15 8 MAJOE P. A. MAcMAHON: MEMOIR ON THE 



We now require the auxiliary theorem 



n _ <L _ 



cx l .1 -- x 2 . 1 -- x 3 . . . 1 -- x p 

 c c c 



1 X l . 1 X&2 . I XiX-s , . . . 1 



so that, eliminating a lt we reach 



-a-, 



and, eliminating 3 , 



2: (l-a 5 ci 1 ..Mw-- J x) 3 (l-a 1 ...a. M - s x)...(I-a 2n .- 3 x) (l-x)' 



Note that for n' = 3 or /6 = 2 this becomes the before obtained expression 



2x 2 -x s 

 Ij-xf 



To eliminate S) we have to substitute for 



(1 a 5 a 7 ... 



the expression 



.r' 



(Y-a b a.,...a, n .-sxf 

 and then put a- a = 1, thus getting 



2dj "t^9 '.../2' 5 ^2n' o ( *5^^7 '('Jn' 3'^' OCfj . t-1j 2ii'&*' "l CJ&y ,..^2' 3^ / *^ 



as the expression to be substituted for 

 in the numerator, which thus becomes 



This, for ?-' = 4 or w = 3, is 

 which may be written 



