1872.] Mr. I. Todhunter on the Attraction of Spheroids. 511 



that results which partly depend on the present investigations are true for 

 all powers of w— r. It seems, therefore, curious that he did not here ex- 

 plain how the terms which involve powers of z above the. third vanish, go 

 as to make (4) absolutely true. To this we now proceed. 

 15. In Art. 11 we see that the coefficient of s' 4 is 



Hence we have to show that 



2(2rc+i)(y+w) I r 4 F„^'=o. 



Jo 



Now, by the nature of Laplace's coefficients, we have 



(*+.>r^{(i-*>g }^5r • • (6) 



Hence, by two integrations by parts, we find that. 



2(2n+.l) (n 2 +n) f ^pF^'sr^ 1) f^F w VfW, 

 Jo Jo 



where, for abbreviation, V is used to denote the operation which, as per- 

 formed on P' n , is expressed on the right-hand side of (6). 



Then, in the same way as (5) is obtained, we have . . . 



2(2n+ 'PVvT 4 dJ WvrV^ 4 ; 



and v£ 4 is zero, for every term involves 'C as a factor. 

 Hence, finally, 



r 4 r n dio'=o. 



16. In Art. 11 it will be found that the coefficient of z'° is zero. 

 The coefficient of z' 6 is 



that is, 

 that is, 

 that is, 



(w + 2) (w + l)n(w— 1) {m-2 f n + 3} 



( w + 2 ) ( w -l)(n a + ti) (2»+ 1) 



(n a + w~2)(n a + «)(2» + l) 

 — g 



y + n) 9 -2( j^HQ (2||+1)> 



