"94 Proceedings of the Royal Irish Academy. 



of terms of the type - — ^-— -, c^ being a constant, and r and s integers ; 



(2 — ny 



if, however, the degree of xp {%) exceeds that of <^(z), then terms of tlie 

 latter type will alone appear. 



3. We now differentiate the expression 2z'"J/(z), and find 



or, substituting for /(s) and/'(s) their values, and integrating between 

 the limits Zj and Za 



2 / 2'- J7(^) 2m/2^,,_i + (2»» - 1) Pl/2«+._2 . . . + Pam-l /r 



= (2m + 2r) /j^+^i + (2wi + 2r - 1 ) Pi/2m+r-2 + • • • + 2rP.,J,_^. 



By assigning to r successively the values 0, 1, 2, ... r, in the 

 above formula, we learn that all integrals such as I^ where r is an 

 integer and greater than 2m - 2, depend on 2m - 1 integrals, namely 

 Iq, Ii, . . . Izm-i, which we shall refer to as the 2m - 1 I^. 



4. We now put . - 



differentiating with regard to n, we find 



(1) ^ ^ /(^) _ _/W_ _ -A^) . 



dn (s - ny (% -n)'^ z-n^ 

 and with regard to z, these results 



(2) dx ^ f{n) /(z) ^ /(z) 

 (?z (z^- «)^ (z - ?^)^ z - » 



Again, we have the identity — 



ym n^y^- m^^ 



d. 



2(z-n)v//(z) (z-w)V/(^)' 



