Boss — Verb- Functions. 55 



(2) Increase every exponent of jS hj 7i + l, and then divide it by n. 



(3) Multiply each combination of p+^y ^+3, p±z . . . by 



1 (-0'-> 



ns ' 



wbere s is the new exponent of /5 of that term, and r is the order of 

 the combination referred to. 



18. Deduction hy " Weights^^ of the Original Coefficients. — There is 

 another rule for obtaining the series for [<^„]"\ by writing out the 

 value of (1 -pi-p% -pz . . .)~^ j that is, 



1 + +P2 +i?3 . . .) + {Pl +i^2 . . + (i^l + ^2 . • 0^ + • • • ■ 



and then attaching the proper elements to the various combinations of 

 Pii V^-) ' • • The same rule applies to the value of [i/'nl'S in which 

 the subscripts of p are negative. 



Suppose we have the combination p^o*"' Pxo"''" and require to attach 

 the proper elements to it. Let w\ and w" be called the weights of the 

 original coefficients ; and suppose that the weight of the whole combi- 

 nation, which we may write P,/, is the sum of the weights of the 

 factors — that is, 



10 = r'w' + r"w" . 



Similarly, the order of Fj is the sum of the exponents of the factors. 

 !N"ow, in examining the series for and given in the 



previous section, we shall see that the power of attached to is 



always /3 " , and that the corresponding coefficient due to operative 

 inversion is always 



1 / 1 + 10 1 



\ -v w\ n ) ]^ 



Hence, after writing out the value of 



(1 -pi-p., . .)-\ 



we have only to attach these elements to each combination of pi,p2 . . ., 

 which already possesses its proper binomial coefficient obtained from 

 the expansions of the successive integral powers of 



(^1 + + . . ■) ; 



and it will be found that the result will agree, after rearrangement of 

 the terms, with the series already given. 



