AND THE FIGURE OP THE EARTH. 85 



8. Returning to the theory of attractions, we have, when the particle is internal, 



V ^.Cfl^f^^ f ^^ (Po + P,7 + . . . + P„ ^ + . .) dj^di^d<^ 



for that portion of the body which is comprised between a sphere of radius r, and the 

 surface of the body, supposed to be a surface of revolution round the axis of % ; for, 

 in that case R, the value of r' at the surface, is independent of ^'. 



9. Suppose, for example, we wish to find the attraction of a homogeneous spheroid, 

 on a point within it. In this case f is constant, and 



a being the semi-major, and c the serai-minor axis. 



First, all the even terms vanish ; for the general even term is 



+ 2^7-1 P2„4-l^^' 



Now Pg^^i consists of odd powers of |U,'; and (^2^* + ^*)-^ can be expanded 

 in even powers of ^' ; therefore the integral of the product (which is an odd function), 

 taken from ^' = - 1 to ^' = 1 , is 0. Alsoyi ^P^^_^^d(i.' = 0*. All the odd terms 

 above the third vanish ; for the (2n -\- l)th term is 



+ ^r^J- 1 ^2n^^^f^'y for it may be shown that^^ ^J'^^'' P, P.' dfji.'d(p' = 0, if / and / 



J" + ^ ) is a rational 



and entire function of y] ^ and, therefore, capable of being expressed in a series of 

 Laplace's coefficients-}-, the highest of which will be of the (2 w — 2)th order ; and 

 therefore no term of this expansion can be of the same order as Pg ^ ; and the inte- 

 gral of the product of any two of different orders, between these limits, vanishes. So 

 the second member of this vanishes. 



* See Pratt. Mec. Phil., § 180. 



t See Pratt. Mec. Phil., § 176. Poisson, 'Ilidorie Math, de la Chal,, chap. viii. Laplace, M6c. C61. liv. iii. 

 chap. ii. 



