AT THE SURFACE OF THE EARTH. 677 



" / . ^ * 1 ^9 is)' (l~ 



- (1 - w, -u,- u, ...) +_+%,.. + sin = = r (9) 



" a a? 2 ^ ' 



The most general Laplace's coefficient of the order is a constant ; and we have 



sin-0 = | + (l -cos'0), 

 of which expression the two parts are Laplace's coefficients of the orders 0, 2, respectively. We 

 thus get from (9), by equating to zero Laplace's coefficients of the same order, 



Fo = "c - ;5 tti'a^ 

 Y, = a Y„u,, 



I'a = a- r„u, - 1 tt,'-a5 (1 _ g^ga q^^ 

 ¥-, = a^y,it„ &c. 



The first of these equations merely gives a relation between the arbitrary constants J';, and c ; 

 the others determine V„ V,,, &c. ; and we get by substituting in (7) 



^=^o[- + ,-5 " + 73 ". + •••) -'oT^'^i- '=°^'^) (10) 



6. Let g be the force of gravity at any point of the surface of the earth, dn an element of 

 the normal drawn outwards at that point ; then g = --—(V + U). Let \|/ be the angle between 

 the normal and the radius vector ; then g cos x|/ is the resolved part of gravity along the radius 

 vector, and this resolved part is equal to - — (F + U). Now \// is a small quantity of the first 

 order, and therefore we may put cos v|/ = 1, which gives 



g= -j- (F+ U), 

 dr 



where, after differentiation, r is to be replaced by the radius vector of the surface, which is given by 

 (8). We thus get 



V V 



^ = -° (I - 2?^, - 271, - 2u, ...) + ~ (2?«, + 3tt, + iu, ...) - ^ai'a U - cos=0) - w'a (|- + 1 - cosW, 



which gives, on putting 



-feo=a = G, -^ = m, (l1) 



and neglecting squares of small quantities, 



g = G {l - ^m {^ -cos'^e) + u., + 2Us+ 3u,...\ (12) 



In this equation G is the mean value of g- taken throughout the whole surface, since we know 



that / / ii„ sin 6 dOd(p = 0, if n be diU'ercnt from zero. The second of equations (II) shews 



that m is tiie ratio of the centrifugal force at a distance from the axis equal to the mean distance to 

 mean gravity, or, which is the same, since the squares of small ipiantitics are neglected, the ratio 

 of the centrifugal force to gravity at the equator. Equation (12) makes known the variation of 

 gravity when the form of the surface is given, the surface being supjiosed to be one of ecjuilibriuni ; 

 and, conversely, equation (S) gives the form of the surface if the variation of gravity be known. 

 It may be observed that on the latter supposition there is nothing to determine ?/,. The most 

 general form of u, is 



a sin cos rf) + /3 sin sin (p + y tos 6, 

 Vol.. Vin. Part V. 4S 



