58 



Proceedings of the Royal Irish Academy, 



Assume it is of the form 



IS^ + t,^' + t,p'^ . . ., 

 and operate on this with 



in succession, and add the partial results. In order that the whole 

 result shall equal ^, the coefficients of every power of /3 except must 

 Tanish. This condition gives us a series of equations, 



n ^.2) 



7iti + a = 0, — t-c + nti + (^e - 1) fl^i + i = 0, etc., 

 L_ 



to solve : from which we obtain the values of ^j, ti, ... already 

 known. But they have now been obtained without assuming the 

 multinomial theorem for fractional indices. 



21. Verifications. — As mentioned in § 15, objections to expansions 

 on the ground of divergency of series are not generally applicable to 

 verb-functions, because these have no quantitative value. For them 

 the expansions are identities, provided only that the whole expansion 

 be considered.^ It is therefore immaterial whether integral or 

 fractional indices be employed. On the other hand, an expansion 

 obtained by an ascending process cannot generally be equated with 

 one obtained by a descending process, as one may possess more 

 potencies than the other. 



The results given above may be shown by many methods to be 

 identities — notably by resolving 



[«-i</,,„ [<^,J[«-^ and [[«-]-\ 



when the issue will be found to be /8 = fi. This is due to the many 

 properties of the multinomial coefficients. 



In order to facilitate the work, it will be useful to give an ex- 

 pression for the r^^ algebraic power of [<^,J'^ and [j/^,J~^ This can be 

 obtained directly by the multinomial theorem ; or by operative 

 division into yS*" instead of into 



since £ = 



9n 



^ Compare Russell: Priaciples of Mathematics, Vol. I., chap. xxiv. 



