52 Proceedings of the Royal Irish Academy, 



Comparing this with (2), we have 



' 2\ n]' ' 3V ^A' ' nj, ' r^l\ n j,' 



that is, the coefficients of /3 in the expansion of are ordinary 



multinomial coefficients divided by an integer, and are therefore 

 already well known. 



The coefficients of /? in the expansion of ['/'n]'^ can be obtained in 

 the same manner, so that we have 



The coefficient ~ written thus for symmetry, and equals 



_ 1 



n 



The following points may be observed : — 



(1) In the coefficients, the quantity within the brackets is the 

 same as the exponent of ^ of the same term, with the sign changed, 



(2) Each term is divided by the numerator of the exponent of /? 

 with its proper sign. 



(3) The numbering of the coefficients as given by the subscripts 

 1, 2, 3, ... is always the same, no matter what the powers of may 

 be. This follows from the fact that the multinomial coefficients are 

 independent of the powers of the variable or base. 



(4) Each power of ;8 is a power of "J/?, so that the expression 

 has n values, and is also an operation performed on ^^S. Let x denote 

 this operation, so that [<^„]"^ = [x] 'lif^- Then 



that is, is the invert of the -^'^ power of </> — which justifies the method 



of solution mentioned in § 15-8. 



17. The General Expression in Detail. — We know that (1) every 

 multinomial coefficient is the sum of various combinations of the 



