Definite Integrals, with Physical Applications. 393 



sphere into m + 1 portions, containing alternately the positive and 

 negative electricities). 



32. It is obvious that in the n + 1 portions into which the 

 spherical surface is thus divided, there will be n— l lines in 

 which the accumulation of the respective electricities is a maxi- 

 mum] beside the points A, B; (represented by the dotted lines 

 in Fig. 5), they possess the remarkable property, that with the 

 transition lines, they divide the surface into belts containing 

 alternately exactly equal quantities of the two electricities. Thus 

 the electricities in the portions E X AF U EiFxbitti are equal and of 

 opposite kinds, the electricities in aJhFsEz, and a.,b,F,E, are 

 similarly equal and opposite and so on. 



For the whole quantity of electricity, from the point A where 

 t = 0, to any value of t as t, 



= f t 4,Tra 2 .A = 4>Trma*(2n + l)(-iyf t P„. 



Now by the theorem of Leibnitz 



d"(uv) d'v dv d'-'.v . 



-dir =H dF +n dt-dl^ +ikc 



i d-'(tt'r 



Lr.t- ^jW ^y .fflrV to. 



the integral being supposed to commence from / = 0. 



„. .. . dP„ 1 d" +1 (tt'Y « + l n _„_■ 



Similarly -^ = ^—.-^I = - _ .-j 



(» + l).w n.(n-l) „-s (n+ l).w.(w-l) n(n- !).(«- 2) .,»- 3 ,.. » 

 + 1.2 • 1 '* ' ~T^3 • O •* ' &C - 



and comparing both, we get f t P n = - ( n + \\ -~dJ ' 



