654 



Proceedings of the Royal Iriah Academy. 



Changing from h' to h in the terms involving % only, by the relation 

 {B), and for convenience denoting the determinant 



x-2, a; - 3, 



y -m-\-2 

 x-m-\-2 



by (234 . . .m- 2)', 



p-2, p-B, . . . p-m + 2 

 the right-hand side of the above equation becomes 



[a^hc ... 0(234 . . . m -2)' - &c.} h,.^ 

 - [a""(i<! . . . 0(234 . . . m-2)'- &c. + «" (5c . . . /)(134 . . . m-2)'-8c(i.]h,^^ 

 + {«"«(5c...0(134...i«-2)'-«S:c. + fl"(5f .. J)(124...m-2)'-&c.jU,_2 



+ (- l)'"-='{fl"+^ {he... 0(123 . . . »e - 4, m - 2)'- &c. 



+ ar{'bc . . . 0(123 . . . m- 3)'- &c.}^,_„,2 

 + (- l)"'-2{a"+i (Jc . . . 0(123 . . . OT-3)'- &c.] A,_„+i, 



where we observe, in respect of the coefficients of the various A's, that 

 the first and last consist each of one series of terms, while of the 

 others, each consists of two series of terms, and where the law of 

 their formation is obvious on inspection. 



But by the case oi m - \ letters and m - 3 general indices, we 

 have 



I', 



&c. 



{hcd... l){2M...tn-2)', 



I, 



and by this, and its Extensions, the foregoing expression is equal to 



«", 







a''+^ 





«-s 







ff"+i. 



«^ 







«^ 





aP^\ 







a^+\ 



«s 







a'. 





a\ 







«3+i, 



: &c. 



K.. 



-2 



: &c. 



K-2 + % 



: &c. 



K-z 



-2 



«•■, &c. 



a, 







a, 





a" 

 a. 







0, 



1, 







1, 





1, 







1, 



A,_i 



