184 PROCEEDINGS OF THE AMERICAN ACADEMY. 



because 



A- 



Xj-Ht){i)-m.^-H 



k — i 

 m — i 



k — i 



k\ '^t^ , fh— i 



=(-»'(■ 2 (-•) 



m 



therefore 



if ^' < Ti and 

 (- Vf if i = h% 



(53) 



p,, = 2* (^ - i)!*'5! (- 1)'" i V^- fo^ ^ ^' 



^ ^ 



A- 



m 



As a consequence of (50) we have 

 (54) G(T)'P,,= Gi- 2 



if G (T^ is any polynomial in T. The ease with which the result of 

 applying any polynomial in T to the P's can be obtained by means of this 

 formula makes the introduction of the P's in place of the p's very advan- 

 tageous. We shall henceforth assume that all functions of the p's have 

 been expressed as functions of the P's by (53), so that the degree of any 

 polynomial in the P's with suffixes as great as 3 is the same as that of the 

 corresponding polynomial in the p's. Equations (50) show also that the 

 operator T, defined by (30) in terms of the p's, can be expressed in terms 

 of the P's thus : 



(05) r=-i^^^-nA. 



We assign to each P a weight equal to its suffix (leaving out of account 

 Pi and Pa, which are and, therefore, nowhere occur) and define the 

 weight of any product of P's to be the sum of the weights of its factors ; 

 any constant may be regarded as a multiple of Pq = 1, and, therefore, as 

 of the weight 0. The weight of any such product as Ps'^s P^n* p^n^ ... is 



(56) - 3 «3 + 4 «4 + 5 ^5 + . . . = «<>, 

 say, so that, by (55) and the last formula of (31), 



(57) T • Pa"' P4"* P5"' . . . = - 'I • P3'" P:'* P,"*' . . . 



