relation to the Figure of the Earth. 333 



terials rearranged, particle from particle. Let p be the density 

 of any point r, 6, co (cos 6 =/ul) of the body; c, 6' } co' the coordi- 

 nates of any external attracted point ; p the cosine of the angle 

 between r and c; r and a the general and mean radii of the sur- 

 face; and let r = a.w, u will be greater than 1 for points out- 

 side the sphere of which the radius is a, and less than 1 for 

 points inside the sphere. 



Our problem is to find the form and law of density of this 

 imaginary body — that is, to find u and p. The potential of the 

 body for the external point 



f 1 C 2 "C r pr* dp, dcodr 

 J-Jo Jo Vc* + r*-2crp 



As the total mass of the imaginary body equals zero, the first 

 term of this integral vanishes of itself. That the external attrac- 

 tion may be zero, the remainder of the integral must be zero, 

 and this for all values of c. Hence 



dcodr. 



J-iJo Jo 



pr i + 2 'P i dfi dcodr = 



for all positive and integral values of i. 



As p is a function of r, and must be independent of all parti- 

 cular standards of measure of r, it must be the same function of 

 r-j-r. Let r-^Y = v. The above condition then becomes 



r rr 



J-iJo Jo 



pv H2 dfi dec dv = 0. 







By successive integration, 



J 



v i+3 v i+ * do 



OV l+z dV = - 7,P~T-- 7TVT- 7V~T + 



Or, if p w , p(\\ pW, . . . represent the values of p and its differen- 

 tial coefficients when v=l, 



pW p 0) p(js) 



J_^ ; + 3 (i + 3)(» + 4) + (i + 3)(i + 4)(i + 5) 



P\ r2n fir 



•1 I pr i+2 ~Pi dfi dcodr 



J—iJo Jo 



J- J, i+sV i+4 + -'-/- 



