Reversion of Power Series. 371 



In order to obtain a complete check, the numerical co- 

 efficients in the above series have been computed twice with 

 Professor McMahon's formula and once with formula (8). 

 Use has also been made of the partial check obtained by 

 putting 



for in this case 



as is otherwise made evident by writing the original and the 

 .reverted equations in the respective forms, 



y=(x)(l-xY l and x=y(l+y)-\ 



"This result suffices to establish a theorem in regard to the 

 coefficients of terms of all orders and of a given weight, viz.: 

 the sum of the numerical coefficients of the terms of even 

 order is greater or less by unity than the sum of the nume- 

 rical coefficients of the terms of odd order according as the 

 weight is even or odd. 



There are a number of special series reducible to the form 

 (1) and therefore capable of reversion in the usual manner. 

 Such are for example equations containing an absolute 

 term c. It is then sufficient to replace y in (1) by ?/ l =y—c. 

 If the power series contains even powers only, it may be 

 written 



y=a X? + a l X*+a 1 X* + ..., 



and this series is reduced to the form (1) by the substitution 

 ,c = X 2 . If the coefficients of the first successive powers of 

 x vanish, 



y=c + a m -ix m + a m a m+1 + ... , 

 and the required transformation is 



There are m reversions in this case corresponding to the 

 m roots of y^. 



Other series containing zero coefficients are reversed by 

 substitution in the complete expansion. Thus, if the series 



2B2 



