text{'}\; \backslash times\; DD\text{'}$, therefore

But the increments being supposed to be continually diminished, $\backslash frac\{PD\}\{a\}$ is the limit of $\backslash frac\{PD\text{'}\}\{a\}$, therefore $\backslash frac\{PD\}\{a\}$ is also the limit of $\backslash frac\{\backslash text\{area\; \}\; DD\text{'}P\text{'}P\}\{a\; \backslash times\; DD\text{'}\}$, and hence (§ 21.)

That is, the fluxion of a curvilinear area is equal to the product of the ordinate, and the fluxion of the abscissa.

62. Before we proceed to investigate the expression for the fluxion of an arch of a curve, it is necessary that we should inquire what is the limiting ratio of an arch of a curve to its chord.

Fig. 2. Let $APB$ be any curve line, all the parts of which are concave towards its chord $AP$. Let $AQ,\; QP$ be tangents at the extremities of the arch, and let $a\; \backslash rho\; q$ be a triangle similar to $APQ$, but having its base $a\; \backslash rho$ of a given magnitude, then

Suppose now the point $P$ to approach to $A$, then the angles at $A$ and $P$, and consequently the angles at $a$ and $\backslash rho$, which are equal to them, will decrease, and may become less than any assignable angles; therefore, the limit of the ratio of $a\; \backslash rho\; +\; q\; \backslash rho$ to $a\; \backslash rho$ is evidently a ratio of equality; hence also the limit of the ratio of $AQ\; +\; QP$ to $AP$ is the ratio of equality; and since the arch $AP$ is less than $AQ\; +\; QP$, but greater than its chord $AP$, the limit of the arch $AP$ to its chord $AP$ must also be the ratio of equality.

63. We proceed now to find the fluxion of an arch of a curve. Let $APP\text{'}$ be a curve line of any kind, and $AB,\; BP$ any two co-ordinates at a point $P$ in the curve. Put $x$ for $AB$, the abscissa, $y$ for $BP$, the ordinate, and $s$ for the curve line $AP$, then $s$ and $y$ may be considered as each a function of $x$. Draw $P\text{'}B\text{'}$ another ordinate, and draw $PM$ parallel to $AB$, meeting $P\text{'}B\text{'}$ in $M$, and draw the chord $PP\text{'}$; then $PM,\; MP\text{'}$ and the arch $PP\text{'}$ are the simultaneous increments of $x,\; y$, and $s$ respectively. Now we have

But $\backslash text\{chord\; \}\; PP\text{'}\; =\; \backslash sqrt\{(PM^2\; +\; MP\text{'}^2)\}\; =\; PM\; \backslash sqrt\{1\; +\; \backslash frac\{MP\text{'}^2\}\{PM^2\}\}$;

therefore,

Suppose now the increments to be continually diminished,

then, as $\backslash frac\{\backslash dot\{s\}\}\{x\}\; =\; \backslash text\{limit\; of\; \}\; \backslash frac\{\backslash text\{arch\; \}\; PP\text{'}\}\{PM\}$, and $\backslash frac\{\backslash dot\{y\}^2\}\{x^2\}\; =\; \backslash text\{limit\; of\; \}\; \backslash frac\{MP\text{'}^2\}\{PM^2\}$ (§ 21.), and $1\; =\; \backslash text\{limit\; of\; \}\; \backslash frac\{\backslash text\{arch\; \}\; PP\text{'}\}\{\backslash text\{chord\; \}\; PP\text{'}\}$ (last §) we have

Hence it appears that the square of the fluxion of a curve line of any kind is equal to the sum of the squares of the fluxions of the co-ordinates.

64. The expression for the fluxion of a solid may be found by the same mode of reasoning as that which we have employed, § 61, to find the fluxion of a curvilinear area. Let $APQ\backslash rho$ be a portion of a solid generated by the revolution of $APB$, a curve line, about $AC$, a line taken in the plane of the curve, as an axis. Let $PD\backslash rho,\; PD\text{'}\backslash rho\text{'}$ be the lines in which $BA,\; b$, a plane passing along the axis $AC$, meets $PQ\backslash rho,\; PQ\text{'}\backslash rho\text{'}$, the planes of two circles formed by sections of the solid perpendicular to its axis. Draw $PM$ and $PN$ parallel to $AD$. Put $AD\; =\; x,\; DP\; =\; y$, let $s$ denote the solid $APQ\backslash rho$, having $y$ for the radius of its circular base, and $x$ for its altitude; put $\backslash pi$ for the number $3.14159\backslash dots$ viz. the circumference of a circle having its diameter $=\; 1$, and let $a$ denote an area, having its numerical measure expressed by unity; then, $DD\text{'}$, or $a\; \backslash times\; DD\text{'}$ being considered as the increment of $x$, the portion of the solid comprehended between the parallel planes $PQ\backslash rho,\; PQ\text{'}\backslash rho\text{'}$ will be the corresponding increment of $s$, which we are to consider as a function of $x$; hence (§ 21.) $\backslash frac\{\backslash dot\{s\}\}\{x\}$ is equal to the limiting ratio of the portion of the solid, comprehended between the planes $PQ\backslash rho$, and $PQ\text{'}\backslash rho\text{'}$ to the solid $a\; \backslash times\; DD\text{'}$. But the former of these solids being evidently greater than a cylinder $Pm$, having the circle $PQ\backslash rho$ for its base, and $DD\text{'}$ for its altitude, that is greater than $\backslash pi\; PD\text{'}\; \backslash times\; DD\text{'}$, and less than a cylinder $N\backslash rho\text{'}$, having the circle $PQ\text{'}\backslash rho\text{'}$ for its base, and $DD\text{'}$ for its altitude, that is less than $\backslash pi\; PD\; \backslash times\; DD\text{'}$;

but the increment $DD\text{'}$ being continually diminished, $\backslash frac\{\backslash pi\; P\text{'}D\text{'}\}\{a\}$, the greater limit of $\backslash frac\{\backslash dot\{s\}\}\{x\}$, approaches continually

to its lesser limit $\backslash frac\{\backslash pi\; PD\text{'}\}\{a\}\; =\; \backslash frac\{\backslash pi\; y^2\}\{a\}$ (because $a=1$) $\backslash pi\; y^2$, so as to come nearer to it than by any assignable difference, therefore $\backslash frac\{\backslash dot\{s\}\}\{x\}\; =\; \backslash pi\; y^2$, and $\backslash dot\{s\}\; =\; \backslash pi\; y^2\; \backslash dot\{x\}$. Now, if we

observe that $\backslash pi\; y^2$ is the area of the circle $PQp$, it will appear, that the fluxion of a solid generated by the revolution of a curve about its axis is equal to the fluxion of the axis multiplied by the general expression for the area of a circle formed by supposing the solid to be cut by a plane perpendicular to its axis.

65. To find the fluxion of the surface of the solid, let us denote that surface by $s$, and let $x$ and $y$ denote as before; then the surface contained between the circles $PQp$ and $P\text{'}Q\text{'}p\text{'}$ will be the increment of $s$, corresponding to $DD\text{'}$ the increment of $x$. Draw the chord $PP\text{'}$; then, the curve line $PP\text{'}$ being supposed to revolve about the axis $AC$, and thus to generate the increment of the surface of the solid, the chord $PP\text{'}$ will generate at the same time the convex surface of a frustum of a cone; now the limiting ratio of the curve line $PP\text{'}$ to its chord $PP\text{'}$ being the ratio of equality, the limiting ratio of the surfaces generated by the revolution of those lines will also be the ratio of equality;

therefore $\backslash frac\{\backslash dot\{s\}\}\{x\}$, which is equal to the limit of

will also be equal to the limit of

but the convex surface of a frustum of a cone is equal to the product of its flant side into half the sum of the circumferences of its two bases (see GEOMETRY), and in the present case these circumferences are equal to $2\; PD$

$\backslash times\; \backslash pi$, and $2\; P\text{'}D\text{'}\; \backslash times\; \backslash pi$, therefore $\backslash frac\{\backslash dot\{s\}\}\{x\}$ is equal to the limit of

but the point $D\text{'}$ being supposed to approach to $D$, the

limit of $PD\; +\; P\text{'}D\text{'}$, will manifestly be $2\; PD\; =\; 2y$; and since $\backslash frac\{PP\text{'}\}\{DD\text{'}\}\; =\; \backslash frac\{\backslash sqrt\{(DD\text{'}^2\; +\; PM^2)\}\}\{DD\text{'}\}\; =\; \backslash sqrt\{1\; +\; \backslash frac\{PM^2\}\{DD\text{'}^2\}\}$, the limit of this expression (if we consider that $PM$ and $DD\text{'}$ are the simultaneous increments of $y$ and $x$) is evidently equal to $\backslash sqrt\{1\; +\; \backslash frac\{y^2\}\{x^2\}\}$, therefore

and consequently

If we now observe that $2\backslash pi\; y$ is the circumference of the circle $PQp$, and $\backslash sqrt\{x^2\; +\; y^2\}$ is the fluxion of the curve line $AP$, § 63, it will appear, that the fluxion of the surface of a solid generated by the revolution of a curve about its axis is equal to the fluxion of the curve line multiplied by the general expression for the circumference of a circle formed by supposing the curve to be cut by a plane perpendicular to its axis.

SECT. III. The Application of the Direct Method of Fluxions.

HAVING explained the principles of the direct method of fluxions at as great a length as we think suitable to the work of which this treatise forms a part, we proceed to shew how the calculus may be applied to the resolution of some general problems in Analysis and Geometry.

Investigation of a general formula for expanding a Function into a Series.

66. In treating of the principles of the method of fluxions, we have from an examination of particular functions, inferred by induction, that $u$ being any function of a variable quantity $x$, which was either actually expressed, or capable of being expressed by a combination of the powers of $x$, then, $x$ being supposed to change its value, and to become $x+h$, the new value which the function $u$ will acquire when $x+h$ is substituted in it instead of $x$ will always be capable of being expanded into a series of this form,

where $p,\; q,\; \backslash dots$ denote functions of $x$ that are quite independent of $h$.

We have shewn that, from the particular form of this development, it happens that the ratio of $p\; h\; +\; q\; h^2\; +\; r\; h^3\; +\; \backslash dots$ the increment of the function, to $h$ the increment of the variable quantity $x$ itself, admits of a limit, which is always expressed by $p$, the coefficient of its second term; and as we have defined this limit to be the expression for the ratio of the fluxions of $u$ and $x$, so that $p\; =\; \backslash frac\{\backslash dot\{u\}\}\{\backslash dot\{x\}\}$, the new value of the function may also be expressed thus

And this expression may be considered as indicating not only the general form of the series, but also the particular relation subsisting between $u$, the original function, and $p$, the coefficient of the second term of the series, the latter being in every case that function of $x$ which results from the operation of taking the fluxion of the former, and dividing by $\backslash dot\{x\}$.

We are now to investigate the relation that subsists between each of the remaining coefficients and the original function.

67. First let us suppose the function $u$ to have the particular form $x^n$, $n$ being a constant number. Then $x$ changing its value to $x+h$, $u$ changes to $u\text{'}=(x+h)^n$, therefore, by the binomial theorem (§ 52.)

But since $u=x^n$, by taking the successive fluxions of $u$, and considering $x$ as constant, we have,

&c.

Let $\backslash frac\{\backslash dot\{u\}\}\{\backslash dot\{x\}\},\; \backslash frac\{\backslash ddot\{u\}\}\{\backslash dot\{x\}^2\},\; \backslash dots$ be now substituted for $x^n,\; n\; x^\{n-1\},\; n(n-1)x^\{n-2\},\; \backslash dots$ respectively, in the series for $u\text{'}$, and we have

68. This manner of expressing the development of $u\text{'}$, or $(x+h)^n$, indicates directly the relation that each of the coefficients of the successive powers of $h$ has to the original function.

The first term of the series is the original function $u$, or $x^n$, itself, or it is what the function $(x+h)^n$ becomes upon the supposition that $h=0$. The second term is $h$, or $\backslash frac\{h\}\{1\}$, multiplied by the coefficient $\backslash frac\{\backslash dot\{u\}\}\{\backslash dot\{x\}\}$, which coefficient is a function of $x$ derived from the original function by the operation of taking its fluxion, and dividing the result by $\backslash dot\{x\}$. The third term is $\backslash frac\{h^2\}\{1\; \backslash cdot\; 2\}$ multiplied by the coefficient $\backslash frac\{\backslash ddot\{u\}\}\{\backslash dot\{x\}^2\}$, that is, by a function of $x$ derived from the preceding coefficient $\backslash frac\{\backslash dot\{u\}\}\{\backslash dot\{x\}\}$ by the same

operation as that coefficient was derived from the original function, namely by taking the fluxion of $\backslash frac\{\backslash dot\{u\}\}\{\backslash dot\{x\}\}$, considering $x$ as constant, and dividing by $\backslash dot\{x\}$. The fourth term is $\backslash frac\{h^3\}\{1\; \backslash cdot\; 2\; \backslash cdot\; 3\}$ multiplied by $\backslash frac\{\backslash ddot\{u\}\}\{\backslash dot\{x\}^3\}$, that is, by a function of $x$ deduced from the third coefficient by the very same operation as that by which the third was derived from the second, or the second from the first. And so on with respect to all the other terms of the series, the $n$th term being the product of $\backslash frac\{h^\{n-1\}\}\{1\; \backslash cdot\; 2\; \backslash cdot\; 3\; \backslash cdot\; \backslash dots\; \backslash cdot\; (n-1)\}$, and the $(n-1)$th fluxion of the function $u$ divided by $\backslash dot\{x\}^\{n-1\}$.

69. Let us now suppose that $u$ denotes any other function of $x$, then, whatever be its nature, it may always be conceived as capable of being expressed by a series, the terms of which are powers of $x$, in this manner;

where $A,\; B,\; C,\; \backslash dots$ $a,\; b,\; c,\; d,\; \backslash dots$ denote constant numbers. Thus we have

Then, $x$ being supposed to become $x+h$, and (in consequence of the change in the value of $x$) $u$ to become $u\text{'}$, we have

Let us now denote $A\; x^a$ by $P$, $B\; x^b$ by $Q$, $C\; x^c$ by $R$, &c. then by last §

&c.

Therefore, substituting these developments in the series expressing $u\text{'}$,

Therefore, substituting for the series to which they are respectively equal,

Hence it appears that $u$ being any function of $x$ whatever, if $x+h$ be substituted in that function instead of $x$, the series expressing the development of this new value of the function will have the general properties which have been shewn, in last §, to belong to it in the case of the function having the particular value $x^n$.

The very general theorem which we have just now investigated is one of the most elegant and important in analysis. It was first published by Dr Brooke Taylor in a work entitled Methodus Incrementorum, which made its appearance about the year 1716. The theorem itself is generally known by the name of Taylor's theorem. It is more general than the celebrated Binomial theorem, inasmuch as this last, and innumerable others, are comprehended in it as particular cases.

70. We shall now give some examples to shew the manner of applying Taylor's theorem, as well as its great utility as an instrument of analysis.

Example 1. Suppose $u\; =\; a^x$, $a$ being constant and $x$ variable. Then $x$ becoming $x+h$, $u$ becomes $u\text{'}\; =\; a^\{x+h\}$. Now from the equation $u\; =\; a^x$ we derive (§ 56.) $\backslash frac\{\backslash dot\{u\}\}\{x\}\; =\; A\; a^x$ (here $A$ denotes $\backslash frac\{\backslash log.\; a\}\{\backslash log.\; e\}$). Again, considering $x$ as constant, and repeating the operation of taking

the fluxion on $\backslash frac\{\backslash dot\{u\}\}\{x\}\; =\; A\; a^x$, we get $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}\; =\; A^2\; a^x$, and hence

again $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}\; =\; A^2\; a^x$, &c. Therefore, substituting for

their values in the general theorem

Suppose now, that $x=0$, then, as in this case $a^x\; =\; 1$, we have

or, exchanging $h$ for $x$

the same result as we formerly obtained in § 54.

Ex. 2. Suppose $u\; =\; \backslash log.\; x$. Then, $x$ becoming $x+h$, $u$ becomes $u\text{'}\; =\; \backslash log.\; (x+h)$. Now from the equation $u\; =\; \backslash log.\; x$, we find (by § 57.) $\backslash frac\{\backslash dot\{u\}\}\{x\}\; =\; \backslash frac\{M\}\{x\}$. Here $M$ denotes the modulus of the system. Again, supposing $x$ constant, we find by § 26, $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}\; =\; -\backslash frac\{M\}\{x^2\}$, $\backslash frac\{\backslash ddot\{u\}\}\{x^3\}\; =\; \backslash frac\{2M\}\{x^3\}$, &c. Therefore, substituting as

before the values of in the general for-

If we suppose $x=1$, and change $h$ into $y$, we have, because $\backslash log.\; x\; =\; \backslash log.\; 1\; =\; 0$,

For the particular method of applying these two series to the calculation of logarithms, see ALGEBRA § 285 to § 291. See also LOGARITHMS.

Ex. 3. Suppose now $u\; =\; \backslash sin.\; x$. Then $u\text{'}\; =\; \backslash sin.\; (x+h)$. From $u\; =\; \backslash sin.\; x$, by the application of the rule in § 59, we deduce $\backslash frac\{\backslash dot\{u\}\}\{x\}\; =\; \backslash cos.\; x$, $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}\; =\; -\backslash sin.\; x$, $\backslash frac\{\backslash ddot\{u\}\}\{x^3\}\; =\; -\backslash cos.\; x$, $\backslash frac\{\backslash ddot\{u\}\}\{x^4\}\; =\; \backslash sin.\; x$, &c. Therefore, substituting

for their values in the general formula as before, we have

or $\backslash sin.\; (x+h)$ is equal to

If we suppose $x=0$, then, as in that case $\backslash sin.\; x\; =\; 0$, the preceding formula becomes

$\backslash sin.$

Ex. 4. Suppose $u\; =\; \backslash cos.\; x$, then $u\text{'}\; =\; \backslash cos.\; (x+h)$, and since $u\; =\; \backslash cos.\; x$, by § 59, $\backslash frac\{\backslash dot\{u\}\}\{x\}\; =\; -\backslash text\{fin.\; \}\; x$, $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}\; =\; -\backslash cos.\; x$, $\backslash frac\{\backslash ddot\{u\}\}\{x^3\}\; =\; \backslash text\{fin.\; \}\; x$, &c. Therefore, substituting as

before these values in the general formula it becomes

which expression, when $x=0$, and therefore $\backslash cos.\; x=1$, $\backslash text\{fin.\; \}\; x=0$, becomes simply

71. It may be remarked that in each of these examples, from the development of $u\text{'}$ the new value of the function $u$, we have been able to deduce a development of $u$ the function itself. But it is easy to see, that by proceeding in the same manner with the general formula as we have done in these particular examples, we shall obtain a general expression for the development of any function whatever.

Now, $u\text{'}$ being the value that $u$ assumes when $x+h$ is substituted in it instead of $x$, if we suppose $x=0$, then $u\text{'}$ becomes the very same function of $h$, that $u$ is of $x$. Let us denote the values which each of the functions $u$, $\backslash frac\{\backslash dot\{u\}\}\{x\}$, $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}$, &c. acquire, when $x=0$, by $U$, $U\text{'}$, $U\text{'}\text{'}$ &c. respectively.

Then ($x$) $u\text{'}$ (considered as the same function of $h$ that $u$ is of $x$) is equal to

Let $x$ be now supposed to be substituted both in $u\text{'}$ and the series which is its development instead of $h$, then $u\text{'}$ becomes $u$, and we have

and in this formula it is to be considered, as already stated, that $U$, $U\text{'}$, $U\text{'}\text{'}$, &c. denote the particular values which the functions $u$, $\backslash frac\{\backslash dot\{u\}\}\{x\}$, $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}$, &c. acquire respectively, by supposing that in each of them $x$ is taken $=0$.

72. As an example of the application of this series let us resume the equation $u\; =\; a^x$, then $\backslash frac\{\backslash dot\{u\}\}\{x\}\; =\; A\; a^x$

Suppose now that $x=0$, then $u$, or $a^x$ becomes $a^0=1$, $\backslash frac\{\backslash dot\{u\}\}\{x\}\; =\; A\; a^x$ becomes $A$, $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}\; =\; A^2\; a^x$ becomes $A^2$, &c. so that $U=1$, $U\text{'}=A$, $U\text{'}\text{'}=A^2$, &c. substituting therefore these values in the general formula, it becomes

Let us next suppose that $u$ is an arch of a circle of which the fine is $x$ (radius being unity), then $x\; =\; \backslash text\{fin.\; \}\; u$. Now the ratio of the fluxion of $u$ to the fluxion of $x$ will be the very same whether we consider $u$ as a function of $x$, or $x$ as a function of $u$; therefore (§ 59.)

($x$) For the sake of illustration let us take a particular example. Suppose $u\; =\; (a+x)^n$, then $\backslash frac\{\backslash dot\{u\}\}\{x\}\; =\; n(a+x)^\{n-1\}$, $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}\; =\; n(n-1)(a+x)^\{n-2\}$, &c. Suppose now that $x=0$, then $u$ becomes $a^n$, $\backslash frac\{\backslash dot\{u\}\}\{x\}$ becomes $n\; a^\{n-1\}$, $\backslash frac\{\backslash ddot\{u\}\}\{x^2\}$ becomes $n(n-1)a^\{n-2\}$, &c. so that in this particular case we have $U=a^n$, $U\text{'}=n\; a^\{n-1\}$, $U\text{'}\text{'}=n(n-1)a^\{n-2\}$, &c.

Taking now the fluxion of $\backslash frac\{1\}\{\backslash sqrt\{(1-x^2)\}\}$, and the fluxion of the result &c. we have

Suppose now that $x=0$, then $u$ becomes 0, $\backslash frac\{\backslash dot\{u\}\}\{x\}$ becomes

$\backslash frac\{\backslash dot\{u\}\}\{x^2\}$ becomes 0, $\backslash frac\{\backslash dot\{u\}\}\{x^3\}$ becomes 1, $\backslash frac\{\backslash dot\{u\}\}\{x^4\}$ becomes 0, &c. so

that $U=0$, $U\text{'}=1$, $U\text{'}\text{'}=0$, $U\text{'}\text{'}\text{'}=1$, $U\text{'}\text{'}\text{'}\text{'}=0$, &c. Therefore, substituting in the general formula, we find

By prosecuting the computations farther, we may find

Application of the Method of Fluxions to the drawing of Tangents.

73. The theory of tangents to curve lines furnishes a good illustration of the truth of the principle which we have considered as the foundation of the method of fluxions, namely, that whatever be the form of a function, the ratio of its increment to the increment of the variable quantity from which the function is formed, is in every case susceptible of a limit.

Let AB, the abscissa of a curve, be the geometrical expression of a variable quantity $x$, and let BP the corresponding ordinate, be the expression for $y$, any function of $x$; then the curve line itself is the locus of the equation expressing the relation between $x$ and $y$. Let PT, a tangent to the curve at P, meet AB the abscissa in T; through P draw any straight line meeting the abscissa in D, and the curve in $\backslash rho$; draw the ordinate $\backslash rho\; b$, and from P draw Pn parallel to the abscissa, meeting the ordinate $\backslash rho\; b$ in $n$.

The triangles DBP, Pn$\backslash rho$ are similar; therefore

Now $\backslash rho\; n$, and $nP$, or $Bb$, are the increments of PB and BA, or of $y$ and $x$ respectively, therefore the ratio of the simultaneous increments of PB and BA, or $y$ and $x$, whatever be their magnitudes, is equal to the ratio of PB to BD. Conceive now the point $\backslash rho$ to approach continually to P, then the angle contained by the straight line $\backslash rho\; PD$, and the tangent PT, will decrease, and the point D will approach to T; at the same time $\backslash rho\; n$, and $nP$, the increments of $y$ and $x$ will be continually diminished; still, however, they will have

to each other the ratio of PB to BD, but this ratio approaches continually to the ratio of PB to BT, and becomes at last more nearly equal to it than any assignable ratio; therefore the ratio of PB to BT is the limit of the ratio of PB to BD, and consequently is also the limit of the ratio of $\backslash rho\; n$, the increment of $y$ to $nP$, the increment of $x$. And as this conclusion does not depend upon the particular nature of the curve, or upon any particular relation supposed to subsist between $x$ and $y$, we may conclude, that whatever be the form of the function, the ratio of the simultaneous increments of the function, and the variable quantity from which it is formed, has a limit to which it approaches when the increments are conceived to be continually diminished.

It is now easy to see how the method of fluxions may be applied to the determination of tangents to curves, for since the ratio of the ordinate PB to the subtangent BT is always the limiting ratio of the increments of the ordinate and abscissa, it is equal to the ratio of their fluxions, that is

Hence in any curve whatever, referred to an axis, the subtangent, (that is, the segment of the abscissa between the ordinate and tangent) is equal to $\backslash frac\{\backslash dot\{x\}\}\{\backslash dot\{y\}\}\; y$ where $x$ denotes the abscissa, and $y$ the ordinate at the point of contact; and the subtangent being found, the position of the tangent is thereby determined.

Let us apply the above general formula to some examples.

Example 1. Let the proposed curve be a circle. It is required to determine the position of PT, a tangent at any point P in its circumference. Fig. 7.

Put $2a$ for AE the diameter, also $x$ for AB the abscissa, and $y$ for BP the ordinate at the point of contact.

From the nature of the curve, we have

Hence taking the relations of the fluxions of $x$ and $y$, we have

from which it appears that BT the subtangent is a third proportional to $a-x$ and $y$, that is, to CB the distance of the ordinate from the centre, and BP the ordinate, agreeing with what is known from the elements of geometry.

Ex. 2. Let the curve be a parabola, required the same as before. Fig. 8.

Put $x$ for AB, the abscissa, and $y$ for BP the ordinate at P the point of contact; also $a$ for the parameter; then, from the nature of the curve

therefore,

therefore, taking the fluxions, we get $a\; \backslash dot\{x\}\; =\; 2y\; \backslash dot\{y\}$, and

from which it appears that the sub-tangent BT is double the abscissa BA.

Ex. 3. Let the curve be an ellipse.

Put $AB\; =\; x$, $BP\; =\; y$, $AC$, the semi-transverse axis $=\; a$, $CH$ the semi-conjugate axis $=\; b$.

The nature of the curve is such, that

and taking the relation of the fluxions,

But from the preceding equation expressing the nature of the curve, $\backslash frac\{a^2\; y^2\}\{b^2\}\; =\; (2a-x)x$; therefore

To this expression for BT, let $BC\; =\; a-x$ be added, and we have

from which it appears that $CB\; :\; CA\; ::\; CA\; :\; CT$.

Ex. 4. Suppose the curve to be a hyperbola, and let it be required to find, as in the preceding examples.

Put $a$ and $b$ as in the case of the ellipse to express the semitransverse and semiconjugate axes, then the equation of the curve is

and taking the relation of the fluxions,

from which we find

or, substituting $(2a+x)x$ for $\backslash frac\{a^2\; y^2\}\{b^2\}$,

Let this expression for BT be subtracted from the expression for CB, that is from $a+x$, and we have

therefore $CB\; :\; CA\; ::\; CA\; :\; CT$.

Ex. 5. Suppose the curve APD to be a cycloid, of Fig. 11. which AE is the axis, and AQE a semicircle described on the axis as a diameter. Suppose AC, the radius, to be unity; put $AB\; =\; x$, $BP\; =\; y$, and the arch $AQ\; =\; v$; then, $AB\; =\; 1\; -\; \backslash cos\; v$, and $BQ\; =\; \backslash sin\; v$. Now, from the nature of the curve, $PB\; =\; \backslash text\{arch\; \}\; AQ\; +\; BQ$; hence we have

and taking the fluxions, by § 59,

therefore,

but from the nature of the circle $\backslash frac\{BQ\}\{EB\}\; =\; \backslash frac\{AB\}\{BQ\}$, there-

fore, $BT\; =\; \backslash frac\{PB\; \backslash times\; AB\}\{BQ\}$, and consequently $BQ\; :\; BA\; ::$

$BP\; :\; BT$, from which it appears that if the chord AQ be drawn, the tangent PT is parallel to the chord QA.

74. If PT be a tangent to the curve AP at the point P, and PC be drawn perpendicular to the tangent, meeting AC the axis of the curve in C, then the line PC is called a normal to the curve at the point P; and BC, the distance between the ordinate and the extremity of the normal, is called the sub-normal.

The triangles TBP, BPC being similar, we have $TB\; :\; BP\; ::\; BP\; :\; BC$; or, since $TB\; :\; BP\; ::\; x\; :\; y$, ($\backslash S$ 73) $x\; :\; y\; ::\; y\; :\; BC$, hence in any curve, BC the sub-normal is equal to $\backslash frac\{y^2\}\{x\}$; and from this expression, we may find the sub-normal in the same way as we have found the sub-tangent in the examples of last $\backslash S$.

75. As by plane trigonometry

and from $\backslash S$ 73, $TB\; :\; BP\; ::\; x\; :\; y$,

hence it appears that $\backslash frac\{y\}\{x\}$ expresses the numeral tangent of the angle T, that is the angle contained by a tangent to the curve, and the axis of the curve. In like manner we have

therefore $\backslash frac\{\backslash dot\{y\}\}\{\backslash dot\{x\}\}$ expresses the tangent of the angle C, that is the angle contained by a normal to the curve and its axis.

76. If a variable quantity be supposed to change its magnitude, then any function of that quantity will also change its magnitude. When the variable quantity is supposed to increase continually so as to acquire successively all degrees of magnitude, there are some functions of such a form, that they either increase continually, or decrease continually; but there are others again which either increase to a certain limit, after which they decrease; or else they decrease to a certain limit, after which they increase.

If, in consequence of the continual increase of a variable quantity, a function of that quantity first increases to a certain limit, and afterwards decreases, when it arrives at that limit it is then said to be a maximum. Or if it decrease to a certain limit, and afterwards increase, when it arrives at that limit, it is then said to be a minimum.

77. Let us consider the function $y\; =\; b\; -\; (x\; -\; a)^2$. If we suppose $x\; =\; 0$, then $y\; =\; b\; -\; a^2$. Suppose now $x$ to be at first very small, and to increase; then as $(x\; -\; a)^2$ will decrease, $y$ will also increase, till $x$ become $=\; a$, and then $y$ will become $=\; b$, when it is a maximum; for $x$ being supposed to become greater than $a$, $y$ will be less than $b$. By supposing $x$ to increase till $(x\; -\; a)^2$ become equal to $b$, then $y$ will decrease to 0; and $x$ being still supposed to increase, $y$ will become negative.

Let us next suppose $y\; =\; b\; +\; (x\; -\; a)^2$. In this case when $x\; =\; 0$, $y\; =\; b\; +\; a^2$; as $x$ increases, $(x\; -\; a)^2$ decreases, and consequently $y$ decreases, till $x\; =\; a$, and then $y\; =\; b$, a minimum; for $x$ becoming greater than $a$, $y$ becomes greater than $b$.

78. Every function that either increases or decreases continually has neither maximum nor minimum; for for whatever value such a function may acquire, in the one case it may always have a greater, and in the other a less value.

The characteristic property of a maximum value of a function, by which it is made the object of analytical inquiry, consists in its being greater than the values immediately preceding, and also greater than the values immediately following it; and that of a minimum consists in its being less than the values immediately preceding, and also less than the values immediately following it.

In some cases a function may increase to a certain limit and then decrease, and afterwards increase again indefinitely; or the contrary. Hence it may happen that such a function may have values greater than its maximum or less than its minimum as they have been here defined. And indeed it is easy to conceive that a function may increase and decrease alternately several times; in such a case it must be considered as having several maxima and minima.

79. Since $y$ any function of a variable quantity $x$ may be considered as the ordinate of a curve, of which $x$ is the abscissa, it is evident that to determine the greatest or least value of such a function, we have only to seek the greatest or least ordinate of the curve which

is the locus of the equation expressing the relation between $x$ and $y$. Let us suppose this curve to be DPE, and that AB is the value of $x$, corresponding to BP the maximum or minimum value of the ordinate $y$; it is evident, that in the case of a maximum, the curve must be concave towards AC, at least to a certain extent, on each side of the point P, as in fig. 12, but that in the case of a minimum it must be convex towards AC, as in fig. 13; and also, that in either case, if a straight line be drawn through P parallel to AC, the curve must be wholly on one side of that line, to a certain extent on each side of the point P, and therefore, that the line PQ must be a tangent to the curve at the point P.

Now when PQ a tangent to a curve at P (fig. 8) meets the axis in T, it has been shewn, § 75, that $\backslash frac\{y\}\{x\}$ is the expression for the tangent of the angle T, radius being unity; but this angle vanishes, when PQ, instead of meeting AC, is parallel to it, as in fig. 12, and fig. 13; therefore, as in this case the tangent of the angle is $=\; 0$,

we have $\backslash frac\{y\}\{x\}\; =\; 0$.

Hence it appears, that to determine the maximum or minimum of $y$, a function of $x$, we must find the fluxion of the function, and divide it by $x$, and put the result equal to 0.

80. We proceed to illustrate this rule by some examples.

Ex. 1. To divide a given number $a$ into two such parts, that their product may be the greatest possible.

Let $x$ denote the one part, then $a\; -\; x$ will be the other part, and $x(a\; -\; x)$ the product of the two parts. Therefore, by the question

hence, taking the fluxion of the function,

therefore, $a\; -\; 2x\; =\; 0$, and $x\; =\; \backslash frac\{1\}\{2\}a$.

Thus it appears that the product of the parts will be the greatest possible, when each is half the given number.

Ex. 2. To find the fraction which shall exceed its cube by the greatest quantity possible.

Let $x$ denote the fraction, then its cube is $x^3$, so that we have

therefore, taking the fluxion of the function,

hence $3x^2\; =\; 1$, and $x\; =\; \backslash sqrt\{\backslash frac\{1\}\{3\}\}$, the fraction required.

Ex. 3. To determine the greatest rectangle that can be inscribed in a given triangle.

Put the base AC of the triangle $=\; b$, and its altitude BD $=\; a$, and let B $n$, the altitude of the rectangle Fig. 14. $p\; q\; r\; s$, considered as variable, be denoted by $x$, then, because

Direct Method. because of the parallel lines $AC,\; pq$, it will be, as $BD\; :\; AC\; ::\; Dn\; :\; pq$, that is $a\; :\; b\; ::\; a\; -\; x\; :\; pq$, hence $pq\; =\; \backslash frac\{b(a-x)\}\{a\}$, and the area of the rectangle, or $pq\; \backslash times\; Bn$, will be $\backslash frac\{b(a-x)x\}\{a\}$, therefore $y\; =\; \backslash frac\{b(a-x)x\}\{a\}$ must be a maximum, and hence $\backslash dot\{y\}\; =\; \backslash frac\{ba\; -\; 2bx\}\{a\}$: thus we have,

and $ba\; -\; 2bx\; =\; 0$, and $x\; =\; \backslash frac\{1\}\{2\}a$; hence it appears that the greatest inscribed rectangle is that whose altitude is half the altitude of the triangle.

81. It is proper to observe that the value of a quantity when a maximum or minimum may often be determined more readily by considering that any given multiple, or part of the function, likewise any power or root of it, must then be also a maximum or minimum. Thus in the preceding example, in which the function to be a maximum is $\backslash frac\{b(a-x)x\}\{a\}$, we may reject the constant multiplier $\backslash frac\{b\}\{a\}$, and then the function to be a maximum is $y\; =\; (a-x)x\; =\; ax\; -\; x^2$, the fluxion of which being taken, we have $\backslash dot\{y\}\; =\; ax\; -\; 2x$ and $\backslash frac\{\backslash dot\{y\}\}\{x\}\; =\; a\; -\; 2x\; =\; 0$, hence $x\; =\; \backslash frac\{1\}\{2\}a$, the same value as before.

82. Ex. 4. Of all the right-angled plane triangles having the same given hypotenuse, to find that whose area is greatest.

Fig. 15. Let $ABC$ be the triangle; put $AB\; =\; x$, and $AC$, the given hypotenuse, $=\; a$, then $BC\; =\; \backslash sqrt\{a^2\; -\; x^2\}$, and consequently the area of the triangle is $\backslash frac\{x\}\{2\}\; \backslash sqrt\{a^2\; -\; x^2\}$, which being a maximum, its square $\backslash frac\{x^2(a^2\; -\; x^2)\}\{4\}$, also four times that square, or $x^2(a^2\; -\; x^2)$, will likewise be a maximum, therefore, $y\; =\; x^2(a^2\; -\; x^2)\; =\; a^2x^2\; -\; x^4$, a max. hence $\backslash dot\{y\}\; =\; 2a^2x\; -\; 4x^3$, and

hence $2x^2\; =\; a^2$, and $x\; =\; a\backslash sqrt\{\backslash frac\{1\}\{2\}\}$.

Ex. 5. To determine the greatest cylinder that can be inscribed in a given cone.

Fig. 16. Put $AC$ the base of the cone $=\; b$, $BH$ its altitude $=\; a$, and $DE$, the diameter of the end of the cylinder $DIGF$ inscribed in the cone, $=\; x$. From the similar triangles $BAC,\; BDE$, we have $AC\; :\; BH\; ::\; DE\; :\; BE$, that is $b\; :\; a\; ::\; x\; :\; BE$, hence $BE\; =\; \backslash frac\{ax\}\{b\}$, and $EH\; =\; BH\; -\; BE\; =\; a\; -\; \backslash frac\{ax\}\{b\}\; =\; \backslash frac\{ab\; -\; ax\}\{b\}$. Put $c$ for the number .78539, then the area of the base of the cylinder is (by the Elements of Geometry) $c\; \backslash times\; DE^2$, and

its solid content $c\; \backslash times\; DE^2\; \backslash times\; EH\; =\; \backslash frac\{cax^2(b-x)\}\{b\}$, hence $\backslash frac\{cax^2(b-x)\}\{b\}$ is to be a maximum, therefore, leaving out

the constant multiplier $\backslash frac\{ca\}\{b\}$, we have $y\; =\; x^2(b-x)\; =\; bx^2\; -\; x^3$, a maximum; taking now the fluxions, we get

hence $3x^2\; =\; 2bx$, and $x\; =\; \backslash frac\{2\}\{3\}b$, and consequently $HE\; =\; \backslash frac\{1\}\{3\}BH$.

Ex. 6. To find the sun's place in the ecliptic, when that part of the equation of time which arises from the obliquity of the ecliptic is a maximum.

Let $EQ$ be the equator, $EC$ the ecliptic, $S$ the sun's place, and $SA$ his declination, then this part of the equation of time is the difference of the sun's longitude $ES$, and right ascension $EA$, turned into time. Put the arch $ES\; =\; x$, the arch $EA\; =\; v$, which is to be considered as a function of $x$, and put $a$ for the cosine of the angle $E\; =\; 23^\backslash circ\; 28\text{'}$. Then, by Spherical Trigonometry, in the right-angled spherical triangle $EAS$, we have $\backslash tan.\; EA\; =\; \backslash cot.\; E\; \backslash times\; \backslash tan.\; ES$; therefore, to determine $x$ we have $\backslash tan.\; v\; =\; a\; \backslash tan.\; x$, and $y\; =\; x\; -\; v$, a maximum.

From the second of these equations we get

and from the first, by § 60,

therefore, $1\; -\; \backslash frac\{a\; \backslash sec^2\; x\}\{\backslash sec^2\; v\}\; =\; 0$, and $\backslash sec^2\; v\; =\; a\; \backslash sec^2\; x$,

or $1\; +\; \backslash tan^2\; v\; =\; a\; +\; a\; \backslash tan^2\; x$, but from the first equation $\backslash tan^2\; v\; =\; a^2\; \backslash tan^2\; x$, therefore,

hence $\backslash tan.\; x\; =\; \backslash frac\{1\}\{\backslash sqrt\{a\}\}\; =\; 1.04416$, the tangent of $46^\backslash circ\; 14\text{'}$, the sun's longitude when this part of the equation is a maximum.

83. We have deduced the rule of § 79, for determining when a function is a maximum, or minimum, from the consideration of curve lines. The whole theory of maxima and minima may however be explained in a manner purely analytical, as follows:

Let us suppose that $y$ is any function whatever of $x$, and that $x$ has acquired the value that produces the maximum or minimum of the function; then, if we suppose $x-h$ and $x+h$ to be substituted successively in the function instead of $x$, the two resulting values ought to be both less than the maximum or both greater than the minimum value.

Let us denote the value of the function that results from the substitution of $x-h$ by $y$ and that which results from the substitution of $x+h$ by $y$, then by the theorem given in § 69,

In each of these values, $\backslash frac\{\backslash dot\{y\}\}\{x\}$, the coefficient of $h$, must either be equal to some quantity, positive or negative, or else it must $\backslash equiv\; 0$. Let us suppose, if possible, that it is equal to some quantity, positive or negative; now as $h$ may be conceived to be so small that the term $\backslash frac\{\backslash dot\{y\}\}\{x\}\; h$, or any other term, shall exceed the sum of all the terms that follow it in each series, (D) if we suppose $h$ to have such a value, then, because of the term $\backslash frac\{\backslash dot\{y\}\}\{x\}\; h$ having the sign — in the one series, and + in the other, it follows that the one value of the function is greater, and the other less than $y$, the maximum or minimum value: But this conclusion is inconsistent with the nature of a maximum or minimum, therefore $\backslash frac\{\backslash dot\{y\}\}\{x\}$ cannot in the case of a maximum or minimum be equal to any positive or negative quantity whatever.

If however we assume $\backslash frac\{\backslash dot\{y\}\}\{x\}\; \backslash equiv\; 0$, so that the two values are,

then as the second term $\backslash frac\{\backslash ddot\{y\}\}\{x^2\}\; \backslash frac\{h^2\}\{2\}$ has the same sign in both, when that term is greater than all the terms that follow

it, we shall have both values greater than $y$ when $\backslash frac\{\backslash ddot\{y\}\}\{x^2\}$ is positive, and both less when it is negative, the first case corresponding to a maximum, and the second to a minimum.

Hence, to determine the maximum or minimum of the function $y$, it appears that we must take the fluxion of $y$, and divide it by $x$, and put the result equal to 0, which agrees with what was shewn in § 79.

84. Although in the case of a function admitting of a maximum or minimum we have always $\backslash frac\{\backslash dot\{y\}\}\{x\}\; \backslash equiv\; 0$, yet we must not conclude that conversely the one or the other of these has place every time that $\backslash frac\{\backslash dot\{y\}\}\{x\}\; \backslash equiv\; 0$. For if it so happens that the value of $x$ that renders $\backslash frac\{\backslash dot\{y\}\}\{x\}\; \backslash equiv\; 0$, causes also $\backslash frac\{\backslash ddot\{y\}\}\{x^2\}$ to vanish, without at the same time making $\backslash frac\{\backslash ddot\{y\}\}\{x^3\}$ to disappear, then we have

and as by giving a proper value to $h$, $\backslash frac\{\backslash ddot\{y\}\}\{x^2\}\; \backslash frac\{h^2\}\{2\; \backslash cdot\; 3\}$ may be rendered greater than the sum of all the following terms in each series, it follows, that $\backslash frac\{\backslash ddot\{y\}\}\{x^2\}$ being supposed to be any quantity either positive or negative, because of its sign being different in the two values, the one of them will be greater, and the other less than $y$, the maximum or minimum value, which result is inconsistent with the nature of a maximum or minimum. If however $\backslash frac\{\backslash ddot\{y\}\}\{x^2\}$ be assumed $\backslash equiv\; 0$, then

(D) If this should not appear sufficiently obvious, let

be such a series, where $A,\; B,\; C,\; D,\; \backslash dots$ denote quantities either positive or negative, but which are independent of $h$. Then, writing the series thus,

it is obvious that if $h$ be conceived to be continually diminished, and at last to become $\backslash equiv\; 0$, the part

will also become $\backslash equiv\; 0$, therefore before it vanishes it will be less than $A$, or any other assignable quantity, therefore $B\; h\; +\; C\; h^2\; +\; D\; h^3\; +\; \backslash dots$ may become less than $A\; h$.

here again, the coefficient of the second term having in both values the same sign, the conditions of the maximum or minimum are fulfilled, and the sign of $\backslash frac\{\backslash dot\{y\}\}\{x^4\}$ shews when the one or the other is to have place.

It must now be sufficiently evident, without proceeding any further, that a function can only admit of a maximum or a minimum when the first of its fluxions that does not vanish, is of an even order (or is its second or fourth fluxion, &c.), and that the sign of that fluxion is negative in the case of a maximum, but positive in the case of a minimum.

85. We shall conclude this theory by applying it to an example. Let $y$ be such a function of $x$ that

and hence $\backslash frac\{\backslash dot\{y\}\}\{x\}\; =\; \backslash frac\{my\; -\; x\}\{y\; -\; mx\}\; =\; 0$,

therefore $my\; -\; x\; =\; 0$, and $y\; =\; \backslash frac\{x\}\{m\}$.

To find the value of $x$, let this value of $y$ be substituted in the original equation, it thus becomes

We must now examine what is the nature of the expression for $\backslash frac\{\backslash dot\{y\}\}\{x^4\}$. Taking the fluxion of the equation

$\backslash frac\{\backslash dot\{y\}\}\{x\}\; =\; \backslash frac\{my\; -\; x\}\{y\; -\; mx\}$, and considering that $\backslash dot\{x\}$ is constant, we have

therefore, dividing by $\backslash dot\{x\}$,

but as in the present case $\backslash frac\{\backslash dot\{y\}\}\{x\}\; =\; 0$, and $y\; =\; \backslash frac\{x\}\{m\}$, this expression becomes simply

VOL. VIII. Part II.

which equation, by putting instead of $x$ its value $\backslash frac\{ma\}\{\backslash sqrt\{1-m^3\}\}$, becomes

and as this result is negative, we conclude that the value which we have found for $y$ is a maximum.

Of the values of fractions, the numerators and denominators of which vanish at the same time.

86. There are some fractional functions of such a nature, that by giving a particular value to the variable quantity, both the numerator and denominator of the fraction vanish, and thus the fraction is reduced to this form $\backslash frac\{0\}\{0\}$, an expression from which nothing can be concluded. We have an example of this in the fraction $\backslash frac\{x-a\}\{x^2-a^2\}$, which, by supposing $x=a$ becomes $\backslash frac\{a-a\}\{a^2-a^2\}\; =\; \backslash frac\{0\}\{0\}$; we must not however conclude that the fraction has no determinate value in this particular case, for if we consider that its numerator and denominator have a common divisor, viz. $x-a$, it is evident that by taking this divisor out of both, the fraction $\backslash frac\{x-a\}\{x^2-a^2\}\; =\; \backslash frac\{x-a\}\{(x-a)(x+a)\}$ becomes $\backslash frac\{1\}\{x+a\}$, an expression, which in the case of $x=a$ is equal to $\backslash frac\{1\}\{2a\}$.

87. In general, if we make $x=a$ in an expression of this form $\backslash frac\{P(x-a)^m\}\{Q(x-a)^n\}$, it becomes $\backslash frac\{0\}\{0\}$; however its true value is either nothing, or finite, or infinite, according as $m\; >\; n$, or $m\; =\; n$, or ; for by taking out the factors common to the numerator and denominator, the fraction becomes $\backslash frac\{P(x-a)^\{m-n\}\}\{Q\}$ in the first case, $\backslash frac\{P\}\{Q\}$ in the second, and $\backslash frac\{P\}\{Q(x-a)^\{n-m\}\}$ in the third; here we suppose that $P$ and $Q$ are such functions as neither become nothing, nor infinite, by the supposition of $x=a$.

88. Therefore, when by giving a particular value to $x$ a function of that quantity assumes the form $\backslash frac\{0\}\{0\}$, to discover the true value of the function in this particular case, we must disengage the factors which are common to the numerator and denominator. This may be done in most cases by finding their common measure (ALGEBRA, § 49.) but the direct method of fluxions furnishes us with another method.

In the expression $P(x-a)$, where $P$ denotes any function of $x$ that is independent of $x-a$, if we suppose $x=a$, then the expression vanishes; the fluxion however of the expression, viz. $(x-a)\backslash dot\{P\}\; +\; P\backslash dot\{x\}$, is a quantity which does not vanish when $x=a$, but is then reduced to its last term, that is to $P\backslash dot\{x\}$.

4 Z

Again, the function $P(x-a)^3$ vanishes by supposing $x=a$, but if we take its fluxion, viz. $(x-a)^3\; \backslash dot\{P\}\; +\; 2(x-a)\; \backslash dot\{P\}\; x^2$, and again the fluxion of this quantity we get

an expression which does not vanish upon the hypothesis of $x=a$, but is reduced to its last term, viz. $1\; \backslash cdot\; 2\; P\; x^3$. By proceeding in this manner, it is easy to see that by taking the fluxion of a function of the form $P(x-a)^m$ $m$ times successively ($m$ being a whole number) we shall finally obtain an expression, all the terms of which, except the last, vanish by supposing that $x=a$; and that the last term will be $1\; \backslash cdot\; 2\; \backslash cdot\; 3\; \backslash dots\; m\; P\; x^m$, an expression free from the factor $(x-a)^m$, and involving only the function $P$.

89. It is not necessary that we should know the number $n$, nor that we should exhibit the factor $(x-a)^n$, in order to determine when the expression $P(x-a)^n$ is freed from that factor. We have only to ascertain after each operation of taking the fluxion, whether the result vanishes or not, when we substitute $a$ instead of $x$; for in the last case the operation is finished, and the result is the quantity $1\; \backslash cdot\; 2\; \backslash cdot\; 3\; \backslash dots\; P\; x^m$. Suppose for example the function to be $x^3\; -\; a\; x^2\; -\; a^2\; x\; +\; a^3$, which vanishes when $x=a$, its first fluxion also vanishes when $x=a$, but not its second fluxion, which is $(6x\; -\; 2a)\; x^2$, hence we may conclude that the function has the form $P(x-a)^3$, which is besides obvious, because

90. In applying these observations to the fraction $\backslash frac\{P(x-a)^m\}\{Q(x-a)^n\}$, it appears, that by repeating the operation of taking the fluxions of its numerator and denominator, they will be freed at once from the factor $x-a$, if $m=n$. If a result, which does not vanish, be obtained first from the numerator, then we may be assured, that the factor $(x-a)$ is found in the numerator raised to a less power than in the denominator, and in this case the fraction is infinite when $x=a$. If on the contrary the first result that does not vanish is found from the denominator, then the numerator contains a higher power of $(x-a)$ than the denominator, and in this case, when $x=a$, the fraction vanishes.

The rule for finding the value of a function which becomes $\backslash frac\{0\}\{0\}$ by giving a particular value to $x$ may therefore be expressed thus. Take the successive fluxions of both the numerator and denominator until a result which does not vanish be obtained from either the one or the other, or from both at the same time; in the first case the function is infinite, in the second it is equal to 0, and in the last case its value is finite.

91. We proceed to illustrate this rule by a few examples.

Ex. 1. The value of the function $\backslash frac\{x^3\; -\; 1\}\{x^2\; -\; 1\}$ is required when $x=1$.

The fluxion of the numerator is $3x^2$, and that of the

denominator is $2x$, neither of which quantities vanish when $x=1$, therefore in this particular case, the value of the fraction is $\backslash frac\{3x^2\}\{2x\}\; =\; \backslash frac\{3\}\{2\}$.

Ex. 2. Suppose the fraction to be $\backslash frac\{a\; x^3\; -\; 2\; a\; c\; x\; +\; a\; c^2\}\{b\; x^3\; -\; 2\; b\; c\; x\; +\; b\; c^2\}$ which vanishes when $x=c$.

By taking the fluxions of the numerator and denominator we obtain $\backslash frac\{2\; a\; x^2\; -\; 2\; a\; c\}\{2\; b\; x^2\; -\; 2\; b\; c\}\; =\; \backslash frac\{a\; x\; -\; a\; c\}\{b\; x\; -\; b\; c\}$, a fraction, the numerator and denominator of which still vanish upon the hypothesis of $x=c$, we therefore take the fluxions a second time, and get $\backslash frac\{2\; a\; x\}\{2\; b\; x\}\; =\; \backslash frac\{a\}\{b\}$ for the value of the proposed fraction in the particular case of $x=c$.

Ex. 3. Suppose the fraction to be

which vanishes when $x=a$. In this example, by taking the fluxions of the numerator and denominator once, we get

an expression, of which only the numerator vanishes upon the supposition of $x=a$, hence we may conclude the true value of the fraction in this case to be 0.

The contrary happens in the fraction

we may therefore conclude that when $x=a$ this last fraction becomes infinite.

92. The rule of § 90 can only be applied when the factors common to the numerator and denominator are integer powers of $x-a$, for as by taking the fluxions, the index of $(x-a)^m$ is diminished by an unit at each operation; when $m$ is a fraction we shall at last arrive at a result containing negative powers of $x-a$, which therefore, when $x=a$, will become infinite. The following mode of proceeding will however apply to all cases whatever.

Let $\backslash frac\{X\}\{X\text{'}\}$ be a fraction of which the numerator and denominator both vanish when $x=a$; by substituting in it $a+h$ instead of $x$, the functions $X$ and $X\text{'}$ may be expanded into a series of this form,

which are ascending, that is, having the exponents of the powers positive and increasing; because the series must become 0, upon the hypothesis that $h=0$. We have therefore

instead of the proposed fraction.

Now

Now if $\backslash alpha\; =\; \backslash alpha\text{'}$, by dividing the numerator and denominator of this expression by the factor $h^\{\backslash alpha\}$, which is common to all the terms of each, it becomes

a quantity which, by supposing $h\; =\; 0$, is reduced to $\backslash frac\{0\}\{A\text{'}\}$, that is to 0. If again $\backslash alpha\; =\; \backslash alpha\text{'}$, the expression for the fraction, after dividing the numerator and denominator by $h^\{\backslash alpha\}$, is

which, by supposing $h$ to be $\backslash infty$, becomes simply $\backslash frac\{A\}\{A\text{'}\}$, a finite quantity. If, however, , then the expression for the fraction is

which, when $h\; =\; 0$, becomes $\backslash frac\{A\}\{0\}$, an expression which may be considered as infinite. Thus it appears that in each case the true value of the fraction depends only on $A$ and $A\text{'}$, the first terms of the series.

The following rule is applicable to every function that can appear under the indeterminate form $\backslash frac\{0\}\{0\}$.

Find the first term of each of the ascending series which express the developments of the numerator and denominator when $a+h$ is substituted in them instead of $x$. Reduce the new function formed of these first terms to its most simple form, and make $h\; =\; 0$; the results shall be the different values of the proposed function when $x$ is made equal to $a$.

Example. Suppose the function to be

which, when $x\; =\; a$, becomes $\backslash frac\{0\}\{0\}$. By substituting $a+h$ instead of $x$, and developing the results into series, the numerator becomes $h^\{\backslash frac\{1\}\{2\}\}\; +\; \backslash frac\{h\}\{2\backslash sqrt\{a\}\}\; +\; \backslash dots$ and the denominator $\backslash sqrt\{2a\}h^\{\backslash frac\{1\}\{2\}\}\; +\; \backslash frac\{h^\{\backslash frac\{3\}\{2\}\}\}\{2\backslash sqrt\{2a\}\}\; +\; \backslash dots$. Taking now the first

term of each series, we have $\backslash frac\{h^\{\backslash frac\{1\}\{2\}\}\}\{\backslash sqrt\{2a\}h^\{\backslash frac\{1\}\{2\}\}\}\; =\; \backslash frac\{1\}\{\backslash sqrt\{2a\}\}$, an expression in which $h$ is not found; therefore the value of the function is $\backslash frac\{1\}\{\backslash sqrt\{2a\}\}$, when $x\; =\; a$.

Of the Radii of Curvature.

93. Let HCF represent a material curve, or mould.

Let a thread be fastened to it at H, and made to pass along the curve, so as to coincide with it in its whole extent from H to F. Let the thread be now unlapped or evolved from the curve, then its extremity F will describe another curve line FAPP'. The curve HCF is called the EVOLUTE of the curve FAP; and the curve FAP is called the INVOLUTE of the curve HCF.

94. From this mode of conceiving the curve to be generated, we may draw the following conclusions.

1st. Suppose PC to be a portion of the thread detached from the evolute, then PC will be a tangent to the evolute at C.

2dly. The line PC will be perpendicular to a tangent to the curve FAP at the point P, or will be a normal to the curve at that point. For the point P may be considered as describing at the same time an element of the curve FAP, and an element of a circle $qPq\text{'}$ whose momentary centre is C, and which has PC for its radius.

3dly. That part of the curve between F and P, which is described with radii all of which are shorter than CP is more incurved than a circle described on P as a centre, with a radius equal to CP. And in like manner PP', the part of the curve on the other side of P, which is described with radii greater than PC, is less incurved than that circle.

4thly. The circle $qPq\text{'}$ has the same curvature as the curve APP' itself has at P: hence it is called an EQUI-CURVE circle, and its radius PC is called the RADIUS of CURVATURE at the point C.

95. We are now to investigate how the radius of curvature at any point in FAP any proposed curve may be found.

Let AB and BP be the co-ordinates at P any point in the curve, and PC its radius of curvature; and let PC meet AB in E. Put the abscissa AB = $x$, the ordinate BP = $y$, the arch AP = $s$, the angle AEP (that is, the arch which measures that angle, radius being unity) = $v$, the radius of curvature PC = $r$. Take P' another point in the curve, and let P'C' be the radius of curvature at that point. Let P'C' meet AB in E', and PC in D, and on D as a centre, with a radius = 1, describe an arch of a circle, meeting the radii PC, P'C' in $m$ and $n$. Then the arch PP' will be the increment of $s$; and since the angle PDP' is the difference of the angles PEA, P'E'A, the arch $mn$ will be the corresponding increment of $v$.

Suppose now the point P' to approach continually to P, then the points C' and D will approach to C, and the ratio of the arch PP', the increment of $s$, to the arch $mn$ the increment of $v$, will approach to the ratio of CP to Cm, that is to the ratio of $r$ to 1; therefore the ratio of $r$ to 1 is the limit of the ratio of PP' to $mn$, or $r\; =\; \backslash lim\_\{P\text{'}\; \backslash rightarrow\; P\}\; \backslash frac\{PP\text{'}\}\{mn\}$, and passing to the ratio of the

fluxions, $r\; =\; \backslash frac\{\backslash dot\{s\}\}\{\backslash dot\{v\}\}$, thus we have obtained a formula ex-

pressing the radius of curvature, by means of the fluxion of the arch of the curve, and the fluxion of the angle which a normal to the curve makes with the line of the abscissas. We proceed to deduce from this formula

Direct Method. other expressions which may involve the fluxions of $x$ and $y$ only.

96. Because PE is a normal to the curve at E, the tangent of the angle PEA or $v$ is equal to $\backslash frac\{\backslash dot\{x\}\}\{\backslash dot\{y\}\}$ (§ 75.), put $\backslash frac\{\backslash dot\{x\}\}\{\backslash dot\{y\}\}\; =\; t$, then because $\backslash tan.\; v\; =\; t$, we have by taking the fluxions (§ 60.), $\backslash dot\{v\}\; \backslash sec^2\; v\; =\; \backslash dot\{t\}$, but $\backslash sec^2\; v\; =\; 1\; +\; \backslash tan^2\; v\; =\; 1\; +\; \backslash frac\{\backslash dot\{x\}^2\}\{\backslash dot\{y\}^2\}\; =\; \backslash frac\{\backslash dot\{x\}^2\; +\; \backslash dot\{y\}^2\}\{\backslash dot\{y\}^2\}\; =\; \backslash frac\{\backslash dot\{z\}^2\}\{\backslash dot\{y\}^2\}$ (§ 63.), therefore $\backslash frac\{\backslash dot\{v\}\; \backslash dot\{z\}^2\}\{\backslash dot\{y\}^2\}\; =\; \backslash dot\{t\}$ and $\backslash dot\{v\}\; =\; \backslash frac\{\backslash dot\{t\}\; \backslash dot\{y\}^2\}\{\backslash dot\{z\}^2\}$.

Substituting now this value of $\backslash dot\{v\}$ in the formula $r\; =\; \backslash frac\{\backslash dot\{z\}^2\}\{\backslash dot\{v\}\}$

If we now recollect that $t\; =\; \backslash frac\{\backslash dot\{x\}\}\{\backslash dot\{y\}\}$, and that $\backslash dot\{z\}^2\; =\; \backslash dot\{x\}^2\; +\; \backslash dot\{y\}^2$, it will appear that this other expression which we have found for $r$ involves in effect the fluxions of $x$ and $y$ only.

97. In computing the values of $t\; =\; \backslash frac\{\backslash dot\{x\}\}\{\backslash dot\{y\}\}$, and $\backslash frac\{\backslash dot\{z\}^2\}\{\backslash dot\{t\}\; \backslash dot\{y\}^2\}$ we may consider any two of the three quantities $x,\; y,\; z$, as a function of the remaining quantity; and upon that hypothesis compute their fluxions.

Thus if we suppose that $y$ and $z$ are functions of $x$, then, as in taking the fluxions of $y,\; t$, and $z$, we must consider $x$ as a given or constant quantity, from the equation $t\; =\; \backslash frac\{\backslash dot\{x\}\}\{\backslash dot\{y\}\}$ we have $\backslash dot\{t\}\; =\; -\backslash frac\{\backslash dot\{x\}\; \backslash dot\{y\}\}\{\backslash dot\{y\}^2\}$ (§ 39.), and substituting this value of $\backslash dot\{t\}$ in $\backslash frac\{\backslash dot\{z\}^2\}\{\backslash dot\{t\}\; \backslash dot\{y\}^2\}$, the value last found for $r$, it becomes

If again, instead of considering $y$ and $z$ as functions of $x$, we consider $x$ and $z$ as functions of $y$, then from the equation $t\; =\; \backslash frac\{\backslash dot\{x\}\}\{\backslash dot\{y\}\}$ (as $\backslash dot\{y\}$ must now be reckoned constant), we get $\backslash dot\{t\}\; =\; \backslash frac\{\backslash dot\{x\}\}\{\backslash dot\{y\}\}$, thus the formula $r\; =\; \backslash frac\{\backslash dot\{z\}^2\}\{\backslash dot\{t\}\; \backslash dot\{y\}^2\}$ becomes

We shall now apply these formulae to some examples.

98. Example 1.—It is required to find the general expression for the radius of curvature of a parabola.

The equation of the parabola is $y\; =\; a^\{\backslash frac\{1\}\{2\}\}\; x^\{\backslash frac\{1\}\{2\}\}$, there-

fore, $\backslash dot\{y\}\; =\; \backslash frac\{1\}\{2\}\; a^\{\backslash frac\{1\}\{2\}\}\; x^\{-\backslash frac\{1\}\{2\}\}\; =\; \backslash frac\{a^\{\backslash frac\{1\}\{2\}\}\; \backslash dot\{x\}\}\{2x^\{\backslash frac\{1\}\{2\}\}\}$, and, making $\backslash dot\{x\}$ constant,

$\backslash frac\{\backslash dot\{x\}\}\{2\}\; \backslash sqrt\{\backslash left(\backslash frac\{4x+a\}\{x\}\backslash right)\}$, and, putting $r$ for the radius of curvature,

If in this general expression, we put $x=0$, we find $\backslash frac\{a^\{\backslash frac\{1\}\{2\}\}\}\{2\backslash sqrt\{a\}\}\; =\; \backslash frac\{1\}\{2\}a$ for the radius of curvature at the vertex of the curve.

Ex. 2. Suppose the curve to be an ellipse, required as in the last example.

Putting $a$ and $c$ to denote the two axes, the equation of the ellipse is $a^2\; y^2\; =\; c^2\; (a\; x\; -\; x^2)$. Hence taking the first and second fluxions, we have $2\; a^2\; y\; \backslash dot\{y\}\; =\; c^2\; x\; (a\; -\; 2x)$, and $2\; a^2\; \backslash dot\{y\}^2\; +\; 2\; a^2\; y\; \backslash ddot\{y\}\; =\; -2\; c^2\; x^2$; whence $\backslash dot\{y\}\; =\; \backslash frac\{c^2\; x\; (a\; -\; 2x)\}\{2\; a^2\; y\}$, and $\backslash ddot\{y\}\; =\; \backslash frac\{a^2\; \backslash dot\{y\}^2\; +\; c^2\; x^2\}\{a^2\; y\}$, which expressions, by substituting the values of $y$ and $\backslash dot\{y\}$ become

therefore,

and

which expression, when $x=0$, becomes simply $\backslash frac\{c^2\}\{2\; a\}$, the radius of curvature at the vertices of the transverse axis; but when $x\; =\; \backslash frac\{1\}\{2\}\; a$, it becomes $\backslash frac\{a^2\}\{2\; c\}$, the radius of curvature at the vertices of the conjugate axis.

99. AS the DIRECT METHOD of fluxions treats of finding the relation between the fluxions of variable quantities, having given the relation subsisting between the quantities themselves; so the INVERSE METHOD treats of finding the relation subsisting between the variable quantities, having given the relation of their fluxions.

Whatever be the relation between variable quantities, we can in every case assign the relation of their fluxions; therefore the direct method of fluxions may in this respect be considered as perfect. But it is not the same with the inverse method, for there are no direct and general rules, by which we can in every case determine from the relation of the fluxions, that of their flowing quantities or fluents. All we can do is to compare any proposed fluxion with such fluxions as are derived from known fluents by the rules of the direct method, and if we find it to have the same form as one of these, we may conclude that the fluents of both, or at least the variable parts of these fluents, are identical.

100. In the direct method we have shewn, that by proper transformations, the finding of the fluxion of any proposed function is reducible to the finding of the fluxions of a few simple functions, and of the sums, or products, or quotients of such functions. In like manner, in the inverse method we must endeavour to transform complex fluxionary expressions into others more simple, so as to reduce them, if possible, to some fluxion, the fluent of which we already know.

101. As when $y$ is such a function of a variable quantity $x$, that $y\; =\; Ax^m\; +\; C$, where $A$, $m$ and $C$ denote constant quantities, we find by the direct method (§ 36. and § 26.) that $y\; =\; mA\; x^\{m-1\}\; \backslash dot\{x\}$, or (putting $a$ instead of $mA$, and $n$ instead of $m-1$), $y\; =\; a\; x^n\; \backslash dot\{x\}$; so on the contrary, as often as we have the fluxional equation

we may conclude that the relation of the fluents is expressed by the equation

for by substituting $mA$ instead of $a$, and $m-1$ instead of $n$ in this equation, it becomes $y\; =\; Ax^m\; +\; C$, the same equation as that from which the fluxional equation was derived.

102. The value of the constant quantity $C$, which is generally called by writers on fluxions, the correction of the fluent, is to be determined from the particular

inquiry in which the fluxional equation $y\; =\; a\; x^n\; \backslash dot\{x\}$ occurs. If it be known that $y\; =\; 0$, when $x$ acquires some known magnitude, which may be denoted by $b$, then the general equation $y\; =\; \backslash frac\{a\; x^\{n+1\}\}\{n+1\}\; +\; C$, becomes in that particular case

Hence, by subtracting each side of this last equation from the corresponding side of the former, we get

an equation that is independent of the constant and arbitrary quantity $C$.

103. By giving particular values to $n$ in the fluxional equation $y\; =\; a\; x^n\; \backslash dot\{x\}$, and in that of the fluents $y\; =\; \backslash frac\{a(x^\{n+1\}\; -\; b^\{n+1\})\}\{n+1\}$, we may obtain particular fluxional equations, and corresponding equations of the fluents. There is, however, one case which requires to be noticed; it is when $n$ is $-1$; then the equation of the fluxions is $y\; =\; a\; x^\{-1\}\; \backslash dot\{x\}\; =\; \backslash frac\{a\; \backslash dot\{x\}\}\{x\}$, and that of the fluents, according to the general formula $y\; =\; \backslash frac\{a(x^\{-1+1\}\; -\; b^\{-1+1\})\}\{-1+1\}$

$=\; \backslash frac\{a(x^0\; -\; b^0)\}\{0\}\; =\; \backslash frac\{a(1-1)\}\{0\}\; =\; \backslash frac\{0\}\{0\}$, but from this expression it is manifest, that nothing can be concluded. The value of the function $\backslash frac\{a(x^\{n+1\}\; -\; b^\{n+1\})\}\{n+1\}$, in the particular case of $n+1=0$ may be found by the rule given in § 90 for determining the value of a function when it assumes the form $\backslash frac\{0\}\{0\}$; but it may be otherwise found by proceeding thus. Put $n+1=m$, and let $\backslash rho\; =\; \backslash frac\{\backslash log.\; x\}\{\backslash log.\; e\}$, and

$q\; =\; \backslash frac\{\backslash log.\; b\}\{\backslash log.\; e\}$; then, by the formula of § 54,

and therefore

Thus we have $\backslash frac\{x^n\; -\; b^n\}\{m\}$, or $\backslash frac\{x^\{n+1\}\; -\; t^\{n+1\}\}\{n+1\}$ expressed generally by a series, all the terms of which, except the first, being multiplied by $m$ or $n+1$, will vanish when $n+1=0$, or when $n=-1$, hence it appears, that the general equation $y\; =\; \backslash frac\{a(x^\{n+1\}\; -\; t^\{n+1\})\}\{n+1\}$, becomes in the particular case of $n=-1$, $y\; =\; a(\backslash rho\; -\; q)$, which, substituting for $\backslash rho$ and $q$ their values, and observing that

where $\backslash log.\; e$, and $\backslash log.\; \backslash frac\{x\}\{b\}$ are to be taken according to the same system, which may be any system of logarithms whatever. So that if we take the Napierian system, in which $\backslash log.\; e\; =\; 1$, then

where $C$ denotes a constant quantity, and where the letter $l$, in this formula, and in others in which it may occur, is put as an abbreviation of the words Napierian logarithm, so that by $a\; \backslash cdot\; l.\; x$ is meant $a$ multiplied by the Napierian logarithm of $x$, &c.

This expression which we have found for the value of $y$ in the particular case of $\backslash dot\{y\}\; =\; ax^\{-1\}\; \backslash dot\{x\}$, or $\backslash frac\{ax\}\{x\}$, coincides with what we might have found by considering that when $y\; =\; l.\; x$, it has been shewn (§ 57.) that $\backslash dot\{y\}\; =\; \backslash frac\{\backslash dot\{x\}\}\{x\}$, so that conversely, when $\backslash dot\{y\}\; =\; \backslash frac\{ax\}\{x\}$, we may conclude that $y\; =\; a\; \backslash cdot\; l.\; x\; +\; C$ where $C$ denotes a constant quantity, to be determined from the particular question in which the fluxional equation may occur.

104. It must now be evident that if

where are constant numbers, then

here $C$ denotes a constant arbitrary quantity that may be considered as the sum of the constant quantities which ought to be added to the terms each being regarded as a distinct fluent.

105. In general, since that when

where denote any functions of a variable quantity, and $C$ a constant quantity, we have (§ 35. and § 36.)

So on the contrary, if we have any fluxional equation of this last form, we may conclude that

And since that when $u\; =\; vt\; +\; C$, where $u,\; v$ and $t$ denote any function of a variable quantity, and $C$ a constant quantity, we have § 37, $\backslash dot\{u\}\; =\; \backslash dot\{v\}t\; +\; v\backslash dot\{t\}$, so on the contrary, if

we may conclude that

and in like manner if we have

we may infer from § 39. that

106. It is often convenient to denote the fluent of a fluxional expression without actually exhibiting that fluent. For this purpose we shall employ the sign $\backslash int$ putting it before the fluxion whose fluent we mean to denote. Thus, by the expression $\backslash int\; ax^n\; \backslash dot\{x\}$, is to be understood the fluent of $ax^n\; \backslash dot\{x\}$; and as this fluent has been found to be $\backslash frac\{ax^\{n+1\}\}\{n+1\}\; +\; C$, we may express this conclusion in symbols shortly thus,

107. Suppose we have $\backslash dot\{y\}\; =\; (ax+b)^m\; \backslash dot\{x\}$, we may expand $(ax+b)^m$ into a series, and multiply the series by $\backslash dot\{x\}$, and find the fluent of each term of the result. But we may also find the fluent of this expression without employing the development of $(ax+b)^m$, by proceeding thus. Put $ax+b\; =\; z$, then $x\; =\; \backslash frac\{z-b\}\{a\}$, and $\backslash dot\{x\}\; =\; \backslash frac\{\backslash dot\{z\}\}\{a\}$. Substitute now these values of $ax+b$, and $\backslash dot\{x\}$, in the expression for $\backslash dot\{y\}$, and it becomes $\backslash dot\{y\}\; =\; \backslash frac\{z^m\; \backslash dot\{z\}\}\{a\}$; hence we have (§ 101.) $y\; =\; \backslash frac\{z^\{m+1\}\}\{a(m+1)\}\; +\; C$, and consequently, by substituting $(ax+b)$ for $z$,

108. Suppose that $\backslash dot\{y\}\; =\; (ax^n\; +\; b)^m\; \backslash dot\{x\}$. By putting as before $ax^n\; +\; b\; =\; z$, we have $na\; x^\{n-1\}\; \backslash dot\{x\}\; =\; \backslash dot\{z\}$, and $x^\{n-1\}\; \backslash dot\{x\}\; =\; \backslash frac\{\backslash dot\{z\}\}\{na\}$; hence $\backslash dot\{y\}\; =\; \backslash frac\{z^m\; \backslash dot\{z\}\}\{na\}$, and $y\; =\; \backslash frac\{z^\{m+1\}\}\{na(m+1)\}\; +\; C$, and, substituting for $z$ its value $ax^n\; +\; b$,

$y\; =$

109. Let us now consider fractional functions, and to begin with a simple case, let us suppose that $y\; =\; \backslash frac\{Ax^n\; x^m\}\{(ax+b)^n\}$. Put $ax+b=x$, then $x\; =\; \backslash frac\{x-b\}\{a\}$, $\backslash dot\{x\}\; =\; \backslash frac\{\backslash dot\{x\}\}\{a\}$, and consequently,

We have now only to find the development of $(x-b)^n$, to multiply each of its terms by $\backslash dot\{x\}$ and divide it by $x^n$, and take the fluent of the result.

Let us take for example the case of $m=3$, and $n=2$, then

Hence, taking the fluents of the several terms, as in § 105, we have

Let us now restore the value of $x$, and then it appears, that when

110. If we suppose that

then, we may write the equation thus,

and take the fluent of each term, in the same manner as

we have found the fluent of $\backslash frac\{Ax^3\; \backslash dot\{x\}\}\{(ax+b)^2\}$.

Of the Fluents of rational Fractions.

111. Every fluxion that is a rational fraction is comprehended under this general formula,

which, by putting $U$ to denote the expression between the parentheses in the numerator, and $V$ the denominator, may be represented by $\backslash frac\{U\; \backslash dot\{x\}\}\{V\}$. Now in the first

place, we remark that the greatest exponent of the powers of $x$ in the numerator may be supposed to be less than that of its powers in the denominator. For if it were not so, by dividing $U$ by $V$, and calling $Q$ the quotient, and $R$ the remainder, we should have $\backslash frac\{U\; \backslash dot\{x\}\}\{V\}\; =\; Q\; \backslash dot\{x\}\; +\; \backslash frac\{R\; \backslash dot\{x\}\}\{V\}$, and

Now, $Q$ being a rational and integer function, $\backslash int\; Q\; \backslash dot\{x\}$ may be found, as in § 101, and it only remains to find

$\backslash int\; \backslash frac\{R\; \backslash dot\{x\}\}\{V\}$, an expression in which the highest exponent of the powers of $x$ in $R$ is less by unity than in $V$; so that the fraction $\backslash frac\{R\; \backslash dot\{x\}\}\{V\}$ may be generally expressed thus,

The general method of finding the fluent of a fractional expression of this form consists in decomposing it into a series of other fractions, the denominators of which are more simple. These fractions may be found by proceeding as follows: By putting the denominator of the proposed fraction equal to 0, we get this equation,

Suppose now that the roots of this equation are found, and that they are denoted by

which quantities we shall suppose in the first place, are all unequal. Then the expression which has been assumed as equal to 0, may (ALGEBRA, Sect. X.) be considered as the product of $n$ factors

Let the proposed fraction $\backslash frac\{R\}\{V\}$ be now assumed as equal to the sum of the simple fractions

having for their denominators the simple factors of the denominator of the proposed fraction, and for their numerators quantities which are constant, but as yet are indetermined.

That we may avoid complicated calculations, and present a determinate object to the mind, let us suppose that the fluxion of which we are to find the fluent is

and that we have by the resolution of the cubic equation $x^3\; +\; A\text{'}x^2\; +\; B\text{'}x\; +\; C\text{'}\; =\; 0$ found

The common denominator of these fractions is the same as that of the proposed fraction, and each of the numerators, as well as their sum, is a function of $x$ of a degree lower than the denominator, that is, in the present case, it is a function of the second degree. By taking the actual products of the factors in the numerators, and adding the results, we find the sum of the fractions equal to

where $V$ denotes the common denominator $(x+a)(x+a\text{'})(x+a\text{'}\text{'})\; =\; x^3\; +\; A\text{'}x^2\; +\; B\text{'}x\; +\; C\text{'}$. Setting aside

the factor $\backslash frac\{\backslash dot\{x\}\}\{V\}$ of the above expression, we are now to compare that part of it which involves the three indeterminate quantities $N,\; N\text{'},\; N\text{'}\text{'}$, with $Ax^3\; +\; Bx\; +\; C$, the numerator of the proposed fraction, thus we obtain these three equations

By these equations, which are all of the first degree, we may determine the values of $N,\; N\text{'}$ and $N\text{'}\text{'}$, and thus

we have the proposed fraction $\backslash frac\{(Ax^3\; +\; Bx\; +\; C)\; \backslash dot\{x\}\}\{x^3\; +\; A\text{'}x^2\; +\; B\text{'}x\; +\; C\text{'}\}$ equal to

where $N,\; N\text{'},\; N\text{'}\text{'}$, and $a,\; a\text{'},\; a\text{'}\text{'}$, are constant and known quantities.

Put $x+a=\backslash infty$, then $\backslash dot\{x\}=\backslash infty$, and the fraction $\backslash frac\{N\; \backslash dot\{x\}\}\{x+a\}$ is transformed to $\backslash frac\{N\; \backslash dot\{x\}\}\{\backslash infty\}$, of which the fluent is $N\; \backslash cdot\; 1\; \backslash cdot\; \backslash infty\; =\; N\; \backslash cdot\; 1$.

$(x+a)$ (§ 103). In like manner we find $\backslash int\; \backslash frac\{N\text{'}\; \backslash dot\{x\}\}\{x+a\text{'}\}\; =\; N\text{'}\; \backslash cdot\; 1\; \backslash cdot\; (x+a\text{'})$, and $\backslash int\; \backslash frac\{N\text{'}\text{'}\; \backslash dot\{x\}\}\{x+a\text{'}\text{'}\}\; =\; N\text{'}\text{'}\; \backslash cdot\; 1\; \backslash cdot\; (x+a\text{'}\text{'})$, and consequently

where by const. is meant a constant quantity.

It is easy to extend this mode of proceeding to the general formula given at the beginning of this §; and it is obvious, that as often as the denominator of a rational fraction can be decomposed into real and unequal factors, the determination of the fluent of that fraction is attended with no other difficulty than this decomposition, which requires the numerical resolution of equations.

112. We have supposed that the factors of the denominator of the proposed fraction are unequal among themselves, and it is only when this is the case that the fraction can be decomposed into others, having all this form $\backslash frac\{N\}\{x+a\}$. If we suppose that the denominator $x^n\; +\; A\text{'}x^\{n-1\}\; +\; B\text{'}x^\{n-2\}\; \backslash dots\; +\; T\text{'}$ has a factor of the form $(x+a)^p$, then the proposed fraction

must be assumed equal to

where $P,\; Q,\; R\; \backslash dots\; Y,\; N\text{'},\; N\text{'}\text{'}$, &c. denotes indeterminate but constant quantities, and $x+a\text{'},\; x+a\text{'}\text{'},$ &c. are the remaining factors of the denominator of the proposed fraction. To determine the quantities $P,\; Q,\; R\; \backslash dots\; Y,\; N\text{'},\; N\text{'}\text{'}$, &c. we must now proceed in all respects as in last §, that is, we must reduce the fractions involving these quantities to a common denominator, which will be the same as the denominator of the proposed fraction; then we must add the numerators, and put the coefficient of each power of $x$ in the sum equal to the coefficient of the same power in the numerator of the proposed fraction. Thus we shall have as many equations as indeterminate quantities, and by resolving these equations, the values of these quantities will be found.

Having thus determined all the quantities $P,\; Q,\; R\; \backslash dots\; Y$, which enter into the fraction

its fluent may be found as shown in § 109. But we may also assume it equal to

and, it is easy to see, that by reducing these fractions to

a common denominator and adding them, the numerator of their sum will have the same form as that of the fraction whose fluent we are seeking; so that the values of the indeterminate quantities will be found by putting the coefficients of the same power of $x$ in both numerators equal to each other. To find the fluent of

$\backslash frac\{M\; \backslash dot\{x\}\}\{(x+a)^p\}$ we may assume $x+a\; =\; z$, then $\backslash dot\{x\}\; =\; \backslash dot\{z\}$, and

In like manner

and so on, all the fluents being algebraic, except the last $\backslash int\; \backslash frac\{M\text{'}\text{'}\; \backslash dots\; \backslash dot\{x\}\}\{x+a\}$ which is $M\text{'}\text{'}\; \backslash dots\; 1\; \backslash cdot\; (x+a)$, a logarithmic function.

113. In resolving the equation

it may happen that some of its roots are imaginary quantities, and then some of the simple factors will be imaginary. These factors always occur in pairs (ALGEBRA, § 179.) and have this form

so that their product

is a real factor of the second degree. As every corresponding pair of imaginary simple factors may be united in this manner into a real factor of the second degree, if these factors are all unequal, we may avoid introducing imaginary quantities into the fluent of the proposed fraction by proceeding thus. Let $x\; +\; \backslash alpha\; +\; \backslash beta\; \backslash sqrt\{-1\}$ and $x\; +\; \backslash alpha\; -\; \backslash beta\; \backslash sqrt\{-1\}$, denote two corresponding imaginary simple factors of the denominator. Instead of the two simple fractions

which would have been assumed if the factors had been real, assume a single fraction

the denominator of which is a real function of $x$ of the second degree, viz. that which is the product of the two imaginary factors. Here $K$ and $L$ denote two constant but indeterminate coefficients, the values of which, as

VOL. VIII. Part II.

also those of the other indeterminate coefficients are to be found as before.

If the denominator of the proposed fraction have several equal factors of the second degree resulting from its imaginary simple factors, so that the product of those equal factors is

then, corresponding to this product, we must, among the fractions having indeterminate coefficients, assume one of this form

where $Q\text{'},\; R\text{'}\; \backslash dots\; Y\text{'}$ denote constant and indeterminate coefficients, the values of which will be found in all respects as those of the others.

We are now to find the fluents of these two fluxional expressions, beginning with the first, viz.

Put $x\; +\; \backslash alpha\; =\; z$, then it becomes

and this again, by putting $L\; -\; K\backslash alpha\; =\; M$, is resolved into these two fluxions

We can immediately find the fluent of the first of these, by putting $z^2\; +\; \backslash beta^2\; =\; v$, for then $z\; \backslash dot\{z\}\; =\; \backslash frac\{v\}\{2\}$, and

With respect to the other fluxion, if we put $z\; =\; \backslash beta\; y$, we have

but we have seen (§ 60.) that $\backslash frac\{\backslash dot\{y\}\}\{1\; +\; y^2\}$ is the fluxion of an arch of which the tangent is $y$, therefore

It is proper to remark that if $\backslash frac\{z\}\{\backslash beta\}$ be the tangent of an arch, then the sine of that arch is $\backslash frac\{z\}\{\backslash sqrt\{z^2\; +\; \backslash beta^2\}\}$, and its

cosine is $\backslash frac\{\backslash beta\}\{\backslash sqrt\{z^2\; +\; \backslash beta^2\}\}$, thus we may express the fluent

Thus we shall obtain

Substituting now this value of $\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^\{q-1\}\}$ in the former equation, we have $\backslash int\; \backslash frac\{M\; \backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^q\}$ equal to

It is easy to see, that like as we obtained an expression for the fluent of $\backslash frac\{\backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^\{q-1\}\}$ by substituting $q-1$ for $q$, and supposing $M=1$, in the equation preceding the last; so by substituting $q-2$ for $q$, we shall obtain an expression for the fluent of $\backslash frac\{\backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^\{q-2\}\}$, which expression will consist of two terms, one an algebraic function of $z$, and the other $\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^\{q-3\}\}$ multiplied by a constant and given coefficient. This value of $\backslash frac\{z\}\{(x^2\; +\; \backslash beta^2)^\{q-2\}\}$ when substituted in the last equation will produce an expression for $\backslash int\; \backslash frac\{M\; \backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^q\}$ consisting of algebraic quantities and $\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^\{q-3\}\}$. By continuing this process it is evident that we shall at last have $\backslash int\; \backslash frac\{M\; \backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^q\}$ expressed by a series of algebraic quantities, and $\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^q\}$, and here we must stop, for if we repeat the process with a view to make the fluent depend on $\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{(x^2\; +\; \backslash beta^2)^0\}$ that is on $\backslash int\; z$, or $z$, we shall find that the coefficient of this quantity becomes infinite. As to the fluent of $\backslash frac\{\backslash dot\{z\}\}\{z^2\; +\; \backslash beta^2\}$ we have exhibited the expression for it in last §.

In comparing together the results which have been obtained in the preceding articles, it must appear that when a fluxion is expressed by a rational fraction, if we grant the resolution of equations, the fluent may always

be assigned either algebraically, or by means of arches of a circle or logarithms; and that to prepare it for a solution, we must decompose the fraction into others, whose denominators may be either binomial or trinomial quantities. This decomposition may always be effected by the method of indeterminate coefficients. There are, however, several analytical artifices by which the labour of calculation may be greatly shortened. These we now proceed to explain.

115. Let us recur to the fraction $\backslash frac\{U\}\{V\}$ and suppose that $x+a$ is one of the unequal factors of the denominator $V$, so that we have $V\; =\; (x+a)Q$; Let us now put $\backslash frac\{U\}\{V\}\; =\; \backslash frac\{A\}\{x+a\}\; +\; \backslash frac\{P\}\{Q\}$, $A$ being supposed a constant quantity, and $P$ an indeterminate function of $x$, but such as not to be divisible by $x+a$. Then we have $U\; =\; AQ\; +\; P(x+a)$, and hence $P\; =\; \backslash frac\{U\; -\; AQ\}\{x+a\}$. As $P$ is an integer function with respect to $x$, it follows from this equation that $U\; -\; AQ$, which is also a rational and integer function of $x$, is divisible by $x+a$, and consequently has $x+a$ for a factor; therefore, the function $U\; -\; AQ$ will vanish when we substitute $-a$ in it instead of $x$, seeing that $-a$ is the value of $x$ that makes the factor $x+a=0$. Let us denote by $u$ and $q$, what $U$ and $Q$ become by this substitution, which however will not affect the indeterminate quantity $A$, because it is independent of $x$. We have therefore $u\; -\; AQ\; =\; 0$, and consequently $A\; =\; \backslash frac\{u\}\{q\}$.

This value of $A$ requires that we should know the function $Q$ given by the equation $V\; =\; (x+a)Q$, and we may always find it by dividing $V$ by $x+a$. The direct method of fluxions affords also a very simple method of determining it. For by taking the fluxion of the above equation we have

if in this result we make $x+a=0$, or $x=-a$, and denote by $v$ what $\backslash dot\{V\}$ becomes by that substitution, we shall have $v=q$, and consequently $A\; =\; \backslash frac\{u\}\{v\}$.

The expression $A\; =\; \backslash frac\{u\}\{q\}$ has always a finite value, for the numerator and denominator can never become $=0$, because we suppose the fraction $\backslash frac\{U\}\{V\}$ reduced to its lowest terms, and consequently, that the numerator $U$ has not for a factor $x+a$, which is a factor of the denominator, but which being contained in it only once does not enter into $Q$.

116. Let us now consider how the numerators of the fractions, into which the proposed fraction $\backslash frac\{U\}\{V\}$ is to be decomposed, are to be found in the case of the denominator

Inverse Method. Numerator V having equal factors of the first degree. In this case we have $V\; =\; Q(x+a)^n$, and we assume

By reducing to a common denominator, we find U equal to

and P equal to

and as P ought to be an integer function of x, the numerator of its value is necessarily divisible n times successively by x+a; therefore, that numerator ought to be equal to 0, when -a is substituted in it instead of x. Now this substitution being made, each of the terms of the numerator which is multiplied by x+a vanishes, so that there remains only U-AQ, but that this quantity may be divisible by x+a it is necessary that $u\; -\; qA\; =\; 0$, where u and q denote the same as in last §, hence $A\; =\; \backslash frac\{u\}\{q\}$.

This value of A changes U-QA into $U\; -\; \backslash frac\{u\}\{q\}Q$, which must be divisible by x+a. Let us, with a view to abridge, put $U\; -\; \backslash frac\{u\}\{q\}Q\; =\; U\text{'}(x+a)$, then, substituting this quantity in the value of P, and dividing both numerator and denominator by x+a, we have P equal to

Now to obtain B we make $x+a=0$, then, putting $u\text{'}$ to denote what $U\text{'}$ becomes by substituting -a in it in place of x, we have $u\text{'}\; -\; qB\; =\; 0$, and $B\; =\; \backslash frac\{u\text{'}\}\{q\}$.

Instead of B let its value be substituted in $U\text{'}\; -\; QB$, and this quantity becomes $U\text{'}\; -\; \backslash frac\{u\text{'}\}\{q\}Q$, which vanishing when $x+a=0$, will have x+a for a divisor; therefore, we may put $U\text{'}\; -\; \backslash frac\{u\text{'}\}\{q\}Q\; =\; U\text{'}\text{'}(x+a)$, then, substituting this last quantity instead of the former in the value of P, and dividing the numerator and denominator by x+a, we have P equal to

By continuing the same mode of reasoning, and the same notation, we find $u\text{'}\text{'}\; -\; qC\; =\; 0$, and $C\; =\; \backslash frac\{u\text{'}\text{'}\}\{q\}$. And so on with the remaining quantities.

Inverse Method. The direct method of fluxions facilitates greatly the preceding operations. For the numerator of P being divisible by $(x+a)^n$ is necessarily of this form $X(a+x)^n$, X being an integer function of x, but which does not contain the factor x+a. Now agreeably to what has been shown in § 88, the successive fluxions of this numerator, as far as the n-1 order inclusive, vanish when x+a is supposed = 0. By giving to the numerator the following form

and observing that the function Q does not contain the factor x+a, it is manifest that it is only the part of this expression between the parentheses which ought to be divisible by $(x+a)^n$. Let us put $\backslash frac\{U\}\{Q\}\; =\; Z$, then the successive fluxions of that part are

and these results ought all to vanish when we put x+a=0. Thus we have

observing that in each of these functions $\backslash frac\{Z\}\{x\},\; \backslash frac\{\backslash dot\{Z\}\}\{2x^2\},\; \backslash dots$ we must substitute -a instead of x.

The most simple way to find the value of Q in this case is to divide V by $(x+a)^n$, but we may also find it by the direct method of fluxions, as in the preceding §; for, since $V\; =\; Q(x+a)^n$, if we take the fluxion of each side of this equation n times, and then make x+a=0, we shall find, § 88, the nth fluxion of V equal to $1\; \backslash cdot\; 2\; \backslash cdot\; 3\; \backslash dots\; n\; Q\; x^n$, and consequently

117. Let us now consider how we are to find the numerator of the fraction which forms a part of $\backslash frac\{U\}\{V\}$ when it has this form

Assume

then, reducing the latter part of this equation to a common denominator, we find

As $P$ is supposed to be an integer function with respect to $x$, it follows that $U\; -\; Q(Ax+B)$ is divisible by $x^3+2ax+a^2+\backslash beta^2$; therefore, the former of these two quantities must contain among its factors those of the latter, and the quantities, which, being substituted for $x$, cause the latter to vanish, must also make the former vanish. But the factors of $x^3+2ax+a^2+\backslash beta^2$ are $x+\backslash alpha+\backslash beta\backslash sqrt\{-1\}$, and $x+\backslash alpha-\backslash beta\backslash sqrt\{-1\}$, and these, being put each $=0$, give us $x\; =\; -(\backslash alpha+\backslash beta\backslash sqrt\{-1\})$, and $x\; =\; -(\backslash alpha-\backslash beta\backslash sqrt\{-1\})$, therefore, each of these values of $x$ being substituted in $U\; -\; Q(Ax+B)$ ought to make that quantity vanish. Let us denote by $u\; \backslash pm\; u\text{'}\backslash sqrt\{-1\}$, and by $q\; \backslash pm\; q\text{'}\backslash sqrt\{-1\}$ what $U$ and $Q$ respectively become when $-(\backslash alpha\; \backslash pm\; \backslash beta\backslash sqrt\{-1\})$ is substituted in each instead of $x$, then, after this transformation, we have

This equation is twofold, because of the sign $\backslash pm$ with which several of its terms are affected, and it is equivalent to those which would be formed by putting the real part equal to 0, and the imaginary part $=0$; from this consideration we have

The function $Q$ may be found as in § 115. For, if we take the fluxions of each side of the equation

Let the two values of $x$, to wit $-(\backslash alpha\; \backslash pm\; \backslash beta\backslash sqrt\{-1\})$, be substituted instead of it in this equation, then, putting $v\; \backslash pm\; v\text{'}\backslash sqrt\{-1\}$ to denote what the expression $\backslash frac\{\backslash dot\{V\}\}\{x\}$ becomes by that substitution, and writing $q\; \backslash pm\; q\text{'}\backslash sqrt\{-1\}$ instead of $Q$, we have

which, by multiplying the terms of the fraction on the latter side of the equation by $\backslash sqrt\{-1\}$, becomes

Hence, by putting the real part of each side of this equation equal to each other, and also the imaginary parts equal to each other, we find

118. If the factor $x^3+2ax+a^2+\backslash beta^2$ is found several times in the denominator of $V$, so that

$V\; =\; Q(x^3+2ax+a^2+\backslash beta^2)^n$, then, § 113, we assume in this case $\backslash frac\{U\}\{V\}$ equal to

reducing this expression to a common denominator, and so ordering the equation as to bring $P$ to stand alone on one side, we find $P$ equal to

By reasoning in this as in the preceding case, it may be concluded that the numerator of this expression ought to vanish when $-(\backslash alpha\; \backslash pm\; \backslash beta\backslash sqrt\{-1\})$ is substituted in it instead of $x$; therefore, putting $u\; \backslash pm\; u\text{'}\backslash sqrt\{-1\}$, and $q\; \backslash pm\; q\text{'}\backslash sqrt\{-1\}$ to denote the same things as before, we deduce from that substitution

the very same equation for the determination of $A$ and $B$, as we have already found in last §.

Having found the values of these quantities, they may be substituted in the numerator of $P$, and the terms $U\; -\; Q(Ax+B)$ becoming divisible by $x^3+2ax+a^2+\backslash beta^2$, the whole expression becomes divisible by the same quantity. Calling therefore $U\text{'}$ the quotient arising from the division of $U\; -\; Q(Ax+B)$ by $x^3+2ax+a^2+\backslash beta^2$, we have $P$ equal to

If in this numerator we substitute instead of $x$ its values deduced from the equation $x^3+2ax+a^2+\backslash beta^2=0$, and put the result $=0$, we may determine $A\text{'}$ and $B\text{'}$ in the very same way that we have already determined $A$ and $B$, and by proceeding in this manner we shall find the remaining coefficients $A\text{'}\text{'}$, $B\text{'}\text{'}$, &c.

This case is quite analogous to that which has been already treated in § 116, and the direct method of fluxions applies to it in the same manner as to the other. For since $Q$ does not contain the factor $x^3+2ax+a^2+\backslash beta^2$, if the numerator of $P$ be divided by the function $Q$, the result, which may be denoted by $r$, ought to be of this form

form $r\; =\; X(x^2\; +\; 2\backslash alpha\; x\; +\; \backslash alpha^2\; +\; \backslash beta^2)^n$ and consequently ought to vanish, as well as all its fluxions, from the first order to the $n-1$ order, inclusively, when $x^2\; +\; 2\backslash alpha\; x\; +\; \backslash alpha^2\; +\; \backslash beta^2\; =\; 0$; this being the case, we have these equations

and so on to the $n-1$ fluxion of $r$, which ought also to be $=\; 0$; each of these equations becomes twofold when we substitute instead of $x$, the values of which it is susceptible in consequence of the equation $x^2\; +\; 2\backslash alpha\; x\; +\; \backslash alpha^2\; +\; \backslash beta^2\; =\; 0$. By putting the real and the imaginary parts separately $=\; 0$, we shall obtain as many equations as are sufficient to determine $A,\; B,\; A\text{'},\; B\text{'},\; \backslash dots$

It may also be remarked, that from the equation

we find $Q$ equal to the quotient arising from the division of the $n$th fluxion of $V$ by the $n$th fluxion of $x^2\; +\; 2\backslash alpha\; x\; +\; \backslash alpha^2\; +\; \backslash beta^2$, observing to assume

119. We shall now give some applications of what has been said relative to the fluxions of rational fractions. Suppose the fraction to be

The factors of its denominator are easily found, for it may be put under this form

the factor $x^2\; -\; 1$ may be decomposed into $x^2\; -\; 1$ and $x^2\; +\; 1$, or $x\; -\; 1,\; x\; +\; 1$, and $x^2\; +\; 1$, thus we have the denominator equal to

therefore (§ 111, § 112, and § 113.) the proposed fraction is to be decomposed as follows

By reducing these fractions to a common denominator, and comparing the numerator of their sum with that of the proposed fraction, we might determine the unknown quantities $A,\; B,\; C,\; \backslash dots$ we shall, however, rather employ the methods that have just been explained.

By comparing this particular example $\backslash frac\{x\}\{x^2+x^3-x^4-x^5\}$

with the general expression $\backslash frac\{Ux\}\{V\}$, it appears that $U=1$, and $V=x^2+x^3-x^4-x^5$. First let us investigate the numerator of the fraction $\backslash frac\{A\}\{x-1\}$, and for this purpose

we employ the formula $A\; =\; \backslash frac\{u\}\{v\}$ (§ 115.). As we have $U=1$, it is evident that $u=1$; and since $V=x^2+x^3$

$-x^4-x^5$, therefore $\backslash frac\{V\}\{x\}\; =\; 8x^7\; +\; 7x^6\; -\; 4x^5\; -\; 3x^4$. If in this expression we substitute $+1$ instead of $x$ (viz. the value of $x$ deduced from the equation $x-1=0$) we find the result to be 8, therefore $v=8$. So that

Let us next investigate the values of $B$ and $C$ in the fractions $\backslash frac\{B\}\{(x+1)^2\}$ and $\backslash frac\{C\}\{x^2+1\}$, by means of the rule of § 116, and that we may make the symbols expressing the quantity under consideration agree with those employed in that formula, let us exchange the letters $B$ and $C$ for $A$ and $B$, so that we are to consider $\backslash frac\{A\}\{(x+1)^2\}$

+ $\backslash frac\{B\}\{x+1\}$. In the first place we have

Put $x+1=0$, then $x=-1$; substituting now this value of $x$ in the value of $Q$, the result is $4=q$, therefore

$A\; =\; \backslash frac\{u\}\{q\}\; =\; \backslash frac\{1\}\{4\}$. Let this value of $A$ be substituted for $A$ in the expression for $U\text{'}$ in the § above cited, and we have

Hence putting $-1$ instead of $x$ in the expression for $U\text{'}$, we have $u\text{'}\; =\; \backslash frac\{9\}\{2\}$ and $B\; =\; \backslash frac\{u\text{'}\}\{q\}\; =\; \backslash frac\{9\}\{8\}$. Thus the two fractions under consideration are found to be $\backslash frac\{1\}\{4\}\; \backslash cdot\; \backslash frac\{1\}\{(x+1)^2\}$ and

$\backslash frac\{9\}\{8\}\; \backslash cdot\; \backslash frac\{1\}\{x+1\}$. We might have deduced the value of $B$ from the formula $B\; =\; \backslash frac\{z\}\{x\}$, § 116, where $Z$ is put for $\backslash frac\{U\}\{Q\}$ for we have

If in this expression we substitute $-1$ instead of $x$, it becomes $\backslash frac\{18\}\{16\}\; =\; \backslash frac\{9\}\{8\}$, the same value for $B$ as before.

Let us now consider the fractions $\backslash frac\{D\}\{x^3\}\; +\; \backslash frac\{E\}\{x^4\}\; +\; \backslash frac\{F\}\{x^5\}$, or exchanging the symbols $D,\; E,\; F$ for $A,\; B,\; C$,

The numerators $A,\; B,\; C$ may all be found from these formulas of § 116.

$A\; =$

observing that in this case $Q\; =\; x^5\; +\; x^4\; -\; x\; -\; 1$; and that we must substitute 0 instead of $x$ in each formula, after taking the fluxions. Now we have

Hence putting $x=0$, we find

There yet remains the fraction $\backslash frac\{Gx+H\}\{x^2+1\}$, or $\backslash frac\{Ax+B\}\{x^2+1\}$ to be considered. It may be found by subtracting the sum of all the others from the proposed fraction; we proceed however to find it directly by the formulas of § 117. In the first place we have $Q\; =\; x^5\; +\; x^4\; -\; x^4\; -\; x^3$; next, the factor $x^2\; +\; 1$ being put $=0$ gives $x\; =\; \backslash pm\; \backslash sqrt\{-1\}$, $x=0$, $x=0$. Hence we find

The equations which determine $A$ and $B$ thus become

therefore $A\; =\; B\; =\; -\backslash frac\{1\}\{4\}$, and

Thus upon the whole, the proposed fraction

$\backslash frac\{x\}\{x^5\; +\; x^4\; -\; x^4\; -\; x^3\}$ is decomposed into the following

The manner of finding the fluent of each of these has been already explained, and the result of taking all the fluents is

The union of all the algebraic terms produces the Inverse Method. fraction $\backslash frac\{2-2x-5x^2\}\{4x^2(1+x)\}$, and that of the logarithmic quantities gives

We have therefore upon the whole $\backslash int\; \backslash frac\{x\}\{x^5\; +\; x^4\; -\; x^4\; -\; x^3\}\; dx$ equal to

120. When a fluxion is a rational fraction having either of these forms

we can always, by the application of a particular theorem in analysis, resolve its denominator into real factors of the first and second degrees. The theorem to which we allude is this. Let $n$ be any positive integer, and let $z$ denote any arch of a circle, of which the radius is unity, then

$(\backslash cos\; z\; \backslash pm\; \backslash sqrt\{-1\}\; \backslash sin\; z)^n\; =\; \backslash cos\; n\; z\; \backslash pm\; \backslash sqrt\{-1\}\; \backslash sin\; n\; z$. We proceed to prove this theorem. Because

If we put $\backslash cos\; z\; =\; \backslash sqrt\{-1\}\; \backslash sin\; z\; =\; v$,

Therefore taking the sum of these two equations,

Now by the arithmetic of fines (ALGEBRA, § 358.)

&c.

Therefore, substituting in the first of these equations $v\; +\; \backslash frac\{1\}\{v\}$ instead of $2\; \backslash cos\; z$, we have

In like manner, substituting in the second equation, $v\; +\; \backslash frac\{1\}\{v\}$ instead of $2\; \backslash operatorname\{cos\}.\; z$, and $v^2\; +\; \backslash frac\{1\}\{v^2\}$ instead of $2\; \backslash operatorname\{cos\}.\; 2z$, we have

Proceeding in the same way with the third and following equations, we find

so that we may conclude in general that

Hence we have this quadratic equation

from which, by completing the square, we find

therefore, by substituting for $v$ the quantity it was put to represent, and observing that $\backslash sqrt\{(\backslash operatorname\{cos\}^2\; nz\; -\; 1)\}\; =\; \backslash sqrt\{(-\backslash sin^2\; nz)\}\; =\; \backslash sqrt\{-1\}\; \backslash sin.\; nz$, we have

as was to be proved.

121. The function $x^n\; =\; a^n$ is transformed to $a^n(y^n\; \backslash pm\; 1)$ by putting $x\; =\; ay$, and to discover its factors, we must resolve the equation

The expression $y\; =\; \backslash operatorname\{cos\}.\; z\; +\; \backslash sqrt\{-1\}\; \backslash sin.\; z$ satisfies this equation, by a very simple determination of the arch $z$;

for we have $y^n\; =\; (\backslash operatorname\{cos\}.\; z\; +\; \backslash sqrt\{-1\}\; \backslash sin.\; z)^n\; =\; \backslash operatorname\{cos\}.\; nz\; +\; \backslash sqrt\{-1\}\; \backslash sin.\; nz$, and as by putting $\backslash pi$ to denote half the circumference, and $m$ any whole number, we have (ALGEBRA, § 352)

where the sign $+$ is to be taken, if $m$ be an even number, but $-$ if it be odd, we have only to suppose $n\backslash pi\; =\; m\backslash pi$, in order to obtain $y^n\; =\; \backslash pm\; 1$.

That we may distinguish the case in which $m$ is even,

from that in which it is odd, we shall write for the first $2m$, and for the second $2m+1$; we therefore make

By the first hypothesis, we find

and by the second

122. By means of the indeterminate number $n$, each of these expressions for $y$ furnishes all the values of which this quantity is susceptible, for we may take successively

The first formula gives

&c.

It is evident that we shall always have different results as far as $m=n-1$. If, however, we suppose $m=n$, then we have $y\; =\; \backslash operatorname\{cos\}.\; 2\backslash pi\; =\; 1$, which is the same as the first of the values already obtained, and if we suppose $m=n+1$, then (ALGEBRA, § 25.)

which is the same as the second value, and so on with respect to the others.

By this mode of proceeding we shall not only obtain the $n$ roots of the equation $y^n\; =\; 1$, or $y^n\; -\; 1\; =\; 0$, but, with a little attention, we shall discover that these roots may be arranged in pairs, by bringing together those that only differ in the sign of the radical $\backslash sqrt\{-1\}$; for since

it follows that

Hence it appears that we may comprehend all the roots of the equation $y^n\; -\; 1\; =\; 0$ in the single expressions

The first and the last of the factors of the second degree are the squares of $y\; -\; 1$, and $y\; +\; 1$, factors of the first degree each of which only enters once into the proposed function; it will therefore be necessary, when we employ the factors of the second degree, to reject the first and last, and take instead of them

The factors of the first degree of the function $y^n\; -\; 1$ are

but it is to be observed that the first factor of the second degree is the square of $y\; -\; 1$, which enters only once into the proposed function.

123. When the function to be decomposed into factors is $y^n\; +\; 1$, the formula

which corresponds to that case (§ 121.) is also susceptible of the double sign $\backslash pm$, provided we stop at the value of $m$ which gives

according as $n$ is odd or even; hence it follows that

When among these last there is found some which are squares, we must take only one of their simple factors, in the same way as in the two preceding examples.

When the function is $y^n\; +\; 1$, the factors of the first degree are

The function $y^6\; +\; 1$ has for factors of the first degree

and those of the second,

124. Such functions as are of this form $x^\{2n\}\; +\; 2x^n\; +\; 1$ may be treated in the same manner as those which consist of only two terms. By putting the function $=\; 0$, and resolving the equation which is thus produced, in the same manner as if it were of the second degree, we find the factors to be

if $\backslash rho^2$ exceed $q$, the second term of these factors is real, and by making

we have functions of the form

to decompose into factors.

When , then we put $\backslash rho\; =\; a^n$, $q\; =\; b^n$, $x\; =\; by$, and the function becomes

but the condition , or , makes , and , therefore $\backslash frac\{a^n\}\{b^n\}$ may be represented by the cosine of a given arch $\backslash delta$, and the proposed function will be reduced to

we have then only to resolve the equation

and we immediately find

we now assume as in § 121,

then we find (§ 120),

which expression for $y^n$, being compared with its other value, gives

These relations will be satisfied if we suppose $n\backslash alpha\; =\; 2m\backslash pi\; +\; \backslash delta$, $m$ being any whole number whatever, for

we have therefore

The factors of the first degree of the function

will consequently be comprehended in this formula

If $\backslash rho$ the coefficient of the second term of the proposed function be negative, the only change necessary is to make $\backslash rho\; =\; -a^n$, and to take the arch $\backslash delta$ greater than a quadrant.

Of the Fluxions of Irrational Functions.

125. When a fluxionary expression involves irrational functions, we must endeavour either to transform it into another that is rational, or to reduce it to a series of irrational terms of this form $Ax^\{\backslash frac\{m\}\{n\}\}$, and then, in either case its fluent may be found by the rules already delivered.

Let us take for example the fluxion $\backslash frac\{(1\; +\; \backslash sqrt\{x\}\; -\; \backslash sqrt[3]\{x^4\})^2\}\{1\; +\; \backslash sqrt[3]\{x\}\}$. It is evident that by putting $x\; =\; x^6$, all the extractions indicated by the radical signs may be effected, and the fluxion may be transformed to $\backslash frac\{6x^5(1\; +\; x^3\; -\; x^4)x\}\{1\; +\; x^2\}$, which, by dividing the numerator by $1\; +\; x^2$, may be otherwise expressed thus,

The fluent of which is

Inverse Method. 126. We shall first consider such fluxions as contain the irrational function $\backslash sqrt\{(A+Bx+Cx^2)\}$, and which have necessarily one or other of these forms

$X$ being put for any rational function of $x$; and it may be remarked, that the latter form comprehends the former, which may be written thus

and here the numerator of the fluxion is a rational function of $x$.

Before we transform the expression $\backslash sqrt\{(A+Bx+Cx^2)\}$ into another that is rational with respect to the variable quantity $x$ contains, we shall put the quantity $A+Bx+Cx^2$ under this form

and, in order to abridge, we shall put

Let us now assume $\backslash sqrt\{(a+bx+x^2)\}\; =\; x+z$, then, squaring both sides of the equation, we find $a+bx\; =\; 2xz+z^2$, hence we get $x\; =\; \backslash frac\{a-z^2\}\{2z-b\}$, and consequently

By means of these values the fluxion $\backslash frac\{X\; \backslash dot\{x\}\}\{\backslash sqrt\{(A+Bx+Cx^2)\}\}$ is transformed into another fluxion $Z\; \backslash dot\{z\}$, where $Z$ denotes a rational function of $z$, which is real when $C$ or $c^2$ is positive; but as when $C$ is negative $c$ becomes imaginary, the fluxion $Z\; \backslash dot\{z\}$ which involves $c$ becomes also imaginary.

In this case we have to consider $\backslash sqrt\{(A+Bx-Cx^2)\}$, and making

it becomes $c\; \backslash sqrt\{(a+bx-x^2)\}$. The quantity $x^2-bx+a$ may always be decomposed into real factors of the first degree; let us represent these factors by $x-a$, and $x-a\text{'}$, then it is evident that

Let us now assume

then, squaring both sides of the equation it becomes divisible by $x-a$, and we have $a\text{'}-x\; =\; (x-a)z^2$, from which we find

values which render the proposed fluxion $\backslash frac\{X\; \backslash dot\{x\}\}\{\backslash sqrt\{(A+Bx+Cx^2)\}\}$ rational.

127. Let us now take for example the fluxion $\backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(A+Bx+Cx^2)\}\}$; by applying to it the first of the preceding transformations it becomes $\backslash frac\{-2\; \backslash dot\{z\}\}\{c(2z-b)\}$, the fluent of which is $-\backslash frac\{1\}\{c\}\; \backslash log(2z-b)\; +\; \backslash text\{const.\}$ Substituting now for $z$ its value $-x\; +\; \backslash sqrt\{(a+bx+x^2)\}$, and for $a,\; b$, and $c$, the quantities they severally represent, the fluent becomes

a result to which we may also give this form

By uniting the constant quantities into one, and observing that the radical quantity $\backslash sqrt\{C\}$ may have the sign $\backslash pm$ prefixed to it, we have at last

128. Let us take for the second example $\backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(A+Bx-Cx^2)\}\}$. By employing the latter transformation of § 126 we have $\backslash frac\{-2\; \backslash dot\{z\}\}\{c(z^2+1)\}$, of which the fluent is

Substituting now instead of $z$ its value $\backslash frac\{\backslash sqrt\{(a\text{'}-x)\}\}\{\backslash sqrt\{(x-a)\}\}$ deduced from the equation $a\text{'}-x\; =\; (x-a)z^2$, and

and putting $\backslash sqrt\{C\}$ for $c$, we get $\backslash int\; \backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(A+Bx-Cx^2)\}\}$ equal to

$x$ and $x\text{'}$ being the roots of the equation

Let us suppose that $A=C=1$ and $B=0$, then the proposed fluxion becomes in this particular case $\backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(1-x^2)\}\}$, and the preceding formula gives for its fluent $-2\; \backslash operatorname\{arc.\}\; \backslash left(\; \backslash tan.\; =\; \backslash frac\{\backslash sqrt\{(1-x)\}\}\{\backslash sqrt\{(1+x)\}\}\; \backslash right)\; +\; \backslash text\{const.\}$ for $x$ and $x\text{'}$ being the roots of the equation $x^2\; -\; 1\; =\; 0$, we must take $x=-1$, and $x\text{'}=1$.

We may, however, give this fluent another form by proceeding thus; Let $v$ be the arch whole $\backslash tan.\; =\; \backslash frac\{\backslash sqrt\{(1-x)\}\}\{\backslash sqrt\{(1+x)\}\}$, then $\backslash tan.\; v\; =\; \backslash frac\{1-x\}\{1+x\}$, and $x\; =\; \backslash frac\{1-\backslash tan.\; v\}\{1+\backslash tan.\; v\}$, then $\backslash frac\{2\}\{1+\backslash tan.\; v\}\; =\; \backslash frac\{2\}\{\backslash sec.\; v\}\; =\; 2\; \backslash cos.\; v\; =\; 1$, but $2\; \backslash cos.\; v\; =\; 1\; =\; \backslash cos.\; 2v$ (ALGEBRA, § 358.) therefore, $x\; =\; \backslash cos.\; 2v$. Put $s$ for the arch whole sine is $x$, and $\backslash pi$ for half the circumference, then $x\; =\; \backslash cos.\; (\backslash frac\{1\}\{2\}\backslash pi\; -\; s)$, therefore $2v\; =\; \backslash frac\{1\}\{2\}\backslash pi\; -\; s$, and since it has been shewn that $\backslash int\; \backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(1-x^2)\}\}\; =\; -2v\; +\; \backslash text\{const.\}$ therefore $\backslash int\; \backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(1-x^2)\}\}\; =\; s\; -\; \backslash frac\{1\}\{2\}\backslash pi\; +\; \backslash text\{const.\}$ or, by including the arch $\backslash frac\{1\}\{2\}\backslash pi$ in the constant quantity, $\backslash int\; \backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(1-x^2)\}\}\; =\; s\; +\; \backslash text\{const.\}$ This conclusion agrees with what has been shewn in § 59.

Instead of finding the fluent of

by first transforming it to a rational expression, we may reduce it directly to an arch of a circle by proceeding as follows. Put $x\; =\; \backslash frac\{b\}\{2\}\; =\; z$, then $\backslash dot\{x\}\; =\; \backslash dot\{z\}$, and the fluxion is transformed to $\backslash frac\{\backslash dot\{z\}\}\{c\backslash sqrt\{(a+\backslash frac\{1\}\{2\}b^2-z^2)\}\}$; again, put $a+\backslash frac\{1\}\{2\}b^2\; =\; g^2$, and $z\; =\; gu$, then $\backslash dot\{z\}\; =\; g\backslash dot\{u\}$, and this last fluxion is transformed to $\backslash frac\{\backslash dot\{u\}\}\{c\backslash sqrt\{(1-u^2)\}\}$, the fluent of which is $\backslash frac\{1\}\{c\}\; \backslash operatorname\{arc.\}\; (\backslash sin.\; =\; u)\; +\; \backslash text\{const.\}$

Of the Fluents of Binomial Fluxions.

129. Let us now consider such fluxions as have this form

and which are sometimes called binomial fluxions. We may here suppose $m$ and $n$ to be whole numbers,

without affecting the generality of the expression.

For if we had $x^\{\backslash frac\{1\}\{2\}\}\; \backslash dot\{x\}\; (a+bx^\{\backslash frac\{1\}\{2\}\})^q$, we may assume $x=z^2$, then $\backslash dot\{x\}=6z\backslash dot\{z\}$, and the fluxion becomes $6z^\{\backslash frac\{3\}\{2\}\}\; \backslash dot\{z\}\; (a+bz)^q$.

We may also suppose $n$ to be positive, for if it were negative, so that the fluxion were $x^\{m-1\}\; \backslash dot\{x\}\; (a+bx^\{-n\})^q$ we have only to assume $x=\backslash frac\{1\}\{z\}$, and the

fluxion is transformed to $-z^\{m-1\}\; \backslash dot\{z\}\; (a+bz^n)^q$.

Let us inquire in what case the fluxion $x^\{m-1\}\; \backslash dot\{x\}\; (a+bx^n)^q$ may become rational. Assume $a+bx^n\; =\; z^p$, then $(a+bx^n)^q\; =\; z^\{pq\}$, $x^n\; =\; \backslash frac\{z^p-a\}\{b\}$, $x\; =\; \backslash left(\backslash frac\{z^p-a\}\{b\}\backslash right)^\{\backslash frac\{1\}\{n\}\}$, and $x^\{m-1\}\; \backslash dot\{x\}\; =\; \backslash frac\{q\}\{nb\}\; z^\{p+q-1\}\; \backslash left(\backslash frac\{z^p-a\}\{b\}\backslash right)^\{\backslash frac\{m-1\}\{n\}\}$; hence the proposed fluxion is transformed to

this expression is evidently rational as often as $\backslash frac\{m-1\}\{n\}$ is a whole number.

There are yet other cases in which the fluxion may become rational, and which may be determined by assuming $a+bx^n\; =\; x^n\; u^q$, thus we have

$x^\{m-1\}\; \backslash dot\{x\}\; =\; \backslash frac\{-q\; a^\{n-1\}\; u^\{q-1\}\; \backslash dot\{u\}\}\{n(u^q-b)^\{\backslash frac\{m-1\}\{n\}+1\}\}$, and because that $(a+bx^n)^q$

$=\; \backslash frac\{a^q\; u^q\}\{(u^q-b)^q\}$ the fluxion $x^\{m-1\}\; \backslash dot\{x\}\; (a+bx^n)^q$ is transformed to

an expression which is rational if $\backslash frac\{m\}\{n\}\; +\; \backslash frac\{p\}\{q\}$ is a whole number.

130. As it is not possible, in every case, to express in finite terms the formula $\backslash int\; x^\{m-1\}\; \backslash dot\{x\}\; (a+bx^n)^q$, we may try to reduce it to its most simple case, as we have done

Inverse Method. done with respect to $\backslash int\; \backslash frac\{\backslash dot\{u\}\}\{(x^2+\backslash beta^2)^2\}$ in § 114, which we

have succeeded in reducing to $\backslash int\; \backslash frac\{\backslash dot\{u\}\}\{x^2+\backslash beta^2\}$. To effect this

reduction, we remark, that, since when $u$ and $v$ denote any functions of a variable quantity, the fluxion of $uv$

is $u\backslash dot\{v\}+v\backslash dot\{u\}$, (§ 37.) therefore $\backslash int\; u\backslash dot\{v\}=u\backslash dot\{v\}-\backslash int\; v\backslash dot\{u\}$. Now if we can decompose the expression $x^\{m-n\}\; \backslash dot\{x\}(a+bx^n)^p$

into two factors such, that we can find the fluent of one of them, then, denoting that factor by $v$, and the other by $u$, the fluent of the proposed fluxion will be made to depend on that of $v\backslash dot\{u\}$, which in some cases will be more simple than the proposed fluxion. That we may

abridge a little the results, we shall write $\backslash rho$ instead of $\backslash frac\{p\}\{q\}$, so that $\backslash rho$ will represent any fraction; the proposed fluxion thus simplified in its form is

Among the different ways of resolving this fluxion into two factors, we shall choose that which diminishes the exponent of $x$ without the parentheses, we therefore write the fluxion thus

now the fluent of the factor $x^\{n-1\}\; \backslash dot\{x\}(a+bx^n)^p$ may always be determined, whatever be the value of $p$, by § 108; let us denote this factor by $v$, then

thus the formula $\backslash int\; u\backslash dot\{v\}=u\backslash dot\{v\}-\backslash int\; v\backslash dot\{u\}$ gives us $\backslash int\; x^\{m-n\}\; \backslash dot\{x\}(a+bx^n)^p$ equal to

Substituting now this last value in the preceding equation, and collecting into one the terms involving

the fluent $\backslash int\; x^\{m-n-1\}\; \backslash dot\{x\}(a+bx^n)^p$, we find

hence at last we get

It is easy to see that, as we have, by this formula, reduced the determination of the fluent of $x^\{m-n-1\}\; \backslash dot\{x\}(a+bx^n)^p$ to that of $x^\{m-n-1\}\; \backslash dot\{x\}(a+bx^n)^p$ we may reduce this last to that of $x^\{m-2n-1\}\; \backslash dot\{x\}(a+bx^n)^p$ by writing $m-n$ in place of $m$ in equation (A), then by changing $m$ into $m-2n$ we may reduce the fluent of $x^\{m-2n-1\}\; \backslash dot\{x\}(a+bx^n)^p$ to that of $x^\{m-3n-1\}\; \backslash dot\{x\}(a+bx^n)^p$, and so on.

In general, if $r$ denote the number of reductions, we shall at last come to

$\backslash int\; x^\{m-rn-1\}\; \backslash dot\{x\}(a+bx^n)^p$, and the last formula will be

It appears by this last formula, that if $m$ is a multiple of $n$, then $\backslash int\; x^\{m-n-1\}\; \backslash dot\{x\}(a+bx^n)^p$ will be an algebraic quantity, for in that case the coefficient $m-rn$ will be $=0$, and therefore the term containing $\backslash int\; x^\{m-rn-1\}\; \backslash dot\{x\}(a+bx^n)^p$ will vanish. This result coincides with what we have already found, § 129.

131. We may also obtain a reduction, by which the exponent $p$ shall be diminished by unity. For this purpose it is sufficient to observe that $\backslash int\; x^\{m-n-1\}\; \backslash dot\{x\}(a+bx^n)^p$ is equal to

and that the formula (A) by changing $m$ into $m+n$, and $p$ into $p-1$ gives

Substitute now this value in the preceding equation, we have

By means of this general formula we may take away successively from $p$ as many units as it contains, and by the application of this formula, and formula (A), we may cause the fluent $\backslash int\; x^\{m-1\}\; \backslash dot\{x\}\; (a+bx^n)^p$ to depend on

$\backslash int\; x^\{m-rn-1\}\; \backslash dot\{x\}\; (a+bx^n)^\{p-s\}$, $rn$ being the greatest multiple of $r$ contained in $m-1$, and $s$ the greatest whole number contained in $p$.

The fluent $\backslash int\; x^s\; \backslash dot\{x\}\; (a+bx^n)^\{\backslash frac\{1\}\{2\}\}$, for example, may, by the application of formula (A) be reduced successively to

and by formula (B) $\backslash int\; x^s\; \backslash dot\{x\}\; (a+bx^n)^\{\backslash frac\{1\}\{2\}\}$ is reduced successively to

132. It is evident, that if $m$ and $n$ were negative, the formulas (A) and (B) would not answer the purpose for which they have been investigated, because, in that case they would increase the exponents instead of diminishing them. If, however, we reverse them, we shall find that they then apply to the case under consideration.

From formula (A) we get

Substitute now $m+n$ in place of $m$, and it becomes

This formula diminishes the exponents without the parentheses, because $m+n-1$ becomes $-m+n-1$, when $-m$ is substituted instead of $m$.

To reverse formula (B) we first take

Then, writing $p+1$ instead of $p$, we find

This formula answers the purpose we have in view, because $p+1$ becomes $-p+1$ when $p$ is negative.

These formulas (A), (B), (C), (D) are inapplicable when their denominators vanish. This is the case with formula (A); for example, when $m=-np$; but, in every such case the proposed fluxion may have its fluent determined either algebraically or by logarithms.

133. Let the fluent be $\backslash int\; \backslash frac\{x^\{m-1\}\; \backslash dot\{x\}\}\{\backslash sqrt\{(1-x^2)\}\}$, $m$ being a whole positive number. Formula (A) immediately applies to this case, so that by putting $a=1,\; b=-1,\; n=2,\; p=-\backslash frac\{1\}\{2\}$ we have

or, substituting $m$ in place of $m-1$,

Let us suppose, for example, that $m=1$, then

Let us next suppose that $m=3$, then

or, substituting for $\backslash int\; \backslash frac\{x\; \backslash dot\{x\}\}\{\backslash sqrt\{(1-x^2)\}\}$ its value,

If we suppose $m=2$, then

But we have already found, § 128. that

therefore, putting $\backslash Lambda$ for $\backslash text\{arc\}(\backslash text\{fin.\}\; =\; x)$,

In the very same way we find that

134. In the case of $m$, a negative number, we must have recourse to formula (C), from which we find

which formula, by writing $-m$ instead of $-m-1$ becomes

We cannot here suppose $m=1$, for that value would render the denominator $=0$; therefore, before we can apply this formula, it is necessary to investigate the fluent of $\backslash frac\{\backslash dot\{x\}\}\{x\; \backslash sqrt\{(1-x^2)\}\}$. We may easily find it from § 126,

or otherwise thus, put $1-x^2=z^2$, then

Therefore

The fluent of the right hand side of this equation is evidently (§ 103.)

or, since $\backslash frac\{1+z\}\{1-z\}\; =\; \backslash frac\{(1+z)^2\}\{1-z^2\}$, the same fluent may be expressed thus

therefore, by substituting $\backslash sqrt\{(1-x^2)\}$ for $z$, and $x$ for $\backslash sqrt\{(1-z^2)\}$ we have

If we suppose $m=2$, the formula becomes

If we suppose $m=3$, then

which expression, by substituting for

$\backslash int\; \backslash frac\{\backslash dot\{x\}\}\{x\; \backslash sqrt\{(1-x^2)\}\}$ its value, becomes

Of Finding Fluents by Series.

135. We can always easily find an expression for the fluent $\backslash int\; X\; \backslash dot\{x\}$, where $X$ denotes any function of $x$; when that function is expanded into a series, each term of which is some power of $x$ multiplied by a constant quantity; thus suppose

then $X\; \backslash dot\{x\}$ is equal to

and taking the fluent of each term by § 101,

If in the development of $x$ there be any term of this form $\backslash frac\{A\}\{x\}$, the fluent corresponding to that term will be $A\; \backslash cdot\; \backslash log\; x$ (§ 103).

136. The most simple function of $x$ that can be expanded into a series is $\backslash frac\{1\}\{a+x\}$, which becomes,

Hence we find

and taking the fluents

Now we know that $\backslash int\; \backslash frac\{\backslash dot\{x\}\}\{a+x\}\; =\; \backslash log(a+x)$ (§ 57.) therefore

To find the value of the constant quantity we have only to make $x=0$, for then the equation becomes $\backslash log\; a\; =\; \backslash text\{const.\}$ therefore

hence, if we subtract $\backslash log\; a$ from each side, and observe that $\backslash log(x+a)\; -\; \backslash log\; a\; =\; \backslash log\backslash left(\backslash frac\{x+a\}\{a\}\backslash right)\; =\; \backslash log\backslash left(1\; +\; \backslash frac\{x\}\{a\}\backslash right)$ we get

From this conclusion we may deduce rules for computing the logarithms of numbers.

137. Let the fluxion be $\backslash frac\{a\; \backslash dot\{x\}\}\{a^2\; +\; x^2\}$, which may be put in

der this form $\backslash frac\{\backslash dot\{x\}\}\{1\; +\; \backslash frac\{x^2\}\{a^2\}\}$, and which consequently belongs

to the arch of which the tangent = $\backslash frac\{x\}\{a\}$ (§ 60). By re-

ducing $\backslash frac\{a\}\{a^2\; +\; x^2\}$ into a series, we find

Hence, multiply both sides by $\backslash dot\{x\}$, and taking the fluent of each term, we get

If we wish to deduce from this equation, the value of the least arch whose tangent is $\backslash frac\{x\}\{a\}$ it is necessary to suppress the arbitrary constant quantity, for when that arch = 0, then $x=0$, thus we have the arch whose tangent is $\backslash frac\{x\}\{a\}$ expressed by the infinite series

Let $\backslash pi$ denote the circumference of a circle whose diameter is unity, or half the circumference of a circle whose radius is unity, then, as the sine of 30 degrees, or $\backslash frac\{1\}\{2\}\backslash pi$, is $\backslash frac\{1\}\{2\}$, and its cosine $\backslash sqrt\{1\; -\; \backslash frac\{1\}\{4\}\}\; =\; \backslash frac\{1\}\{2\}\backslash sqrt\{3\}$, we have $\backslash tan.\; \backslash frac\{\backslash pi\}\{6\}\; =\; \backslash frac\{\backslash sin.\; \backslash frac\{1\}\{2\}\backslash pi\}\{\backslash cos.\; \backslash frac\{1\}\{2\}\backslash pi\}\; =\; \backslash sqrt\{\backslash frac\{1\}\{3\}\}$. Let $\backslash sqrt\{\backslash frac\{1\}\{3\}\}$ be substituted instead of $x$ in the above series, and $a$ be supposed = 1, thus we get

and therefore

by taking the sum of about fifteen terms of this series, we shall find $\backslash pi\; =\; 3.1415927$. The determination of this number is of great importance in every branch of mathematics.

138. By proceeding with the fluxion $\backslash frac\{x^m\; \backslash dot\{x\}\}\{a^n\; +\; x^n\}$, in the same manner as we have done with $\backslash frac\{a\; \backslash dot\{x\}\}\{a^2\; +\; x^2\}$ we get

This series proceeds by the positive powers of $x$, or is an ascending series, but we may also expand $\backslash frac\{1\}\{a^n\; +\; x^n\}$ into a series proceeding by the negative powers of $x$, and which will therefore be called a descending series. Thus because

we have, after multiplying both sides by $x^\{m-1\}\; \backslash dot\{x\}$ and taking the fluents

This last series will be convergent when $x$ is greater than $a$, and at the same time , and $n\; >\; 1$. But besides, that it may contain algebraic terms only, it is necessary that none of the divisors $n-m-1$, $2n-m-1$, $3n-m-1$, &c. become $\backslash equiv\; 0$; this circumstance will take place as often as $m+1$ is a multiple of $n$, and in which case

the series $\backslash frac\{x^\{m-1\}\; \backslash dot\{x\}\}\{x^n\; +\; a^n\}\; +\; \backslash dots$ which is the development of the fluxion will contain a term of this form $\backslash frac\{x^\{r-n\}\; \backslash dot\{x\}\}\{x\}$, the fluent of which is $a^r\; \backslash ln\; x$.

If in this result we put $m=0$, $n=2$, and $a=1$ we get

But although the expression $\backslash frac\{\backslash dot\{x\}\}\{1+x^2\}$ is the fluxion of the arch having $x$ for its tangent, we must not conclude that this series is the development of that arch, for $x$ being supposed $\backslash equiv\; 0$, each of the terms of the series becomes infinite.

The consideration of the constant quantity added to the fluent will remove this apparent difficulty, if we remark, that to know the true value of a series, it is always necessary to begin with the case in which it is convergent. Now the series

converges so much the faster as $x$ is greater, and it vanishes when $x$ is infinite; but in this extreme case the equation

becomes simply $\backslash text\{arc\}\; \backslash frac\{\backslash pi\}\{2\}\; =\; \backslash text\{const.\}$ where $\backslash pi$ denotes half the circumference of the circle; therefore, substituting this value of the constant quantity, we have

VOL. VIII. Part II.

We may easily find an expression for the fluent of the rational fraction $\backslash frac\{U\; \backslash dot\{x\}\}\{V\}$ (§ 111.) by expanding the quantity $\backslash frac\{U\}\{V\}$ into a series, but the result thus obtained is in general very complicated, and seldom convergent; besides, this manner of finding the fluent is hardly of any use, since it may be expressed by means of arches of a circle and logarithms, both of which are readily obtained from the common trigonometrical tables.

139. The fluent of $x^\{m-1\}\; \backslash dot\{x\}\; (a+b\; x^n)^\{\backslash frac\{p\}\{q\}\}$ is easily obtained by first expanding the quantity $(a+b\; x^n)^\{\backslash frac\{p\}\{q\}\}$ into a series by the binomial theorem, then multiplying each term of that series by $x^\{m-1\}\; \backslash dot\{x\}$, and taking the fluents of the results by § 101. Thus we have $\backslash int\; x^\{m-1\}\; \backslash dot\{x\}\; (a+b\; x^n)^\{\backslash frac\{p\}\{q\}\}\; =$

This is an ascending series, but to get a descending series we must divide $(a+b\; x^n)^\{\backslash frac\{p\}\{q\}\}$ by $x^\{\backslash frac\{p\}\{q\}\}$ and multiply $x^\{m-1\}\; \backslash dot\{x\}$, the remaining part of the fluxional expression by the same quantity, thus the fluxion is transformed to

the fluent of which, by proceeding as in the former case, is

either of these series may be employed if $a$ and $b$ are both positive, or $q$ an odd number, but if $q$ be an even number, the first formula becomes imaginary on account of the factor $a^\{\backslash frac\{p\}\{q\}\}$ if $a^\{\backslash frac\{p\}\{q\}\}$ be negative, and the same thing happens to the second formula if $a^\{\backslash frac\{p\}\{q\}\}$ be negative.

140. Let it be required to express by a series the fluent of $\backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{1-x^2\}\}$. That we may develop the radical quantity $\backslash frac\{1\}\{\backslash sqrt\{1-x^2\}\}$ we put it under this form

Invert Method. $(1-x^2)^\{-\backslash frac\{1\}\{2\}\}$, an expression which when expanded by the binomial theorem is

therefore, multiplying each term of this series by $\backslash dot\{x\}$, and taking the fluent, we get

If we suppose $x$ to denote the sine of an arch, then $\backslash sqrt\{(1-x^2)\}$ is its cosine, and $\backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(1-x^2)\}\}$ is the fluxion of the arch itself (§ 59.); therefore the series which we have just found, expresses the length of the arch of a circle, radius being unity, and the sine of the arch $x$. If we suppose the series to express the smallest arch that corresponds to the sine $x$, then, as when the sine of that arch $=0$, the arch itself $=0$, the series expressing the arch must vanish when $x=0$, therefore we must suppress the constant quantity added to complete the fluent; or suppose it $=0$. The same series has already been found by the direct method of fluxions in § 72.

Let $\backslash pi$ denote the same as in § 137, then, as the sine of 30 degrees, or of $\backslash frac\{1\}\{6\}\backslash pi$, is $\backslash frac\{1\}\{2\}$, we have, by substituting $\backslash frac\{1\}\{2\}$ instead of $x$ in the preceding series, and multiplying both sides by 6,

by means of this series, which involves only rational numbers, we may compute (but with more labour), the value of $\backslash pi$ as before.

Suppose the fluxion to be $\backslash dot\{x\}\backslash sqrt\{ax-x^2\}$, which may be otherwise expressed thus, $\backslash dot\{x\}a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}(1-\backslash frac\{x^2\}\{a\})^\{\backslash frac\{1\}\{2\}\}$. By the binomial theorem $(1-\backslash frac\{x^2\}\{a\})^\{\backslash frac\{1\}\{2\}\}\; =$

Let each term of this series be multiplied by $a^\{\backslash frac\{1\}\{2\}\}x^\{\backslash frac\{1\}\{2\}\}\backslash dot\{x\}$, and the fluent taken by § 101, thus we get $\backslash int\; x\backslash sqrt\{ax-x^2\}\; =$

141. By resolving a fluxion into an infinite series, the object in view is to transform it into a series of other fluxions, each of which may have its fluent determined by known methods; but it is not always neces-

sary that the terms of the series should be each simply a power of $x$ multiplied by $\backslash dot\{x\}$ and constant quantities.

If for example we have this fluxion

in which $e$ is supposed to denote a small constant quantity, we may expand $\backslash sqrt\{(1-e^2x^2)\}$ or $(1-e^2x^2)^\{\backslash frac\{1\}\{2\}\}$ into a series, which will thus become

and the fluxion $\backslash frac\{\backslash dot\{x\}\backslash sqrt\{(1-e^2x^2)\}\}\{\backslash sqrt\{(1-x^2)\}\}$ will be transformed to

the series will converge very fast when $e$ is small, for that $\backslash sqrt\{(1-x^2)\}$ may be a real quantity, $x^2$ must be less than 1. We must now multiply each term of the series

by the common factor $\backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(1-x^2)\}\}$ and take the fluents, which being all contained in the general expression

$\backslash int\; \backslash frac\{\backslash dot\{x\}\backslash sqrt\{(1-e^2x^2)\}\}\{\backslash sqrt\{(1-x^2)\}\}$, will be found by § 133. Thus, putting $A$ to denote an arch of which $x$ is the sine, we have

Of the Fluents of such Fluxions as involve Logarithmic and Exponential functions.

142. Let it be required to find the fluent of $x^m\; \backslash dot\{x\}(1.x)$, where $1.x$ denotes the Napierian logarithm of $x$. In this case, as well as in some following examples, we shall have recourse to the principle already employed in § 130, namely, that if $v$ and $z$ denote any functions of a variable quantity $x$, then

Let us therefore assume $x^m\; \backslash dot\{x\}\; =\; \backslash dot\{v\}$, and $1.x\; =\; z$, then, (§ 101.), $\backslash frac\{x^\{m+1\}\}\{m+1\}\; =\; v$, and (§ 57.), $\backslash frac\{\backslash dot\{x\}\}\{x\}\; =\; \backslash dot\{z\}$, therefore, substituting

Inverse Method. substituting these values of $v,\; z,\; \backslash dot\{v\},\; \backslash dot\{z\}$ in the formula it becomes

Let us next suppose that the proposed fluxion is $x^m\; \backslash dot\{x\}\; (1.x)^n$. Put $x^m\; \backslash dot\{x\}\; =\; v$, and $(1.x)^n\; =\; z$, then (§ 101.) $\backslash frac\{x^\{m+1\}\}\{m+1\}\; =\; v$, and (§ 57.) $\backslash frac\{n\; x\; (1.x)^\{n-1\}\}\{n\}\; =\; \backslash dot\{z\}$, therefore, substituting as before these values in the formula $\backslash int\; z\; \backslash dot\{v\}\; =\; v\; z\; -\; \backslash int\; v\; \backslash dot\{z\}$, we get

It is evident that by this formula the determination of the fluent of $x^m\; \backslash dot\{x\}\; (1.x)^n$ is reduced to that of $x^m\; \backslash dot\{x\}\; (1.x)$ which we have already found, and in like manner that the determination of the fluent of $x^m\; \backslash dot\{x\}\; (1.x)^2$ is reduced to that of $x^m\; \backslash dot\{x\}\; (1.x)^3$, and so on, from which it appears that the fluent of $x^m\; \backslash dot\{x\}\; (1.x)^n$ is expressible in finite terms when $n$ is a whole positive number. The formula however will not apply when $m\; =\; -1$, because of the denominator $m+1\; =\; -1+1=0$. But in this case we have

If $n$ be negative or fractional the fluent of $x^m\; \backslash dot\{x\}\; (1.x)^n$ can only be expressed by an infinite series.

143. As an example of an exponential function, let it be required to find the fluent of $a^x\; \backslash dot\{x\}$. Here we may put $v$ for $x^m\; \backslash dot\{x\}$, and $z$ for $a^x$, then we have $v\; =\; \backslash frac\{x^\{m+1\}\}\{m+1\}$, § 101, and $\backslash dot\{z\}\; =\; (1.a)\; a^x\; \backslash dot\{x\}$, (§ 56.) therefore, substituting these values of $v,\; z,\; \backslash dot\{v\},\; \backslash dot\{z\}$ in the formula $\backslash int\; z\; \backslash dot\{v\}\; =\; v\; z\; -\; \backslash int\; v\; \backslash dot\{z\}$ we get

therefore, substituting $n\; =\; -1$ every where instead of $n$,

hence, bringing $\backslash int\; a^x\; \backslash dot\{x\}$ to stand alone one side of the equation

If we suppose $n\; =\; 1$, then, observing that $\backslash int\; a^x\; \backslash dot\{x\}\; =\; \backslash frac\{a^x\}\{1.a\}$ Inverse Method.

(§ 56.),

If $n\; =\; 2$, then

In this expression we substitute the value of $\backslash int\; a^x\; \backslash dot\{x\}$ just found, thus it becomes

Proceeding in this way we may find the fluent when $n\; =\; 3$, or when $n\; =\; 4$, or in general, when $n$ is any integer number whatever, the number of terms in the fluent being in this case always finite; it is not so however when $n$ is either negative or fractional.

Of the Fluents of such Fluxions as contain functions related to a circle.

144. Let us suppose that $x$ is an arch whose fine is $x$, and that it is required to find the fluent of $x\; \backslash dot\{x\}$. Put $x\; \backslash dot\{x\}\; =\; v$, then

but since $v\; =\; x\; \backslash dot\{x\}$, we have $v\; =\; \backslash frac\{x^\{n+1\}\}\{n+1\}$, § 101, and since $x\; =\; \backslash text\{fin.\; \}\; x$, we have $\backslash dot\{x\}\; =\; \backslash dot\{z\}\; \backslash cos.\; x\; =\; \backslash dot\{z\}\; \backslash sqrt\{1-x^2\}$ (§ 59.), and therefore $\backslash dot\{z\}\; =\; \backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{1-x^2\}\}$; thus we have

hence the determination of the proposed fluent is reduced to $\backslash int\; \backslash frac\{x^\{n+1\}\; \backslash dot\{x\}\}\{\backslash sqrt\{1-x^2\}\}$ which we have already considered in § 133. By the same mode of reasoning we may determine the fluent when $x$ denotes the cosine of the arch $x$.

145. It appears from § 59. that $n\; x$ being put to denote any arch of a circle to radius unity, the fluxion of the fine of that arch is $n\; \backslash dot\{x\}\; \backslash cos.\; n\; x$; therefore on the contrary,

In like manner from the formulas of § 59. and § 60. we find

146. By the second of these expressions we find the fluent of

to be

and from the first expression we find the fluent of

to be

147. It has been shewn in the Arithmetic of fines, (see ALGEBRA, § 356.) that

therefore, by what has been shewn in § 145,

It has also been shewn that

therefore, multiplying each term of this expression by $\backslash dot\{z\}$, and taking the fluents,

In the same manner may the fluent of $\backslash dot\{z\}\; \backslash sin.\; nz$ be found, $n$ being any positive integer number whatever.

Again, it has been shewn (ALGEBRA, § 356.), that

therefore

and because

therefore, multiplying by $\backslash dot\{z\}$, and taking the fluents,

and proceeding in this way we may find the fluents of $\backslash dot\{z\}\; \backslash cos^n\; z$, $n$ being any positive integer number.

148. The fluents of $\backslash dot\{z\}\; \backslash sin^n\; z$, and $\backslash dot\{z\}\; \backslash cos^n\; z$ may be expressed under another form, by proceeding as in § 142. Thus, beginning with $\backslash dot\{z\}\; \backslash sin^n\; z$, and resolving it into $\backslash dot\{z\}\; \backslash sin.\; z\; \backslash times\; \backslash sin^\{n-1\}\; z$, if we put $\backslash dot\{z\}\; \backslash sin.\; z\; =\; v$, and $\backslash sin^\{n-1\}\; z\; =\; t$, we have by § 145, $v\; =\; -\backslash cos.\; z$, and (by § 26 and § 59) $t\; =\; (n-1)\; \backslash dot\{z\}\; \backslash cos.\; z\; \backslash sin^\{n-2\}\; z$, therefore, substituting in the formula $\backslash int\; t\; v\; =\; v\; t\; -\; \backslash int\; v\; t$ we have

but $\backslash cos^2\; z\; =\; 1\; -\; \backslash sin^2\; z$, therefore $\backslash int\; \backslash dot\{z\}\; \backslash sin^n\; z$ is equal to

which expression, by bringing together the terms containing $\backslash int\; \backslash dot\{z\}\; \backslash sin^n\; z$ becomes

By giving particular values to $n$ we have

We may proceed in this way as far as we please, deducing the fluent of $\backslash dot\{z\}\; \backslash sin^4\; z$ from that of $\backslash dot\{z\}\; \backslash sin^3\; z$, and the fluent of $\backslash dot\{z\}\; \backslash sin^5\; z$ from that of $\backslash dot\{z\}\; \backslash sin^4\; z$, and so on.

If in the general formula we substitute every where $2-n$ instead of $n$, it becomes

an expression which, by bringing $\backslash int\; \backslash dot\{z\}\; \backslash sin.\; -n\; z$ or $\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{\backslash sin.\; n\; z\}$ to stand on one side of the equation, becomes

This formula is not applicable to the case of $n=1$, because then each of the terms of the fluent is divided by $n-1=0$, and therefore becomes infinite. In order to obtain the expression for the fluent in this particular case, we proceed thus. It is evident that $\backslash frac\{1\}\{\backslash sin.\; z\}\; =$

$\backslash frac\{1\}\{1-\backslash cos.\; z\}$, but $\backslash frac\{1\}\{1-\backslash cos.\; z\}\; =\; \backslash frac\{1\}\{2(1-\backslash cos.\; z)\}\; +\; \backslash frac\{1\}\{2(1+\backslash cos.\; z)\}$ as will be found by reducing the fractions to a common denominator, therefore $\backslash frac\{1\}\{\backslash sin.\; z\}\; =\; \backslash frac\{\backslash sin.\; z\}\{2(1-\backslash cos.\; z)\}\; +$

$\backslash frac\{\backslash sin.\; z\}\{2(1+\backslash cos.\; z)\}$ and consequently,

but if it be considered that the fluxion of $\backslash cos.\; z$ is $-\backslash dot\{z\}\; \backslash sin.\; z$ (§ 59.) it will appear by § 103 that $\backslash int\; \backslash frac\{\backslash dot\{z\}\; \backslash sin.\; z\}\{1-\backslash cos.\; z\}$

$=\; 1.\; (1-\backslash cos.\; z)$, and that $\backslash int\; \backslash frac\{\backslash dot\{z\}\; \backslash sin.\; z\}\{1+\backslash cos.\; z\}\; =\; -1.\; (1+\backslash cos.\; z)$, therefore

If in the general formula for $\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{\backslash sin.\; n\; z\}$ we suppose $n=2$ we have

$\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{\backslash sin.\; z\}\; =\; -\; \backslash frac\{\backslash cos.\; z\}\{\backslash sin.\; z\}\; =\; -\backslash text\{cotan.\; \}\; z\; +\; \backslash text\{const.\}$ which agrees with what has been already observed (§ 145.), and if we suppose $n=3$, then

In this way we proceed as far as we please, deducing

$\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{\backslash sin.\; z\}$ from $\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{\backslash sin.\; z\}$, and so on.

149. If in the general formulas for $\backslash int\; \backslash dot\{z\}\; \backslash sin.\; n\; z$ and

$\backslash int\; \backslash frac\{\backslash dot\{z\}\}\{\backslash sin.\; n\; z\}$ we substitute $\backslash frac\{1\}\{2\}\; \backslash pi\; -\; z$ instead of $z$, (where $\backslash frac\{1\}\{2\}\; \backslash pi$ denotes a quadrant), and observe that $\backslash sin.\; (\backslash frac\{1\}\{2\}\; \backslash pi\; -\; z)\; =\; \backslash cos.\; z$, $\backslash cos.\; (\backslash frac\{1\}\{2\}\; \backslash pi\; -\; z)\; =\; \backslash sin.\; z$, and that the fluxion of $(\backslash frac\{1\}\{2\}\; \backslash pi\; -\; z)$ is $-\backslash dot\{z\}$ we shall immediately obtain

and in like manner from the formula expressing the fluent of $\backslash frac\{\backslash dot\{z\}\}\{\backslash sin.\; z\}$ we deduce

150. It has been shewn in ALGEBRA, § 357, that $16\; \backslash cos.\; z\; \backslash sin.\; z\; =\; -\backslash sin.\; 5\; z\; +\; \backslash sin.\; 3\; z\; +\; 2\; \backslash sin.\; z$, therefore

The same mode of finding the fluent will apply to any fluxion of this form $\backslash dot\{z\}\; \backslash sin.\; m\; z\; \backslash cos.\; n\; z$; or by resolving the fluxion into two parts, the determination of its fluent may be reduced to that of a fluxion in which the exponents $m$ and $n$ are less than in the proposed fluxion, by the method of proceeding already employed in § 148.

151. Let us now denote $\backslash sin.\; z$ by $x$, then $\backslash cos.\; z\; =\; \backslash sqrt\{(1-x^2)\}$ and since $\backslash dot\{z\}\; \backslash cos.\; z\; =\; \backslash dot\{x\}$ (§ 59.) therefore $\backslash dot\{z\}\; =\; \backslash frac\{\backslash dot\{x\}\}\{\backslash sqrt\{(1-x^2)\}\}$; these values being substituted in any function involving $\backslash dot\{z\}$, $\backslash sin.\; z$ and $\backslash cos.\; z$ will immediately reduce it to an algebraic form. Thus for example, we shall have $\backslash dot\{z\}\; \backslash sin.\; m\; z\; \backslash cos.\; n\; z$ transformed to

an expression which may have its fluent determined by the formulas of § 133 and § 134.

SECT. II. Application of the Inverse Method of Fluxions to the Resolution of Problems.

To find the Areas of Curves.

152. It has been shewn in § 61, that if the abscissa of a curve be denoted by $x$, the ordinate by $y$, and the area

Inverse Method. area by $s$, then $s\; =\; yx$. Therefore the general formula expressing the area of any curve will be

Hence, to find the area of any curve, we must either find from the equation of the curve the value of $y$ in terms of $x$, or else the value of $x$ in terms of $y$, and $y$, and either the one or the other of these being substituted the above formula, and the fluent found by the methods already explained, we shall have a general expression for the curvilinear area as required.

Fig. 19. N^o 1. Ex. 1. Let it be required to find the area of any curve of the parabolic kind, of which, putting the abscissa $AB\; =\; x$, and the ordinate $BP\; =\; y$, the equation is $y^2\; =\; ax^2$.

Then we have $y\; =\; a^\{\backslash frac\{1\}\{2\}\}\; x^\{\backslash frac\{1\}\{2\}\}$, and

where $C$ denotes the constant quantity that may be required to complete the fluent. As in the present case, it is required to find the area of the portion of the curve next its vertex, so that when $s\; =\; 0$, then $x\; =\; 0$, there-

fore, also $C\; =\; 0$, and the area is simply $\backslash frac\{2\}\{5\}\; a^\{\backslash frac\{1\}\{2\}\}\; x^\{\backslash frac\{5\}\{2\}\}$.

If it be required to find the area comprehended between two ordinates $PB,\; pb$, put $AB\; =\; d$, then when $s\; =\; 0$, we have $x\; =\; d$, therefore the general expression

and consequently the area $BPpb$, or $s$, is equal to

When $n$ is an even number, the expression for the

area, viz. $\backslash frac\{2\}\{5\}\; a^\{\backslash frac\{1\}\{2\}\}\; x^\{\backslash frac\{5\}\{2\}\}$ may be considered as negative as well as positive, on account of the radical quantity $x^\{\backslash frac\{1\}\{2\}\}$, or $\backslash sqrt\{x^\{\backslash frac\{1\}\{2\}\}\}$, which has then a twofold value, it may therefore have the sign $\backslash pm$ prefixed to it; but in this case the same abscissa $AB$ belongs to two branches of the curve $APp$ and $AP\text{'}p\text{'}$ as in fig. 19. N^o 1, there-

fore the two values of the expression $\backslash pm\; \backslash frac\{2\}\{5\}\; a^\{\backslash frac\{1\}\{2\}\}\; x^\{\backslash frac\{5\}\{2\}\}$ may be considered as indicating the two areas $\backslash Delta\; PB,\; \backslash Delta\; P\text{'}B$,

on each side of the axis, corresponding the one to the positive, and the other to the negative ordinates.

When the exponents $m$ and $n$ are both odd numbers, the quantity $x^\{\backslash frac\{m+n\}\{2\}\}$ has only one sign and remains always positive whatever be the sign of $x$, but in this case one of the two branches of the curve has its abscissas and its ordinates negative at the same time (as in fig. 19. N^o 2.) it follows therefore that the areas corresponding to the negative abscissas and ordinates ought to be regarded as positive.

If $n$ alone is odd, then the quantity $x^\{\backslash frac\{m+n\}\{2\}\}$ becomes negative at the same time as $x$, but in this case the two branches of the curve are on the same side of the line in which the abscissas are taken (as in fig. 19. N^o 3.) and the ordinates remain always positive.

Upon the whole it may be concluded, that the area of a curve is positive when the abscissas and the ordinates have the same sign, and negative when they have contrary signs.

If we suppose $m\; =\; 1$, and $n\; =\; 2$, then the curve is the common parabola, the area of which from the general formula is found to be $\backslash frac\{2\}\{3\}\; a^\{\backslash frac\{1\}\{2\}\}\; x^\{\backslash frac\{5\}\{2\}\}\; =\; \backslash frac\{2\}\{3\}\; xy$; hence it

appears that the parabola is $\backslash frac\{2\}\{3\}$ of its circumscribing parallelogram.

Ex. 2. Suppose the curve to be a circle. Put $AB\; =\; x$, $BP\; =\; y$, the diameter $AD\; =\; a$, the area $\backslash Delta\; BP\; =\; s$. From the nature of the circle $y^2\; =\; a^2\; -\; x^2$, therefore $y\; =\; \backslash sqrt\{a^2\; -\; x^2\}$, and

In this case the fluxion is not of such a form as to admit of an algebraic fluent in finite terms, we must therefore have recourse to the method of series, but we have already found the fluent in this way in § 140, therefore, from the series there brought out we have

this expression does not require a constant quantity to be added to it, because when $x\; =\; 0$ we must also have $s\; =\; 0$.

If we suppose the arch $AP$ to be $\backslash frac\{1\}\{4\}$ of the quadrant $AE$, then it is known that $PB\; =\; \backslash frac\{1\}\{4\}$ the rad. $AC\; =\; \backslash frac\{1\}\{4\}a$, therefore, if we suppose the radius $=\; 1$, we have in this case $BC\; =\; \backslash frac\{1\}\{4\}\backslash sqrt\{3\}$, and $AB\; =\; 1\; -\; \backslash frac\{1\}\{4\}\backslash sqrt\{3\}\; =\; 0.1339746$ nearly. If this number be substituted instead of $x$, and a few terms of the series computed, we shall find the area $\backslash Delta\; BP\; =\; 0.452931$; to this add the triangle $CBP\; =\; \backslash frac\{1\}\{4\}\; \backslash times\; \backslash sqrt\{4\}\; =\; 0.2165063$, and we have the sector $ACP\; =\; 0.6694374$, which number when multiplied by 3 gives $1.9983122$ for the area of the quadrant. This number also expresses the area of a circle of which the diameter is 1.

Ex. 3. Suppose the curve to be an ellipse. Put $AB\; =\; x$, $BP\; =\; y$, the

the transverse axis $AD=a$, the conjugate axis $2CE=b$, also $AB=x$, $BP=y$; then by the nature of the curve $y\; =\; \backslash frac\{b\}\{a\}\; \backslash sqrt\{(a^2\; -\; x^2)\}$, and $\backslash dot\{s\}\; =\; y\; \backslash dot\{x\}\; =\; \backslash frac\{b\}\{a\}\; \backslash dot\{x\}\; \backslash sqrt\{(a^2\; -\; x^2)\}$; but if a circle be described on $AD$ as a diameter, and $BP$ the ordinate of the ellipse be produced to meet the circle, it appears from last example that $\backslash dot\{x\}\; \backslash sqrt\{(a^2\; -\; x^2)\}$ is the fluxion of $AQB$ the segment of the circle corresponding to the elliptic area $APB$ or $s$; therefore, putting $v$ for the segment $AQB$, we have $\backslash dot\{s\}\; =\; \backslash frac\{b\}\{a\}\; \backslash dot\{v\}$, and $s\; =\; \backslash frac\{bv\}\{a\}$, here the constant quantity $c$ must be suppressed because $s$ and $v$ must vanish together. Hence it appears that the area of any segment of an ellipse is to the area of the corresponding segment of its circumscribing circle as the lesser axis of the ellipse is to the greater; therefore the whole ellipse must be to the whole circle in the same ratio.

Ex. 4. Let the curve be a hyperbola, of which $C$ is the centre. Put the semi-transverse axis $CA=a$, the semiconjugate axis $=b$, $CB=x$, $BP=y$, the area $APB=s$. From the nature of the curve $y\; =\; \backslash frac\{b\}\{a\}\; \backslash sqrt\{(x^2\; -\; a^2)\}$, therefore

But it appears from formula B (§ 131.) that

and again by § 127,

therefore

To discover the value of the constant quantity $c$ we must consider that when $x=a$, then $s=0$, and in this extreme case the general equation just found becomes

hence $c\; =\; \backslash frac\{ab\}\{2\}\; \backslash frac\{1\}\{a\}\; \backslash cdot\; a$, and consequently, observing that

we get

It immediately follows from this formula that

but if a straight line be drawn from $C$ to $P$ so as to form the triangle $CBP$, it is manifest that $\backslash frac\{b\}\{2a\}\; x\; \backslash sqrt\{(x^2\; -\; a^2)\}$ is equal to $\backslash frac\{1\}\{2\}\; CB\; \backslash times\; BP$, that is to the triangle $CBP$, therefore the excess of the triangle $CBP$ above the area $s$, that is the hyperbolic sector $CAP$ is equal to the logarithmic function

Ex. 5. Suppose the curve to be an equilateral hyperbola, that is a hyperbola whose axes are equal, and that it is required to find the curvilinear area $DCBP$ comprehended between $DC$, a perpendicular from $D$ (a given point in the curve) to one of the asymptotes, and $PB$, a perpendicular from any other point in the curve to the same asymptote.

Let $A$ be the centre, put $AC=a$, $CD=b$, $AB=x$, $BP=y$, the area $DCBP=s$. From the property of the asymptotes we have $xy=ab$, and therefore $y\; =\; \backslash frac\{ab\}\{x\}$, hence (§ 103.)

To find the value of $c$, let us suppose $x=a$, then $s=0$ and the general formula becomes in this case

therefore

If we suppose $a=b=1$, then $s=1\; \backslash cdot\; x$, from which it appears that in this case the hyperbolic area $DCBP$ represents the Napierian logarithm of the number $x$; it was from the consideration of this property that the logarithms originally invented by Napier were called hyperbolic logarithms.

But the logarithms of any other system may also be represented by areas of the same hyperbola; for this purpose it is only necessary to determine the magnitudes of $a$, and $b$, so that $\backslash frac\{b\}\{a\}\; =\; M$, where $M$ denotes the modulus of the system, thus we shall have $ab\; =\; a^2\; M$, and $s\; =\; a^2\; M\; \backslash frac\{1\}\{a\}\; \backslash cdot\; \backslash frac\{x\}\{a\}$, or, putting $a=1$, $s\; =\; M\; \backslash frac\{1\}\{a\}\; \backslash cdot\; x$, an expression for the logarithm of $x$ according to any system whatever of which the modulus is $M$ (ALGEBRA, § 287.).

Ex. 6. Let the curve be the cycloid of which $AE$ is the axis and $A$ the vertex, let a semicircle be described on $AE$ as a diameter, draw $AG$ perpendicular to the axis, and from any point in the curve draw $PB$ perpendicular to $AG$ and $PD$ perpendicular to $AE$, meeting the circle in $Q$, and draw $QC$ to $C$ the centre of the circle. Put $AC=a$, $AB=x$, $BP=y$, the area $ABPQ=s$, and put $v$ for the angle $ACQ$, that is for the arch of a circle which measures $ACQ$, the radius of

of that circle being unity, then $AD\; =\; a(1\; -\; \backslash cos.\; v)$, $DQ\; =\; a\; \backslash sin.\; v$, and arch $AQ\; =\; av$, and since from the nature of the curve, $PD\; =\; \backslash text\{arch\; \}\; AQ\; +\; DQ$, therefore $PD\; =\; av\; +\; a\; \backslash sin.\; v\; =\; a(v\; +\; \backslash sin.\; v)$; hence

and here no constant quantity is wanted to complete the fluent; because upon the supposition that $AQ\; =\; 0$ both sides of the equation vanish as they ought to do; now it is obvious that $\backslash frac\{1\}\{2\}\; AC\; \backslash times\; AQ\; =\; \backslash text\{area\; of\; sector\; \}\; ACQ$, and $\backslash frac\{1\}\{2\}\; CD\; \backslash times\; DQ\; =\; \backslash text\{area\; of\; triangle\; \}\; DCQ$, therefore

Let $AG$ be the greatest value of $x$; complete the parallelogram $AGFE$, then from the general expression for the cycloidal area, it follows that the whole cycloidal space $APFG$ is equal to the semicircle $AQE$; but from the nature of the curve, $EF$ is equal to $AQE$, half the circumference, therefore the rectangle $EG$ is equal to four times the semicircle $AQE$; from these equals take away the external cycloidal space $AGF$, and the semicircle $AQE$, which have been shown to be equal, and the remainders, viz. the internal cycloidal space $APFE$, and three times the semicircle $AQE$ are equal to each other.

153. In some cases it is more convenient to refer a curve line to a fixed point than to an axis. Thus instead of expressing the nature of the circle by the equation $y^2\; =\; ax\; -\; x^2$, where $y$ denotes a perpendicular from any point of the curve upon $a$ the diameter, and $x$ the distance of that perpendicular from one end of the diameter, we may otherwise express it by the equation $z\; =\; r\; v$, where $z$ denotes a variable arch of the circle reckoned from one end of its diameter, $r$ its radius, and $v$ the angle contained by a line drawn from the centre of the circle through the extremity of $z$, which angle is measured by an arch of a circle having its radius unity.

The nature of the different conic sections may be defined in the same manner. Let $P$ be any point in a conic section, of which $F$ is one focus, and $FA$ a part of the axis; let $DC$ the directrix of the section meet $FA$ in $C$, join $PF$, and draw $PB$ perpendicular to the axis, and $PD$ to the directrix; then from the nature of the curve (CONIC SECTIONS) $PF$ has a given ratio to $PD$, that is to $FC\; :\; FB$; put $FC\; =\; a$, $FP\; =\; r$, the angle $PFC\; =\; v$, and suppose $PF\; :\; PD\; ::\; n\; :\; 1$, then $PF\; =\; n\; \backslash cdot\; PD\; =\; n\; \backslash cdot\; FC\; =\; n\; \backslash cdot\; FB$, hence, observing that $FB\; =\; FP\; \backslash times\; \backslash cos.\; v$, we get $r\; =\; a\; n\; -\; n\; r\; \backslash cos.\; v$, and $r\; =\; \backslash frac\{an\}\{1\; +\; n\; \backslash cos.\; v\}$ which equation expresses generally the nature of a conic section.

154. The formula which we have found for the flux-

ion of a curvilinear area, in § 61, is not immediately applicable when the nature of a curve is expressed in this way, we shall therefore investigate another formula suited to this particular manner of considering curves.

Let us suppose that $APR$ is a curve the position of Fig. 26. any point $P$ of which is determined by $PF$, its distance from a given point $F$, and the angle which $PF$ makes with $AF$ a line given by position. Let a circle be described on $F$ as a centre with a rad. $=\; 1$, then $FP$, as also the area $FAP$ may be considered as functions of $BD$ the arch of that circle which measures the angle $PFA$. From $F$ draw $FP\text{'}$ to any other point $P\text{'}$ of the curve meeting the circle $BD$ in $D\text{'}$. Put $FP\; =\; r$, the area $FAP\; =\; s$, the angle $AFP$, or the arch $BD\; =\; v$, then the area $FPP\text{'}$, and the arch $DD\text{'}$ will be the corresponding increments of $s$ and $v$, therefore § 21,

here $DD\text{'}$ the increment of $v$ is multiplied by $FD\; =\; 1$, to render the terms of the ratio homogeneous. On $F$ as a centre, with $FP$ as a radius, describe an arch of a circle meeting $FP\text{'}$ in $Q$, then, as the sectors $FDD\text{'}$, $FPQ$ are similar, we have

but the point $P\text{'}$ being supposed to approach continually to $P$, it is manifest that the limit of $\backslash frac\{\backslash text\{area\; \}\; FPP\text{'}\}\{\backslash text\{sect.\; \}\; FPQ\}$ is unity, or 1, therefore

155. By means of this formula we may find the areas of that class of curves called spirals. Let us take for example the spiral of ARCHIMEDES, which may be defined thus. Conceive a straight line $FR$ to revolve Fig. 27. about $F$ the centre of a given circle, departing from a given position $FB$; conceive also a point $P$ to move in the revolving line, so that $PF$ its distance from the centre may be to $BD$ the arch of the circle passed over by the revolving line, as $m$ to $n$, then the point $P$ will generate the spiral.

Put $BF\; =\; a$, the angle $BFR\; =\; v$, the line $FP\; =\; r$, and the area generated by the line $FP\; =\; s$, then the arch $BD\; =\; av$, and since from the nature of the curve

therefore,

this fluent does not require a constant quantity to be added,

added, as both $s$ and $r$ evidently vanish at the same time.

156. As the general expression for a curvilinear area BCPD is $\backslash int\; y\; \backslash dot\{x\}$, where $x=AB$ the abscissa reckoned from a given point A in the axis and $y=BC$ the ordinate, it follows that $X$ being put to denote any function of a variable quantity $x$, the fluent of $X\; \backslash dot\{x\}$ may always be exhibited by means of a curvilinear area. Thus let CP be a curve of such a nature that AD and DP the co-ordinates being denoted by $x$ and $y$, the equation of the curve is $y=X$, then, assuming any ordinate BC as given by position, we have evidently

As the ordinate BC (which is assumed as given by position) may be taken any where, the fluxion of the area being the same wherever it is taken, it appears, as has been already observed (§. 101) that the function $\backslash int\; X\; \backslash dot\{x\}$ may be considered as indeterminate, for it admits of innumerable values corresponding to any particular value of $x$, and in this respect it differs from an algebraic function, which for a given value of $x$ has always a determinate number of values. If however $x$ be supposed to increase from any determinate magnitude $a$, to any other determinate magnitude $a\text{'}$, then, taking the abscissa AD = $a$, and $A\; d\; =\; a\text{'}$, and drawing the ordinates DP, $d\; p$, we have

therefore, while $x$ increases from $a$ to $a\text{'}$, or receives the increment $a\text{'}\; -\; a$, the function $\backslash int\; X\; \backslash dot\{x\}$ increases from area CBPD to area CB $d\; p$, and thus receives the increment area PD $d\; p$, which is of a determinate magnitude as the ordinates PD, $p\; d$ have both a determinate position. Hence it appears that in assigning the fluent of $X\; \backslash dot\{x\}$, we only determine the change that takes place in the value of the function $\backslash int\; X\; \backslash dot\{x\}$ while $x$ passes from one particular value to another particular value.

157. As there are general and known methods by which an approximate value of any curvilinear area may be found, when a fluent is expressed by such an area, those methods may be applied to find an approximate value of the fluent. Let PD $d\; p$ be a curvilinear area, supposed to represent the fluent $\backslash int\; X\; \backslash dot\{x\}$ between the limits of $x=AD$ and $x=A\; d$. Conceive D $d$ to be divided into a number of equal parts DD', D'D'', D'$d$, and the ordinates P'D', P''D'', D'$d$, and the two sets of parallelograms DE, D'E', D''E'' and D'e, D''e', D''e'' to be completed, the former constituting a rectilinear figure circumscribed about the curvilinear space DPP'P'$d$, and the latter a rectilinear figure inscribed in that space; then as the circumscribed figure must necessarily be greater than the curvilinear space, that is, greater than

$\backslash int\; X\; \backslash dot\{x\}$ taken between the limits of $x=AD$ and $x=A\; d$, and the inscribed figure must be less, it follows that if we compute the areas of the circumscribed and inscribed figures we shall obtain two limits, the one greater, and the other less than $\backslash int\; X\; \backslash dot\{x\}$. And as by increasing the number of equal parts into which D $d$ is divided we may bring the circumscribed and inscribed rectilinear figures as near to a ratio of equality as we please, it is always possible to find two limits which shall differ from each other, and consequently from $\backslash int\; X\; \backslash dot\{x\}$ (which lies between them) by less than any assignable quantity.

158. If we join P, P', P'', $p$, the tops of the ordinates, the rectilinear space formed by the trapeziums DPP'D', D'P''D'', D''P'$p$ will be more nearly equal to the curvilinear area, than the circumscribed rectilinear figure formed by the parallelograms DE, D'E', D''E''; therefore the sum of those trapeziums being found, it will be equal to the fluent $\backslash int\; X\; \backslash dot\{x\}$ nearly.

Suppose, for example, that it is required to find the value of $\backslash int\; \backslash frac\{x\}\{1+x^2\}$ between the limits of $x=0$, and $x=1$. In this case $X\; =\; \backslash frac\{1\}\{1+x^2\}$, so that the equation of the curve P $p$ is $y\; =\; \backslash frac\{1\}\{1+x^2\}$; let us suppose D $d$ the distance between the extreme ordinates to be divided into ten equal parts, then putting 0, 1, 2, &c. to 1 instead of $x$ in the formula $y\; =\; \backslash frac\{1\}\{1+x^2\}$, we obtain eleven successive values of $y$, or eleven equidistant ordinates, the numeral values of which will be as follows,

By the elements of geometry the area of the rectilinear 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 that sum by the breadth of one of the trapeziums; now the sum of the ordinates, with the exception of the first and last, together with half the sum of the first and last, is 7.84981, and the common breadth

of the trapeziums is .1, therefore $\backslash int\; \backslash frac\{x\}\{1+x^2\}\; =\; 7.84981\; \backslash times\; .1\; =\; .785$ nearly, as required. It is evident from § 137,

that $\backslash int\; \backslash frac\{x\}\{1+x^2\}$ taken between the limits of $x=0$, and $x=1$, is accurately equal to an arch of $45^\backslash circ$, radius being unity, which arch being $\backslash frac\{1\}\{4\}\backslash pi$ will be found = .7854 nearly.

If we recur to the series which has been found to express the above fluent in § 137, and put $a=1$, and $x=1$, we shall have $\backslash int\; \backslash frac\{x\}\{1+x^2\}$ taken between the prescribed limits equal to

It is impossible however to compute the fluent simply by the addition of the terms of this series, on account of the slowness of its convergence.

159. As the curvilinear area $DPd$ is the limit of the sum of the parallelograms $DP\text{'},\; DP\text{'}\text{'},\; \backslash dots$ which constitute the rectilinear figure inscribed in that area; as also the limit of the sum of the parallelograms $DE,\; DE\text{'},\; \backslash dots$ which constitute the circumscribed figure, the number of parts into which $Dd$ is divided being in either case conceived to be increased indefinitely, so that each may be continually diminished (and this being the case it is of no consequence whether those parts be conceived as all equal or as unequal) so from analogy we may conclude that if $x\text{'},\; x\text{'}\text{'},\; x\text{'}\text{'}\text{'},\; \backslash dots\; x^\{(n)\}$ be put to denote successive values of a variable quantity $x$, and $X\text{'},\; X\text{'}\text{'},\; \backslash dots\; X^\{(n)\}$ the corresponding values of $X$ any function of $x$, the limit of the sum of the products $X(x\text{'}\; -\; x),\; X(x\text{'}\text{'}\; -\; x\text{'}),\; X(x\text{'}\text{'}\text{'}\; -\; x\text{'}\text{'}),\; \backslash dots$ when the number of successive values of $x$ and $X$ is continually increased, so that the difference between any two of them immediately following each other may be continually diminished, is equal to $\backslash int\; X\; dx$, the fluent to be taken between the two extreme values of $x$.

160. It was in this manner that the first writers on the differential calculus conceived a fluent; as the difference between any two of its succeeding values is the product of the function $X$ by $x\text{'}\; -\; x$ the increment of $x$, they called that product (when $x\text{'}\; -\; x$ was conceived to be infinitely diminished) the Differential of the fluent; and as the fluent is the sum of all the products, or differentials, instead of calling it a fluent they called it an Integral; the process by which an integral is found from its differential or fluxion they called Integration. The terms Integral and Integration are sometimes employed by writers in applying the method of fluxions to mathematical enquiries.

To find the Lengths of Curves.

161. It has been shewn in § 63, that if the abscissa of a curve be denoted by $x$, the ordinate by $y$, and the curve line by $z$, then $z\; =\; \backslash sqrt\{(x^2\; +\; y^2)\}$, hence the general formula for finding the length of a curve is

Therefore, if from the equation expressing the nature of a curve we find the value of $\backslash dot\{y\}$ in terms of $x$ and $\backslash dot\{x\}$, or else the value of $\backslash dot\{x\}$ in terms of $y$ and $\backslash dot\{y\}$, and substitute the one or the other in the above general formula, we shall obtain a fluxion the fluent of which will be the length of the curve.

Ex. 1. Suppose the curve to be a parabola, and that $AB\; =\; x$, $BP\; =\; y$, the arch $AP\; =\; z$, the parameter $=\; a$, then the equation of the curve being $ax\; =\; y^2$, we have

By comparing this expression with formula (B) in § 131, we find

again by § 127,

therefore

To determine the value of $C$, we must consider that when $y\; =\; 0$, then $z\; =\; 0$, so that then the general formula will become simply

therefore, substituting the value of $C$, and bringing together the logarithmic quantities,

Ex. 2. Suppose the curve to be a circle, and that $C$ Fig. 20. is its centre, and $AE$ a quadrant of the circle. Put $CB\; =\; x$, $BP\; =\; y$, the arch $EP\; =\; z$, the radius of the circle $=\; a$, then $x^2\; +\; y^2\; =\; a^2$, and $y\; =\; \backslash sqrt\{(a^2\; -\; x^2)\}$, and $\backslash dot\{y\}\; =\; \backslash frac\{-x\backslash dot\{x\}\}\{\backslash sqrt\{(a^2\; -\; x^2)\}\}$, hence

This fluent can only be expressed by an infinite series, under which form it has been already exhibited in § 140, the radius being there supposed unity.

Ex. 3. Let the curve be an ellipse, and let it be required to find the length of the curve between $E$ the vertex of the lesser axis, and $P$ any point in the curve. To simplify the calculation, let us suppose that the semi-transverse axis $AC\; =\; 1$; put the semi-conjugate axis $CE\; =\; b$, the eccentricity (that is $\backslash sqrt\{(1\; -\; b^2)\}\; =\; e$), the abscissa $CB\; =\; x$, the ordinate $PB\; =\; y$, the arch $EP\; =\; z$; then, the equation of the curve being $y^2\; =\; b^2(1\; -\; x^2)$, we have

Inverse Method. have $y\; =\; b\backslash sqrt\{1-x^2\}$, and $\backslash dot\{y\}\; =\; \backslash frac\{-bx\}\{\backslash sqrt\{1-x^2\}\}$, and therefore

This fluent can only be expressed by means of an infinite series, and it has been already given in this form in § 141.

If we take $x=1$, then all the quantities in that series which are multiplied by $\backslash sqrt\{1-x^2\}$ will vanish, but in this particular case $x$ is the elliptic quadrant $EA$, and $A$ is a quadrant of the circumscribing circle, or $\backslash frac\{1\}\{4\}\backslash pi$, therefore the elliptic quadrant is equal to

This series converges very fast if $e$ be a small fraction.

Ex. 4. Suppose the curve to be a cycloid. Let a circle be described on its axis meeting the ordinate $PB$ in $Q$, and draw $CQ$ to the centre of the circle. Put $AB=x$, $BP=y$, the cycloidal arch $AP=x$, the radius $AC=a$, the angle $ACQ=\backslash nu$, then $AB=a(1-\backslash cos\; \backslash nu)$, $BQ=a\; \backslash sin\; \backslash nu$, the circ. arch $AQ=a\; \backslash nu$, so that $x=a(1-\backslash cos\; \backslash nu)$, and from the nature of the curve $y=a(\backslash nu\; +\; \backslash sin\; \backslash nu)$, therefore (§ 59)

but $2\; +\; 2\; \backslash cos\; \backslash nu\; =\; 4\; \backslash cos^2\; \backslash frac\{1\}\{2\}\; \backslash nu$ (ALGEBRA, § 356.), therefore

but as when $\backslash nu=0$, then $z=0$, therefore $C=0$, and $z=4a\; \backslash sin\; \backslash frac\{1\}\{2\}\; \backslash nu$; but if the chord $AQ$ be drawn, $2a\; \backslash sin\; \backslash frac\{1\}\{2\}\; \backslash nu\; =\; \backslash text\{chord\; \}\; AQ$, therefore $z=2\; \backslash text\{\; chord\; \}\; AQ$.

162. The formula $z\; =\; \backslash int\; \backslash sqrt\{\backslash dot\{x\}^2\; +\; \backslash dot\{y\}^2\}$ not being applicable in its present form to curves of the spiral kind, we shall here investigate another suited to that particular class of curves.

Let $APR$ be a curve of such a nature that the position of any point $P$ in the curve is determined by $PF$, its distance from a given point $F$, and by the angle which $PF$ makes with $AF$ a line given in position. We shall employ the same construction and notation here as in § 154, with the addition of drawing the chords $DD\text{'}$, $PQ$, $PP\text{'}$, and putting the arch $AP=x$; then it is manifest that the simultaneous increments of $\backslash nu$, $z$, and $r$ will be the arches $DD\text{'}$, $PP\text{'}$, and the straight line $PQ$ respectively. Hence

but it is evident from § 62, that the limiting ratio of those arches must be the same as that of their chords, therefore,

now the limit of the angle $PQD$ being evidently a right angle, we have

but $\backslash frac\{PQ^2\}\{DD\text{'}^2\}\; =\; \backslash frac\{FP^2\}\{FD^2\}\; =\; r^2$, and $\backslash text\{lim.\; \}\; \backslash frac\{PQ^2\}\{DD\text{'}^2\}\; =\; \backslash frac\{r^2\}\{v^2\}$, therefore

Let us apply this formula to the spiral of Archimedes, Fig. 27, the equation of which (§ 155.) is $a\; m\; \backslash nu\; =\; n\; r$, and therefore

This fluent may be found by formula B, § 131, and it is worthy of remark that the fluxion has the same form as that which we have found in § 161 for an arch of a parabola; thus the length of any portion of the spiral of Archimedes may be exhibited by means of an arch of a parabola.

To find the Content of Solids.

163. If $AD$ the abscissa of a curve be denoted by $x$, Fig. 4, and $PD$ the ordinate by $y$, and the solid generated by the revolution of the curve $APD$ about $AD$ as an axis by $s$, it has been shewn, in § 64, that $s\; =\; \backslash pi\; y^2\; x$, therefore the general formula for finding the content of a solid is

Ex. 1. Suppose the solid to be a paraboloid, or that which is generated by the revolution of a parabola about its axis; in this case $y^2\; =\; ax$, and taking the fluent so that when $x=0$, then $s=0$,

or $s\; =\; \backslash frac\{1\}\{2\}\; \backslash pi\; x^2\; y^2$; but $\backslash pi\; x\; y^2$ is the content of a cylinder having $y$ for the radius of its base and $x$ for its altitude, therefore the content of a paraboloid is half that of a cylinder having the same base and altitude.

Ex. 2. Suppose the solid to be a parabolic spindle, Fig. 31, which is generated by the revolution of $APB$ an arch of a parabola about $AC$ an ordinate to its axis. In this case let $AD=x$, $DP=y$, $AB=b$, the parameter of the

Inverse
Method. axis $=\; a$, then from the nature of the parabola $AD\; \backslash times\; DB$
$=\; a\; \backslash times\; PD$, that is $x(b-x)\; =\; ay$, hence $y\; =\; \backslash frac\{(b-x)x\}\{a\}$, and
taking the fluent, so that $s$ and $x$ may vanish together;

which expression (by supposing $x\; =\; AC\; =\; \backslash frac\{1\}\{2\}b$, and putting $d$ for $CE$, the greatest value of $y$) gives $\backslash frac\{4\backslash pi\; d^2\; b\}\{15\}$ or the content of half the solid generated by the curve $AEB$, therefore the entire spindle is $\backslash frac\{8\backslash pi\; a^3\; b\}\{15\}$, or (by observing that $\backslash pi\; d^2\; b$ is the content of a cylinder having $d$ for the radius of its base and $b$ for its length) it is $\backslash frac\{1\}\{3\}$ of the circumscribing cylinder.

Ex. 3. Suppose the solid to be a spheroid produced by the revolution of an ellipse about either of its axes; put $a$ for $\backslash frac\{1\}\{2\}AB$ the axis round which the curve revolves, $b$ for $\backslash frac\{1\}\{2\}EF$ the other axis, $x$ for $AD$ the height of any segment made by a plane perpendicular to the axis of the solid, $y$ for $PD$ the radius of its base, and $s$ for its content. Then, from the nature of the curve $y^2\; =\; \backslash frac\{b^2\}\{a^2\}(2ax\; -\; x^2)$, therefore, taking the fluent upon the supposition that $s$ and $x$ vanish together,

To find the content of the whole spheroid we have only to take $x\; =\; 2a$, thus the formula becomes $s\; =\; \backslash frac\{4\backslash pi\; b^2\; a\}\{3\}$, and as $2\backslash pi\; b^2\; a$ expresses the content of a cylinder having $2b$ for the diameter of its base, and $2a$ for its height, it follows that the content of a spheroid is $\backslash frac\{2\}\{3\}$ that of its circumscribing cylinder.

It is obvious that what has been found for the spheroid will apply also to the sphere, by supposing the axes equal, or $a\; =\; b$.

164. If instead of supposing the solid $APQ\backslash rho$ to be formed by the revolution of a curve round its axis (in which case it is called a solid of revolution) we had supposed it to have any figure whatever, then by referring the solid to some straight line $AC$, given by position, as an axis, and in which $A$ is a given point, and supposing $PQ\backslash rho$ to be a section of the solid made by a plane

perpendicular to that axis, meeting it in $D$, and putting $AD\; =\; x$, and the variable solid $APQ\backslash rho$ (considered as a function of $x$) $=\; s$, by proceeding as in § 64, we would have found the limit of $\backslash frac\{\backslash text\{increment\; of\; \}\; s\}\{\backslash text\{increment\; of\; \}\; x\}$, and con-

sequently $\backslash frac\{\backslash dot\{s\}\}\{\backslash dot\{x\}\}$, equal to the area of the section of the solid made by the plane $PQ\backslash rho$; therefore, putting $V$ for that function of $x$ which expresses the area of the section, we have $\backslash dot\{s\}\; =\; V\backslash dot\{x\}$, and $s\; =\; \backslash int\; V\backslash dot\{x\}$.

Let us suppose for example that $AEFG$ is a solid bounded by any plane figure $EFG$ as a base, and by the surface which will be generated if we suppose a straight line drawn from $A$ any given point above that plane to revolve in the circumference of the base.

Let $AC$ be a perpendicular drawn from the vertex of the figure to its base, and let $PQ\backslash rho$ be a section of the solid by a plane parallel to the base, meeting the perpendicular in $D$. Put $a$ the area of the base of the solid, $V$ the area of the section $PQ\backslash rho$, $b\; =\; AC$ the altitude of the whole solid, $x\; =\; AD$ the altitude of the part cut off by the plane $PQ\backslash rho$, and $s$ the content of that part; then, as from the nature of the solid it is pretty evident that the part of it cut off by the plane $PQ\backslash rho$ is similar to the whole, and as the bases of similar solids are as the squares of their altitudes, we have

this expression for $s$ does not require the addition of any constant quantity, for by putting $x\; =\; 0$, we have $s\; =\; 0$ as it ought to be. Suppose now $x\; =\; b$, then $s\; =\; \backslash frac\{a\; b^3\}\{3b^2\}\; =\; \backslash frac\{1\}\{3\}ab$, from which it appears that the content of the whole solid is $\backslash frac\{1\}\{3\}$ of the product of the base by the perpendicular. It is evident that pyramids and cones are solids of the kind we have been considering.

To find the Surfaces of Solids.

165. The altitude $AD$ of a solid, generated by revolution of a curve about $AD$ as an axis, being as before denoted by $x$, and $PD$ the radius of its base by $y$, let us now put $s$ to denote the curved surface of the solid, then, as it has been shewn, § 65, that $s\; =\; 2\backslash pi\; y\; \backslash sqrt\{(x^2\; +\; y^2)\}$, we have

as a general formula for the surface of a solid.

Ex. 1. Suppose the solid to be a sphere, generated by the revolution of a circle about its diameter $AB$, put the radius of the sphere $=\; a$, then, $AD$ being denoted by $x$, and $PD$ by $y$, we have from the nature of the curve $y^2\; =\; 2ax\; -\; x^2$, therefore

therefore, $y\backslash sqrt\{(\backslash dot\{x\}^3\; +\; \backslash dot\{y\}^3)\}\; =\; a\backslash dot\{x\}$, and taking the fluent, so that when $x=0$, then $s=0$,

now if it be considered that $2\backslash pi\; a$ is the circumference of a great circle of the sphere, it will immediately appear that the surface of a segment of a sphere is equal to the circumference of a great circle of the sphere multiplied into the height of the segment. Hence it follows that the whole surface of the sphere is equal to four times the area of one of its great circles.

Ex. 2. Suppose the curve to be a parabola, then putting $AD=x$, $DP=y$, the parameter of the axis $=a$, we have (§ 161. example 1.)

To discover the value of the constant quantity $C$, we must observe that when $x=0$, then $y=0$, and $s=0$, therefore, putting 0 instead of $s$ and $y$, the above equation becomes $0\; =\; \backslash frac\{\backslash pi\; a^3\}\{6\}\; +\; C$, hence $C\; =\; -\backslash frac\{\backslash pi\; a^3\}\{6\}$, and

To find the Centre of Gravity of any Line, Surface, or Solid.

166. It belongs to the theory of MECHANICS to explain what is meant by the centre of gravity, and to demonstrate its general properties, and here it is only necessary to shew how the method of fluxions may be applied to deduce from some one of those properties rules for finding that centre in any proposed case.

The property of centres of gravity which we shall employ as the foundation of the application of the method of fluxions to its determination may be enunciated shortly thus.

Let $C$ be the centre of gravity of a mass of matter denoted by $M$, and $c$ the centre of gravity of another mass $m$, and $D$ the centre of gravity of the two masses $M$ and $m$, from these points let perpendiculars $CA,\; c\; a,\; DE$ be drawn to any straight line $PQ$, then

167. Let us now suppose that $AP$ is any curve line

(having weight,) of which the centre of gravity is required, and that $PB,\; PD$ are co-ordinates drawn from any point in the curve perpendicular to $AB,\; AD$ two axes at right angles to each other; let the arch $AP$ receive any increment $P\backslash rho$, let $C$ be the centre of gravity of $AP$, $G$ the centre of gravity of $P\backslash rho$, and $C\text{'}$ the centre of gravity of $AP\backslash rho$. From $C$ and $G$ draw $CE,\; CF,\; GH,\; GK$ perpendicular to the axes $AB,\; AD$. Put $PD=x,\; PB=y,\; CF=Y,\; AP=\backslash infty$, also let the arch $AP\backslash rho=\backslash infty\text{'}$, and let the distances of $C\text{'}$ its centre of gravity from the axes $AD,\; AB$ be denoted by $X\text{'},\; Y\text{'}$ respectively; then, observing that the arch $P\backslash rho=\backslash infty\text{'}\; -\; \backslash infty$, by the proposition in last §,

If we now suppose the arch $P\backslash rho$ to be continually diminished, and observe that $\backslash infty\text{'}\; X\text{'}\; -\; \backslash infty\; X$, and $\backslash infty\text{'}\; -\; \backslash infty$ are the simultaneous increments of $\backslash infty\; X$ and $\backslash infty$, it will appear (§ 23.) that

By the very same way of reasoning we find

but the point $\backslash rho$ approaching to $P$, it is manifest that the point $G$ will also approach to $P$, so that the limit of $GK$ is $PD$ or $x$, and the limit of $GH$ is $PB$ or $y$, hence

Taking now the fluents of each side of these equations, and dividing by $\backslash infty$,

It is evident that by these two equations the position of $C$ the centre of gravity is determined.

168. Let us next suppose that it is required to find $C$ Fig. 37. the centre of gravity of the plane area $APB$. As the arch $AP$ was in last § supposed to receive the increment $P\backslash rho$, so let the area $APB$ now receive the increment $BP\backslash rho\; b$, and let $C,\; C\text{'}$ and $G$ (which in the former case were supposed to be the centres of gravity of the arches $AP,\; AP\backslash rho$, and $P\backslash rho$ respectively) now be supposed to be the centres of gravity of the areas $APB,\; AP\backslash rho\; b$, and $BP\backslash rho\; b$; put the area $APB=s$, the area $AP\backslash rho\; b=\backslash infty\text{'}$, and let $X,\; Y,\; X\text{'},\; Y\text{'}$ denote as before. Then, reasoning exactly as in last case, we have (by § 166.),

and the point $\backslash rho$ being supposed to approach to $P$, so that $s\text{'}X\text{'}\; -\; sX$, $s\text{'}Y\text{'}\; -\; sY$, and $s\text{'}\; -\; s$, the simultaneous increments of $X$, $Y$ and $s$, may be continually diminished,

but as the ordinate $\backslash rho\; b$ approaches to $PB$, it is manifest that the ultimate position of $G$ the centre of gravity of the area $BP\backslash rho\; b$ will be the middle of $PB$, therefore the limit of $GK$ is $x$, and the limit of $GH$ is $\backslash frac\{1\}\{2\}y$, thus we have

and consequently,

or since $\backslash dot\{s\}\; =\; y\; \backslash dot\{x\}$, and $s\; =\; \backslash int\; y\; \backslash dot\{x\}$ (§ 61.),

169. Let it next be required to find the centre of gravity of the surface of a solid generated by the revolution of the curve $AP$ about $AB$ as an axis. Let the surface of the solid be conceived to receive an increment generated by $P\backslash rho$ an arch of the curve. In this case it is evident that the centres of gravity of the surface generated by the curve $AP$, the surface generated by the curve $AP\backslash rho$, and the surface generated by the arch $P\backslash rho$, will each be in $AB$, the axis of the solid; suppose them to be at $E$, $E\text{'}$ and $H$ respectively. Put $AE\; =\; X$, $AE\text{'}\; =\; X\text{'}$, also put $s$ for the surface generated by $AP$, and $s\text{'}$ for the surface generated by $AP\backslash rho$, then as before (from § 166.) we have