126 Mr MURPHY'S SECOND MEMOIR ON THE 



l_i 

 follows that if we divide this function by T"^ , and then substitute f 



for T, we shall obtain the value of {p, §-)„ ; we have thus, 



_ ip + \){p + l+q){p^-\ + 2q)....\p + 1+{n-l).q] n 



ia+g){l + 2q)....{l + (n-l).g\ l' 



, (2p + l)(2p + l+q)....{2p-i-l + (n-l).q} n.( ?i-l) 



l.{l+q)....{l+{n-l).q} ' 1.2 ^ .'^^• 



13. The functions {p, q\ and (5-, jo)„ may be termed reciprocal func- 

 tions, and possess the remarkable property, that if n and «' are any 

 different integers, then shall 



ft(p,q)n.{q,p)n' = 0. 



For if n>n' then {q, p)„' is a rational and entire function of t^ of 

 less than n dimensions, and therefore by the preceding Article the 

 integral of the product must vanish ; again if n' > n, then {p, q)„ is a 

 function of f^ of less than n' dimensions, and therefore when multiplied 

 by (q,p)„' the integral ought to vanish. 



To determine the value of the same integral when 71 = n', it is 

 evident by the nature of the function {p, q)„ that we need only attend 

 to the last term in the expansion of {q,p)n, namely 



. {nq^l){nq + l+p)....{nq^-l + {n-\).p} 



^ "->•'' 1.0.+p)....{l^{n-l).p} 



