SECTION I.

INTRODUCTION.

The determination of the figure and dimensions of the earth is a problem of very great importance in astronomy, inasmuch as it is in reference to the earth's diameter that the distances of the planets from the sun and from each other are estimated. It is also a problem of very great interest and curiosity; and has accordingly attracted the attention of mankind since the earliest dawn of civilization.

There are two points of view under which this great question may be considered. The figure of the earth may be regarded as a fact to be determined by investigation and experience, like any other phenomenon or law of nature; in which case it is necessary to find, by the actual measurement and comparison of different portions of the terrestrial surface, the nature of its curvature, and the magnitude of its diameters. Under the second point of view, the question is one of pure theory. The earth may be regarded as a congeries of material particles, attracting each other with forces reciprocally proportional to their mutual distances, and endowed with a rotatory motion about a fixed axis; and the problem is to determine the form the whole mass would assume in virtue of the attractive and centrifugal forces by which the particles are impelled. Viewed in this light, the actual figure of the earth becomes one of the series of consequences resulting from the universal gravitation of matter, and depending on the same laws which regulate its motion in its orbit about the sun.

It would be a waste of time to inquire what were the notions of the figure of the earth which were or might have been entertained by its earliest and most ignorant inhabitants. A very slight attention to the most common phenomena renders the fact of its general roundness almost palpable to the senses. The uniform level appearance of the sensible horizon in every situation in which a spectator can be placed,—the depression of the circumpolar stars as he advances toward the south, and their elevation as he proceeds in a contrary direction,—the disappearance of a ship standing out to sea,—the projection of the earth as seen in a lunar eclipse,—and a number of other familiar appearances,—put the globular figure of our planet beyond all manner of doubt. Reasoning from such appearances, the earliest astronomers universally regarded the earth as a sphere; and their attention was solely directed, in their various measurements and computations, to ascertain its dimensions. Modern science has discovered that its figure deviates slightly from that of a sphere, being compressed or flattened at the extremities of its axis of rotation; and the object of the astronomer, at the present time, is to determine not only its dimensions, but also the exact amount of its compression.

Attempts to estimate the magnitude of the earth were made at a very early date; for Aristotle relates that the mathematicians prior to his time had found the circumference to be 400,000 stadia. But Eratosthenes appears to have been the first who entertained an accurate idea of the principles on which the determination of the figure of the earth really depends, and attempted to reduce these principles to practice. His results, in consequence of the imperfect data from which they were deduced, were very inaccurate; but his method is the same as that which is followed at the present day, depending, in fact, on the comparison of a line actually measured on the surface of the earth with the corresponding celestial arc. He had remarked, or been informed, that at Syene in Upper Egypt, on the day of the summer solstice, at noon, objects cast no shadows; whence he concluded that the sun was exactly in the zenith at mid-day. On the same day at Alexandria he observed the sun's meridional distance from the zenith to be $7^\circ 12'$, or a fifth part of the circumference. Then, assuming Syene to be exactly under the meridian of Alexandria (the error in this assumption was about $3^\circ$), and the distance between the two places, measured in a straight line, to be 5000 stadia, he had $5000 \times 50 = 250,000$ stadia for the whole circumference of the earth. It is easy to see how very imperfect this operation must have been. Without mentioning smaller errors, the neglect of the solar diameter would alone occasion an uncertainty as to the sun's declination, and consequently as to the length of the celestial arc, amounting to half a degree on the observation at Syene; and there is no reason to suppose that that at Alexandria was more exact. The terrestrial distance between the two places was assumed on equally, or probably still more loose and inaccurate determinations.

The next attempt to ascertain the dimensions of the earth was made by Posidonius. This astronomer adopted a method which differed from that of Eratosthenes only in determining the celestial arc by means of the altitude of a star, instead of the sun's zenith distance. At Rhodes the bright star Canopus, when on the meridian, barely appears above the horizon. At Alexandria the same star was observed to have a meridional altitude of a quarter of a sign, or seven and a half degrees, which, therefore, was the celestial arc intercepted between the zenith of Alexandria and Rhodes. The terrestrial distance between the two places was estimated, like that between Alexandria and Syene, at 5000 stadia, and they were both supposed to be under the same meridian. Hence, since seven and a half degrees is the forty-eighth of the circumference, we have $5000 \times 48 = 240,000$ stadia, for the whole circumference of the globe.

It is impossible to form any correct opinion of the degree of approximation attained in these ancient measures, as the length of the stadium is not known with any certainty. That it varied in different places, and at different times, is sufficiently obvious from the statement of Ptolemy, who, in his work on geography, assigns the length of the degree at 500 stadia, and consequently the whole circumference at 180,000, differing from the determination of Posidonius in the proportion of three to four, and still more from that of Eratosthenes. Ptolemy remarked that it was not necessary that the line measured should lie exactly in the meridian: it was sufficient to know its inclination to the meridian, or the azimuthal angle, together with the latitudes of its extreme points, in order to compare it with the meridional arc. The determination of the azimuth is, however, an operation of considerable difficulty; and Ptolemy has given no details of the method by which he proposed to estimate it. He has been equally silent in respect of the means by which the mean length of a degree was ascertained to be 500 stadia, so that the result which he has recorded is still less satisfactory than those of the two more ancient astronomers.

The active curiosity of the Arabians, which was exerted so successfully in promoting practical astronomy, did not FIGURE OF THE EARTH.

The Caliph Almamoon, who began his reign in the year 814, ordered a company of astronomers to measure a degree on the level plain of Mesopotamia. Dividing themselves into two parties, the one proceeded northward, and the other southward, in the direction of the meridian, through a degree of latitude, and measured with rods the itinerary distance as they proceeded. The perfect agreement of their conclusion with that of Ptolemy throws it open to great suspicion; and when it is considered that their operation was repeated at a different place, with exactly the same result; there can be no doubt that they blindly adopted the statement of the Greek astronomer, either from inability to execute the task assigned to them, or because they had no confidence in their own determination.

From the time of Almamoon, the problem of determining the dimensions of the earth was neglected, till the revival of astronomy with general learning in Europe. The first attempt to solve it was made by Fernel, who, about the middle of the sixteenth century, measured the distance from Paris to Amiens along the high road, by observing the number of revolutions made by his coach-wheel in the journey between these two cities. Supposing them to be under the same meridian, which is nearly true, and having ascertained the difference of their latitudes, Fernel found by this means the length of the degree to be 57,070 French toises, or about 364,960 English feet. A degree was measured long afterwards at the same place by Lacaille, in a far more adequate and scientific manner, and he found it to be 57,074 toises. This agreement is rendered less extraordinary by the circumstance that the toise of Fernel was not exactly of the same length as that of Lacaille. After all, it must be allowed that Fernel made a fortunate guess. (Delambre, Astronomie, tom. iii. chap. xxxv.)

But the first who had the merit of attempting to execute the geodetic operations that are indispensably necessary to effect the accurate measurement of a long line on the surface of the earth, was Willebrord Snell, a native of Holland, and a teacher of mathematics. Having established a chain of triangles between Alkmaar and Bergen-op-Zoom, and observed the angles of each triangle by means of a quadrant of five and a half feet radius, he measured a base on the frozen surface of the meadows between Leyden and the village of Soeterwoud, and determined the distance between the two places by trigonometrical computation. The length of the degree which he found was 28,500 Rynland feet, or about 55,020 toises, which is about 2050 toises too small. This result was published by Muschenbroek, who in fact revised or calculated the observations from the original papers a century after the death of Snell.

Norwood in the year 1635 attempted to measure a degree in England nearly in the same manner as Fernel. He measured the distance between London and York along the public road, taking the bearings and reducing the direction to the meridian in a rough way. The difference of latitudes he found by observations of the solstice to be 2° 28', and thence concluded the length of the degree to be 367,176 English feet. Like the measurement of Fernel, this has been found to be a much nearer approximation than the method employed would have led us to expect.

The application of the telescope to circular instruments gave a far higher degree of precision to geodetic operations. Picard, to whom practical astronomy is indebted for this capital improvement, was the first who measured an arc of the meridian with such precautions and care as the delicate nature of the operation requires. With a standard scale he measured a base of about six English miles in length; and observed the angles of his triangles with a quadrant, having a telescope adapted to it with cross wires in its focus. He even calculated the error produced by the instrument being placed out of the centre of the station, and determined the zenith distance of a star in the constellation Cassiopeia with a sector, for the purpose of obtaining the differences of latitude. The distance between Amiens and Malvoisine was found to be 78,850 toises, and the difference of latitude 1° 22' 55", whence the result gave for the degree at Amiens 57,060 toises; but as the aberration and nutation were unknown at that time, and the refraction was not taken into account,—causes of error to which it is indispensably necessary to have regard,—a determination which agrees so nearly with the results of recent measures could only have arisen from a fortunate compensation of errors. In fact, his toise was somewhat shorter than that which has since been adopted as the standard; and the error occasioned by this circumstance nearly compensated that which was committed in determining the celestial arc, so that in recalculating all the observations, the degree is found to be very nearly the same as was found by Picard.

Hitherto geodetic operations had been confined to the determination of the magnitude of the earth; but a discovery made by Richer turned the attention of mathematicians to its deviation from the spherical form. This astronomer having been sent, by the Academy of Sciences of Paris, to the island of Cayenne in South America, for the purpose of determining the amount of terrestrial refraction, and other astronomical objects, observed that his clock, which had been regulated at Paris to beat seconds, lost about two minutes and a half daily at Cayenne, and that, in order to bring it to measure mean solar time, it was necessary to shorten the pendulum by more than a line. This fact, which appeared exceedingly curious, and was scarcely credited till it had been confirmed by the subsequent observations of Varin and Deshayes, was first explained in the third book of the Principia, by Newton, who showed that it could only be referred to a diminution of gravity arising from one of two causes—a protuberance of the equatorial parts of the earth, and consequent increase of the distance from the centre, or from the counteracting effect of the centrifugal force, occasioned by the rotation of the earth. The former could not, on any reasonable supposition regarding the figure of the earth, be regarded as adequate to produce the effect; but the latter, which would produce a retardation of the pendulum at Cayenne in the ratio of the square of the sine of 6° to that of 49° (the respective latitudes of Cayenne and Paris), might amount to 1·46 seconds. This was the first direct proof that had been obtained of the diurnal rotation of the earth.

From this time the exact determination of the figure of the earth began to assume a degree of importance which had not formerly attached to it. The centrifugal force arising from the diurnal rotation completely set aside the idea of perfect sphericity. Newton, assuming that the earth had been originally fluid, and supposing its density to be the same throughout the whole mass, and supposing moreover that its constituent molecules attract one another in proportion to the inverse square of the distance, demonstrated that it would assume, in consequence of the rotation, the form of a spheroid flattened at the poles; and that the proportion of its equatorial to its polar axis would be 230 to 231. But the supposition of the equal density of the earth is obviously very improbable, and consequently the ratio of the equatorial and polar diameters must be different from that now mentioned. Newton erroneously concluded that if the density is greater in the interior of the earth than at the centre, the compression would be greater than in the case of a spheroid of equal den- Figure of the Earth.

This mistake was pointed out by Huygens, who, in order to determine the amount of compression from theory, reasoned in this way. Suppose two tubes to be united at the centre of the earth, forming a right angle with each other at that point, and extending to the surface, one in the plane of the equator, and the other along the polar axis, and filled with a homogeneous fluid. Now the fluid contained in the polar branch exerts a pressure on the centre equal to the whole of its weight, while the pressure of that in the other tube will be diminished by the centrifugal force. The second column, therefore, if of the same length, will be less heavy than the first; and in order to restore the equilibrium, it is necessary that the equatorial tube shall have gained as much in length as it has lost in weight through the effect of rotation. Hence the sea in the equatorial regions must be higher, or at a greater distance from the centre, than the polar sea, and consequently the earth must have a flattened form. Calculating from the supposition that the density increases regularly from the surface to the centre, where it is infinite, Huygens found the ratio of the diameters to be that of 578 to 379. This investigation is given in his work *De Causa Gravitatis*, published in 1690.

The theoretical determinations of the form of the earth by Newton and Huygens were at variance with the results of geodetic operations that had been carried on in France under the superintendence of the first Cassini, from 1680 till 1716, for the purpose of making a geometrical survey of that country. Cassini found the degree of the meridian to the south of Paris to be 57,092 toises, while on the north of that city it was only 56,960 toises. This result led to the conclusion that the earth is a protracted spheroid, or elongated at the poles; a conclusion entirely inconsistent with the principles of hydrostatic equilibrium, and the deductions of Newton and Huygens. The question, however, was of too great importance to astronomy to be allowed to remain undecided. Accordingly, the Academy of Sciences of Paris determined to apply a decisive test, by the measurement of arcs at a great distance from each other. For this purpose some of the most distinguished members of their body undertook the measurement of two meridional arcs, one in the neighbourhood of the equator, and the other in a high latitude. In 1735 Godin, Bouguer, and La Condamine, proceeded to Peru, where they were joined by two Spanish officers, Don Georges Juan and Antonio d'Ulloa, and, after ten years of laborious exertion, they measured an arc of above three degrees, between the parallels of 2° north and 3° 4' 32" south latitude. The other party, consisting of Mauupertuis, Clairaut, Camus, Lemonnier, Outhier, and Celsius, were in some respects more fortunate, inasmuch as they completed the measurement of an arc near the polar circle, of 57 minutes, and returned to Europe within sixteen months from the period of their departure.

The measurement of Bouguer was executed with great care; and, on account of the locality (the extremities being on different sides of the equator), as well as the excellent manner in which all the details were conducted, it has always been regarded as a most valuable determination. The original base was measured twice, and the difference between the two measures was scarcely two and a half inches. A second base of 5259 toises was measured with the most scrupulous attention by Bouguer, Condamine, and Ulloa, near the southern extremity of the arc, and the result differed less than a toise from the length calculated from the triangles. The latitudes at the two extremities of the arc were determined by simultaneous observations of the same star with zenith sectors. In fact, as the nutation of the earth's axis was then unknown, and as a considerable time necessarily elapsed before a large instrument could be transported from one extremity of the arc to the other, the results of successive observations with a single instrument presented little accordance. By the simultaneous observations, the greatest deviation from the mean did not exceed three or four seconds.

In consequence of a misunderstanding that unhappily sprung up among the principal persons engaged in this memorable expedition, their operations were conducted separately, and we have accordingly three independent results, but all agreeing very nearly with each other. The first is that of Bouguer, who found, after the various reductions, the length of the degree of the meridian at the equator, and reduced to the level of the sea, to be 56,753 toises. Condamine found 56,749 toises, and the Spanish officers 56,768 toises.

The party of Mauupertuis, though their labours were sooner completed, had also to contend with very great difficulties. At their departure from France they had hoped to find a sufficient number of stations among the small islands situated in the Gulf of Bothnia; but on their arrival it was found impracticable to carry on a triangulation among the islands, consequently they were obliged to penetrate into the forests of Lapland. They commenced their operations at Tornea, a city situated on the mainland near the extremity of the gulf, and which formed the southern extremity of their arc. From this place they carried a chain of triangles northward to the mountain of Kittis, lat. 65° 51'. Here they commenced the observations of latitude by observing the distance of their zenith from the star δ Draconis; and having obtained a sufficient number of observations, they returned to Tornea, to determine the latitude of their station there in the same manner. At Tornea they observed not only the star δ Draconis, but also α Draconis; and some doubt having arisen respecting the accuracy of the determination at Kittis, they again repaired to that station, to observe the last star there also. The amplitude of the celestial arc was found, by taking the mean of the observations, to be 57° 29' 6".

It now remained only to measure a base, and thereby determine the terrestrial distance between Tornea and Kittis. This was accomplished on the frozen surface of the river, by two parties who measured separately, and the difference between their results amounted only to about four inches. From this it resulted that the terrestrial arc between Tornea and Kittis was 55,023 toises, giving 57,437 toises to the degree, and consequently exceeding the degree measured by Picard in France by 337 toises.

As all these determinations concurred in proving that the degrees of the meridian increase very sensibly in length from the equator to the high latitudes, no doubt could any longer be entertained that the earth is compressed at the poles. The operations of the two Cassinis in France alone gave results leading to an opposite conclusion; and it therefore became desirable to ascertain the cause of the anomaly. For this purpose Mauupertuis and his former associates undertook the verification of Picard's operations; and it resulted from the remeasurement of the arc between Paris and Amiens that the degree was equal to 57,183 toises. But a more extensive series of operations was undertaken some time after by Cassini de Thury (the third of that name) and Lacaille. With a view to determine more accurately the variation of the degree along the French meridian, they divided the whole arc between Dunkirk and Collioure into four partial arcs, embracing about 2° each, by observing the latitudes at five different stations. While engaged in this measurement, they discovered that the toise which had been used by Picard was shorter by about a thousandth part of the whole than that which formed their standard, and that his final results were accordingly too great by about a thousandth part. They also found the mean length of a degree between Paris and Bourges to be 57,071 toises; between Bourges and Rhodes only 57,040 toises; and between Rhodes and Perpignan 57,048 toises; thus showing on the whole a very sensible diminution as we proceed towards the south. The last portion which they examined was that between Paris and Dunkirk; and after a very careful determination of the latitudes, and the measurement of a base of verification near Dunkirk, they found the degree to be 57,074 toises, greater than any of the former, as ought to be the case on the hypothesis of a polar compression.

Another important operation was likewise undertaken by Cassini and Lacaille, namely, the measurement of a degree of a parallel of latitude. The place they selected was near the mouths of the Rhone, under the forty-third parallel. One of the observers took his station on the mountain Sainte Victoire, near Aix, and the other on Saint Clair, near Clette. On the tower of the church of Sainte Marie, a village situated between the two stations, a signal was made by firing a quantity of loose gunpowder; and the instant of the flash was noted by each of the observers. The difference of sidereal time thus determined was 7 minutes 33-25 seconds, corresponding to a difference of meridians of 1° 53' 19". The distance between the two stations, which was ascertained by triangulation, being reduced to the distance on the parallel intercepted between the two meridians (at the latitude of 43°), was found to be 78,663 toises, whence the length of the degree was computed to be 41,651 toises. This exceeded by about 250 toises the length of a degree of the same parallel computed on the hypothesis of the earth's being a perfect sphere. The details of all these operations are given in the Meridienne de Paris Vérifiée.

About the middle of the last century several arcs of meridian were measured in various countries, which, though of inferior importance in comparison of the more extensive surveys which have since been undertaken, are nevertheless deserving of enumeration. In 1751 Lacaille measured an arc at the Cape of Good Hope, whither he had gone for the purpose of determining the lunar parallax, and making other astronomical observations. At the latitude of 38° 18' 4" he found the degree of the meridian to be 57,037 toises. This result was nearly the same as had been obtained in France, 10° farther from the equator; and clearly proved either the existence of great local irregularities in the form of the earth, or the dissimilarity of the two terrestrial hemispheres. Considerable disposition has been manifested by later observers to question the accuracy of this result; fortunately the details of the operations have been preserved, and, on a careful investigation of them by Delambre, no error of any importance was found either in the methods or the calculations. If there is any considerable inaccuracy, it must have proceeded from the use of inadequate instruments. But, after making all reasonable allowance for this cause of error, the Cape degree still continues to appear anomalous; and it would be desirable to have the whole of Lacaille's arc re-measured.

In 1751 the measurement of a terrestrial arc was undertaken in the Roman states by the Jesuits Maire and Boscovich. It extended nearly two degrees, between Rome and Rimini, and it was found that the degree of meridian between these parallels, namely, 42° and 44°, contained 56,973 toises. The details are given at length by Boscovich, in a work of great elegance, and entitled De Litteraria Expeditione per Pontificiam ditionem, &c. Romae, 1755.

Liesganig, a Jesuit, in 1762 also executed two measures Figure of a meridional degree, one in Hungary and the other in the Austrian states; but it has been shown by Baron Zach, in his Correspondence Astronomique, vol. vii., that the results merit no confidence, and, in fact, would lead to certain error if employed as elements in determining the figure of the earth.

About the same time, in 1764, an arc of meridian was measured in North America, on the peninsula between the Chesapeake and Delaware bays, by two Englishmen, Charles Mason and Jeremiah Dixon. They employed no triangulation, but measured the line with deal rods along the whole extent of the arc, the mean latitude of which was 39° 12'. Their rods were afterwards compared with the five-feet brass rods made by Bird. The latitudes were determined with a zenith sector. The length of the degree, after the necessary corrections and reductions were made, was found to be 60,625 English fathoms, or 56,888 toises. There is no doubt that great care was bestowed on this operation; it is, however, easy to see that the measurement of so long a line by means of rods is liable to many causes of error from which the method of triangulation is exempt.

In 1777 Beccaria undertook to measure a degree in the plains of Piedmont. He found the degree of the meridian at the latitude of 44° 44' to contain 57,024 toises; but great uncertainty remained respecting the correctness of the latitudes, the extreme points of the arc being in the near neighbourhood of immense ranges of mountains, which could not fail to produce a very considerable deviation of the plumb line. It was supposed that as both ends of the arc were terminated by mountain ranges, whereas Boscovich's arc had been carried across the Appenines and terminated at the sea coast, the errors of the two measures occasioned by the local attraction, being of opposite kinds, would neutralize each other, and give a correct mean result.

Amidst the rapid advances of mathematical science towards the end of last century, the determination of the figure of the earth was not overlooked. In the year 1785 a memorial was presented to the British government by Cassini de Thury, stating the important advantages that would result to astronomy and navigation, from having the difference of longitude of the Greenwich and Paris observatories determined by a geodetic measurement. Fortunately this proposal was agreed to. The English operations were placed under the superintendence of General Roy, who to active and indefatigable zeal united great skill and experience in practical astronomy and surveying. In the summer of 1784 a base of rather more than five miles was measured on Hounslow Heath. In the measurement of this base, deal rods were at first employed; but as these were found to warp, and be affected with the variations of the hygrometrical state of the atmosphere, glass tubes were substituted; and in 1791 the same base was measured with a steel chain carefully made by Ramsden, yet the difference from the former measure was found to be only three inches. The mean result was 27,404-2 feet reduced to the level of the sea, and the scale being taken at the temperature of 62° of Fahrenheit. A chain of 32 triangles, in connection with this base, extended over the country to Dover and Hastings; and two more, stretching across the channel, connected them with the French signals on the opposite side. The instruments employed in this survey were of the most excellent description, and far superior to any that had ever been employed in similar operations. The angles of each triangle were measured by a large theodolite constructed by Ramsden; and it was this splendid instrument that first exhibited the spherical excess, or the minute quantity by which, on ac- Figure of count of the sphericity of the earth, the sum of the three angles of a triangle on the earth's surface exceeds 180°.

The French part of this great operation was conducted with equal ability by Cassini (the fourth of that name), Mechain, and Delambre. The angles were measured with the repeating circle of Borda; an instrument of a very different description from the theodolite, but which in geodetic operations may fairly be allowed to give, if not equally, at least sufficiently correct results, while in practice it is much more commodious. The result of the combined measures showed the meridian of Paris to be 2° 19' 51" east of Greenwich, or 9" less than had been determined by Dr Maskelyne.

Soon after this time a series of geodetic measurements was commenced both in France and England, which, in point of extent, as well as minute accuracy, far surpassed all the operations which had yet been undertaken with a view to determine the figure of the earth. In 1791 the National Convention of France having agreed to remodel the system of weights and measures, determined to adopt a standard taken from nature, which might be universally applicable in all countries, and capable of being restored in any future age, if by accident it should happen to be lost. Two such standards were proposed,—namely, the length of the pendulum, which makes a given number of vibrations in a given latitude; and the quadrant of the terrestrial meridian. Of these the pendulum is by far the most easy to be determined; but it was objected, that as the length of the pendulum varies at different latitudes, and also depends in some degree on the geological character of the country where it is measured, its length, if it should happen to be lost, could not be recovered, without knowing the precise place at which it had formerly been determined. The length of the quadrant of the meridian is, however, invariable, and, if the earth is a regular spheroid of revolution, must be the same at all places. Accordingly, the Convention chose the ten millionth part of the meridian from the equator to the pole as the unit of their new scale; and in order that this unit might be determined with the greatest possible precision, it was resolved to remeasure the arc of the meridian of Paris, and to extend it from Dunkirk to Barcelona, a distance comprehending altogether an arc of about nine degrees. The practical execution of this undertaking was confided to two astronomers of distinguished ability, Delambre and Mechain, by whom the requisite operations were carried on during the years 1792, 1793, and 1794, amidst all the dangers and difficulties arising from the disorganized state of the country, with a resolution and courage of which the annals of science afford few examples. The triangles amounted to 115 in number. Each of the three angles of every triangle was separately observed with the repeating circle. The different observations, with the original registers and remarks of the observers, were compared by commissioners, among whom were some of the ablest men in France. A form was drawn up, after which all the calculations were made. The calculations of the triangles, as well as of the azimuths, were examined by Tralles, Van Swinden, Legendre, and Delambre himself. The triangles were connected in the neighbourhood of Paris with a base of upwards of seven miles in length, being 6075-9 toises, at the temperature of 16½° centigrade, or 61½° of Fahrenheit. A base of verification of 6006-25 toises was measured by Mechain, near Perpignan, at the southern extremity of the arc; and the measured length was found to differ by less than a foot from the length deduced by calculation from the first base, though the distance was more than 436 miles. A line of this length, measured with extreme precision, is obviously quite sufficient to enable us to infer, with all the requisite exactness, the length of the quadrantal arc; but the French astronomers resolved to extend the triangulation still farther. Accordingly, Mechain repaired again to Spain, and in the year 1805 continued the chain of triangles from Barcelona to Tortosa, on the coast of the Mediterranean. At this place his labours were prematurely terminated by an epidemic fever. The prolongation of the arc was, however, committed to two philosophers of distinguished reputation, Biot and Arago. An immense triangle, one of the sides of which exceeded 100 miles, connected the coast of Valencia with the small island of Formentera, the most southerly of the Balearic group, and distant no less than 12° 22' 13" from Dunkirk, the northern extremity of the arc. The result of the whole gave a value of the quadrantal arc, differing somewhat from that determined by Delambre and Mechain, but so little that the length of the teise would be scarcely affected by four times the millionth part of itself. The details of this magnificent operation are given at length in the four volumes of the Base Metroique.

The English survey, which had been interrupted by the death of General Roy in 1790, was resumed in 1793 under the direction of Colonel Mudge. The original triangles between Greenwich and Dover were extended along the coast to Dunnose in the Isle of Wight, and thence through Devonshire and Wiltshire, and connected with a base of verification measured on Salisbury Plain. The length of this base was found, after the proper reductions, to be 36574-4 feet, differing scarcely one inch from the length deduced by calculation from the base on Hounslow Heath. So near a coincidence, though probably owing in some degree to a compensation of errors, affords a convincing proof of the extreme accuracy with which every part of the operation had been conducted. In 1802 the triangulation was carried into Yorkshire, and a meridian arc measured from Dunnose to Clifton. The latitudes at the terminal points were determined with Ramsden's zenith sector. The arc was afterwards extended to Burleigh Moor, comprehending nearly four degrees. It may be remarked, that both the French and English arcs present this singular anomaly, that when portions of them, at particular places, are considered separately, the length of the degree appears to increase on going southward.

The survey has been continued, with occasional interruptions, up to the present time, under the able directions of Lieutenant-Colonel Colby; and the triangulation has been carried to the remotest parts of Scotland, and over a considerable part of Ireland. In the course of the operations, several important improvements, both in respect of the instruments employed, and in the method of conducting geodetic measurements in general, have been introduced into practice. A base of upwards of seven miles has been measured near Londonderry; and it now only remains to determine the latitudes of some stations, to give us the elements of a new and greatly prolonged arc.

In the years 1801, 1802, and 1803, Maupertuis' Swedish arc was re-measured by Svanberg, and extended nearly 40' to the south. The methods were the same as had been employed by Delambre. The extremities of the new arc were at Malörn and Pahatwara. The distance was found to be 92,778 toises, and the difference of the latitudes 1° 37' 20"3; whence 1° = 57,196 toises. This agrees much better than the result of Maupertuis (57,422 toises) with other measures; but the difference, which implies an error of 12" in the latitude of Kittis as determined by the French academicians, has not been satisfactorily accounted for; so that there is still some doubt about the length of the degree in that latitude. (See Svanberg's Exposition des Operations faites en Lapponie, &c. Stockholm, 1805.)

Since the beginning of the present century, two arcs of The first was in the neighbourhood of Madras, and comprehended only 1° 32'. The second, however, is the longest which has yet been measured. The first, and a large part of the second, was accomplished under the direction of Colonel Lambton; and the instruments and methods of observation and calculation were exactly the same as those that had been employed by Colonel Mudge in the English survey. The southern extremity of the second arc was at Punnar, near Cape Comorin, latitude 8° 9' 32" 51'; and the northern at Daumerigida, latitude 18° 3' 16" 07'. The amplitude is consequently 9° 53' 43" 56', and the distance between the extremities was found to be 598,629-98 fathoms (about 680 miles), giving 60,495 fathoms, or 362,970 feet, for the length of the degree. Several bases were measured, and the whole of the operations appear to have been conducted with great skill and accuracy. This arc has since been extended by Captain Everest to Kullampoor, latitude 24° 7' 11" 8'; so that the whole length now includes very nearly sixteen degrees. The details of Colonel Lambton's operations are given in the different volumes of the Asiatic Researches (see vols. viii. x. xii.), and those of Captain Everest, in his "Account of the Measurement of an Arc of the Meridian between the Parallels of 18° 3' and 24° 7," printed at the expense of the East India Company, but not yet published.

Various geodetic operations on a less extensive scale have been recently executed, which are better adapted, perhaps, to give information respecting the local curvature than the general form of the earth. Beccaria's arc has been remeasured by Plana and Carlini; the results clearly demonstrate the existence of some errors in the original measurement, but they are not yet altogether satisfactory, and the country is very unfavourable. The distance between Göttingen and Altona has been measured by Gauss; and the amplitude of the corresponding celestial arc is known with the utmost precision, from observations of the latitude made at the respective observatories of the two places. The amplitude, however, is only about two degrees, and there is some doubt about the exact length of the iron bars with which the base was measured. A more extensive arc has been measured in Russia by Struve. It extends at present to three and a half degrees, and it is understood that it is in contemplation to prolong it still further. Many new methods have been employed in this measurement; and it acquires additional value from its high latitude, and the acknowledged skill and accuracy of the observer.

The above are the principal arcs of meridian, but some arcs of parallel have also been measured. Theoretically speaking, the figure of the earth may be determined from the measurement of arcs of parallel, as readily as from meridional arcs; and the geodetical operations in the one case differ in no respect from those in the other. But the great, and, we fear, insurmountable difficulty, is to determine with sufficient precision the difference of astronomical longitudes. In a subsequent part of this article we shall have again occasion to mention Cassini's measurement of an arc of parallel across the mouth of the Rhone; of the English arc between Beachy Head and Dunnose; and that recently made from Marennos to Padua.

The problem of determining the earth's figure from the laws of hydrostatic equilibrium has not yet received the complete solution which the present advanced state of analytical science might lead us to expect. We have already mentioned the results found by Newton and Huygens. Maclaurin first demonstrated that the ellipsoid of revolution is a figure which satisfies the conditions of equilibrium in the case of a revolving homogeneous body, whose particles attract one another according to the law of the inverse square of the distance; and he moreover determined the amount of the attraction in a particle situated anywhere on the surface of such a body. This was an important step towards the solution of the problem. A few years afterwards, Clairaut published his Théorie de la Figure de la Terre, which, among other results, demonstrated with uncommon elegance, contained a very remarkable theorem which establishes a relation between the oblateness of the earth and the variations of gravity at different points of its surface, and consequently gives the means of determining the earth's figure by means of observations of the length of the seconds pendulum at different latitudes. Maclaurin's discovery gave the intensity of the gravitating force on the surface of the spheroid, or in its interior; but it was necessary to assign the force with which an exterior point is attracted. This was partly accomplished by D'Alembert, who proved that the attraction of any ellipsoid whatever upon a point situated in the prolongation of one of its axes, is to the attraction of a similar spheroid having the same centre, and passing through the attracted point, as the mass of the first spheroid is to the mass of the second. Legendre found an approximate expression by means of series, for the attraction on a point placed anywhere without a spheroid of revolution; and Laplace extended the solution to ellipsoids in general. But the complete solution was reserved for Mr Ivory, who obtained in finite terms a very simple expression for the attractive force of an ellipsoid on an exterior point, and thereby completed the theory of Maclaurin. It may be stated in a general way, that notwithstanding all the researches that have been made respecting this very intricate subject, the only positive results which theory has yet supplied are the three following:—1st, Maclaurin's demonstration that a homogeneous mass revolving about an axis will be in equilibrium with the figure of an ellipsoid of revolution; 2d, Clairaut's theorem for determining the ellipticity by means of pendulum observations; and, 3d, the expression for the attraction of ellipsoids on exterior particles, found by Mr Ivory. A complete theory would determine whether there are or are not other figures besides the ellipsoid, under which the equilibrium could subsist; but this has not yet been accomplished. Indeed it has not even been proved that the sphere is the only figure which could be assumed by a body at perfect rest, and whose molecules are subjected to no other forces than their mutual attractions.

Our limits do not permit us to enter into details respecting the numerous experiments that have been made of late years to determine the figure of the earth by measuring the variations of gravity at different places by means of the pendulum. The most valuable series of observations of this kind we yet possess are those of Captain Foster, reduced under the direction of Mr Baily, and published in the Memoirs of the Royal Astronomical Society, vol. vii. But a discovery recently made by Bessel proves that less accuracy has been obtained by this method than was supposed. It has been found, that a pendulum, when vibrating, drags along with it a portion of air, the precise effect of which can be ascertained in no other way than by actual experiment in vacuo with each individual pendulum. The probable correction which it would be necessary to apply to the results that have already been found, cannot be satisfactorily determined.

The mean of the pendulum experiments gives rather a higher value of the ellipticity than the results of geodetic measures; but there are many elements, particularly the irregular constitution of the exterior crust of the earth, and the density of the strata surrounding the station, which can scarcely be determined, and which yet affect materially the results of the experiments. This subject will be resumed under the article Pendulum. Besides the methods which have now been alluded to, physical astronomy furnishes other means of arriving at a knowledge of the figure of the earth. The precession of the equinoxes, and nutation of the earth's axis, are phenomena depending on the compression of the earth; and as their amount is now ascertained, from astronomical observations, with the utmost accuracy, we can reciprocally deduce from them a knowledge of the compression. They do not, however, give us an absolute value of the amount of compression, but they make known the limits within which it must necessarily be confined. These limits are $\frac{1}{27}$ and $\frac{1}{35}$. But a more delicate measure of the same element is furnished by some irregularities in the moon's motion to which it gives rise; and as the lunar theory has now attained a very high state of perfection, and as the small irregularities which cause so much perplexity in geodetical measures here entirely disappear, this is perhaps the most satisfactory method of all of determining the ellipticity of the earth. The equations into which the irregularity in question enters were discovered by Laplace; and the ellipticity necessary to produce the observed effect was found, on calculation, to be $\frac{1}{35}$; confirming in a most remarkable manner the deductions from the measurement of arcs and the observations of the pendulum.

SECTION II.

OF THE MEASUREMENT OF TERRESTRIAL ARCS.

The determination of the figure of the earth, being founded on the comparison of lines actually measured on its surface with the corresponding arcs of the celestial sphere, requires a combination of geodetical and astronomical operations, which it will be our object in the present section to explain. Theoretically considered, the geodetical line to be measured may lie in any direction, because it can always be reduced to the meridian or to any other given direction; but, for practical reasons, which will be pointed out hereafter, it ought to be taken, as nearly as the nature of the country in which the measurement is made will permit, either in the direction of the meridian, or of a parallel of latitude. Differences of latitude, however, can be determined with greater precision than differences of longitude; the meridional arcs are consequently the best adapted to the purpose. Satisfactory results can only be obtained from the measurement of lines of very considerable length, for example, of several degrees; and this circumstance renders the geodetical operations both laborious and complicated. In no place of the world, probably, and certainly in no populous country, could it be possible to find a territory so free from obstructions as to admit of the direct measurement of a line of two or three hundred miles in the same direction. Hence, when an operation of this nature is to be undertaken, the country in which the direction of the line to be measured lies is divided into triangles; the angles of each triangle are accurately observed with a theodolite, or some other equivalent instrument; and the side of one of them being actually measured, all the other sides are computed from it by the rules of trigonometry. In this manner the length of the whole line is ascertained, and on comparing it with the celestial arc, the data are obtained for deducing the dimensions and form of the earth.

It appears, then, that in order to determine the figure of the earth by the measurement of degrees, four distinct sets of operations are necessary: 1st, The measurement of one side of a triangle, or of the base as it is called; 2d, the observation of the angles of each triangle, together with the height of the station, in order that the triangles may be reduced, if necessary, to the horizontal plane, and figured to the level of the sea; 3d, the observation of the azimuths of the sides of the triangles, or the angles they make with the meridional line; 4th, the latitudes of the extreme points of the line. It is immaterial in what order these operations are undertaken. We shall begin with the measurement of the base.

The length of the base must be regulated in some degree by the nature of the locality; but being the fundamental element from which the sides of all the triangles are deduced, it should not be less than five or six miles. A piece of ground must therefore be selected, as nearly level as possible, and admitting a line of that length to be described on it. The direction of the base is first marked out by pickets; and, in order to place these in the same straight line, or rather in the same vertical plane, a telescope, mounted in the manner of a transit instrument, is set up at one extremity, by means of which the observer is enabled to direct an assistant, by signals, to the exact position in which the picket is to be placed, and also to bring their tops exactly into the same level. As a farther guide, a rope may then be stretched between the pickets; and, for greater security in the subsequent operations, the whole extent may be approximately measured by rods or chains, in the usual manner. The extremities of the base ought to be permanently marked by points. The centres of metal tubes, cannon for example, let into the ground to a considerable depth, answer this purpose very conveniently.

These preliminary operations being accomplished, the next step is to proceed to the more delicate operation of measuring the exact distance between the terminal points, or extremities of the base.

Various methods have been practised for this purpose. In the measurement of the first base of the trigonometrical survey on Hounslow Heath by General Roy, deal rods were at first employed, as had been done by La Caille, Lepomnier, and others; but though every precaution was used in selecting the best seasoned timber, and every means employed to secure the rods from flexure, it was discovered, in the course of the operations, that they were liable, from variations in the state of the atmosphere as to moisture or dryness, to sudden and irregular changes of such magnitude as to destroy all confidence in the results. They were therefore laid aside; and glass rods, consisting of straight tubes twenty feet in length, and about an inch in diameter, and enclosed in wooden cases, were substituted. The only alteration to which these were subject arose from the expansion and contraction from variations of temperature; and as such alterations follow constant and ascertainable laws, their effects could be easily computed, and a correction made for them. But the apparatus ultimately adopted, both in the measurement of this base, and the other bases subsequently measured for the purpose of verifying the operations in the English survey, consisted of two steel chains, made with great care by Ramsden, of a hundred feet in length, and composed each of forty links, joined in the manner of a watch chain. The chains were supported by deal coffers, and after being placed in a straight line, were stretched with a weight of fifty-six pounds. Their temperature and inclination to the horizon were carefully noted at the same time, and the proper corrections subsequently made.

This method has been objected to, on account of the uncertainty of giving the chain always the same degree of tension, and of the rubbing and wear of the joints; but the best proof of the accuracy of the results obtained by it consists in the fact, that the two measurements of the base on Hounslow Heath, first by glass rods, and secondly The use of chains has been confined to the English and Indian surveys. Bouguer, in the Peruvian measurement, employed an iron rod of about a toise in length, which, from the frequent comparisons that have since been made of it with other standards, has been particularised by the name of toise of Peru. For the determination of the meridian of Dunkirk, Borda employed rules of platina, exactly two toises in length, and furnished with a very ingenious contrivance for ascertaining the expansion. Along one side of the rule there was a slip of brass, firmly fixed at one extremity, but at liberty to move along the rule at the other end, according to the relative expansion of the two metals. The relative expansion being read off by means of a micrometer, gave a ready means of computing, from tables previously constructed, the absolute expansion of the platina rule. Four rules were made use of, three of which were always placed on the ground at once; and in order to prevent any derangement, they were not brought into absolute contact. A small interval of a few millimetres was left between them, which was measured by means of a slider attached to the preceding end of each rule, and which was pushed out till it came into contact with the following end of the preceding rule. See Delambre, Astronomie, tome iii. p. 538, or Base du Système Métrique.

Various other contrivances have been used for the measurement of the base line. MM. Plana and Carlini, in the re-measurement of Beccaria's arc, employed measuring rods of wood, twelve metres in length, of the form of hollow parallelepipeds. For the Hanoverian base measure, between Altona and Gottingen, Gauss employed three rods of hammered iron, each twelve French feet in length. But the most ingenious apparatus which has yet been invented for this purpose, is that which was made by our celebrated artist Troughton for the Irish base. As this instrument is probably destined to supersede all others previously used, we shall briefly describe the principle on which it is constructed.

Let AB and DE (Plate CCXLIV. fig. 1) represent two bars of different metals, firmly united to a transverse bar at the middle C, but entirely free to move independently of each other at the extremities, according to their different expansions. AP and BQ are two tongues of steel attached to the extremities of the two rods AD and BE, by double conical joints, around which they are capable of making a small angle with the lines perpendicular to AB or DE. At a certain given temperature the bars are exactly of the same length, when, consequently, AP and BQ are parallel to each other, and perpendicular to the bars. Let us now conceive the bars to receive an increase of temperature, and that the dilatation of AB is greater than that of DE. In consequence of the increased temperature, both bars will be lengthened, but their lengths will no longer be equal; consequently the two tongues AP and BQ will no longer be parallel; for, suppose AB to become ab, and DE to become de, then AB being by hypothesis a more expansive metal than DE, ab will be longer than de, and the steel tongues will take the positions ax and by, inclined to each other. But if the point P is taken at such a distance from the bars that PA is to PD in the ratio of the dilatation of AB to DE, then in the new position ax, the point P will not sensibly deviate from the perpendicular AP, at least for moderate variations of temperature. In the same manner, the point Q will not sensibly deviate from its first position in the perpendicular BQ, so that Figure of the distance between P and Q remains sensibly the same.

It is easy, however, to see that the dilatation of the bars must be confined within narrow limits; for, as the length of the tongue is invariable, P will not be found exactly in the line AP after the expansion has taken place; it will be on the same side of AP on which are the points a and d, and its distance from AP will be accurately measured by the quantity AP × (tan. APa — sin. APd); and as the same thing takes place with regard to Q, the distance between P and Q will be increased by twice this quantity. But as the angle APa must in all cases be extremely small, the difference between its tangent and its sine is altogether inappreciable. It is unnecessary to remark, that the same consequences (mutatis mutandis) take place when the temperature falls below the assumed standard.

In the instruments made use of in the measurement of the Irish base, the bar AB was of brass, and DE of iron. The length of each was ten feet, and the distance between them two inches. The length of the tongue is hence easily found from the relative dilatation of the two metals. Suppose the expansion of brass to be to that of iron in the ratio of 83 to 53 for an increase of 1° of temperature, and let PD = x inches, we shall have \(x : x + 2 = 53 : 83\); whence \(x = \frac{53}{15} = 3\frac{1}{3}\) inches very nearly. But the precise position of the points P and Q must be determined by actual experiment; for the expansion or contraction of no two bars, even of the same metal, are ever found to be exactly the same.

In practice, five or six sets of bars, constructed in the manner now described, and placed in strong deal boxes supported on trestles, are laid along the line to be measured, and accurately levelled. They are placed at a small distance from each other, and the distance between the dots on the adjacent steel tongues of two succeeding bars is accurately measured by an ingenious micrometrical apparatus constructed so as to form a compensating instrument of exactly the same nature as the measuring bars.

The conception of this elegant apparatus belongs to Lieutenant-Colonel Colby; and the practical results it gives correspond with its theoretical excellence. It is computed that the greatest possible error of the base measured on the eastern shore of Lough Foyle, in the county of Londonderry, cannot exceed two inches, though the length is very nearly eight miles.

The reduction to the level of the sea is made as follows: Let \(r\) = radius of earth, \(x\) = height above the level of the sea, \(a\) = measured length of the base, and \(a'\) = its reduced length; then

\[ a' = a \left( \frac{r}{r+x} \right) = a \left( \frac{1}{1+\frac{x}{r}} \right) = a \left\{ 1 - \frac{x}{r} + \left( \frac{x}{r} \right)^2 - \ldots \right\} \]

a series of which all the terms after the second may be neglected, by reason of their smallness.

It is easy to see, that as the base is not, correctly speaking, a straight line, but an arc of a circle, or rather ellipse, the sum of the lengths of all the rods or chains horizontally applied to it is not precisely the length of the arc, but of the circumscribing polygon. In order to determine how far this circumstance can affect the result, let us suppose the length of the measuring rod to be ten feet; when laid on the ground, exactly in the horizontal plane, it forms two tangents to the surface, of five feet each, and therefore each half length of the rod ought to be diminished by the difference between the tangent whose length is five feet, and the arc to which it belongs. Let A be the arc, then (Algebra, Sect. XXV. H. 2) whence tan \( A - A = \frac{1}{3} \tan \frac{A^3}{r^2} = \frac{125}{3r^2} \).

This must be multiplied by the number of times that the half rod, or five feet, is contained in the whole base. Supposing the base to be five miles, or 5280 times the length of the half rod, and the radius of the earth to be 4000 miles, or 4000 × 5280 feet, we should have tan \( A - A = \frac{125}{3 \times 4000^2 \times 5280} \) feet; a quantity which is altogether insensible.

We come now to the triangulation. The first thing to be done is to select proper stations for the vertices of the triangles. The selection will necessarily depend on the nature of the country through which the survey is made; but there are certain principles which ought to be observed, as nearly as the circumstances will permit. The stations should be chosen, so that each angle of the principal triangles should approach as nearly as possible to 60°, because a small error in the estimation of a very acute or obtuse angle will greatly affect the length of the opposite side computed from it. Excepting in unavoidable cases, no angle ought to be less than 30°. The sides cannot be too long, while the signal at one extremity is visible from the other. The signals may be towers, spires, or other conspicuous objects conveniently situated; but such signals can seldom be well bisected by the wire of the telescope. Flag-staffs are better when the distances are not too great. Delambre preferred signals erected on purpose. They were generally constructed of wood, and of the form of a truncated pyramid. One considerable advantage arises from the use of signals constructed in this manner, namely, that the instrument can be placed exactly at the centre of the station. Frequently Bengal lights, reverberating lamps, and white lights, are employed; but these of course can only be seen during the night. The best signal of this sort was invented by Lieutenant Drummond. It is formed by directing a stream of hydrogen gas on a small piece of ignited quicklime; an intense light is produced, which may be seen at very great distances. Care should be taken to place the signals, if they are to be observed by day, in such positions, that they may be projected on the sky; if projected on forests or mountains, they will seldom be distinctly visible. Svanberg constructed his signals with openings, to permit the light of the sky to be seen through them.

The triangles being marked out, and the signals prepared, the next object is to measure the angles. When the theodolite is used for this purpose, no correction is required on account of the different altitudes of the signals, the reduction being effected by the instrument itself; but when a sextant or repeating circle is employed, the observed angles must be reduced to the plane of the horizon.

Let P, Q, and R (fig. 2), be the three stations, PQR the observed angle oblique to the horizon; produce PQ and PR till they meet Q'R', a line parallel to the plane of the horizon; through Q'R' let a plane be conceived to pass parallel to the horizontal plane, meeting PZ, the vertical passing through P, in P'; then Q'P'R' is the horizontal projection of QPR, and the angles ZPQ, ZPR are the complements of the inclinations of the line PQ and PR to the horizon.

With P as a centre, and a radius = 1, let a sphere be described, intersecting PZ, PQ, and PR in the points A, B, and C respectively; the angle BAC is evidently equal to the horizontal angle Q'P'R'. Now, in the spherical triangle ABC, we have given by observation the side AB = the zenith distance of Q, AC = the zenith distance Fig. of R, and BC the oblique angle at the centre of the sta- tion, from which we are to find the angle BAC = Q'P'R'.

Let BC = a, AC = b, AB = c, and BAC = A. By a well-known formula of spherical trigonometry,

\[ \cos A = \frac{\cos a - \cos b \cos c}{\sin b \sin c} \]

but \(\cos A = 1 - 2 \sin^2 \frac{1}{2} A\),

therefore \(2 \sin^2 \frac{1}{2} A = 1 - \frac{\cos a - \cos b \cos c}{\sin b \sin c}\),

and \(2 \sin^2 \frac{1}{2} A = \frac{\cos (b-c) - \cos a}{\sin b \sin c}\).

Now, if p and q denote two arcs, we have (Algebra, XXV. D),

\[ \cos p - \cos q = -2 \sin \frac{1}{2} (p+q) \sin \frac{1}{2} (p-q), \]

therefore

\[ \cos (b-c) - \cos a = -2 \sin \frac{1}{2} (a+b-c) \sin \frac{1}{2} (b-c-a) \]

\[ = 2 \sin \frac{1}{2} (a+b-c) \sin \frac{1}{2} (a-b+c), \]

and the equation becomes

\[ \sin^2 \frac{1}{2} A = \sin \frac{1}{2} (a+b-c) \sin \frac{1}{2} (a-b+c) \]

Let \(\frac{1}{2} (a+b+c) = s\); then \(\frac{1}{2} (a+b-c) = s - c\), and \(\frac{1}{2} (a-b+c) = s - b\); consequently

\[ \sin \frac{1}{2} A = \left( \frac{\sin (s-c) \sin (s-b)}{\sin b \sin c} \right)^{\frac{1}{2}}. \]

From this formula the value of A is easily computed, when the inclinations of the lines PQ and PR to the horizon are not very small. In practice, however, it frequently happens that these inclinations are very small, or that the angles b and c differ very little from right angles. When this is the case, the logarithmic calculation of the formula becomes troublesome, particularly if it is required to determine A with great exactness; and it is better to adopt a different method of proceeding.

Instead of seeking to find the projected angle Q'P'R', we may find its difference from the observed angle QPR, which difference will seldom amount to more than a few seconds. Since the arcs b and c are supposed to differ little from 90°, if we make \(b = 90^\circ - \beta\), \(c = 90^\circ - \gamma\), the angles \(\beta\) and \(\gamma\) will be very small. Substituting these values of b and c in the formula

\[ \cos A = \frac{\cos a - \cos b \cos c}{\sin b \sin c}, \]

it becomes

\[ \cos A = \frac{\cos a - \sin \beta \sin \gamma}{\cos \beta \cos \gamma}; \]

but

\[ \sin \beta = \beta - \frac{\beta^3}{6} + \text{etc}, \quad \sin \gamma = \gamma - \frac{\gamma^3}{6} + \text{etc}. \]

\[ \cos \beta = 1 - \frac{\beta^2}{2} + \text{etc}, \quad \cos \gamma = 1 - \frac{\gamma^2}{2} + \text{etc}; \]

therefore, on substituting, and rejecting all terms above those of the second order, we have

\[ \cos A = \frac{\cos a - \beta \gamma}{1 - \frac{1}{2} (\beta^2 + \gamma^2)} = \cos a - \beta \gamma + \frac{1}{2} (\beta^2 + \gamma^2) \cos a. \]

Let \(A = a + x\). It is evident that when \(\beta\) and \(\gamma\) both vanish, \(x\) also vanishes, and we get \(A = a\). Hence if \(\beta\) and \(\gamma\) be both very small, \(x\) is also very small, and consequently we may suppose its cosine = 1, and its sine = \(x\). In this case

\[ \cos A = \cos (a + x) = \cos a \cos x - \sin a \sin x, \]

becomes

\[ \cos A = \cos a - x \sin a. \]

Equating this with the former value of \(\cos A\), we get

\[ x \sin a = \beta \gamma - \frac{1}{2} (\beta^2 + \gamma^2) \cos a, \] In order to adapt this fraction to logarithmic computation, let both its numerator and denominator be multiplied by 2; then, because \(\cos^2 \frac{1}{2}a = \sin^2 \frac{1}{2}a\), and \(2\beta = 2\gamma \cos^2 \frac{1}{2}a + 2\gamma \sin^2 \frac{1}{2}a\), we find, on multiplying and reducing,

\[x = \frac{\beta - \gamma}{\sin a} \left( (\beta + \gamma) \sin^2 \frac{1}{2}a - (\beta - \gamma)^2 \cos^2 \frac{1}{2}a \right)\]

but \(\sin a = 2 \sin \frac{1}{2}a \cos \frac{1}{2}a\); therefore, by substitution,

\[x = \left( \frac{\beta + \gamma}{2} \tan \frac{1}{2}a - \left( \frac{\beta - \gamma}{2} \right)^2 \cot \frac{1}{2}a \right) \sin 1^\circ.\]

Here the arcs \(\alpha, \beta, \gamma\) are expressed in parts of the radius; but they are given by observation in minutes and seconds.

Now an arc of 1° may be considered as equal to its sine; hence the arc in seconds is to the arc in parts of the radius as 1° to sin 1°; consequently the arc in feet is equal to the arc in seconds \(\times\) sin 1°. Multiplying, therefore, by sin 1°, we get

\[x = \left( \frac{\beta + \gamma}{2} \tan \frac{1}{2}a - \left( \frac{\beta - \gamma}{2} \right)^2 \cot \frac{1}{2}a \right) \sin 1^\circ.\]

Another reduction is necessary. Unless the signals are constructed on purpose, it will rarely happen that the instrument can be placed exactly in the centre of the station. In this case it must be placed at some other point near the centre of the station, and the observed angle reduced to the centre by calculation. Suppose, for example, it were required to determine the angle ACB (fig. 3), which the objects A and B subtend at C, and that the instrument cannot be placed exactly at C. Let P be the point, at a short distance from C, where the instrument is actually placed, and APB be the observed angle; then, since CP can be measured, and CA, CB are supposed to be known, approximately at least, the problem is, having given the angle APB, and the distances CA, CB, and CP, it is required to determine the angle ACB.

Let AC and BP intersect in E. Then, since AEB = ACB + CBP = APB + CAP, we have

\[ACB = APB = CAP = CBP.\]

But \(\sin CAP = \frac{CP}{CA} \sin CPA\), and likewise \(\sin CBP = \frac{CP}{CB} \sin CPB\); therefore, making AC = m, BC = n, CP = d,

\[ACB = C, APB = P, CBP = p,\] then CPA = P + p, and by substitution,

\[\sin CAP = \frac{d}{m} \sin (P + p), \text{ and } \sin CBP = \frac{d}{n} \sin p,\] whence, dividing by \(\sin 1^\circ\) to reduce the expression to seconds,

\[CAP = \frac{d}{m} \sin (P + p), \text{ and } CBP = \frac{d}{n} \sin p;\] therefore (since CAP = CBP = ACB = APB = C = P),

\[C = P = \frac{d}{m} \sin (P + p) - \frac{d}{n} \sin p,\] which gives C, or the angle ACB, expressed in seconds of a degree.

This expression is exact; but an approximation more convenient for calculation, and sufficient in almost every case that can occur in practice, was generally followed by Delambre. Let a circle be described about the triangle ABC, intersecting BP in D, and let CD, DA, and AB be joined. We have then ACB = APB = ADB = APB = DAP. Let the angle CDB (= CAB, which is known by observation) be denoted by A. In the triangle DPA,

\[\sin DAP = \frac{DP}{AD} \sin APB = \frac{DP}{AD} \sin P,\] and in the triangle DPC,

\[DP = PC \frac{\sin PCD}{\sin CDB} = PC \frac{\sin (CDB - CPB)}{\sin CDB},\] or \(DP = d \frac{\sin (A - p)}{\sin A};\) therefore

\[\sin DAP = \frac{d}{AD} \frac{\sin (A - p)}{\sin A} \sin P.\] But AD may be considered as equal to AC = m; whence

\[\sin DAP = \frac{d \sin (A - p) \sin P}{m \sin A},\] and consequently, when expressed in seconds,

\[DAP = C - P = \frac{d \sin (A - p) \sin P}{m \sin A \sin 1^\circ}.\]

This expression vanishes when \(A - p = 0\), or when the points P and D coincide, that is, when the instrument is placed on the circumference of the circle, circumscribing the triangle ACB. And it may be so placed, unless obstacles intervene, by moving it along the line PB, till the angle CDB is observed to be equal to BAC.

When the three observed angles of each triangle have been reduced to the horizon, they represent the angles of a spherical triangle, the sides of which are intercepted by the verticals of the three stations, and the sum of the three angles of each triangle exceeds 180° by a quantity which is called the spherical excess. The sides ought therefore to be calculated by the rules of spherical trigonometry; but as the sides of the triangles are in all cases very small in comparison of the radius of the earth, the calculation made in this way (for which indeed the existing tables are not well adapted) becomes exceedingly tedious. Instead, therefore, of calculating directly the spherical triangle, it is more convenient to compute its deviation, which is always very small, from a plane triangle. The method employed by Delambre was the following:

In the first place, the spherical angle, or that which is formed by tangents at the surface, must be reduced to the corresponding plane angle formed by the chords. This is easily accomplished by the help of the theorem above given for reducing the observed angles to the horizon; for the angle formed by the chord and the tangent of an arc is equal to the angle at the centre subtended by half the arc; consequently when the lengths of the two sides of a spherical angle are known nearly, the angle of depression of the chords is also known; and therefore the preceding theorem for reducing angles to the horizon can be immediately applied. For example, suppose the sides of a spherical angle to be respectively p and q miles, and the circumference of the earth to be 25,000 miles, then

\[25,000 : p :: 360^\circ : \frac{360^\circ p}{25,000}, \text{ or } \frac{21,600^\circ p}{25,000} = \frac{108 p}{125} \text{ seconds}\] of a degree, which is the angle subtended at the centre by the whole arc; and consequently one half of it, or \(\frac{54^\circ p}{125}\), is the depression of the chord. In the same manner, \(\frac{54^\circ q}{125}\) is the depression of the chord of q; and by substituting these in the formula

\[x = \left( \frac{\beta + \gamma}{2} \tan \frac{1}{2}a - \left( \frac{\beta - \gamma}{2} \right)^2 \cot \frac{1}{2}a \right) \sin 1^\circ,\] we obtain the angles made by the chords, by means of which the triangle may be computed in the same manner as if it were a plane triangle.

To make this reduction, it is necessary to know, ap- Figure of approximately at least, the diameter of the earth as well as the Earth the distances of the stations; but the result will be little affected by a moderate error in either of these data. In fact, it is only when the sides of the triangles are very large that the effects of the earth's curvature become at all sensible.

But a more elegant manner of estimating the effects of the earth's curvature, though not always of so easy application, was proposed by Legendre. This illustrious geometer discovered the following very remarkable property of spherical triangles; namely, that when the sides are very small in comparison of the radius of the sphere, if from each of the angles there be subtracted one third part of the quantity by which the sum of the three angles exceeds two right angles, or $180^\circ$, the angles thus diminished may be regarded as the angles of a plane triangle, the sides of which are equal in length to those of the proposed spherical triangle. The excess of the sum of the three angles above two right angles is proved by trigonometry to be equal to $\frac{a}{r^2}$, $a$ being the area of the triangle, and $r$ the radius of the sphere; so that the theorem of Legendre may be enumerated as follows:

"If the angles of a spherical triangle whose surface is small in comparison of the surface of the whole sphere, be denoted by $A$, $B$, and $C$, and the opposites by $a$, $b$, and $c$, the triangle may be calculated as a plane triangle, the sides of which are $a$, $b$, and $c$, and the opposite angles $A - \frac{1}{3} \epsilon$, $B - \frac{1}{3} \epsilon$, $C - \frac{1}{3} \epsilon$, $\epsilon$ being the excess of the sum of the three angles of the proposed spherical triangle above two right angles."

In applying the above theorem, it is necessary first to calculate $\epsilon$ or $\frac{a}{r^2}$, which can always be done a priori from the known parts of the spherical triangle considered as rectilinear. This being done, we have only to deduct $\frac{1}{3} \epsilon$ from each of the observed angles, and then compute the remaining parts, as in the case of a plane triangle.

By this method, the distances between the several signals may be computed; and as the angles at all the stations are given by observation, the inclinations of all the sides of the triangles to any one of them are also known, so that it is only necessary to determine very accurately the inclination of one side of a triangle to the meridian, in order to ascertain the inclinations of all the other sides, and thence to compute the meridian itself.

The determination of the azimuth of a signal is accomplished by means of astronomical observation, and may be performed in various ways, but all depending on the same principle. It is necessary, however, in any case to be provided with the means of determining the time with very great precision. The azimuth of the sun, or a star, at any given instant, can be determined with sufficient accuracy. Let the observer, therefore, take his station at one of the signals, and observe the angle formed between the other signal and the sun, or a star, when nearly in the horizon, and let him note the instant of time at which the observation was made. Knowing the error of his clock or chronometer, he knows also the true time, and can consequently calculate the azimuth of the observed celestial body. Taking the sum or difference, as the case may be, of this and the observed angle, he obtains the azimuth of the distant signal, or the angle which the straight line joining the two signals makes with the meridian. The refraction will scarcely affect the result; but a small error with respect to the time would lead to considerable errors. If the sun is observed, the error of the clock must be determined by observations of his meridional transits; but most frequently a circumpolar star, as Capella, is preferred.

Sometimes the observation is made on the pole star itself; but in this case it is necessary to know the latitude of the station very accurately. Another method is frequently resorted to. Having set up a mark very nearly in the meridian, adjust a transit instrument upon it, and then, by means of the transits of stars at different polar distances, or other means known in practical astronomy, determine the deviation of the instrument from the meridian. This gives the direction of the mark with respect to the meridian, and consequently the angle to be added to or subtracted from the angle between the mark and the signal. For greater security, the azimuths may be observed at several signals. They ought at least to be observed at each extremity of the chain of triangles.

The triangles being calculated and reduced to the horizon, we are now in a condition to compute the length of the arc between the parallels of the extreme stations. Let ABCDEFG, &c. (fig. 4), be a chain of triangles lying nearly in the direction of the meridian AZ, and traced on a spheroidal surface, supposed to be formed by the continuation of the ocean. Let L be the last station, and LX the perpendicular passing through L and meeting AZ in the point X; it is required to determine AX.

By means of the previous calculations, all the sides of the triangles are supposed to have been computed, as well as the azimuth of C, or the inclination of AC to the meridian. Let CD be produced to M. In the triangle ACM there are given the side AC, and the two angles CAM, ACM. To find the third angle AMC, first compute the spherical excess $\epsilon$; from each of the angles CAM, ACM, subtract $\frac{1}{3} \epsilon$; take the sum of the two remainders from $180^\circ$, and there will be left the value of AMC. The angles being thus given, and also the side AC, the other two sides AM and CM are computed in the same manner as if the triangle were rectilinear.

In the quadrilateral MDFN there are given the two opposite angles at M and F (the first from the previous calculation, and the second by observation), together with the two sides MD and DF. Draw the diagonal MF; then in the triangle DMF there are given the two sides DM, DF, and the included angle D. From D deduct $\frac{1}{3} \epsilon$, the excess $\epsilon$ being computed by an approximate value of the area of the triangle, and compute, as in a plane triangle, the side MF and the two angles DMF, DFM. To each of these add $\frac{1}{3} \epsilon$, and the two sums being deducted from the angles DMN and DFN, there will remain the two angles FMN and MFN. Hence, since MF has been already computed, we can find MN, FN, and the angle MNF, in the usual manner.

Now, in the triangle NPH, NH is known (for FH and FN have been calculated), and the adjacent angles PNH, NHP are given by observation; therefore we can compute NP, PH, and the remaining angle NPII.

To find PX we may produce IL to Z, and resolve the two triangles PIZ and ZLX. In the first of these, the side PI, and the two adjacent angles PIZ, IPZ, are already known; whence we find IZ, PZ, and the angle IZP. We have then in the right-angled triangle LZX, the side LZ, and the angle LZX to find ZX. From PZ subtract ZX, and there remains PX, the quantity sought.

In proceeding by the method now indicated, the spherical excess $\epsilon$ must be first computed for each of the triangles to be resolved; then each observed spherical angle must be diminished by $\frac{1}{3} \epsilon$, in order to allow the calculations to be made by the rules of plane trigonometry; and when the result has been found, we must again add to each angle the small quantity $\frac{1}{3} \epsilon$, in order to have the true spherical angle. (See Delambre, Methodes Analytiques pour la Determination d'un Arc du Meridien.)

It is necessary to remark, that the point X, determined FIGURE OF THE EARTH.

If by drawing a perpendicular from the last station to the meridian, is not situated exactly on the same parallel of latitude with L. Its latitude is a little greater than that of L; and unless the distance of the station from the meridian is very small, the difference will be sensible, and requires to be calculated. Conceive another point, L' (fig. 5), on the same parallel of latitude with L, and at the same distance from the meridian on the opposite side. The point X is on the circumference of a great circle, perpendicular to the meridian PE, and passing through L and L'; but the small circle, or parallel of latitude of L and L', intersects the meridian in a different point l. Now let PE (fig. 6) be the meridian, XC the intersection of its plane with that of the great circle passing through the points LL', and In the intersection of the plane of the meridian and the plane of the small circle or parallel of latitude passing through the same points LL', so that El is the latitude of L, EX that of the point X, and IX the distance between the perpendicular and the small circle passing through L. Let In and XC intersect in m, then IX being a small arc, may be regarded as a straight line; consequently XI = mX tan. Xl. But Xl = XCE = latitude of L very nearly, consequently, taking λ to denote the latitude of L, we have XI = mX tan. λ. Now mX is the versed sine of the arc whose chord is LX; therefore, making d = earth's diameter, mX = \(\frac{LX^2}{d}\), and consequently Xl = \(\frac{LX^2}{d} \tan. λ\), which is the quantity to be subtracted from the calculated meridional arc.

Delambre employed a different method of computing the distance between the parallels of the terminal points of the measured arc. It requires, however, the dimensions of the earth to be previously known with tolerable accuracy. Let AB (fig. 7) be the arc measured on the surface of the spheroid PAB; then, having observed the latitude of the station A, and the azimuth of AB, that is the angle BAP, we have sufficient data, supposing the dimensions of the earth to be nearly known, to find not only the distance between the parallels of A and B, but also the latitude and longitude of B, and the azimuth of BA, as observed from B. Through A and B draw the normals AM and BN, meeting the axis in M and N; join BM, AN; and about M as a centre, with an arbitrary radius, describe the circular arcs bp, pa, ob, forming the spherical triangle pab. Now, if we compare the triangle PAB on the spheroid with pab on the sphere, we find pa = PA in degrees, each being the colatitude of A; pb = PB, each being the colatitude of B; the angle apb = APB being the difference of longitude of A and B, and also the angle pab = PAB, each measuring the inclination of the planes PMA, AMB. But the angle pba is in general not equal to PBA; for as the normals do not meet the axis in the same point, the planes AMB and ANB do not coincide; and pba is the measure of the inclination of PMB and ANB, whereas PBA is the measure of the inclination of PMB and ANB. Now, in the triangle pba there are given the angles at p and a, so that if the side ab can be determined in terms of AB, the other parts of the triangle may be computed, and thence the difference between pb and pa, or the distance between the parallels, which being found in terms of BA, will be expressed in feet.

We have therefore to find an expression for ab, or the angle AMB. Now, in the triangle ABM there are given AB (the measured arc), and AM the normal at A (the dimensions of the earth being nearly known). But

\[ BM = BN \frac{\sin. MNB}{\sin. NMB}; \]

and if we make \( l = PNB \) the co-

latitude of B, and \( x = NBM \), we shall have NMB = PNB Figure of \( -x = 90° - l - x \); therefore sin. NMB = cos. (l + x); the Earth. we have also sin. MNB = sin. (90° - l) = cos. l; where-

fore BM = BN \(\frac{\cos. l}{\cos. (l + x)}\) = \(\frac{\cos. l}{\cos. l \cos. x - \sin. l \sin. x}\) = BN \(\frac{1}{\cos. x - \tan. l \sin. x}\)

= BN \((1 + \tan. l \sin. x + 2 \sin^2 \frac{x}{2}) + \text{etc.}\) Now \( x \) is a very small angle; its sine is in fact expressed in terms of the square of the eccentricity, so that terms multiplied by it may be neglected without any sensible error. We may therefore assume BM = BN. In the triangle ABM we have then given the three sides, whence the angles may be computed, and consequently ab = AMB becomes known.

Having found ab, we have given in the triangle abp the two sides pa, ab, and the observed angle pab, from which to compute the remaining parts of the triangle, viz. pb the latitude of B, apb its longitude, and pba the azimuth of A, as seen from B. Make \( pa = pa \), and \( pb \) to PB, then \( ba = pb - pa \) is the distance between the parallels, the radius being unit; therefore, since \( 1 : MA :: ab : Ab \), we have \( Ab = AM \times ab \), that is, the distance between the parallels of the extreme stations on the spheroid.

When the distance between the parallels of the extreme points of the arc has been ascertained, it only remains to determine the difference of their latitudes, or the length of the corresponding celestial arc. This is the most difficult part of the whole operation. The error of a single second in the difference of the latitudes corresponds to about a hundred feet on the terrestrial meridian; and when it is considered that the latitudes of the best determined spots on the earth, even of the Observatories of Greenwich and Paris, are still uncertain to the amount of at least half a second, it will be easily apprehended that the errors of the latitudes are more to be feared than any which can affect the measurement of the base, or the angles, or the direction of the meridian. In the British survey, that of India, and some on the Continent, the latitudes were observed with the zenith sector, an instrument peculiarly adapted to this purpose. Ramsden's zenith sector, made expressly for the determination of the arc of meridian between Dunnose and Clifton, carried a telescope of eight feet in length. With this superb instrument, for a description of which we must refer to the second volume of the Trigonometrical Survey, the zenith distances of several of the northern stars were observed at both extremities of the arc; and the difference of two zenith distances of the same star is evidently the same as the difference of the latitudes of the two stations. It may be remarked, that the result of this observation is independent of the declination of the star; and if the observation is made at the second station within a short time after it was made at the first, the result is also nearly independent of the nutation.

For the determination of the French arc of meridian, the latitudes were observed with the repeating circle. This instrument, on account of its portability, is of the greatest use in geodetic operations; but it may be questioned whether it can be safely relied on for the determination of so very important an element as the latitude. Besides the want of power in the telescope, it is found to be affected in some unaccountable way by a constant error, which, however carefully its amount may have been determined, leaves a degree of uncertainty with regard to the results. Mechain mistook the latitude of Barcelona by about 3°. It is to be regretted that the latitudes of Dunkirk and Formentera, the extremities of the French arc, Figure of have not been determined with the zenith sector. All the Earth, the other operations connected with the measurement of that extensive arc have been executed in a style of such decided superiority, that a more satisfactory, if not more accurate, determination of the latitudes is alone wanting to render it the most valuable application of science that has ever been made with a view to ascertain the magnitude and exact figure of our globe.

The measurement of an arc of parallel usually forms part of the operations connected with the survey of a large extent of country. It will frequently happen that the sides of some of the great triangles lie in a direction nearly perpendicular to the meridian, in which case the arc of the terrestrial parallel may be easily computed from an approximate knowledge of the earth's dimensions; and if the difference of longitude of the two stations, or the angle which their respective meridians make with each other, has been accurately determined, the comparison of the corresponding celestial and terrestrial arcs will give the length of the degree on the parallel. By comparing this result with the length of a degree measured on a different parallel, or on the meridian, we can easily deduce the ellipticity.

Perhaps the simplest way of conveying an accurate idea of the nature of the operations necessary to be undertaken in the measurement of an arc of parallel, will be to describe the method that was actually followed in the British trigonometrical survey, in determining the length of an arc of parallel from the measured distance between Beachy Head and Dunnose.

Let BW (fig. 8) be an arc of the great circle perpendicular to the meridian of Beachy Head at B, meeting that of Dunnose in W; and let DR be an arc of the great circle perpendicular to the meridian of Dunnose at D, meeting that of Beachy Head in R; and let BL and DE be the parallels of latitude passing through B and D. It is supposed that the latitudes of the two places have been determined, together with the angles PBD and PDB, which they reciprocally make with each other and the pole; and also that the distance between them on the arc of the great circle has been measured: the question is then to find the difference of their longitudes, or the angle BPD, and the distance in feet between the meridians on the arcs of the small circles BL and DE passing through the places.

In the small spheroidal triangle WBD there were given by observation the two angles WBD and WDB, which being reduced to the angles made by the chords, gave the two angles of the corresponding plane triangle. The side BD was found by triangulation = 339,397-6 feet; whence the chord of the perpendicular arc BW was found = 336,115-6 feet. In like manner the chord of DR was found = 336,980 feet.

The terrestrial distances being thus found, it was next necessary to find the difference of longitude, or the angle P, and the lengths of the arcs of parallel BL and DE in degrees. For this purpose there are given PB—the colatitude of B, PD the colatitude of D, the angle PBD, which is the azimuth of D as seen from B, and PDB the azimuth of B as seen from D.

The rigorous solution of this problem on the spheroid is attended with considerable difficulty. An approximation is however easily obtained from a property of spheroidal triangles (for the demonstration of which we must refer to the Phil. Trans., vol. Ixxx.), namely, that the sum of the horizontal angles on a spheroid (or indeed on any surface differing little from that of a sphere) is nearly the same as the sum of those which would be observed on a sphere, the latitudes and difference of longitudes being the same on both figures. Assuming therefore the sum of the two angles PDB and PBD to be the same as the sum of two spherical angles, we have, from Napier's analogies (see the Trigonometry),

\[ \cos \frac{1}{2} (PD + PB) : \cos \frac{1}{2} (PD - PB) = \cot \frac{1}{2} P : \tan \frac{1}{2} (PDB + PBD), \]

whence

\[ \tan \frac{1}{2} P = \frac{\cos \frac{1}{2} (PD - PB)}{\cos \frac{1}{2} (PD + PB)} \times \cot \frac{1}{2} (PDB + PBD); \]

that is to say, the tangent of half the difference of longitudes is equal to the cotangent of half the sum of the azimuthal angles multiplied into the ratio of the cosine of half the difference of the colatitudes to the cosine of half their sum. In the case under consideration the angle P was found = 1° 26' 47" 93'.

We have now given, in the right-angled triangle PBW, which may be considered as spherical, the side PB and the angle P, whence BW was found = 54' 56" 21'. And from the triangle PDR, in which the side PD and the angle P are given, DR was found = 55' 4" 74'.

The chords of the two perpendicular arcs BW and DR, whose radius is the radius of the earth, are found, from an approximate knowledge of the earth's diameter, to be three feet and a half shorter than the arcs themselves, whence the arc BW = 336,119-1 feet, and DR = 336,983-5 feet. Hence the length of the degree of the great circle perpendicular to the meridian, at the middle point between W and B, is found by proportion = 367,096-8 feet, and in the middle point between R and D = 367,090-8 feet. Therefore the mean, or 367,093-8, is the length of a degree of the great circle perpendicular to the meridian at latitude 50° 41', which is nearly that of the middle point between Beachy Head and Dunnose.

Now, by reason of the short distance between the two stations, and their small difference in latitude, the two radii of curvature at B and D may be regarded as intersecting each other on the axis at M (fig. 9). Therefore DM is the radius of the great circle perpendicular to the meridian, and DE is the radius of the parallel at D. Therefore 1° of the great circle : 1° of the parallel as DM : DE; and DM : DE = 1 : cos. latitude of D; whence 1 : cos. 50° 44' 24" = 367,094 : 232,914 feet for the degree of parallel at Beachy Head, and 1 : cos. 50° 37' 7" = 367,094 : 232,914 feet for the degree of parallel at Dunnose. (Trig. Survey, vol. i. p. 113.)

The length of the degree of the great circle perpendicular to the meridian, deduced in the manner now explained, is about 1250 feet greater than on the spheroid which corresponds with the measurements of the meridional arcs. This discrepancy gave rise to a suspicion of errors in the observation of the azimuthal angles; and, in fact, the difference of longitude, in order to agree with other determinations, ought to be about 18° greater than that which was determined in the survey. With this correction the length of the degree of the perpendicular circle is found = 365,838 feet. Assuming the dimensions of the earth deduced from the survey, the length of the degree perpendicular to the meridian at the mean latitude is 365,844 feet. Hence 1° parallel = 231,801 feet.

In the Philosophical Transactions for 1824 an account is given of some experiments performed by Dr Tirkas for determining the difference of longitude of Dover and Falmouth. Twenty-four chronometers were transported by sea, three several times, from the one place to the other, by which means the difference of the apparent times was determined. The difference of longitude was thus found to be 6° 22' 6"; and the length of the parallel, as found from the survey, being 1,474,672 feet, we have the length of a degree of parallel at latitude 50° 44' 24" = 231,653 feet.

The most extensive arc of parallel which has yet been measured is that between Marennes (near Bordeaux) and Padua. The details of the operations, which were performed in 1822 and 1823, are given in the *Connaissance de Tems* for 1829. The terrestrial arc was determined by triangulation in the usual manner, and the astronomical amplitude by fire signals observed from station to station at five intermediate stations. There are consequently six independent arcs; and the final result is affected with the accumulated errors at all the stations. The terrestrial distance was found to be 1,010,996 metres, or 3,316,976 English feet; and the difference of time determined by chronometers 51 min. 56'24 sec., corresponding to 12° 59' 3"75 of longitude. The mean length of the degree, found from the partial arc between Marennes and Geneva, is 255,546 feet, and from the whole arc between Marennes and Padua 255,470 feet, both of which results are greater than the degree on the regular spheroid, which is found to represent most nearly the meridional arcs under the same parallel of latitude, namely, 45° 43' 12". No great reliance, however, can be placed on this determination, on account of the uncertainty that exists respecting some of the longitudes, the determination of which was found to be attended with great difficulty. If the difference of longitude of Marennes and Padua were found by good astronomical observations, the discrepancies would probably be diminished.

**SECTION III.**

**DETERMINATION OF THE FIGURE OF THE EARTH FROM GEODETIC MEASURES.**

Assuming the figure of the earth to be that of an elliptic spheroid of revolution, the magnitude and ratio of its equatorial and polar diameters may be determined from the comparison of the lengths of lines measured between determined points on its surface, with the corresponding arcs of the celestial sphere, by means of theorems which we shall now proceed to demonstrate.

1. To express the radius of curvature of a meridian in terms of the latitude.

Let PDQ (fig. 10) be the meridian, CP half the polar axis, CQ the radius of the equator, DM a perpendicular to the tangent at D, meeting CP in M, and CQ in N; and let

\[ a = \text{CP half the polar axis}, \] \[ b = \text{CQ the radius of the equator}, \] \[ n = \text{DN the normal at D}, \] \[ r = \text{radius of curvature at D}. \]

Join CD; let CD' be the diameter conjugate to CD, meeting DM in G; and through D draw DE and DF respectively perpendicular to CP and CQ. It is demonstrated in the article *Conic Sections* (Part IV, Sec. 2, Prop.V.Cor.I.), that \( r \times DG = CD^2 \); and (Prop. XVII, Part II.) \( CD' \times DG = CP \times CQ = ab \); whence \( CD^2 = a^2 b^2 - DG^2 \); and consequently \( r \times DG^2 = a^2 b^2 \). But (Prop. XXI, Part II.) \( DN \times DG = CP^2 \); that is, \( n \times DG = a^2 \); whence \( n^2 \times DG^2 = a^2 \), and therefore \( DG^2 = a^2 - n^2 \), consequently

\[ r = \frac{b^2 n^2}{a^2} \quad \text{(1.)} \]

Again (Prop. XXI, Part II.) \( DN : DM = a^2 : b^2 \); and the triangles NCM, NFD being similar, \( DN : DM = NF : CF \), whence \( CF = \frac{b^2}{a^2} NF \). Now let the angle DNF or the latitude of D = \( l \); then DF = \( n \sin l \), NF = \( n \cos l \),

and \( CF = \frac{b^2}{a^2} n \cos l \). But DF = CE = \( x \) and CF \( y \); substituting therefore \( n \sin l \) for \( x \), and \( \frac{b^2}{a^2} n \cos l \) for \( y \) in the equation of the ellipse, namely,

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \]

we find

\[ \frac{n^2 \sin^2 l}{a^2} + \frac{b^2 n^2 \cos^2 l}{a^2} = 1, \]

whence

\[ n^2 = \frac{a^2}{\sin^2 l + (1 + e)^2 \cos^2 l}. \]

Let \( b = a (1 + e) \) (\( e \) denoting the ellipticity or the ratio of the difference of the semiaxes to the polar semi-axis, that is, \( e = \frac{b - a}{a} \); by substituting, we get

\[ n^2 = \frac{a^2}{\sin^2 l + (1 + e)^2 \cos^2 l}. \]

but since the ellipticity \( e \) is very small, all terms multiplied by the square or higher powers of \( e \) may be rejected without sensible error. Hence \( (1 + e)^2 = 1 + 2e \), and therefore \( n^2 = \frac{a^2}{1 + 2e \cos^2 l} \) or \( n = a (1 + 2e \cos^2 l) - \frac{1}{2} \);

whence, on developing and rejecting terms containing \( e^2 \),

\[ n = a (1 - e \cos^2 l) \quad \text{(2.)} \]

Now, from equation (1), \( r = \frac{b^2}{a^2} n^3 = \frac{1 + 2e}{a^2} n^3 \); and from equation (2), \( n^2 = a^2 (1 - 3e \cos^2 l) \); therefore

\[ r = a (1 + 2e)(1 - 3e \cos^2 l) = a (1 + 2e - 3e \cos^2 l) \];

or, finally,

\[ r = a (1 - e + 3e \sin^2 l) \quad \text{(3.)} \]

2. To express the radius of a circle parallel to the equator in terms of the latitude.

Suppose the parallel to pass through D, then DE = \( y \) is the radius required. But it was shown in the last Prop. that \( CF (= DE) = \frac{b^2}{a^2} n \cos l \); therefore, substituting \( a^2 (1 + 2e) \) for \( b^2 \), we have \( y = n \cos l (1 + 2e) \).

Substituting in this the value of \( n \) given by equation (2), we have \( y = a \cos l (1 + 2e) (1 - e \cos^2 l) \); whence

\[ y = a \cos l (1 + e + e \sin^2 l) \quad \text{(4.)} \]

3. To find an expression for the length of an arc of the meridian, in terms of the latitudes of its extreme points.

If the arc is small, it may be regarded as coinciding with the osculating circle at its middle point. Let \( z \) be the arc, and \( r \) the radius of curvature at its middle point; then \( z \) being supposed small we have \( z = r \sin z \).

But \( z : \sin z = 1 : 1 \); therefore \( \sin z = z \sin 1 \),

and the length of the arc in feet \( = r \sin z \).

Now let \( l \) and \( l' \) be the latitudes at the extremities of the arc, then \( \frac{1}{2} (l + l') \) is the latitude of its middle point; and on substituting for \( r \) its value found in equation (3), the expression for the length of a meridional arc, in terms of its extreme latitudes \( l \) and \( l' \), is

\[ z = a \left\{ 1 - e + 3e \sin^2 \frac{1}{2} (l + l') \right\} (l - l') \sin 1 \quad \text{(5.)} \]

But if the arc is of considerable extent, this expression will not be sufficiently exact, and it becomes necessary to find by integration the lengths of the arcs from the equator to each extremity of the given arc; the difference of these two will give \( z \) itself. At the latitude \( l \), \( dz = r dl \); therefore, from equation (3),

\[ dz = a (1 - e) dl + 3ae \sin^2 l dl, \] Figure of and integrating,

\[ z = a(1 - e) + 3ae \int \sin^2 l \, dl. \]

But \( \int \sin^2 l \, dl = \frac{1}{2}l - \frac{1}{4}\sin 2l \); therefore

\[ z = a\left\{ (1 + \frac{1}{2}e)l - \frac{3}{4}e\sin 2l \right\}. \]

(6.)

No constant is necessary, because at the equator \( z \) and \( l \) vanish together.

Let \( z' \) be another arc reckoned from the equator to the point whose latitude is \( \ell \); then, similarly,

\[ z' = a\left\{ (1 + \frac{1}{2}e)(\ell - l) - \frac{3}{4}e\sin(\ell + l)\sin(\ell - l) \right\}; \]

and consequently

\[ z' - z = a\left\{ (1 + \frac{1}{2}e)(\ell - l) - \frac{3}{4}e\cos(\ell + l)\sin(\ell - l) \right\}; \]

but \( \sin 2l = \sin 2l = 2\cos(\ell + l)\sin(\ell - l) \); therefore

\[ z' - z = a\left\{ (1 + \frac{1}{2}e)(\ell - l)\sin(\ell - l) - \frac{3}{4}e\cos(\ell + l)\sin(\ell - l) \right\}. \]

(7.)

4. To express the length of an arc of parallel in terms of the radius.

Let \( y \) be the radius, \( D \) the amplitude of the arc in seconds, and \( V \) its length in feet. Then, by what is shown above, the length of the arc in feet is \( yD \sin 1^\circ \).

Hence, from the value of \( y \) given in equation (4), the length of the arc of parallel at latitude \( l \) is

\[ V = a\cos l(1 + e + e\sin^2 l)D\sin 1^\circ. \]

(8.)

5. From the measured length of two degrees of meridian at different latitudes, to determine the axis and ellipticity of the spheroid.

Let \( Z \) and \( Z' \) be the measured lengths of two degrees, and the latitudes of their middle points be respectively \( L \) and \( L' \); then (since in this case \( \ell - l = 1^\circ = 3600'' \)) we have, by equation (5),

\[ Z = a(1 - e + 3e\sin^2 L)3600\sin 1^\circ; \] \[ Z' = a(1 - e + 3e\sin^2 L')3600\sin 1^\circ; \]

therefore

\[ \frac{Z'}{Z} = \frac{1 - e + 3e\sin^2 L'}{1 - e + 3e\sin^2 L}; \]

whence, performing the division, and neglecting terms multiplied by \( e^2 \),

\[ \frac{Z'}{Z} = 1 + 3e(\sin^2 L' - \sin^2 L), \]

from which we find the ellipticity

\[ e = \frac{3}{Z}(Z' - Z)(\sin^2 L' - \sin^2 L). \]

(9.)

The accuracy of this expression will depend on the magnitude of the denominator, or the difference between \( L' \) and \( L \). If \( L \) is nearly equal to \( L' \), the denominator becomes very small, and the value of \( e \) in consequence uncertain. The most favourable determination, therefore, that can be obtained is, when one of the degrees, \( Z' \), for example, is measured near the pole, and \( Z \) at the equator, where \( L \) is zero. As an example of this method, we may deduce the value of \( e \) from a comparison of the lengths of a degree at the equator and in England. Bouguer found the length of the degree at the equator to be 362,932 English feet; and from the British arc, between Dunmore and Clifton, the length of the degree is 364,970 feet, at the latitude of \( 52^\circ 35' 45'' \). We have therefore \( Z = 362,932 \), \( Z' = 364,970 \), \( L = 0 \), \( L' = 52^\circ 35' 45'' \), by substituting which in equation (9), there results

\[ e = \frac{3}{362,932} \times \sin^2(52^\circ 35' 45''). \]

whence \( e = \frac{1}{346} \).

Having found the value of \( e \), that of the polar semi-diameter, or \( a \), is obtained as follows. At the equator, where \( L \) vanishes, we have \( Z = a(1 - e)3600\sin 1^\circ \), whence

\[ a = \frac{Z}{(1 - e)3600\sin 1^\circ} = \frac{Z \times 346}{345 \times 3600\sin 1^\circ}; \]

and substituting 362,932 for \( Z \), we find

\[ a = 20,853,000 \text{ feet nearly.} \]

The method of deducing the ellipticity and dimensions of the earth which has now been explained, may be conveniently employed when the measured arcs are small; but when the arcs are considerable (and it is necessary that they should extend over several degrees, in order to give results deserving of confidence), we must employ the more accurate expression of the arc found in equation (7); that is to say, we must employ the whole arc measured, instead of the resulting length of a single degree. Equation (7), when reduced to numbers, takes the form

\[ A = am + acn. \]

(10.)

in which \( m \), \( n \), and the arc \( A \), are numbers determined by observation. Another arc will give a similar equation,

\[ A' = am' + acn', \]

and by combining the two equations we get

\[ e = \frac{Am' - Am}{An' - An} = \frac{Am' - Am}{mn' - mn}. \]

(11.)

If the terrestrial meridian were a regular ellipse, the ellipticity deduced from two measured arcs would be the same at every latitude. But it is found that no two different measures that have yet been effected concur in giving exactly the same figure to the meridian. Hence it becomes necessary to seek the most probable mean among the different results. Each new measure affords a new equation of condition, of the form of equation (10); and as the number of unknown quantities remains the same, whatever may be the number of equations, recourse must be had to the method of least squares, or some of the other methods of combination employed by astronomers. In this way each individual measurement serves to correct the results given by the former ones.

6. From the measured lengths of an arc of meridian, and an arc of parallel, to determine the ellipticity of the spheroid.

Let \( z \) be the arc of meridian between the latitudes \( l \) and \( \ell \), \( V \) the length of the arc of parallel in feet at the latitude \( L \), and \( D \) the difference of longitude in seconds between its extreme points. From equation (5) we have

\[ z = a\left\{ 1 - e + 3e\sin^2 \frac{1}{2}(\ell + l)\right\}(\ell - l)\sin 1^\circ, \]

and from equation (8),

\[ V = a(1 + e + e\sin^2 L)\cos L \times D\sin 1^\circ, \]

whence

\[ \frac{V}{\cos L \times D\sin 1^\circ} = \frac{z}{(\ell - l)\sin 1^\circ} = ae\left\{ 2 + \sin^2 L - 3\sin^2 \frac{1}{2}(\ell + l)\right\}, \]

and, consequently,

\[ e = \frac{V}{\cos L \times D\sin 1^\circ} - \frac{z}{(\ell - l)\sin 1^\circ} = \frac{a}{2 + \sin^2 L - 3\sin^2 \frac{1}{2}(\ell + l)}. \]

(12.)

When the two arcs have been measured at the same This combination may be considered as giving a correct representation of the curvature at any particular place, or of the dimensions and eccentricity of the osculating spheroid at that place. But, speaking generally, the exact determination of the differences of longitude is attended with so much uncertainty, that the results obtained from the comparison of arcs of parallel, either with one another or with arcs of meridian, cannot be relied on with much confidence. It is consequently unnecessary to point out the equations to be employed in deducing the ellipticity from the comparison of two arcs of parallel.

We shall now proceed to apply the above formulas to the results of the principal geodetic measurements that have been executed in different countries. The following table contains twenty arcs of meridian, between the Figure of equator and 66° of north latitude; and as they have all the Earth been determined with the utmost attention to every circumstance which could be supposed to affect their accuracy, they may be regarded as decidedly the best elements we yet possess for the solution of the problem of the figure of the earth. We have excluded all the ancient arcs except that of Bouguer, and also some recent ones, where the locality was unfavourable. They are all reduced to the level of the sea. In reducing the foreign measures, the metre has been assumed (according to Captain Kater's determination) = 3'280899 feet, and the toise = 6'394596 feet, of Schuckburgh's scale. The original documents are given in the various works to which reference has been made in the first section of this article.

| No. | Extremities of Arc | Latitude | Amplitude | Length in English Feet | |-----|-------------------|----------|-----------|------------------------| | | PERUVIAN ARC | | | | | 1 | Tarqui | -3° 4' 30" | 3° 7' 3"1 | 1,131,057 | | | Catchesqui | +0 2' 32" | | | | | 1ST INDIAN ARC | | | | | 2 | Trivandeporum | +11 44' 52"9 | 1 34' 56"4 | 574,369 | | | Paudree | 13 19' 49"02 | | | | | 2D INDIAN ARCS | | | | | 3 | Punna | +8 9' 32"51 | 2 50' 10"5 | 1,029,171 | | | Putchapollium | 10 59' 43"05 | 4 6' 11"3 | 1,489,198 | | | Namthabad | 15 5' 54"33 | 2 57' 21"7 | 1,073,409 | | | Daumerigida | 18 3' 16"07 | 3 2' 35"9 | 1,105,499 | | | Takal Khera | 21 5' 51"94 | 3 1' 19"9 | 1,097,320 | | | Kulliampoor | 24 7' 11"85 | | | | | MODERN FRENCH ARCS| | | | | 8 | Formentera | +39 39' 56"11 | 2 41' 50"5 | 982,247 | | | Montjouy | 41 21' 46"58 | 1 51' 7"7 | 674,623 | | | Carcassone | 43 12' 54"31 | 2 57' 48"2 | 1,073,706 | | | Evaux | 46 10' 42"54 | 2 40' 6"8 | 973,853 | | | Pantheon | 48 50' 49"37 | 2 11' 19"1 | 798,971 | | | Dunkirk | 51 2' 8"50 | 0 26' 31"5 | 161,412 | | | Greenwich | 51 28' 40"00 | | | | | ENGLISH ARCS | | | | | 14 | Dunnose | +50 37' 8"60 | 1 36' 20"0 | 586,319 | | | Arbury Hill | 52 13' 28"59 | 1 14' 3"4 | 450,018 | | | Clifton | 53 27' 31"99 | 1 6' 49"7 | 406,516 | | | Burleigh Moor | 54 34' 21"70 | | | | | HANOVERIAN ARC | | | | | 17 | Göttingen | +51 31' 47"85 | 2 0' 57"4 | 736,426 | | | Altona | 53 32' 45"27 | | | | | RUSSIAN ARCS | | | | | 18 | Jacobstadt | +56 30' 4"64 | 1 52' 42"8 | 686,022 | | | Dorpat | 58 22' 47"41 | 1 42' 22"5 | 623,719 | | | Hochland | 60 5' 9"90 | | | | | SVANBERG'S SWEDISH ARC | | | | | 20 | Mallörn | +65 31' 31"06 | 1 37' 20"3 | 593,279 | | | Pahtawara | 67 8' 51"41 | | | Substituting in equation (7) the values of \( z - z_1 \), \( l \), and \( p \), for each of the arcs in the above table, we obtain the following equations of condition:

1. \( a \times 0.544111 = ae \times 0.5426 = 1131057 \) 2. \( a \times 0.976169 = ae \times 0.92371 = 574368 \) 3. \( a \times 0.495037 = ae \times 0.4536 = 1029171 \) 4. \( a \times 0.716119 = ae \times 0.6058 = 1489198 \) 5. \( a \times 0.515924 = ae \times 0.3891 = 1073409 \) 6. \( a \times 0.531157 = ae \times 0.3519 = 1105499 \) 7. \( a \times 0.527472 = ae \times 0.2933 = 1097320 \) 8. \( a \times 0.470778 + ae \times 0.1131 = 982247 \) 9. \( a \times 0.523260 = ae \times 0.1159 = 674623 \) 10. \( a \times 0.517210 = ae \times 0.2504 = 1079706 \) 11. \( a \times 0.465751 = ae \times 0.2940 = 973853 \) 12. \( a \times 0.381990 + ae \times 0.2893 = 798971 \) 13. \( a \times 0.077158 + ae \times 0.0636 = 161412 \) 14. \( a \times 0.0280221 + ae \times 0.2335 = 586319 \) 15. \( a \times 0.215422 + ae \times 0.1900 = 450018 \) 16. \( a \times 0.194397 + ae \times 0.1875 = 406516 \) 17. \( a \times 0.351849 + ae \times 0.3132 = 736426 \) 18. \( a \times 0.327868 + ae \times 0.3708 = 686022 \) 19. \( a \times 0.297797 + ae \times 0.3618 = 632719 \) 20. \( a \times 0.283097 + ae \times 0.4293 = 593278 \)

From any two of these equations we may deduce values of \( a \) and \( e \); but, as might be expected, the results are found to be very different according as different arcs are selected for comparison. We may also add any number of the equations together, for instance Nos. 3, 4, 5, 6, and 7, the sum of which forms the continuous arc measured in India by Colonel Lambton and Captain Everest; or Nos. 8, 9, 10, 11, and 12, which form the French arc from Dunkirk to Formentera. But as it is impossible to satisfy at once all the equations by any value whatever that can be assigned to \( a \) and \( e \), in order to exclude all arbitrary hypotheses, it is necessary to suppose all the observations to have been alike good, and deduce values of \( a \) and \( e \) from a combination of the whole. The best method of effecting the combination is that of minimum squares, the principle of which is contained in the following rule: Multiply all the terms of each of the proposed equations by the co-efficient of \( a \) in that equation, taken with its proper sign, and make the sum of all the products equal to zero. This will give an equation in which the sum of the squares of the errors is a minimum in respect of \( a \). Multiply then each equation by the co-efficient of \( ae \) in that equation, and make the sum of all the products \( = 0 \). This will give a second equation, in which the sum of the squares of the errors is a minimum in respect of \( ae \). From the two equations thus obtained, the values of \( a \) and \( ae \), and consequently \( e \), are found in the usual manner. The operation is extremely laborious when the equations are numerous, but it is attended with no difficulty.

By combining the above twenty equations in the manner now described, M. Schmidt obtained the following results, which are given in No. 213 of Schumacher's Astronomische Nachrichten, p. 371; and they are unquestionably entitled to be regarded as the best determination of the magnitude and figure of the earth which has been found from the operations that have been undertaken up to the present time for the measurement of meridional arcs.

\[ \begin{align*} \text{Radius of equator} &= b = 20921665 \text{ Eng. feet.} \\ \text{Radius of pole} &= a = 20852994 \\ \text{Ellipticity} &= e = \frac{b-a}{a} = \frac{1}{301.02} = 0.0032555 \\ \text{Degree at equator} &= 362732 \\ \text{Degree at latitude } 45^\circ &= 364543.5 \end{align*} \]

In order to discover how nearly the different measures agree with a spheroid having the dimensions now given, we shall compare the length of a degree, as given by each of the above twenty arcs, with its length, computed from these values of \( a \) and \( e \) at the same latitudes.

| No. | Latitude of Middle Point | Measured Length of Degree | Computed Length of Degree | Error in Measure | |-----|--------------------------|---------------------------|---------------------------|-----------------| | 1 | 1° 31' 0" | 362,809 | 362,736 | + 73 | | 2 | 12° 32' 21 | 362,988 | 362,905 | + 83 | | 3 | 9° 34' 43 | 362,863 | 362,834 | + 29 | | 4 | 13° 2' 54 | 362,873 | 362,919 | - 46 | | 5 | 16° 34' 42 | 363,125 | 363,029 | + 96 | | 6 | 19° 34' 34 | 363,257 | 363,139 | + 118 | | 7 | 22° 36' 32 | 363,084 | 363,268 | - 184 | | 8 | 40° 0' 52 | 364,152 | 364,233 | - 81 | | 9 | 42° 17' 21 | 364,289 | 364,376 | - 137 | | 10 | 44° 41' 48 | 364,347 | 364,528 | - 181 | | 11 | 47° 30' 46 | 364,962 | 364,706 | + 256 | | 12 | 49° 56' 29 | 365,052 | 364,859 | + 193 | | 13 | 51° 15' 24 | 365,116 | 364,940 | + 176 | | 14 | 51° 25' 18 | 365,208 | 364,952 | + 256 | | 15 | 52° 50' 30 | 364,625 | 365,036 | - 411 | | 16 | 54° 0' 56 | 365,002 | 365,109 | - 107 | | 17 | 52° 32' 17 | 365,301 | 365,019 | + 282 | | 18 | 57° 26' 26 | 365,203 | 365,310 | - 107 | | 19 | 59° 13' 58 | 365,551 | 365,412 | + 139 | | 20 | 66° 20' 11 | 365,697 | 365,777 | - 80 |

From this table we see that the degrees increase gradually from the equator to the pole; but the increase is by no means regular. It occurs in some instances that the degrees appear to diminish on going northward; but these anomalies must be ascribed either to errors in the observations or to local irregularity of form or density. The most probable source of error is in the latitudes, first, on account of the difficulty of the observation to the requisite degree of accuracy; and, secondly, on account of the irregularities in the density of the exterior crust of the earth, which cause a deflexion of the plumb-line from the true zenith. Hence the longest arcs are the best; for the probable error in the determination of the difference of latitudes is the same whether the arcs are great or small. An error of 1° in the latitude corresponds to about a hundred feet on the ground.

If we compare the degree found by some of the measures which have been rejected from the foregoing table, with the degree computed for the same latitudes, we shall find the errors or anomalies much greater.

| Arcs | Lat. of Middle Point | Measured Length of Degree | Computed Length of Degree | Difference | |-----------------------|----------------------|---------------------------|---------------------------|------------| | Lacaille's arc at Cape of Good Hope | 33° 18' 30" | 364,712 | 363,826 | + 886 | | North American arc, by Mason and Dixon | 39° 12' 00" | 363,786 | 364,181 | - 395 | | Roman arc, by Boscovich | 42° 59' 00" | 364,262 | 364,418 | - 156 | | Piedmontese arc, by Plana and Carlini | 44° 57' 30" | 368,242 | 364,543 | + 3699 | | Maupertuis' Swedish arc | 66° 19' 37" | 367,086 | 365,774 | + 1312 | Two of these differences are considerably greater than any in the preceding table. Lacaille's arc corresponds to one in the northern hemisphere at a higher latitude by 11°, whence it has been inferred that the two hemispheres are not exactly similar. The measurement of Plana and Carini appears to have been excellent; but both extremities of the arc were in the immediate vicinity of lofty mountain ranges, and the very great deviation from the mean result of the other measures can only be accounted for by the disturbing effects of local attraction. Maupertuis' arc was never reckoned of much value, as it was evident from the first that there was an error in the determination of the latitudes.

Taking the results of the four arcs of parallel mentioned before, and computing the corresponding degree from the formula \(a \cos l (1 + e + e \sin^2 l) \times 3600 \sin l\) (supposing \(a\) and \(e\) to have the same values as above), we get the following table:

| Arcs of Parallel | Latitude | Measured Degree | Computed Degree | Difference | |-----------------------------------|----------------|-----------------|-----------------|------------| | Lacaille's arc across the mouth of the Rhone | 43° 31' 50" | 265,345 | 265,154 | + 1191 | | Roy's arc from Beachy Head to Dunnose | 50° 44' 24" | 232,331 | 231,542 | + 789 | | Arc from Dover to Falmouth | 50° 44' 24" | 231,579 | 231,542 | + 37 | | Arc from Padua to Marennes | 45° 43' 12" | 255,480 | 255,370 | + 110 |

It is remarkable that all these errors are affected with the same sign; and the circumstance might seem to strengthen a conclusion indicated by a comparison of the errors of the meridional arcs, namely, that the meridional curve is not exactly an ellipse, but protuberant between the latitudes of 40° and 52°; in consequence of which the degrees of meridian are shorter, and the degrees of parallel longer between those latitudes, than if the earth were a regular ellipsoid. There is no reason to infer that the meridians are not similar, or that the earth is not a solid of revolution.

The six partial arcs into which the whole arc from Marennes to Padua was divided, give the following results (Connaissance des Temps, 1829).

| Arcs | Astronomical Amplitudes | Length in Feet | Length of Degree | |-----------------------------|-------------------------|----------------|------------------| | 1 Marennes to St Preuil | 0 3 48-99 | 244,123 | 255,865 | | 2 St Preuil to Sauvagnac | 0 6 23-09 | 407,429 | 255,248 | | 3 Sauvagnac to Jason | 0 6 51-39 | 437,493 | 255,228 | | 4 Isson to Geneva | 0 11 57-82 | 764,786 | 255,686 | | 5 Geneva to Milan | 0 12 9-57 | 776,646 | 255,486 | | 6 Milan to Padua | 0 10 45-38 | 686,549 | 255,308 |

The mean of these results, or the most probable value of the degree of parallel at the latitude of the stations (45° 43' 12"), is 255,470 feet, exceeding the computed length only by a hundred feet; and the ellipticity which they indicate is \(\frac{1}{282}\). This is considerably greater than the ellipticity indicated by the measures of meridional arcs, but agrees pretty nearly with that which we shall see is given by the pendulum observations. The results, however, though valuable as confirming the results obtained by other methods, are not by any means worthy of the same confidence as those deduced from the measurement of meridional arcs.

We shall conclude this section with a re-statement of the dimensions of the earth, resulting from the principal measures of arcs of meridian.

\[ \begin{align*} \text{Equatorial diameter} & = 41843330 = 7924873 \\ \text{Polar diameter} & = 41704788 = 7898634 \\ \text{Difference of diameters, or } \Delta & = 138542 = 26239 \\ \text{Ratio of diameters} & = 302:026 : 301:026 \\ \text{Ellipticity} & = \frac{b-a}{a} = \frac{1}{301:026} \\ \text{Length of degree at equator} & = 362732 \text{ feet} \\ \text{Length of degree at lat. 45°} & = 364543.5 \text{ feet} \end{align*} \]

SECTION IV.

THEORETICAL INVESTIGATION OF THE FIGURE OF THE EARTH FROM THE LAWS OF HYDROSTATICS.

In attempting to deduce the figure of the earth from the general laws of hydrostatic equilibrium, we suppose the whole mass to have been originally fluid, or in such a state, that all its molecules were at liberty to obey the forces by which they are impelled. These forces are, 1st, the attraction of the molecules on one another, according to the Newtonian law; and, 2dly, the centrifugal force generated by the revolution of the whole mass about a fixed axis; and the problem is, to determine the form which such a body, under the influence of these forces, would ultimately take. In order to obtain a solution of the question thus generally enunciated, it is necessary to know, a priori, the attractions of the different parts of the fluid body on one another. But the attraction of its different parts depends on their mutual arrangement; that is to say, on the internal constitution of the body or the variations of its density, and also on its form. Of the density in the interior of the earth we know little; and its form is the very element we are seeking to determine. This mutual dependence of the attraction of the mass on its form, and of its form on its attraction, renders it necessary to have recourse to certain arbitrary assumptions respecting the primitive figure of the earth and its internal structure, in order to deter- Figure of mine the relation of the forces with which its different parts attract each other, and thence assign its figure, and the variations of gravity at the different points of its surface.

A fluid mass whose particles are impelled by no other forces than those which result from their mutual attractions, would assume (it is natural to suppose) a spherical form. Such at least would be a figure of equilibrium; for the surface of the sphere being everywhere at right angles to the directions of the attracting forces, which in this case all pass through the centre of gravity, a particle placed on its surface would have no tendency to move along the surface in any direction. But if the mass is made to revolve, the rotation gives rise to an extraneous force, tending to cause every particle to recede from the axis, and which acts with an energy on each particle directly proportional to the distance of the particle from the axis. Hence the attraction of the mass on each particle, in the direction perpendicular to the axis of rotation, is diminished, and the diminution is greatest towards the equatorial parts, where the distance from the axis is greatest. In order, therefore, that the equilibrium may be restored, an accumulation of matter must take place about the equator, so that the mass will bulge out in that quarter, and become flattened at the poles, where the force of attraction is not counteracted. This consideration led Newton to suppose the figure of the earth to be that of an oblate spheroid, or the figure that is generated by the revolution of an ellipse about its shorter axis.

Before proceeding with the investigation, it will be proper to inquire what are the conditions necessary to ensure the equilibrium of a mass of fluid matter, the particles of which are acted on by their mutual attractions and the centrifugal force of rotation. Conceive the particles of a body A to be solicited by accelerating forces of any kind. Suppose dm (which may be regarded as a rectangular parallelepipedon, having its faces parallel to the planes of the co-ordinates) to be one of the molecules of the mass, and x, y, z, the co-ordinates of the solid angle nearest the origin; then, putting k = the density of the mass, we shall have dm = k · dx dy dz. Let X, Y, Z be the accelerating forces acting on dm in the direction parallel to the respective axes of the co-ordinates, then the motive forces in the direction of the same axes will be respectively Xdm, Ydm, Zdm.

Now let p = the pressure in the direction of the axis of x, referred to the unit of surface; then p · dydz is the whole pressure on the face dydz of the element dm. But p may be regarded as a function of x, y, z; therefore at the point whose co-ordinates are x + dx, y, z, the pressure on the unit of surface is p + dp/dx dx, and consequently the whole pressure on the face of dm opposite to dx dy is \((p + \frac{dp}{dx} dx)\) dydz. The particle dm is thus urged in the direction of the axis of x by the two forces \(p \cdot dydz\) and \((p + \frac{dp}{dx} dx)\) dydz, or by a force equal to their difference, namely, \(\frac{dp}{dx} dx dydz\). In order, therefore, that dm may remain at rest, this last force or pressure must be exactly balanced by the motive force acting in the direction of the same axis; that is to say, we must have \(\frac{dp}{dx} dx dydz = Xdm\). In the same manner, we must have \(\frac{dp}{dy} dx dydz = Ydm\), and \(\frac{dp}{dz} dx dydz = Zdm\); whence, since dm = k · dx dy dz, we deduce

\[ \frac{dp}{dx} = kX, \quad \frac{dp}{dy} = kY, \quad \frac{dp}{dz} = kZ. \tag{a} \]

Let the first of these equations be multiplied by dx, the second by dy, and the third by dz, the sum of the products gives

\[ dp = k(Xdx + Ydy + Zdz). \tag{b} \]

The first member of this equation being an exact differential, it is necessary, in order that the equilibrium be possible, that the second member be an exact differential likewise. This, therefore, is a condition which must be satisfied.

At the surface of the mass the pressure p vanishes, and equation (b) becomes

\[ 0 = k(Xdx + Ydy + Zdz), \]

which expresses that the resultant of the three forces X, Y, Z, is perpendicular to the surface.

If the fluid is homogeneous, the density k is constant, and may be represented by unit. In this case equation (b) becomes

\[ dp = Xdx + Ydy + Zdz, \]

and the equation of the surface is

\[ \int (Xdx + Ydy + Zdz) = \text{constant}. \]

It will be remarked that this last equation does not belong merely to the exterior surface or to the homogeneous fluid: by giving a different value to the constant it will form the equation of any surface in the interior of a mass of heterogeneous fluid, at every point of which the pressure or the density is the same. For let the second member of equation (b) be a complete differential, and equal to kdf. We have then dp = kdf, whence k is necessarily a function of p and f. But on integrating the equation, f will be given in a function of p, consequently k must be a function of p. When therefore k is constant, p must be constant, or dp = 0; hence

\[ \int (Xdx + Ydy + Zdz) = \text{constant}. \]

The different surfaces defined by this equation all possess the common property of intersecting at every point the resultant of the accelerating forces X, Y, Z at right angles; hence they are denominated level surfaces.

It follows, therefore, that when the above equation is satisfied, two conditions necessary to the equilibrium are fulfilled; namely, that the resultant of all the forces urging any particle in the interior of the mass is perpendicular to the level surface passing through that particle; and that the force acting on any particle of the surface is perpendicular to the surface. When the fluid is homogeneous, or the density equal throughout the mass, the first of these conditions is always satisfied, and the second is of itself sufficient. Writers on this subject have therefore in general considered that all the conditions necessary to ensure the equilibrium of a fluid mass, whether homogeneous or heterogeneous, revolving about a fixed axis, are comprehended in the single equation

\[ \int (Xdx + Ydy + Zdz) = \text{constant}; \]

and this is the view which has been taken of the question by Clairaut, D'Alembert, Lagrange, Laplace, Legendre, and Poisson. It has, however, been demonstrated by Mr Ivory, in an elaborate paper in the Philosophical Transactions for 1824, that when, as in the case of the earth, a mutual attraction exists among the constituent molecules of the mass, another condition must be fulfilled; and that the equilibrium will not necessarily take place, unless the figure of the mass is such, that any "interior body of the fluid, bounded by a level surface, be in equilib- FIGURE OF THE EARTH.

Within the limits to which the present article must necessarily be confined, it would be in vain to attempt to give a general solution of this very intricate problem. We shall therefore content ourselves with demonstrating that all the conditions of equilibrium will be satisfied, if the figure of the earth (assumed to be fluid) is that of a spheroid of revolution of small ellipticity. We shall, first, suppose the fluid to be homogeneous; and, secondly, that its density is variable.

PROP. I.—A particle placed anywhere within a hollow solid of homogeneous matter, generated by the annular space comprised between two similar and similarly situated concentric ellipses revolving about their common axis, is attracted by the solid equally in all directions.

For let \( p \) (fig. 11) be such a particle, and \( ab \) any straight line passing through \( p \), meeting the exterior surface in \( a \) and \( b \), and the interior in \( c \) and \( d \); then if \( ab \) be bisected in \( e \), \( cd \) will also be bisected in \( e \), because the figures are similar and similarly situated. Therefore \( ae = db \). Now conceive \( ab \) to be the axis of two opposite pyramids having their vertices at \( p \), and terminated by the surface at \( a \) and \( b \), and let \( mn \) be an infinitely thin slice of the pyramid \( ab \), formed by surfaces parallel to that of the given spheroid. The attraction of \( mn \) on \( p \) is directly as the surface \( mn \) and inversely as \( pm^2 \); but the surface \( mn \) is proportional to \( pm^2 \); therefore the attraction of \( p \) on the pyramidal slice \( mn \) is constant; hence the attraction of \( p \) on the whole frustum between \( a \) and \( c \) is proportional to the number of such slices, or to \( ac \). In the same manner the attraction of \( p \) on the frustum between \( d \) and \( b \) is proportional to \( db \); but \( ac = db \); therefore \( p \) is attracted equally in both directions. The same thing being true of all other pyramids which have their vertices at \( p \), it follows that the particle \( p \) is attracted equally in all directions, or remains at rest.

This property, which is true also of the sphere, and forms one of the necessary conditions of equilibrium, was first demonstrated by Newton in the Principia.

PROP. II.—To find the ratio of the axes of a homogeneous spheroid of revolution, when a fluid column from the equator to the centre balances the column from the pole to the centre, the spheroid being supposed to revolve about its minor axis, and its particles to attract one another with forces varying in the inverse ratio of their mutual distances.

Let \( PQ \) (fig. 12) be a meridian of the spheroid, of which the centre is \( C \). Let us assume in the following propositions,

\[ P = \text{the attraction of the spheroid at the pole, or gravity}, \] \[ Q = \text{the attraction at the equator}, \] \[ \varphi = \text{the centrifugal force at the equator}, \] \[ Q' = Q - \varphi = \text{the gravitation at the equator, or the attraction of the spheroid there diminished by the centrifugal force}. \]

Let also

\[ a = CP \text{ half the polar axis}, \] \[ b = CQ \text{ the radius of the equator}. \]

In \( CP \) take any point \( E \), and make \( CE = x \). It is demonstrated in the article Attraction (art. 8), that the force with which a particle at \( E \) is attracted towards \( C \), is to that with which a particle at \( P \) is attracted in the same direction as \( CE \) to \( CP \), or \( x \) to \( a \); therefore the force at \( E = P \cdot \frac{x}{a} \).

Now if we suppose the area of a section of the column perpendicular to the axis \( = 1 \), the mass of the element of the column will be \( 1 \times dx = dx \), and its pressure will consequently be \( P \cdot \frac{x}{a} \). Let this quantity be represented by \( du \). On integrating, and observing that while \( x \) increases \( u \) diminishes, we find

\[ u = \text{const.} - P \cdot \frac{x^2}{2a}. \]

At the surface \( u = 0 \) and \( x = a \); therefore

\[ 0 = \text{const.} - P \cdot \frac{a^2}{2a}; \]

whence, by subtraction, \( u = P \cdot \frac{a^2 - x^2}{2a} \),

and consequently at the centre, where \( x = 0 \), \( u = P \cdot \frac{a}{2} \).

In like manner, if in \( CQ \) we take a point \( F \), and make \( CF = y \), the force at \( F \) resulting from the attraction of the spheroid will be \( Q \cdot \frac{y}{b} \). But the centrifugal force at \( F \) is to that at \( Q \) as \( CF \) to \( CQ \), or as \( y \) to \( b \); therefore the centrifugal force at \( F = \varphi \cdot \frac{y}{b} \). Hence the force with which a particle at \( F \) is attracted towards the centre is \( (Q - \varphi) \cdot \frac{y}{b} \)

\[ = Q' \cdot \frac{y}{b}, \]

and the pressure of the element of the column

\[ = Q' \cdot \frac{y}{b} \cdot dy. \]

Putting this \( = du \), and integrating as before, we find, at the centre, \( u' = Q' \cdot \frac{b}{2} \). But, by hypothesis, at the centre \( u' = u \); therefore \( P \cdot \frac{a}{2} = Q' \cdot \frac{b}{2} \), whence \( P : Q' = b : a \); that is to say, the attraction at the pole is to the gravitation at the equator as the radius of the equator to the radius of the pole.

PROP. III.—If the radius of the equator is to the radius of the pole as the attraction at the pole is to the gravitation at the equator, the resultant of all the forces which urge a particle situated anywhere on the surface of the spheroid is perpendicular to the surface.

Let \( D \) (fig. 10) be the particle, and \( PDQ \) the meridian passing through it. As the plane of the meridian divides the solid into two parts exactly alike in every respect, it is obvious that the particle can have no tendency to move out of that plane; we shall therefore only consider the direction of the force urging it in the plane of the meridian. Let \( DE \) be perpendicular to \( CP \), and \( DF \) to \( CQ \); and make \( CE = x \), \( CF = y \). It is demonstrated in the article Attraction (21 and 22), that the force which attracts \( D \) in the direction \( DF \), perpendicular to the plane of the equator, is to the force at the pole as \( DF \) to \( CP \), or as \( x : a \); and that the force attracting \( D \) in the direction \( DE \) perpendicular to the axis is to the attraction at \( Q \) as \( DE \) to \( CQ \), or as \( y : b \); but the centrifugal force at \( D \), which also acts in the direction \( DE \), is to that at \( Q \) as \( DE \) to \( CQ \); therefore, taking \( X \) to represent the whole force at \( D \) in the direction parallel to the axis, and \( Y \) that in the direction perpendicular to the axis, we have

\[ X : P = x : a, \quad Y : Q' = y : b. \] whence \( X = P \frac{x}{a} Y = Q \frac{y}{b} \). In QC take the point N such that FN : FD = Y : X; then FN : FD = Q' \(\frac{y}{b}\) : P \(\frac{x}{a}\) or FN \(\cdot\) x : FD \(\cdot\) y = Q'a : Pb, whence, since FD = x, and FC = y, FN : FC = Q'a : Pb. But by hypothesis Q' : P = a : b, therefore FN : FC = a² : b², whence by the well-known properties of the ellipse FN is the subnormal, and DN the normal or perpendicular to the surface of the spheroid at D.

Corollary. If the attraction at the pole is represented by CP, then the force at any point D on the surface, in the direction perpendicular to the surface, is proportional to DN the normal passing through D. It is also proportional to DM (the normal produced till it meets the axis); for, by conic sections, DM : DN = b² : a²; that is, DM has to DN a constant ratio.

The proposition which has now been demonstrated is sufficient for the equilibrium of a homogeneous fluid mass revolving about an axis. It may, however, be shown directly, that when the above condition is fulfilled, any particle in the interior of the mass will be equally urged in all directions, or will remain at rest. Thus—

Prop. IV.—If the radius of the equator is to the radius of the pole as the attracting force at the pole is to gravitation at the equator, the pressure which any point D in the interior of the spheroid sustains from the fluid in canals of any form, extending from that point to the surface, is the same for every canal.

Let DD'd (fig. 13) be a canal drawn from any point D in the interior of the spheroid to D' and d in the surface. Let \(x'\), \(y'\), \(z'\) be the rectangular co-ordinates of D, and \(x\), \(y\), \(z\) those of D'. Take F and f, any two points in the canal indefinitely near each other, and let \(x\), \(y\), \(z\) be the co-ordinates of F; then \(x + dx\), \(y + dy\), and \(z + dz\) are those of f. Now X, Y, and Z being the resultants of the forces acting on F in the directions of the co-ordinates, the forces X, Y, Z will be reduced to the direction of the canal at F by multiplying them respectively by the cosines of the angles which that direction makes with the axes, that is, by \(\frac{dx}{du}\), \(\frac{dy}{du}\), \(\frac{dz}{du}\), (\(u\) representing the canal).

Hence the accelerating force acting on F in the direction of the canal is \(X \frac{dx}{du} + Y \frac{dy}{du} + Z \frac{dz}{du}\); and supposing the area of a section of the canal = 1, the pressure produced by the action of this force on the fluid contained in the portion of the canal Ef is \(du \left( X \frac{dx}{du} + Y \frac{dy}{du} + Z \frac{dz}{du} \right)\).

But, by what has been demonstrated in the article Attraction (9), \(X = P \frac{x}{a} Y = Q \frac{y}{b}, Z = Q' \frac{z}{b}\); therefore, denoting the whole pressure of the canal on F by \(p\), and observing that \(p\) diminishes as F is nearer the surface, we have

\[-dp = du \left( P \frac{x}{a} \frac{dx}{du} + Q \frac{y}{b} \frac{dy}{du} + Q' \frac{z}{b} \frac{dz}{du} \right).\]

and integrating,

\[p = \text{const.} - P \frac{x^2}{2a} - Q \frac{y^2}{2b}.\]

At the point D this becomes

\[p = \text{const.} - P \frac{x^2}{2a} - Q \frac{y^2}{2b},\]

and at D', where \(p = 0\),

\[0 = \text{const.} - P \frac{x'^2}{2a} - Q \frac{y'^2}{2b}.\]

therefore, by subtraction,

\[p = P \frac{x^2}{2a} + Q \frac{y^2}{2b} + z^2.\]

Now the equation of the surface of the spheroid gives

\[x^2 = \frac{a^2}{b^2} \left( b^2 - (y^2 + z^2) \right),\]

whence

\[P \frac{x^2}{2a} = P \frac{a}{2} - P \frac{(y^2 + z^2)}{2b},\]

and, by hypothesis, \(P : Q' = b : a\), or \(Pa = Q'b\); therefore

\[P \frac{x^2}{2a} = P \frac{a}{2} - Q' \frac{y^2 + z^2}{2b};\]

and on substituting this in the above expression of the value of \(p\), we get, finally,

\[p = P \frac{a}{2} - P \frac{x^2}{2a} - Q' \frac{y^2 + z^2}{2b}.\]

This equation being independent of the form of the canal, and of the situation of the point D', will necessarily be the same for every canal extending from D to the surface of the spheroid; consequently the point D sustains the same pressure in all directions.

From what has now been demonstrated, it appears that a fluid spheroid of uniform density, revolving about its lesser axis, and whose particles attract one another with forces varying in the inverse ratio of the distances, will be in equilibrium when the gravity at the pole is to the gravity at the equator diminished by the centrifugal force there, as the equatorial axis to the polar axis. The equation of equilibrium of such a spheroid is therefore \(Pa = Q'b\); and from this we shall now proceed to determine the ratio of the axes, or of \(a : b\), corresponding to a given velocity of rotation.

Let \(e\) be the eccentricity, that is, let \(e^2 = \frac{b^2 - a^2}{a^2}\), whence \(b^2 = a^2 (1 + e^2)\). It is demonstrated in the article Attraction (23 and 24), that

\[P = 4\pi a^2 \frac{1 + e^2}{r^2} (1 - \text{arc tan. } e),\]

\[Q = (Q' + \varphi) = 2\pi b \frac{1 + e^2}{r^2} \left( \text{arc tan. } e - \frac{e}{1 + e^2} \right),\]

the density of the mass being 1, and \(r\) the semicircumference of a circle whose radius is unit. Now if \(t\) denote the time of a revolution, we shall have, from the theory of central forces, \(\varphi = \frac{4\pi^2}{T^2} b\); and making \(\frac{T}{2\pi} = m\), we get \(\varphi = 4\pi bm\), whence

\[Q' = 2\pi b \left\{ \frac{1 + e^2}{r^2} \left( \text{arc tan. } e - \frac{e}{1 + e^2} \right) - 2m \right\}.\]

Substituting these values of \(P\) and \(Q'\) in the equation \(Pa = Q'b\), it becomes

\[4\pi a^2 \frac{1 + e^2}{r^2} (1 - \text{arc tan. } e) = 2\pi b \left\{ \frac{1 + e^2}{r^2} \left( \text{arc tan. } e - \frac{e}{1 + e^2} \right) - 2m \right\};\]

whence, putting \(a^2 (1 + e^2)\) instead of \(b^2\), and neglecting the common factors, we get

\[\text{arc tan. } e = \frac{3 + 2me^2}{3 + e^2}.\]

In order to discover whether this equation, which is transcendental, has a real root, or how many, we may put

\[\beta = \frac{3 + 2me^2}{3 + e^2} = \text{arc tan. } e.\] and suppose the curve to be described, of which \( x \) is the abscissa, and \( y \) the ordinate. This curve will evidently cut the axis at the origin of the co-ordinates; for when \( x = 0 \), we have also \( y = 0 \); but the root \( x = 0 \) has no concern with the question; for the velocity, and consequently \( v \), is supposed to be a real quantity. By changing the sign of \( x \), the value of \( y \) undergoes no alteration, whence the curve has two equal and similar branches on opposite sides of the origin; we need therefore attend only to the positive values of \( x \). If then we suppose \( x \) to increase from nothing to infinity, the value of \( y \) will begin and end with being positive, whence between these two values of \( x \) the curve will either not cut the axis at all, or cut it an even number of times; or, which is the same thing, the equation has either no roots, or an equal number.

To find the values of \( x \) corresponding to the greatest and least values of \( y \), we must make \( dy/dx = 0 \). Differentiating equation (2), we get, after arranging the terms,

\[ \frac{dy}{dx} = \frac{2x^3 + (10m - 2)x^2 + 9m}{(1 + x^2)(3 + x^2)}. \]

whence, making \( dy/dx = 0 \),

\[ mx^4 + (10m - 2)x^2 + 9m = 0, \]

the two roots of which are

\[ x^2 = \frac{5 - \frac{1}{m}}{\pm \sqrt{(5 - \frac{1}{m}) - 9}}. \]

Since these are the only roots of equation (3), it follows that there can be only one maximum and one minimum value of the ordinate \( y \) on each side of the origin of the abscissa. Hence we infer that the curve intersects the positive abscissa only in two points exclusive of the origin; so that there are two, and only two, real and positive roots of the equation (1). Thus there are two forms of the oblate spheroid, which, for a given velocity of rotation, satisfy the condition of equilibrium; a curious result, which was first made known by Legendre.

The limits within which the roots of equation (1) are possible will be obtained by determining the values of \( m \) and \( n \) at the point where the curve touches the axis of the abscissa without cutting it. At this point \( y = 0 \), and \( dy/dx = 0 \). When \( dy/dx = 0 \), equation (3) gives

\[ m = \frac{2x^2}{x^4 + 10x^2 + 9} = \frac{2x^2}{(1 + x^2)(9 + x^2)}. \]

substituting this in (2), and making \( dy/dx = 0 \), we get, after reduction,

\[ \tan^{-1} x = \frac{7x^2 + 30x^2 + 27x}{(3 + x^2)(1 + x^2)(9 + x^2)}. \]

or, dividing the terms by \( (3 + x^2) \),

\[ \tan^{-1} x = \frac{7x^2 + 9x}{(1 + x^2)(9 + x^2)}; \]

an equation which can have only one positive root besides \( x = 0 \). On solving it by approximation we easily find, on a few trials,

\[ x = 2.5292; \]

whence, from equation (4), the corresponding value of \( m \) is

\[ m = 0.11234. \]

From this we conclude that, when the value of \( m \) is smaller than 0.11234, the equation (1) has two unequal positive roots; that when \( m \) is equal to this number, the two roots become equal, or the two intersections of the curve with the axis pass into a contact; and that when \( m \) is greater than 0.11234, equation (1) has no real roots, or the equilibrium is impossible.

When the eccentricity \( e \) is a very small fraction, as in the case of the earth, the values of the quantities denoted by \( P \) and \( Q \) may be developed in series proceeding according to the powers of \( e \) and converging rapidly. Because (see ALGEBRA, sect. xxv. (II 2))

\[ \tan^{-1} x = \frac{x^2}{3} + \frac{x^4}{5}, \ldots \text{etc.} \]

and because

\[ \frac{x^2}{1 + x^2} = x^2 + x^4 + \ldots \text{etc.} \]

the preceding expression for \( P \) becomes

\[ P = 4\pi a \left( \frac{1 + x^2}{3} + \frac{x^4}{5} + \ldots \right); \]

whence, on multiplying, and rejecting all terms involving higher powers of \( e \) than the square,

\[ P = 4\pi a \left( \frac{1 + x^2}{3} + \frac{2}{15}x^4 \right) = \frac{4\pi a}{3} \left( 1 + \frac{2}{15}x^4 \right). \]

Making the same substitutions in the value of \( Q \), we find

\[ Q = 2\pi b \left( \frac{1 + x^2}{3} + \frac{2}{15}x^4 + \ldots \right), \]

and multiplying as before,

\[ Q = 2\pi b \left( \frac{1 + x^2}{3} + \frac{2}{15}x^4 \right) = \frac{4\pi b}{3} \left( 1 - \frac{1}{15}x^4 \right). \]

Now, if instead of \( x \) we introduce the ellipticity \( e \), and make \( e = \frac{b-a}{a} \), we shall have \( b = a(1+e) \). But we have already assumed \( b = a^2(1+e^2) \); whence \( 1+e = \sqrt{1+e^2} \); and on rejecting the second and all the higher powers of \( e \) (which may be done because \( e \) is supposed to be a very small fraction), we have \( 1+2e=1+e^2 \), and consequently \( 2e=e^2 \). Substituting this in the above formulas, and observing that \( b = a(1+e) \), we get

\[ P = \frac{4\pi a}{3} \left( 1 + \frac{4}{5}e \right); \quad Q = \frac{4\pi a}{3} \left( 1 + \frac{3}{5}e \right); \]

and since \( \varphi = 4\pi bm = 4\pi am(1+e)m = 4\pi am \) (for \( m \) and \( e \) being both very small fractions, their product may be neglected), we have also

\[ Q' = Q - \varphi = \frac{4\pi a}{3} \left( 1 + \frac{3}{5}e - 3m \right). \]

Substituting these values of \( P \) and \( Q' \) in the equation of equilibrium, \( Pa = Q'b = Q'a(1+e) \), there results

\[ 1 + \frac{4}{5}e = (1+e)\left(1 + \frac{3}{5}e - 3m\right); \]

whence, rejecting terms multiplied by \( e^2 \) and \( em \), we find

\[ e = \frac{15}{4}m. \]

The ratio of the centrifugal force to gravitation at the equator is \( \frac{P}{Q'} \) or \( \frac{3m}{1 + \frac{3}{5}e - 3m} = 3m \); hence \( e = \frac{5}{4}Q' \), that is to say, the ellipticity of a homogeneous spheroid revolving about an axis, and whose form does not differ greatly from that of a sphere, is equal to the fraction formed by dividing five times the centrifugal force at the equator by four times the gravitation at the equator. By comparing the length of the arc described by a point on the earth's surface in a second of time, with the descent of falling bodies, Newton found this ratio in the case of the earth to be \( \frac{1}{289} \), whence \( m = \frac{1}{3 \times 289} = \frac{1}{867} \).

and consequently \( e = \frac{15}{4} \times \frac{1}{3 \times 289} = \frac{1}{231} \); so that, supposing the earth to be homogeneous, the ratio of its polar to its equatorial axis ought to be 230 to 231 very nearly. Figure of The ellipticity deduced from the actual measurements of the Earth degrees, as well as from the variations of gravity indicated by pendulum observations, is considerably less than $\frac{1}{231}$; consequently the earth is not homogeneous.

In what precedes, we have supposed generally $m = \frac{\pi}{\varepsilon}$, and found for the limit which $m$ cannot exceed, $m = 0.11234$. Let $m'$ be what $m$ becomes in the case of the earth, and $t'$ be the time (expressed in parts of a mean solar day) in which the earth makes one revolution about its axis; $t$ being the time in which a spheroid of the same mean density as the earth would make a revolution when $m = 0.11234$. We have then $m' = \frac{\pi}{\varepsilon'}$, whence $m':m = t':t$, whence $t = \frac{t'}{m'}$. But $t = 23 h. 56 m. 4 sec.$

$= .99727$ day; and we have already found $m' = \frac{1}{867}$; therefore $t = .99727 \times \frac{1}{867 \times 0.11234} = 0.1009$ day = 2 hours 25 minutes 26 seconds. Hence a fluid mass of the same density as the earth could not be in equilibrium with the figure of an ellipsoid of revolution if its time of rotation were less than 2 h. 25 m. 26 sec. If the time of rotation is greater, there are two elliptic spheroids, and not more, which, with the same velocity of rotation, give a figure of equilibrium.

Hitherto we have considered only one of the roots of equation (1), and seen that it gives $230 : 231$ as the ratio of the polar to the equatorial axis. Let us now inquire into the value of the other root.

As the value of $\varepsilon^2$ in the former case was very small, it will now be large; we may therefore proceed as follows:

The identical equation

$$\tan \varepsilon = \frac{1}{2} \tan \frac{1}{\varepsilon}$$

gives

$$\tan \varepsilon = \frac{1}{2} - \frac{1}{\varepsilon} + \frac{1}{3\varepsilon^3} - \frac{1}{5\varepsilon^5} + \ldots$$

whence equation (1) becomes

$$\frac{3\varepsilon + 2m\varepsilon^3}{3\varepsilon^3} = \frac{1}{2} - \frac{1}{\varepsilon} + \frac{1}{3\varepsilon^3} - \frac{1}{5\varepsilon^5} + \ldots$$

Reverting this series, in order to find $\varepsilon$ in a series proceeding by the powers of $m$, we get

$$\varepsilon = \frac{\pi}{2m} - \frac{8}{\pi} + \frac{12m}{\pi} \left(1 - \frac{64}{3\pi^2}\right) + \ldots$$

or

$$\varepsilon = 785398 \frac{1}{m} - 2546479 - 4436656 m.$$

Relatively to the earth, it has been seen that $m = \frac{1}{867}$; by substituting which in the above equation, it gives $\varepsilon = 680$ nearly; so that $\sqrt{1 + \varepsilon^2}$, the ratio of the equatorial to the polar axis, is 680 : 1. In this case the spheroid is extremely flattened.

In the preceding propositions it has been demonstrated that a fluid mass of uniform density revolving about a fixed axis will be in equilibrium, if its form is that of an oblate spheroid, provided the velocity of rotation does not exceed a certain limit, which has been ascertained. But it has been remarked, that if the same relation exists between the attraction and the centrifugal force as exists in the case of the earth, the ellipticity of such a spheroid Fig. 6 would be different from the actual ellipticity of the earth, as determined by observation. It follows that the terrestrial spheroid must be heterogeneous, or of variable density; and this supposition is rendered more probable by what is otherwise known of the nature of the earth. The difficulties of the problem are however greatly increased in the case of a heterogeneous fluid; indeed it is only brought within the power of analysis by assuming that the form of the revolving mass differs very little from that of a sphere, and that it is regularly composed of concentric layers, increasing in density from the surface towards the centre, according to a determinate law. Admitting these assumptions, it may be demonstrated that the oblate spheroid of revolution is in this case also a figure of equilibrium.

The demonstration will be greatly simplified by means of the two following propositions respecting the attraction of homogeneous spheroids of small ellipticity, and which are easily deduced from the properties demonstrated in the article Attraction.

Prop. V.—To find the measure of the attractive forces that urge a particle situated anywhere within or in the surface of a homogeneous oblate spheroid of revolution, in the directions parallel and perpendicular to the axis, the ellipticity being small.

Let $D$ be the particle, $x$ and $y$ its rectangular co-ordinates ($x$ being taken along the axis, and $y$ perpendicular to the axis, in the meridian plane passing through $D$); let also $X$ and $Y$ denote the attractive forces soliciting the particle in those directions, and $k$ the density of the spheroid.

By Attraction (25), $P:X = a:x$, and $Q:Y = b:y$;

therefore $X = P \frac{x}{a}$ and $Y = Q \frac{y}{b}$. But it has been already shown, that when the ellipticity is small,

$$P = \frac{4\pi}{3} a \left(1 + \frac{4}{5} \varepsilon\right), \quad Q = \frac{4\pi}{3} a \left(1 + \frac{3}{5} \varepsilon\right),$$

therefore

$$X = \frac{4\pi}{3} x \left(1 + \frac{4}{5} \varepsilon\right), \quad Y = \frac{4\pi}{3} y \left(1 + \frac{3}{5} \varepsilon\right).$$

Introducing the density $k$, to which the attraction is directly proportional, and observing that $\frac{a}{b} = \frac{1}{1 + \varepsilon}$, these expressions become

$$X = \frac{4\pi}{3} kx \left(1 + \frac{4}{5} \varepsilon\right), \quad Y = \frac{4\pi}{3} ky \left(1 + \frac{3}{5} \varepsilon\right).$$

Prop. VI.—To find the measure of the forces with which an oblate spheroid, whose ellipticity is small, attracts a particle situated anywhere without it, in the direction parallel to the axis, and in the direction perpendicular to the axis, the density being uniform.

Let PQ (fig. 14) be the given spheroid, of which the axes are $a$ and $b$, D the given point without the spheroid, and $x', y'$ the co-ordinates of D. From the two assumed equations $b^2 - a^2 = b^2 - a^2$, and $\frac{x'^2}{a^2} + \frac{y'^2}{b^2} = 1$, find $a'$ and $b'$, and conceive another spheroid PQ' to be described whose polar and equatorial semidiameters are respectively $a'$ and $b'$, whose centre coincides with that of the given spheroid, and which has its equator in the same plane. The surface of this spheroid, in consequence of the equation $\frac{x'^2}{a'^2} + \frac{y'^2}{b'^2} = 1$, will necessarily pass through the point D. Make $x = \frac{ax'}{a'}, y = \frac{by'}{b'}$; then, by what is demonstrat- Substituting these expressions in the above values of X and Y, writing \( r \cos \theta \) for \( x' \) and \( r \sin \theta \) for \( y' \), we get

\[ X = \frac{4\pi}{3} k \frac{a^2}{r^2} \left[ 1 + 2e - \frac{3a^2}{5r^2} (2 - 5 \sin^2 \theta) \right] \cos \theta, \]

\[ Y = \frac{4\pi}{3} k \frac{a^2}{r^2} \left[ 1 + 2e - \frac{3a^2}{5r^2} (4 - 5 \sin^2 \theta) \right] \sin \theta. \]

Thus the expression of the force with which a homogeneous spheroid attracts a point placed without it has been found in finite terms. The ingenious process of analysis by which this result, of very great importance in the theory of the figures of the planets, has been obtained, is one of the many discoveries for which mathematical science is indebted to Mr Ivory.

PROP. VII.—To find the measure of the forces with which an oblate spheroid whose ellipticity is small, attracts a particle situated anywhere on its surface, in the direction parallel to the axis, and in the direction perpendicular to the axis, the density being variable.

In the solution of this problem we suppose the spheroid to be composed of infinitely thin concentric layers, bounded by spheroidal surfaces of different ellipticities, and that the density is uniform for a single layer, but variable from one layer to another.

Let \( PQ' \) (fig. 15) be the given spheroid, \( a', b' \) its polar and equatorial radii, and \( D \) the attracted point. Let \( D' \) be any point within the spheroid \( PQ' \), and \( PQD' \) be a spheroidal surface at every point of which the density is the same, \( a, b \) its polar and equatorial radii, and \( e \) its ellipticity. Also let \( p, q \) be another spheroidal surface indefinitely near \( PQ \), of which the polar radius \( Cp = a + da \), the equatorial radius \( Cq = b + db \), and the ellipticity \( e + de \). Between the two surfaces \( PQ \) and \( pq \) the density \( k \) may be supposed constant.

In order to find the attraction of the whole spheroid on \( D \), we must first find expressions in terms of \( a \) for the forces with which \( D \) is attracted by the elementary layer \( PQpq \); the integration of these expressions from \( a = 0 \) to \( a = a' \) will give the attractions of the spheroid.

Conceive for a moment the density of the spheroid \( CPQ \) to be uniform and \( = k \). The forces \( X \) and \( Y \) with which this spheroid attracts \( D \) are given by the last proposition, and the forces with which \( Cpq \) attracts \( D \) will be found by substituting \( a + da \) for \( a \), and \( e + de \) for \( e \) in the same expressions for \( X \) and \( Y \). Now, as the ellipticity varies from \( C \) to \( P' \), \( e \) may be regarded as a function of \( a \); its variation is consequently included in the variation of \( a \), and the attractions of the spheroid \( Cpq \) hence become

\[ X + \frac{dX}{da} da, \quad Y + \frac{dY}{da} da. \]

The difference between the attractions of the spheroids \( Cpq \) and \( CPQ \) is the attraction of the layer \( PQpq \), which is therefore \( \frac{dX}{da} da \) and \( \frac{dY}{da} da \). Hence, if we represent the attractions of the whole heterogeneous spheroid \( CPQ' \) by \( X' \) and \( Y' \), we shall have

\[ X' = \int \frac{dX}{da} da, \quad Y' = \int \frac{dY}{da} da, \]

which integrals must be taken from \( a = 0 \) to \( a = a' \), the density \( k \), which enters as a factor into \( X \) and \( Y \), being regarded as a function of \( a \).

Let us first consider the force \( X' \). By last proposition we have

\[ X = \frac{4\pi}{3} k \frac{a^2}{r^2} \left[ 1 + 2e - \frac{3a^2}{5r^2} (2 - 5 \sin^2 \theta) \right] \cos \theta, \] Figure of whence

\[ \frac{dX}{da} = \frac{4\pi k}{3r^2} \left\{ d(1+2e^2) - \frac{3}{5r^2}(2-5\sin^2\theta)d(a^2e) \right\} \cos \theta. \]

Integrating, and regarding \( k \) as variable, and a function of \( a \), and going through the same process with respect to \( Y \), we get for the attractions of the spheroid \( CPQ' \) in the directions respectively parallel and perpendicular to the axis,

\[ X' = \frac{4\pi}{3} \left\{ \frac{1}{r} \int kd(1+2e^2) - \frac{3}{5r^2}(2-5\sin^2\theta)d(a^2e) \right\} \cos \theta, \]

\[ Y' = \frac{4\pi}{3} \left\{ \frac{1}{r} \int kd(1+2e^2) - \frac{3}{5r^2}(2-5\sin^2\theta)d(a^2e) \right\} \sin \theta. \]

Prop. VIII.—To find the measure of the forces with which a heterogeneous layer, included between two level surfaces, attracts a particle situated anywhere within it.

Let \( D' \) (fig. 15) be the given particle situated within the spheroidal layer \( PQQP' \), \( x' \) and \( y' \) the co-ordinates of \( D' \), \( a \) polar semiaxis, and \( e \) eccentricity of the inner surface \( PQ \) passing through \( D' \), \( a' \) polar semiaxis of the exterior surface \( P'Q' \), and \( k \) density at the interior surface.

Suppose the interior of the spheroid \( PQ \) to be filled with matter of uniform density \( k \), then by Proposition V., the attractions of that spheroid on \( D' \) are

\[ X = \frac{4\pi}{3} k \left( 1 + \frac{4}{5} e \right) x', \quad Y = \frac{4\pi}{3} k \left( 1 - \frac{2}{5} e \right) y'; \]

or, changing \( x' \) into \( r \cos \theta \), and \( y' \) into \( r \sin \theta \),

\[ X = \frac{4\pi}{3} rk \left( 1 + \frac{4}{5} e \right) \cos \theta, \quad Y = \frac{4\pi}{3} rk \left( 1 - \frac{2}{5} e \right) \sin \theta. \]

Now the attraction of another similar spheroid \( Cpq \), whose polar semiaxis is \( a+da \), will be found by substituting \( a+da \) for \( a \) and \( e+de \) for \( e \) in the above expressions; and the attraction of the elementary layer \( PQpq \) on \( D \) will be the difference of these attractions. But

\[ \frac{dX}{da} = \frac{4\pi}{3} \cdot \frac{4}{5} rhde \cos \theta, \]

\[ \frac{dY}{da} = -\frac{4\pi}{3} \cdot \frac{2}{5} rhde \sin \theta, \]

\( e \) being regarded as a function of \( a \). The integral of these expressions will give the attraction of the layer \( PQpq \) on \( D' \); but as this attraction diminishes while \( a \) increases, if we take \( A \) to represent what \( \int kdde \) becomes when \( a \) becomes \( a' \), and denote by \( X'' \) and \( Y'' \) the attractions of the whole layer \( PQQP' \), we shall have

\[ X'' = \frac{4\pi}{3} \cdot \frac{4}{5} r(A - \int kdde) \cos \theta, \]

\[ Y'' = -\frac{4\pi}{3} \cdot \frac{2}{5} r(A - \int kdde) \sin \theta. \]

Prop. IX.—To find the attraction of a heterogeneous spheroid on a point within it.

Let \( D' \) (fig. 14) be the given point, and \( CPQ' \) the given spheroid, \( r \cos \theta \) and \( r \sin \theta \) the co-ordinates of \( D' \), \( a' \), and \( e' \), the polar semiaxis and ellipticity of the given spheroid \( CPQ' \); and \( a \) and \( e \) those of the spheroidal surface passing through \( D' \) and all the points of equal density.

The whole force urging \( D' \) may be conceived as made up of two parts; that arising from the spheroid \( CPQ \), the surface of which passes through \( D \), and that of the spheroidal shell \( PQQP' \). The first of these forces is given by the expressions represented by \( X' \) and \( Y' \) in Prop. VII., and the second by the expressions represented by \( X'' \) and \( Y'' \) in Prop. VIII. Therefore, if we denote by \( X'' \) and \( Y'' \) the attractions of the whole spheroid on the particle \( D' \), in the direction parallel to the axis, and in the direction perpendicular to the axis, we shall have

\[ X'' = X' + X'', \quad Y'' = Y' + Y''. \]

and if the spheroid revolves about its polar axis, \( Y'' \) must be diminished by the amount of the centrifugal force \( = \)

\[ \frac{4\pi}{3} \cdot \frac{3\pi}{\ell^2} r \sin \theta; \]

therefore the attractions of the given spheroid on the point \( D' \) are

\[ X'' = X' + X'', \]

\[ Y'' = Y' + Y'' - \frac{4\pi}{3} \cdot \frac{3\pi}{\ell^2} r \sin \theta. \]

Prop. X.—To find the ratio of the ellipticity to the density of any level surface, when the component of all the forces urging any point on the surface is perpendicular to the surface.

The forces urging the particle \( D' \) (fig. 14) being \( X'' \) and \( Y'' \), in order that their resultant may be perpendicular to the surface, we must have

\[ X'' : Y'' = ME : ED \] (fig. 10).

But \( ME \) (the subnormal) \( = \frac{b^2}{a^2} x' = (1+2e)x' \), and \( ED = y' \);

therefore \( X'' : Y'' = (1+2e)x' : y' \), whence, substituting \( r \cos \theta \) for \( x' \) and \( r \sin \theta \) for \( y' \),

\[ \frac{X''}{\cos \theta} = \frac{Y''}{\sin \theta} (1+2e) = 0. \]

Now, if we substitute for \( X'' \) in this equation the values of \( X' \) and \( X'' \) given by Propositions VII. and VIII., and for \( Y'' \) the values of \( Y' \) and \( Y'' \) multiplied by \( (1+2e) \), and neglect terms involving \( e^2 \), we shall find, on changing the signs and having regard to the centrifugal force,

\[ \frac{2e}{r^2} \int kd(1+2e^2) - \frac{6}{5r^2} \int kd(a^2e) - \frac{6}{5} \left( A - \int kdde \right) - \frac{3\pi}{2\ell^2} = 0. \]

Dividing by \( 2r \), and substituting \( a \) for \( r \) (which may be done without sensible error, because all the terms of the equation are small, and because \( r^2 \) differs from \( a^2 \) only by a quantity depending on the ellipticity \( e \)), we shall find

\[ \frac{e}{a^2} \int kd(1+2e^2) - \frac{3}{5a^2} \int kd(a^2e) - \frac{3}{5} \left( A - \int kdde \right) - \frac{3\pi}{2\ell^2} = 0. \]

From this we infer that if \( k \) and \( e \) have the relation to each other indicated by this equation, and that \( e \) is very small from \( a=0 \) to \( a=a' \), each of the level surfaces will be intersected at every point by the resultant of the attractive forces under an angle which will differ from a right angle only by an infinitesimal of the second order, supposing the centrifugal force to be very small in comparison of gravity. Hence, neglecting quantities of the second order, the elliptical spheroid (supposing the equation possible) is a figure of equilibrium.

In order to discover whether the equation which has now been obtained is possible, it must be differentiated twice in order to eliminate \( \int kd(a^2e) \) and \( \int kdde \), both of which contain the ellipticity \( e \). Observing that as the first term is multiplied by \( e \), we may suppose, without sensible error,

\[ \frac{e}{a^2} \int kd(1+2e^2) = \frac{e}{a^2} \int kd(a^2) = \frac{3e}{a^2} \int ka^2da, \]

and the first differentiation gives, on multiplying by \( a^2 \),

\[ \frac{de}{da} \int ka^2da - 3a^2e \int ka^2da + \int kd(a^2e) = 0. \]

Differentiating again, and considering \( da \) as constant, we shall find, after reduction,

\[ \frac{de}{da} + \frac{2ka^2}{\int ka^2da} \cdot \frac{da}{da} + \left( \frac{2ka}{\int ka^2da} - \frac{6}{a^2} \right) e = 0. \] This is the equation given by Clairaut (Figure de la Terre, p. 276). It can be integrated by the usual methods when \( k \) is expressed in terms of \( a \); consequently the equilibrium is always possible if the density is a function of the distance from the centre. The integration being performed, an equation will result between the densities and ellipticities of the different spheroidal layers, by means of which, when one of these elements is supposed to be given, the other can be determined.

Prop. XI.—To find the whole force urging a particle situated at any point on the surface of a heterogeneous spheroid revolving about its shorter axis.

By Prop. VII. the force urging the particle in the direction of the axis is \( X' \), and in the direction perpendicular to the axis \( Y' \); but in consequence of the centrifugal force, the latter becomes \( Y' - \phi \), therefore the whole force urging \( D \) in the direction of the normal

\[ \sqrt{X^2 + (Y - \phi)^2}. \]

Neglecting the terms which involve \( e^2 \) and \( e \phi \) (\( \phi \) being small as well as \( e \)), and putting \( \int h d(1 + 2e \cdot a^2) = M \), and

\[ \int h d(a^2 e) = N, \]

and substituting \( \frac{4}{3} r \sin \theta \) for \( \phi \), we get

\[ X^2 = \left( \frac{4}{3} r \right)^2 \left\{ M^2 \cos^2 \theta - \frac{6(2 - 5 \sin^2 \theta)}{5r^2} MN \cos^2 \theta \right\}, \]

\[ (Y - \phi)^2 = \left( \frac{4}{3} r \right)^2 \left\{ M^2 \sin^2 \theta - \frac{6(4 - 5 \sin^2 \theta)}{5r^2} MN \sin^2 \theta \right\}, \]

whence, putting \( X^2 + (Y - \phi)^2 = U^2 \), there results

\[ U^2 = \left( \frac{4}{3} r \right)^2 \left\{ M^2 - \frac{6(2 - 3 \sin^2 \theta)}{5r^2} MN \right\}, \]

and extracting the square root,

\[ U = \frac{4}{3} \left\{ M - \frac{3(2 - 3 \sin^2 \theta)}{5r^2} N - \frac{3 \pi r \sin^2 \theta}{\ell^2} \right\}. \]

But the equation of the ellipse gives

\[ r^2 \cos^2 \theta + \frac{r^2 \sin^2 \theta}{1 + 2e} = a^2; \]

whence we deduce \( r^2 = a^2(1 + 2e \sin^2 \theta) \); therefore, substituting and dividing, and neglecting terms in \( Ne \) and \( e^2 \),

\[ U = \frac{4}{3} \left\{ M \left( 1 - 2e \sin^2 \theta \right) - \frac{3}{5} \left( 2 - 3 \sin^2 \theta \right) \frac{\pi r \sin^2 \theta}{\ell^2} \right\} N. \]

Let the centrifugal force at the equator \( = m \times \) gravity; then, if \( m \) is a very small fraction (as in the case of the earth, where \( m = \frac{1}{289} \)), terms involving the product \( me \) may be neglected, and the last equation, multiplied by \( m \), will become

\[ mU = \frac{4}{3} \frac{M}{a^2} m; \]

but the centrifugal force at the equator \( = \frac{4}{3} a(1 + e) \),

which does not differ sensibly from \( \frac{4}{3} a \); therefore

\[ \frac{4}{3} \frac{M}{a^2} m = \frac{3}{5} \frac{Mm}{a^2}. \]

Now, at the surface the equation of equilibrium in Prop. X. becomes

\[ \frac{e}{a^2} M - \frac{3}{5a^2} N - \frac{3}{2a^2} = 0, \]

therefore

\[ \frac{e}{a^2} M - \frac{3}{5a^2} N - \frac{Mm}{2a^2} = 0, \]

whence

\[ \frac{3N}{5a^2} = \frac{M}{a^2} \left( e - \frac{m}{2} \right). \]

Substituting these values of \( \frac{3}{a^2} \) and \( \frac{3N}{5a^2} \) in the above value of \( U \), we get, after reduction,

\[ U = \frac{4}{3} \frac{M}{a^2} \left\{ 1 - 2e + m - \left( \frac{5}{2} m - e \right) \sin^2 \theta \right\}. \]

At the equator \( \sin \theta = 1 \), and this equation becomes

\[ Q = \frac{4}{3} \frac{M}{a^2} \left\{ 1 - e - \frac{3}{2} m \right\}; \]

and at the pole, where \( \sin \theta = 0 \), it gives

\[ P = \frac{4}{3} \frac{M}{a^2} \left\{ 1 - 2e + m \right\}; \]

consequently the difference between the force of gravity at the pole and that of gravitation at the equator is

\[ \frac{4}{3} \frac{M}{a^2} \left( \frac{5m}{2} - e \right). \]

Dividing this by the polar gravity \( P \), we get for the quotient \( \frac{5m}{2} - e \), which, therefore, expresses the ratio of the excess of the polar gravity above the gravitation at the equator to the gravity at the pole.

Let this ratio \( = n \); we have then

\[ n = \frac{5}{2} m - e; \]

a very remarkable relation between the gravity and ellipticity, which was discovered by Clairaut, and is generally called Clairaut's theorem.

If we substitute \( 1 - \cos^2 \theta \) for \( \sin^2 \theta \) in the equation

\[ U = \frac{4}{3} \frac{M}{a^2} \left\{ 1 - 2e + m - \left( \frac{5}{2} m - e \right) \cos^2 \theta \right\}, \]

we get

\[ U = \frac{4}{3} \frac{M}{a^2} \left\{ 1 - e - \frac{3}{2} m + \left( \frac{5}{2} m - e \right) \cos^2 \theta \right\}; \]

or, neglecting quantities of the order \( m e \),

\[ U = \frac{4}{3} \frac{M}{a^2} \left( 1 - e - \frac{3}{2} m \right) \left\{ 1 + \left( \frac{5}{2} m - e \right) \cos^2 \theta \right\}; \]

whence we have \( U = Q(1 + n \cos^2 \theta) \). Now in this equation the angle \( \theta \) may be taken for the complement of the latitude, from which indeed it differs only by a quantity of the second order, and the equation then becomes

\[ U = Q(1 + n \sin^2 l); \]

whence we infer that, on going from the equator towards the pole, the increase of gravity is proportional to the square of the sine of the latitude.

Let \( p \) be the length of the seconds pendulum at the equator, \( p' \) its length at any latitude \( l \), and \( U \) the intensity of the gravitating force at the same latitude \( l \). When the time of oscillation is constant, the length of the pendulum is directly proportional to the intensity of gravity; therefore \( p' : p = U : Q \), and consequently

\[ \frac{p' - p}{p} = \frac{U - Q}{Q}. \]

But \( U = Q(1 + n \sin^2 l) \),

therefore,

\[ \frac{p' - p}{p} = n \sin^2 l = \left( \frac{5m}{2} - e \right) \sin^2 l. \]

At the pole \( \sin l = 1 \), and \( \frac{p' - p}{p} = \frac{5m}{2} - e \); therefore,

since \( e = \frac{b-a}{a} \),

\[ \frac{p' - p}{p} + \frac{b-a}{a} = \frac{5m}{2}. \] Figure of Now it has been shown that in the case of a homogeneous spheroid the ellipticity is $\frac{5}{4}$; hence we have this theorem:

In the case of a heterogeneous spheroid, the excess of the length of the pendulum at the pole above its length at the equator divided by its length at the equator, and the excess of the axis of the equator above the polar axis divided by the polar axis, form two fractions, of which the sum is constant, and equal to twice the ellipticity the spheroid would have if homogeneous.

SECTION IV.

ON THE FIGURE OF THE EARTH AS DETERMINED BY OBSERVATIONS OF THE PENDULUM.

As the various methods of determining the length of a pendulum which makes a given number of vibrations in a mean solar day, and the corrections that are required to reduce the observations to the same circumstances in respect of altitude, temperature, barometric pressure, &c., form the subject of a separate article, we shall here limit ourselves to a brief abstract of the results of some of the latest and best observations.

The ellipticity of the earth is deduced from the observed number of vibrations made by an invariable pendulum of a given length at different latitudes, by means of the theorem of Clairaut. Let $N$ = the number of oscillations made by a pendulum at the equator in a mean solar day, $N'$ = the number of oscillations made by the same pendulum at the latitude $l$, then $Q$ representing, as before, the gravitation at the equator, and $U$ the gravitation at latitude $l$, we have, by the last proposition in the preceding section,

$$U = Q (1 + n \sin^2 l).$$

But by the property of the invariable pendulum, the square of the number of vibrations is directly proportional to the force of gravitation; therefore $U : Q = N^2 : N'^2$, whence $N'^2 = N^2 (1 + n \sin^2 l)$.

In this equation $n = \frac{5}{2} m - e = \frac{5}{2 \times 289} - e$; therefore, as $N^2$ is given by observation at any station, we can, by combining the results at two different latitudes, determine $N$ and $n$, and consequently $e$. The pendulum gives no information respecting the magnitude of the earth.

Instead of $N$ and $N'$, we might substitute in the above equation the lengths of the seconds pendulum at the equator, and the latitude $l$ (the length of the seconds pendulum being also directly proportional to the intensity of gravity); but as the invariable pendulum is now almost universally employed for the measurement of the variations of gravity, the formula under the above form is more immediately applicable to the results of observation. The absolute length of the pendulum (which is not very easily determined) has no concern with the present question.

For the purpose of exhibiting the variations of gravity, we shall confine our attention to the results that have been found from observations with the invariable pendulum. The results given by pendulums of a different kind are probably not less exact, but they are not directly comparable, on account of the great uncertainty that remains respecting the correction that ought to be applied for the buoyancy of the atmosphere.

In the seventh volume of the Memoirs of the Royal Astronomical Society, Mr. Baily has given the results of fourteen different sets of observations with invariable Figure pendulums. They are as follows:

1. The observations of Captain Kater at different stations in Great Britain. These are detailed in the Philosophical Transactions for the year 1819. 2. Those of Mr. Goldingham at London and Madras, and of Mr. Lawrence and Mr. Robinson with the same pendulum at the small island of Pulo Guansah Lout, on the western coast of Sumatra, and almost immediately under the equator. (Phil. Trans. 1822.) 3. Those of Captain Hall at the Galapagos Islands, San Blas, Rio Janeiro, and London. (Phil. Trans. 1823.) 4. Those of Sir Thomas Brisbane at London and Paramatta. (Phil. Trans. 1823.) 5. Those of Captain Sabine in various parts of the Atlantic Ocean and the North Sea, in the years 1822-4. (Sabine's Account of Experiments, &c. 1825.) 6. Those of Captain Foster at Greenwich, London, and Port Bowen. (Phil. Trans. 1826.) 7. Those of Mr. Fallows at the Cape of Good Hope. The same pendulum was swung in London previous to its departure. 8, 9, 10. Those of Captain Sabine for the purpose of determining the difference in the number of oscillations at London and Paris (Phil. Trans. 1828), London and Greenwich (Phil. Trans. 1829), and at Greenwich, London, and Altona (Ibid. 1829 and 1830). 11. Those of Captain Freycinet, who commanded an expedition fitted out by the French government in 1817, for the purpose of making scientific observations in a voyage round the world. The stations at which the experiments were made were the island of Rawak (near the coast of Guinea), Guam (one of the Ladrones), the Isle of France, Mowi (one of the Sandwich Islands), Rio Janeiro, Port Jackson, Cape of Good Hope, Paris, and the Falkland Islands. (Voyage autour du Monde, par M. Freycinet.) 12. Those of Captain Duperrey, who commanded another French expedition. The stations were, island of Ascension, Isle of France, Port Jackson, Toulon, Paris, and the Falkland Islands. (Connaissance des Tems for 1830.) 13. Those of Captain Leutke, a Russian officer, under whose orders a ship of war was dispatched to the South Seas, for the purpose, among other things, of swinging the pendulum at various places. The stations at which experiments were made were Ualan (one of the Caroline Isles), Guam, St Helena, Bonin Island (off the south-east coast of Japan), Valparaiso, Greenwich, Petropaulovski, Sitka (off the north-west coast of America), and Petersburg. (Mr Baily's Report in the Memoirs of the Royal Astronomical Society, vol. vii.) 14. Those of Captain Foster, in his last voyage, which commenced in 1828, and which form perhaps the most valuable series of the whole. The observations were made at fourteen different stations, principally in the southern hemisphere, from the equator to the latitude of 63°. Captain Foster carried out with him four pendulums, two of brass, one of iron, and one of copper, and at seven of the stations all four were swung. The details are published at full length in the volume of the Astronomical Society's Memoirs to which we have already referred, with an excellent Report by Mr. Baily.

The results of all these observations are contained in the following table, in which the observed number of vibrations at each station is corrected, for any deviation from the standard temperature of 62°, or from the true correction for the buoyancy of the atmosphere, and afterwards reduced to a comparison with Captain Foster's mean pendulum, assumed as making 86,400 vibrations in a mean solar day at London. | No. | Station | Latitude | Vibrations | Observer | Computed | Difference | Observer | |-----|--------------------------|----------------|------------|----------|----------|------------|----------| | 1 | Rawak | 0° 1' 34" S | 86261-46 | 86264-06 | -3-40 | Freycinet | | 2 | Pulo Guansah Lout | 0° 1 49 N | 86260-64 | 86264-86 | +1-73 | Goldingham | | 3 | St Thomas | 0° 24 41 | 86268-84 | 86264-87 | +3-97 | Sabine | | 4 | Galapagos | 0° 32 19 | 86264-96 | 86264-83 | -0-33 | Hall | | 5 | Para | 1° 27 0 S | 86260-61 | 86265-00 | -4-39 | Foster | | 6 | Maranham | 2° 31 35 | 86258-74 | 86255-30 | -6-56 | Foster | | 7 | Ditto | 2° 31 43 | 86259-10 | 86255-30 | -6-11 | Sabine | | 8 | Fernando de Noronha | 3° 49 59 | 86271-20 | 86255-06 | +5-34 | Foster | | 9 | Ulan | 5° 21 16 N | 86275-44 | 86266-78 | +8-06 | Leutke | | 10 | Ascension | 7° 55 23 S | 86272-26 | 86269-06 | +3-20 | Foster | | 11 | Ditto | 7° 55 48 | 86272-96 | 86269-08 | +2-98 | Duperrey | | 12 | Ditto | 7° 55 48 | 86272-56 | 86269-08 | +3-48 | Sabine | | 13 | Sierra Leone | 8° 00 30 N | 86277-54 | 86269-70 | -2-16 | Sabine | | 14 | Porto Bello | 9° 32 39 | 86270-96 | 86270-96 | +1-03 | Foster | | 15 | Trinidad | 10° 38 55 | 86267-24 | 86272-42 | -5-18 | Foster | | 16 | Ditto | 10° 38 56 | 86266-70 | 86272-42 | -5-18 | Sabine | | 17 | Bahia | 12° 50 21 S | 86272-38 | 86276-07 | -3-69 | Golzegham | | 18 | Madras | 13° 4 9 N | 86272-36 | 86276-19 | -3-83 | Leutke | | 19 | Guam | 13° 25 21 | 86280-64 | 86276-34 | +3-30 | Freycinet | | 20 | Ditto | 13° 27 51 | 86282-98 | 86276-90 | +6-03 | Leutke | | 21 | St Helena | 15° 54 59 S | 86282-29 | 86281-54 | +0-75 | Leutke | | 22 | Ditto | 15° 56 7 | 86280-29 | 86281-54 | +0-75 | Foster | | 23 | Jamaica | 17° 56 7 N | 86284-66 | 86285-90 | -1-24 | Sabine | | 24 | Isle of France | 20° 9 23 S | 86297-60 | 86291-20 | +6-40 | Duperrey | | 25 | Ditto | 20° 9 56 | 86290-08 | 86291-23 | +6-65 | Freycinet | | 26 | Mowt | 20° 52 7 N | 86297-52 | 86293-00 | +4-52 | Freycinet | | 27 | St Blas | 21° 32 24 | 86295-80 | 86294-77 | -5-97 | Hall | | 28 | Rio Janeiro | 22° 55 13 S | 86293-43 | 86289-52 | -5-04 | Freycinet | | 29 | Ditto | 22° 55 22 | 86294-90 | 86298-52 | -3-62 | Hall | | 30 | Ross Island | 4° 12 N | 86310-61 | 86301-61 | +11-23 | Leutke | | 31 | Valparaiso | 33° 2 3 | 86329-16 | 86329-53 | +2-36 | Leutke | | 32 | Paramatta | 33° 48 43 | 86331-49 | 86332-53 | +2-07 | Brisbane | | 33 | Port Jackson | 33° 51 34 | 86334-06 | 86333-68 | -0-74 | Freycinet | | 34 | Ditto | 33° 51 40 | 86332-04 | 86333-68 | -0-74 | Duperrey | | 35 | Cape of Good Hope | 33° 54 37 | 86331-33 | 86333-99 | +2-57 | Foster | | 36 | Ditto | 33° 55 15 | 86331-53 | 86333-93 | +2-37 | Freycinet | | 37 | Ditto | 33° 55 56 | 86332-56 | 86333-93 | +1-42 | Fallows | | 38 | Monte Video | 34° 54 26 | 86334-26 | 86337-48 | +3-12 | Foster | | 39 | New York | 40° 42 43 N | 86358-06 | 86359-22 | -1-16 | Sabine | | 40 | Toulon | 43° 7 20 | 86367-16 | 86368-48 | -1-32 | Duperrey | | 41 | Paris | 48° 50 14 | 86388-01 | 86390-64 | -2-63 | Freycinet | | 42 | Ditto | 48° 50 14 | 86388-01 | 86390-54 | -2-24 | Sabine | | 43 | Ditto | 48° 50 14 | 86388-56 | 86390-54 | -1-96 | Duperrey | | 44 | Shanklin Farm | 50° 37 24 | 86396-40 | 86397-32 | -0-92 | Kater | | 45 | Greenwich | 51° 20 40 | 86400-00 | 86400-88 | +1-68 | Foster | | 46 | Ditto | 51° 28 40 | 86400-24 | 86400-88 | +1-34 | Leutke | | 47 | Ditto | 51° 28 40 | 86400-48 | 86400-88 | +1-12 | Foster | | 48 | Ditto | 51° 28 40 | 86400-67 | 86400-88 | +0-86 | Sabine | | 49 | Ditto | 51° 28 40 | 86400-72 | 86400-53 | +0-14 | Sabine | | 50 | London | 51° 31 8 | 86400-76 | 86400-74 | +0-98 | Fallows | | 51 | Ditto | 51° 31 8 | 86400-90 | 86400-74 | +0-84 | Foster | | 52 | Ditto | 51° 31 8 | 86400-90 | 86400-74 | +0-84 | Sabine | | 53 | Ditto | 51° 31 8 | 86400-90 | 86400-74 | +0-84 | Foster | | 54 | Ditto | 51° 31 8 | 86400-90 | 86400-74 | +0-84 | Sabine | | 55 | Ditto | 51° 31 8 | 86400-90 | 86400-74 | +0-84 | Sabine | | 56 | Ditto | 51° 31 8 | 86400-90 | 86400-74 | +0-84 | Sabine | | 57 | Ditto | 51° 31 8 | 86400-90 | 86400-74 | +0-84 | Sabine | | 58 | Ditto | 51° 31 8 | 86400-90 | 86400-74 | +0-84 | Sabine | | 59 | Ditto | 51° 31 8 | 86400-90 | 86400-74 | +0-84 | Sabine | | 60 | Ditto | 51° 31 17 | 86400-00 | 86400-75 | +0-75 | Foster | | 61 | Falkland Island | 51° 31 44 S | 86400-00 | 86400-78 | +0-94 | Duperrey | | 62 | Ditto | 51° 35 18 | 86400-74 | 86400-78 | +0-94 | Freycinet | | 63 | Arbury Hill | 52° 12 55 N | 86400-00 | 86400-86 | +1-86 | Kater | | 64 | Euppanoucski | 53° 0 53 | 86400-00 | 86400-86 | +1-86 | Kater | | 65 | Clifton | 53° 20 45 | 86407-42 | 86407-90 | +0-51 | Kater | | 66 | Altom | 53° 22 45 | 86408-08 | 86408-23 | +0-70 | Sabine | | 67 | Staten Island | 54° 46 23 S | 86415-22 | 86412-20 | +2-42 | Foster | | 68 | Cape Horn | 55° 51 20 | 86417-96 | 86416-72 | +1-26 | Foster | | 69 | Leith Fort | 55° 53 41 N | 86418-02 | 86417-16 | +0-65 | Kater | | 70 | Sitka | 57° 2 56 | 86420-54 | 86420-66 | -0-12 | Leutke | | 71 | Portsoy | 57° 40 59 | 86424-70 | 86423-21 | +1-49 | Kater | | 72 | Petersburg | 59° 56 31 | 86432-20 | 86430-94 | +1-26 | Leutke | | 73 | Unst | 60° 45 28 | 86435-40 | 86433-64 | +1-76 | Kater | | 74 | South Shetland | 62° 56 11 S | 86444-52 | 86440-65 | +3-87 | Foster | | 75 | Drontheim | 63° 25 54 N | 86435-64 | 86442-20 | -3-96 | Sabine | | 76 | Hammerfest | 70° 40 5 | 86461-14 | 86462-23 | -1-09 | Foster | | 77 | Fort Bowen | 73° 13 39 | 86470-48 | 86469-06 | +2-42 | Foster | | 78 | Greenland | 74° 32 19 | 86470-72 | 86470-75 | -0-03 | Sabine | | 79 | Spitzbergen | 79° 49 58 | 86483-25 | 86479-53 | +3-70 | Sabine | Taking the mean results of the observed vibrations given in col. 4 of the above table, and forming with each of them the equation of condition $N^2 = N^2 (1 + n \sin^2 I)$, seventy-nine equations are formed, from which, on deducing the values of $N^2$ and $n$ by the method of least squares, Mr Baily obtained the following results:

$$N^2 = 7441625711$$

$$n = 0.0514491$$

$$e = \frac{1}{285.26}.$$

By means of these values of $N^2$ and $n$, the number of vibrations made by a pendulum which beats seconds at London, at any latitude $I$, supposing the earth to be a regular spheroid, is given by the formula

$$N = 86246.8 \left(1 + 0.0514491 \sin^2 I\right)^{\frac{1}{2}};$$

and from this the results given in col. 5 of the table have been computed.

Although the differences between the observed and computed vibrations are in some instances considerable, the observations are on the whole well represented by supposing the ellipticity $= \frac{1}{285.26}$. To be satisfied of this, it will be sufficient to remark, that as the differences in the table present twenty-four alternations of sign, the curve which represents all the observations intersects the meridian having that ellipticity no less than twenty-four times along an arc of $152^\circ$ (from Spitzbergen to South Shetland), or, on an average, once for every $6^\circ$ degrees; and (with the exception of the island of Bonin) the deviations on the opposite sides are nearly equal.

If, instead of seeking an ellipticity by the fusion (if we may so speak) and agglomeration of all the observations by the method of least squares, we consider the results of the different voyagers separately, we shall find marked differences among them. Taken by themselves, Captain Sabine's observations give $e = \frac{1}{288.4}$; Captain Foster's, $e = \frac{1}{289.5}$; Duperrey's, $e = \frac{1}{266.4}$; Freycinet's, $e = \frac{1}{267.6}$; and Leutke's, $e = \frac{1}{267.7}$. The singular agreement of the results of the foreign voyagers with each other, and their difference from those of Captains Sabine and Foster, which also agree well with each other, have not been satisfactorily explained.

In one of the numbers of the *Annales des Sciences de l'Observation* (tome i. No. 3), M. Saigey has computed the ellipticity from fifty-one different observations, combined by the method of least squares. Captain Foster's observations are of course not included in the table; but it contains those of Biot, made with a variable pendulum at different stations along the French and British arcs of meridian, and more recently in Italy and the Lipari Islands. The result gives $e = \frac{1}{286.5}$.

The differences between the observed and computed values are, in many instances at least, too great to be ascribed entirely to errors of observation. They must, therefore, be occasioned by real differences in the intensity of gravity, arising either from variations in the density of the materials which compose the exterior crust of the earth, or from local deviations of its form from that of the regular spheroid. On attending to the preceding table, it will be perceived that the differences are generally in excess where the station is on small islands, and especially if situated at a great distance from the main land. From this it may be inferred, that such islands are composed of denser materials than the general crust of the earth; but it is not improbable that part of this effect may be occasioned by a difference of the level of the surface of the ocean. On the shores of elevated continents, the direction of gravity is altered by the action of the high land; and the surface of the water, which is perpendicular to gravity, deviates from the regular curvature, and becomes elevated above the general level of the ocean. Hence the apparent elevation is less than the real elevation of the station, and the observed intensity of gravity is consequently less than it was calculated to be. How far the disturbing influence of this cause may affect the results, will be better understood when we shall be furnished with numerous pendulum observations, made in the interior of the large continents, and at a great distance from the sea.

All the pendulum experiments agree in giving a greater ellipticity to the earth than that which is deduced from the comparison of arcs of meridian. To what cause this discrepancy is to be assigned is by no means apparent. The geological character of the country in the immediate neighbourhood of the station probably exerts a considerable influence in accelerating or retarding the pendulum; but the experiments have now been made at so many points, that we might expect to find the effects of local irregularities almost entirely eliminated, and a much nearer agreement between the results of the two methods of investigation than actually exists. We can, however, have no difficulty in giving the preference to the results of the geodetic measures.

It was mentioned in the first section of this article that the ellipticity of the earth may be deduced from certain inequalities in the motions of the moon, to which it gives rise; but this part of the subject belongs to the theory of the lunar perturbations. The ellipticity so deduced is about $\frac{1}{300}$, which agrees well with that given by the meridional arcs; and the result has the advantage of being entirely independent of local disturbance.

One conclusion cannot fail to be drawn from the results of this enquiry. However irregular the surface of the earth may be in its details, its general form agrees so nearly with the figure of hydrostatic equilibrium, that the agreement cannot be regarded as fortuitous or accidental. The regular increase of gravity from the equator to the poles, indicated by the pendulum experiments, also proves that it is symmetrically constituted, or that the materials in its interior are disposed about the centre of gravity in regular elliptical strata, and arranged according to the order of density. The earth must therefore have taken its present form while its particles were at liberty to arrange themselves in obedience to the forces arising from their mutual attractions and from the rotation; in other words, it must have existed at some period of time in a state of fluidity. This inference, which so many geological facts tend to confirm, may be extended to all the other planets.