1886.] e\ log e (1 + x), (1 + x) m . 419 



The % in each case denotes summation for all possible solutions 

 of the corresponding equations in a,, a 2 ...a r , in which each of 

 these quantities have any positive integral values between and r 

 inclusive. 



Remembering that r = la 1 + 2a 2 + . . . + ra r , 



multiplying both sides of the equation for c r by 



r _!«!+ 2« 2 + ... 4- ro. 



the summation extending to terms corresponding to all possible 

 solutions of the equations in a v a 2 ... a r , where these quantities have 

 any positive integral values between and r inclusive. 



Suppose that in one set of values of the quantities a lt « 2 ... a r , 

 which satisfy the equations, the value of a t is not zero (the value 

 of any or of all the remaining quantities cq ... a t _ li a t+J ... a r may be 

 zero), then the corresponding term is 



a t \..,a t _ x \ a t \ a t+1 \ ...a r \ I* .'.'. (t-l) at -H at {t+ l) a ** a ...-r* 

 This breaks up into the sum of terms, of which 



ti t 

 cq! ... a t _ x \ a t \ a t+1 \ ... a r \ l a i... [t — l) a < H at [t+ I)*'* 1 ... r ar - 



is a type ; and this may be expressed thus 



1 



a,! ... a^J a t - 1 ! a m ! ... a r ! l a > ... (i - l) a '-i^- 1 (i + l) a <+* ... r ar ' 



Now putting 



a t = lt . .. a,_j = /^ a ( - 1 = ft, a, +1 = ft +1> ... or, = ft 



29—2 



