AND THE FIGURE OP THE EARTH. 87 



11. When the particle attracted is external, the series in (3.) does not give a finite 

 expression. Instead of taking it separately, I make it a case of a general theorem 

 which follows. 



12. To find the attraction of a spheroid on a point within it, the density being any 

 function of the distance from the centre, and the eccentricity being small. 



Let f <p r' represent the law of density ; then the value of V, for the portion com- 

 prised between the surface and a sphere of radius r, is 



2'?X''X'i^'' •'' (Po + Pi^ + ••• + P.P + ••) -^ '''''''• 

 Integrating by parts we have 



Therefore the {n-\- l)th term of V is 

 Now R = I ^ + (^, - ^) f.'2 j " * = a (1 + e2 1»'2)" * = a (l - ^ fi'2), rejecting ^ 



a e* /Lt" 



and higher powers of e ; and ^/^x R = p^^-^ a — <f>^m-\) ^ • — 2 — *^ ^^^ same degree of 

 accuracy. The last member of the expression for V need only be calculated when 

 n = ; for all the rest of the terms (involvingy_ ^ P^ c?^' where n ^ Oj vanish. 



The first member need only be calculated when n = 0, and when w = 2 ; for when 

 n is odd, it vanishes as before ; and also when n is even and greater than 2 : for the 

 functions of R involve no higher powers of (JtJ than the square ; and consequently 

 they vanish, when multiplied by P4, Pg, . . &c., and integrated with respect to fji,', from 

 — 1 to + 1*. 



When w = 0, the term is 



2 ^ §yli (R ?>/ R - <P, R) ^1^' - 2 T^y^j (rp^r- p^,r) d(^\ 



= 4T§[a(p,a - (p„a - ^ a^ <f> a - {r (l),r - (p„r)). 



* See Art. 9. 



