602 E A R T H upon a particle within its mass, whose co-ordinates are/, 0, Ji, are x - " p We take into account the rotation of the earth by subtract ing the centrifugal f orce /w 2 = F f rom X. Now, the sur face of constant density upon which the point /, 0, h is situated gives (1 - 2e)fdf+ hdk = ; and the condition of equilibrium is that (X - F)df+ Zdh = 0. Therefore, , which, neglecting small quantities of the order e 2 and putting w 2 < 2 = 4?r 2 , gives Here we must put now c for c p c for r, and 1 + 1e under the first integral sign may be replaced by unity. Two integrations lead us to the following very important differential equation : d^e 2pc 2 de dc? fpc*dc dc 2pc 6 cj e ~ When p is expressed in terms of c, this equation can be integrated. We infer then that a rotating spheroid of very small ellipticity, composed of fluid homogeneous strata such as we have specified, will be in equilibrium; and when the law of the density is expressed, the law of the corresponding ellipticities will follow. If we put M for the mass of the spheroid, then M-TJ and m "M and putting c = c in the equation expressing the condition of equilibrium, we find M(2-m)= |.A /"p o 5c ^X Making these substitutions in the expressions for the h 2 forces at the surface, and putting r=l + e- e , we get " Here G is gravity in the latitude </>, and a the radius of the equator. Since sec <jf> = j il+e + e-^ , M f. 3 /5 . ) G = ll--m +1 - m-e ] sin- rf> J , ac 2 V 2 / r j which expression contains the theorems we have referred to as discovered by Clairaut. The theory of the figure of the earth as a rotating ellipsoid has proved an attractive subject to many of the greatest mathematicans, Laplace especially, who has devoted a large portion of his Mecaniqiie Celeste to it. In English the principal existing works on the subject are Sir George Airy s Mathematical Tracts, where the subject is treated in the lucid style so characteristic of its author, but without the use of Laplace s coefficients, Archdeacon Pratt s Attrac tions and Figure of the Earth, and O Brien s Mathematical Tracts ; in the last two Laplace s coefficients are used. In the Cambridge Transactions, vol. viii., is a valuable essay by Professor Stokes, in which he proves, without making any assumption whatever as tc the ellipticity of internal strata, or as to the past or the present fluidity of the earth that if the external form of the son. imagined to percolate the land by canals be a spheroid with small ellipticity, then the law of gravity will be that found above. 1 An important theorem by Jacobi must not be overlooked. He proved that for a homogeneous fluid in rotation a spheroid is not the only form of equilibrium ; an ellipsoid rotating round its least axis may with certain proportions of the axes and a certain time of revolution be a form of equilibrium. 2 Local Attraction. In speaking of the figure of the earth, we mean the surface of the sea imagined to percolate the continents by canals. That this surface should turn out, after precise measurements, to be exactly an ellipsoid of revolution is a priori improbable. Although it may be highly probable that originally the earth was a fluid mass, yet in the cooling whereby the present crust has resulted, the actual solid surface has been left in form the most irregular. It is clear that these irregularities of the visible surface must be accompanied by irregularities in the mathematical figure of the earth, and when ve consider the general sur face of our globe, its irregular distribution of mountain masses, continents, with oceans and islands, we are prepared to admit that the earth may not be precisely any surface of revolution. Nevertheless, there must exist some spheroid which agrees very closely with the mathematical figure of the earth, and has the same axis of rotation. We must conceive this figure as exhibiting slight departures from the spheroid, the two surfaces cutting one another in various lines; thus a point of the surface is defined by its latitude, longitude, and its height above the spheroid of reference. Call this height for a moment n ; then of the actual magnitude of this quantity we can generally have no information, it only obtrudes itself on our notice by its variations. In the vicinity of mountains it may change sign in the space of a few miles; n being regarded as a function of the latitude and longitude, if its differential coefficient with respect to the former be zero at a certain point, the normals to the two surfaces then will lie in the prime vertical; if the differential coefficient of n with respect to the longitude be zero, the two normals will lie in the meridian ; if both coefficients are zero, the normals will coincide. The comparisons of terrestrial measurements with the corresponding astronomi cal observations have ever been accompanied with discrepancies. Suppose A and B to be two trigonometrical stations, and that at A there is a disturbing force drawing the vertical through an angle 8, then it is evident that the apparent zenith of A will be really that of some other place A , whose distance from A is r8, when r is the earth s radius ; and similarly if there be a disturbance at B of the amount 8 , the apparent zenith of B will be really that of some other place B , whose distance from B is r8 . Hence we have the discrepancy that, while the geodetical measure ments deal with the points A and B, the astronomical observations belong to the points A , B . Should 8, 8 be equal and parallel, the displacements AA , BB will be equal and parallel, and no discrepancy will appear. The non- recognition of this circumstance often led to much per plexity in the early history of geodesy. Suppose that, through the unknown variations of n, the probable error of an observed latitude (that is, the angle between the normal to the mathematical surface of the earth at the given point and that of the corresponding point on the spheroid of reference) be e, then if we compare two arcs of a degree 1 See also a paper by Professor Stokes, in the Cambridge and Dublin Mathematical Journal, vol. iv. 1849. 2 See a paper in the Proceedings of *J>* ftr.ual Society, No. 123
1870, by I. Todhunter, M.A., F.R.S.