510 Mr. I. Todhunter on the Attraction of Spheroids. [June 20, 



11. Put*'forw-r. Then 



_n+2+n-l , 2 ( n + 2)(n+l)-(n-l)n 



* "2 Z + : (3 Z 



_^ (n + 2) ( W +1) W -K»— !) *(*+ 1) , 4 



_ L (w + 2)(n+l)n(»-l)-(n-l)n(n+l)(n + 2) , 5 



+ ________ 



. . + . . . . 



_2n+ 1 „t 1 jnj_ (2* + l) (» 2 +» ) , 4 , 



2~* + + |T + 



12. Let £' be a discontinuous function of p and ^ such that £' is always 

 equal to z when z' is positive, and always zero when z is negative. Then, 

 for all values of m, we have 



This is a very important step in Poisson's process ; and he explains it with 

 adequate care. We may suppose that £' is expressed by means of a series 

 of Laplace's functions. 



"13. As we may also suppose £' 2 expanded in a series of Laplace's func- 

 tions, it will follow, from the well-known properties of such functions, that 



^{2n+\)V\"¥ n dJ=A^\ (5) 



where I is the value of £' when 0'==0 and \j/'=\p. But, by supposition, £ 

 is zero. Hence 



_(2«+i) prp'u^=o, 



and therefore 



x(2?i+i) jy 2 p;,<7a/=o. . 



In precisely the same manner we have 



2(2w + l) (V 3 P' w ™'=0. 

 •/ 



14. Thus far Poisson carries his process. His words are, on his page 

 368 : — " Pour simplifier la question, on neglige ici les puissances de £' supe- 

 rieures a la troisieme, ou autrement dit, on borne 1' approximation aux 

 quantites de l'ordre a 3 inclusivement." 



I am not certain whether Poisson himself had carried his investigation 

 beyond this point. In the later part of his memoir he certainly implies 



