512 Mr. I. Todhunter on the Attraction of Spheroids. [Juue 20, 

 Hence we have to show that 



Z(V + ny (2n + 1) f*V F n dJ - 22(» 2 + n) {2a + 1 ) f*' £' 8 P' M ^' = 0. 

 Jo Jo 



The second term we see vanishes by the process of Art. 15. As to the 

 first term, we must apply that process twice ; and we shall then transform 

 this term into 



S(2n+1) f *P' M V(Vf •)<*«'; 

 Jo 



and, as before, this is equal to 47rV(V£ 6 ), which vanishes, because every 

 term will have £ 2 as a factor. 



17. In Art. 11 it will be found that the coefficient of z n is 



-^{(n 2 + ny-2(n 2 + n)}(2n+\), 



and hence this term may be treated as the term was in the preceding 

 Article. 



18. Generally the coefficient of z r in U will be found to be 



(n + 2)(n+l). . .(n-r + 4) , , y (n-l)n . . .(» + r-3 ) 



(r K } t ; 



and hence, in order to carry on the process like that in Art. 1G, we must 

 show that this coefficient will take the form 



where N is some rational integral function of w 2 + w. 

 This may be etsablished inductively. 



Assume that the required theorem holds for a certain value of r, and 

 also for the value r+ 1 ; then it will hold for the value r + 2. 

 For let it be assumed that 



(n + 2)(n+ 1) . . . (n-r+4) + (- l) p (»-l)n . . . (w+y~3)=(2» + l)N ll 



and also that 



(n + 2)(n + l) . . . (n-r + 3)-(-iyO-l)tt . . . (n + r- 2)=(2»+'l)N* 



where N x and N 2 are rational integral functions of « 2 +rc ; then we require 

 to show that 



(» + 2)(n+l). . . (n-r+?) + (~l) r (n-l)» . . .(w + r— 1) 



will take a similar form. 



We may denote our two assumed results thus : — 



P+(-.l)'Q=(2«+ 1)N X , 



P(n - r + 3) - ( ~ 1 )'Q( w +f*-2)«= (2n + 1 )N 2 ; 



