108 Prof. W. E. Story on Partial 



of the same product of powers of the *'s in co be denoted 



and 



where 2^ denotes the sum of the terms following it 

 lor all possible combinations of all possible numbers 

 j/i» </2j ^3» • • • j #„_i of which each is a positive integer or 0. 

 For the sake of brevity, we shall write a ( '°.. instead 

 a a a a a ,» an( l c instead of c 



Any coefficient with a 7 written among the suffixed dots 

 with -f 1 or — 1 attached shall denote the result of increasing 

 or diminishing that particular suffix by 1, without altering 

 the other suffixes. Thus, a {r) a -. denotes the coefficient of 



w r that is derived from a {r) by writing- </,— 1 



instead of y k for the &-th .suffix, and c..g .+1.. denotes the 

 coefficient of o> that is derived from Cg v g v g 9t ...,ff K -i u y 

 writing # fc + 1 instead of g k for the k-ih suffix. 



Equation (35) is equivalent to the set of k — 1 homo- 

 geneous linear partial differential equations 



SQ'i — "«) , 5 (" 2— "*) , , BK - i- u K ) __ d<u 1 



* ^Zr " + ~ 2 J* +»-+*«-l ^ ~a*r[(43) 



(r = l,2,3,...,/c-l). j 



Substituting the expressions (41) and (42) for the u's and 

 &> and equating the coefficients of like products of powers of 

 the z's in the two members of each of the equations (43), we 

 have, for each set of values of* cr l9 g 2 , #3, . . • 5# K _i» 



^J,{a.. H _ 1 ..-a.. g _,..) = OrC-.y (44) 

 (»'=1, 2, 3,...,«-l), j 



where 2 denotes the sum of the terms following it for 

 all values of k from 1 to k— 1. Namely, the term actually 



