172 MAJOR P. A. MAcMAHON: MEMOIR ON THE 



To establish the expression it is merely necessary to verify that the coefficient of 



, vz. : 



_ 

 c\blcl (a+l)!(6-l)!c! a! (6+1)1 (c-1)! 



c)\ (a + 6 + c)! __ (a + 6 + c)! 



(a+2)! (6-1)! (c-1)! (o+l)! (6+1)! (c-2)! (a+2)! 6! (c-2)! 



reduces to the value of (6c;) above given. 



This is easy because it appears at once that a 6 + 1, 6 c+1, and a c + 2 are 

 factors. 



It is easy now to conjecture the form of the general result. 



The truth of the generating function for (abc;) is, perhaps, best seen by writing 



6+1/V a + 2/' 



where ( ) stands for ' J- . 



\ a, 6 / a ! 6 ! c ! 



24. I will now establish the result 



x fi L-i_ > i 



_L_l--- l- 



" 



f 1 '-fn 1 \ / i ^n 



X 1 - 



by showing that this expression satisfies the difference equation 

 (a,,a 2 , a s , ...;) = (i-l, a 2 , 03. ..;) + (a 1; a a -l, a s , ...;) + ... + ( 

 Writing the co-factors of the multinomial coefficients in 



(a,, a 2 , a 3 , ... a n ;) and (a 1( a 2 , ... a,-l, ... a n ;) 

 as 



C and C, respectively, 





