146 Mr MURPHY'S SECOND MEMOIR ON THE 



24. The functions which have been all along designated by {p, q)„ and 

 {q, p)„, have been already shewn to be reciprocal one to the other; putting 

 p = q, the resulting function {p, p)„ must be reciprocal to itself; that is, 

 ft{p, p)„{p, p),„ = when m and n are unequal positive integers; when 

 p = l the function {p, j)),, is then identical with that denoted by P„ , which 

 has been before shewn to be reciprocal to itself; again, the function T„ or 



n n.(n~\) {hA.ty ^ n.{n-l) .{n-^) (h. 1. If 



is reciprocal to itself, for if we mviltiply by (h. 1. ty, and integrate, we get 



j;r„(h.l.0"' = 1.2.3...«.(-ir{l-f^-^!^ (^±i)^_&e.}. 



The expression between the brackets is the term independent of h in the 



product (1+/^)"(1 + t) , or the coefficient of //-('"+'> in (1+A)"-'"-'; 



it is therefore zero when n>m, but when n = m its value is ( — I)'", 

 and when n<m, its value is 



, _ (?w + l -n){m + 2-n)...m 

 ^~ ' ' 1 .2 ...n '■ 



Hence fi T„ 7'„ = 0, when m and n are unequal, and 



1.2. ..n 



25. Put h. 1. (^) = T, and substituting in T„, we have 



1.2...W J'„e'^ = e"|l.2...M + w.2.3...Wx+^^^^^\3.4...WT^ + &C.| 

 (dw c?"-'t" n.(n-l) d"-W „ 1 



_ d"{e'^r'') 

 „ _, 6-^d" (e-T") 



Hence 7; = -^—p^ t-^ . 



1.2. ..war" 



