the Distribution of Electncity upon Spherical Surfaces. 123 



increase, becomes a very laborious one, and the law of the terms 

 which destroy each other is not in anywise exhibited thereby. 



I have succeeded in obtaining a complete demonstration, 

 founded on Herschel's theorem for the development of a function 

 of e', and the expression thereby given for Bernoulli's numbers. 

 The theorem in question is, that for any function of e' which 

 admits of development in positive integer powers of t, 



where the right-hand side denotes the series the general term 

 whereof is 



172^/(1 + ^)0"' 



and /(I + A) is of course to be developed in powers of A, and 

 the different terms A, A^ A^ &c. applied to the symbol 0" 

 (viz. AO"=l"-0", A20" = 2"-2 . l" + 0% &c.). This gives 

 t _ log(l+A) ^,,„_ 



e'-l A 



And comparing the development of the right-hand side with the 

 development 



we find 



log(l + A) o_ 

 ^ u-i, 



log(l + A) 1 



A ~ 2' 



It is now easy to obtain the equation 



0.=^^^^(^{(i+o(i+i)y -(-0(1+6)7}. 



In fact, the first two terms of the development of the expression 

 on the right-hand side agree with those of the foregoing expres- 

 sion for ©;. For any even power 2x (except, when i is even, the 

 power 2x=i) the term is 



