144 Mr MURPHY'S SECOND MEMOIR ON THE 



But if we put for (q, p)„ the value above found, and observe that the 

 operations A and fi are with respect to different variables h and t, and 

 therefore their order is transmutable, we have also, 



= {-iy A"^ H,,p<p{qh), by hypothesis. 

 Comparing this value of the integral with that already found, we get 



'" ^ ' {pqT I'l" 2 ■ 2 • 3 • 3 ■■■ 



l + {m-l).p \ +{m-\) .q 

 "mm 



X A" JZ", p {qh), when h = 0, 

 from whence we have finally 



At) = {p, q). (qh) - ip, q), . ^+f/^ • T ' T " ^ ^V. <l> W 

 , , 1 + 2(0 + 0) 1 1 1+p 1+q .,„„ , , 



_(« «N l+^Ci>+g) 1 1 l+£ l+i 1±2£ 1+22 A3 W^'" ri.r«M 

 ^^'^'°- (pqf ri- 2 •^^- 3 •—^■^■^'>f't>'^W 



+ &c &c. 



h being put =0, after the operation, and H', H", H', &c. being the 

 values of Hp,, when m = \, 2, 3, &;c. successively. 



Cor. 1. Multiply by t\ and then integrate from ^ = to ^=1; for 

 Itf{t).t' put its value <p{x), and for ft{p,q)mt'' its value 



/ ,v„„,„ 1-2. 3. .-^^ xix-q)...{x-{m-l).q\ 



^ 'P ■ 1 (1 + g)(l + 2y)...{l -!-(»«- 1| .^)*(a;+l)(a;+jt) + l)...(a; + »w^+l)' 



