56 Proceedings of the Royal Irish Academy. 



Of course, the weights of the original coefficients p^, . . . are 

 the same as their subscripts, which are the same as the exponent of 

 the attached power of minus n. Thus, for inversion by this method 

 the operation 



should be written 



The result will agree with that of § 15-4. iN'egative weights are dealt 

 with in the same way as the positive weights ; and the ordinaiy alge- 

 braic expansion of a multinomial can be written out by means of a 

 similar rule. 



The name ''weights" is appropriate for the following reason: — 



Let V = and consider the combination Qp_\P_2P-z- The attached 

 power of V must be ; so that the whole term may be written 



5 [3 V v2 v3 



Similarly, every term in the series for consists of the same 



ratios multiplied by v and coefficients. Thus, the coefficient p,„ is 

 always associated with yS'" in the original operation, and with v'" in 

 its invert. 



iSow, let be a symbolic distributive operator which denotes that 

 the coefficients indicated by 



1 / 1 1 



1 + ^^ \ 71 J 



are to be attached to the various combinations of pi, p^ . - . ] then 



= i - f 7 ^^^^ 5 • • > • • f ff • ■ J- •; ' 



where v = ^/jS. 



