1872.] Mr. I. Todhunter on the Attraction of Spheroids. 



513 



and then we have to investigate the form of 



PO-r+ 3) (n-r + 2)+ (— l) r QO+r-2) 1). 



Now the two following identities may be verified :— 



(n-r + 3)(n-r+2)=n 2 + n-(r-3)(r-2)-2(n-r + 3)(r-2), 

 (n + r—2) (n + r—l) = n 2 + n—(r—3) (r-2) + 2(w + r— 2) (r— 2). 



Hence 



PO-r + 3) (w— r +2) + (-l) r Q(w + r-2) (n + r-l) 



== { w 2 4. w _( r _3) (r~2)}(2w+ 1)^-2^—2) (2w+l)N a 

 = (2rc+l)N, 



where N is a rational integral function of n 2 -\-n. Hence, as we have seen 

 by actual inspection that for integral values of r up to 7 inclusive the re- 

 quired form is obtained, it follows that this form will be obtained for all 

 positive integral values of r. 



19. We may collect our results into two propositions, one of elementary 

 algebra and one of the theory of Laplace's functions. 



Let f(z') stand for 



{( ,. +zT+ 3_ r » + a } + jl_ {(r+zr «V- r -»«}, 



and suppose r greater than z, so as to ensure convergent series when the 

 binomials are expanded in powers of z ; then the coefficient of every power 

 of *' is the product of 2n + 1 into some rational integral function of n 2 + n. 



Let '( be a Laplace's function of the usual variables // and xp', and £ the 

 same function of /x and \p ; and suppose r greater than £' ; then 



is a function of £ and its differential coefficients, which involves £ 2 as a factor, 

 and so vanishes when £ vanishes. 



20. I have not proposed to examine any difficulties which a reader may 

 find in Poisson's process, but only to show that it can be made to furnish 

 a general result instead of the result merely to the third order. Poisson's 

 memoir has been much used by Bowditch in his translation of the ' Meca- 

 nique Celeste,' with a commentary (see vol. ii. p. 185) ; but Bowditch con- 

 fines himself to the same order of approximation in the theorem as Poisson. 



May 3, 1872. 



VOL. XX. 



2p 



