60 Transactions of the Royal Canadian Institute 



L 2! 3! J 



L 2! 3! J 



and its verification is effected immediately by the use of (15). 



We shall now state the general theorem indicated by the preceding 

 analysis. 



Theorem. — Reverting to the general expression (12), the numerator of 

 the reduced fraction 



v'^iv+n^-l) (i; + w + 2) .... (v + 2w-l) 

 -w(v+2)'"+^ {v+n + 2) (i' + w+3) .... (v+2w-l) 



+ "^^^^.+ 1) (>^+4)'"+^ (v+w+3) .... {v + 2n-\) 



+ (-l)"(v+l) (i'+2)....(v+w-l) {v + 2nY, 

 tho^igh apparently of the order m+n — l, is in reality of the order m — l. 



It vanishes for v=— 1, — 2, , — (2« — 1) provided m is even; if 



the order m — l of the polynomial is less than 2n — l, the polynomial vanishes 

 identically. Accordingly the expression (12) is zero when m takes any one 

 of the n values 2, 4, . . . ., 2w — 2. The subsystems (i.), (ii.), (iii). of 

 (9) correspond to the cases n = 2, 3, 4. 



This theorem furnishes the answer to the second of the two questions 

 that we have had in view. The sub-systems of (9) are complete. 



It appears to be difficult to provide an absolutely direct algebraic 

 proof of the reduction of the numerator's order from m-\-n — \ to m — l, 

 but we can convince ourselves of the truth of the theorem by the use of 

 the method employed by Nielsen (I.e. p. 301). 



Nielsen defines X^ to mean 



\2r-\ 



x:. = n ^^^ + 



v.v-\-l . . . v-\-n v-\-l.v-{-2. . .v-\-n-{-l 



n{n-l) (v+4) 



>r-l 



2! V + 2.VVZ.. .v^n+2 



- (16) 



where r ranges from 1 to n. 



