178 PROCEEDINGS OP THE AMERICAN ACADEMY 





therefore. 



ds dpz dpi dpi 



^l^^+X-l;/^''' 



3 



'^k dpt j ju 





Equating the two expressions for 8/ thus found, omitting the common 

 factor — > we obtain 



a diiferential equation for/" which may be written 



where 



(30) T=hy.[k(k-1) p,_, - kp,] ^ 



^k dpk 



is a differentia] operator that acts upon i only through the ps with suffixes 

 as great as 3 ; the first {)art of that term of T for which ^' = 3 vanishes 

 because pi = 0. We have omitted the upper limit to the sum in y be- 

 cause that depends upon the number of p's involved in f, a number as 

 yet unknown. 



VII. The Operator T. The P's and ^'s. 



The operator T is so important for the theor}- that it deserves par- 

 ticular study. By a co7istant we shall henceforth mean a quantity that 

 is independent of the p's with suffixes as great as 3 as well as of s. If 

 u and V are any two operands and C is any constant, we have 



