826 Mr MURPHY'S THIRD MEMOIR ON THE 



and passing to the variable cp, since 1 — 2^=cos^; therefore ^ = sin — 



id 



and /' = cos-^, whence #'^ +\/ — 1 #* = cos^ + \/ — 1 sin^ by substi- 

 2 2 ~" 2 -^ 



tuting which we obtain 



„ 1.3.5....(2w-l) , 



^'-^ 2.4.6....2/. •^"^^'^' 



1.3.5....(2«4-J) . , ,, ^ 



'?''=2:476::::(2¥T2)-''"^''^'^-'^' 



which values are the same with those in Art. 8. 



The numerical coefficients in these formuhe may be rejected as having 

 no importance in self-reciprocal functions ; it is also observable that q„ 

 contains a different multiple arc from that in Q„, the reason of which 

 is that Q„, <7„ are to be self-reciprocal functions for all entire values 

 of n from to + oo, and then f,j,q„q,n = except when 7i = »i, this ex- 

 ception (on which the main value of reciprocal functions depends) would 

 not hold universally true if q„ were of the form sin(«0), for then 5-0 = 0, 

 and therefore f^qo.qo=f> contrary to the principle of the exception, 

 but in the form above found this irregularity does not occur. 



10. From the results found in Art. 9, it follows that if we put 

 the real functions Q„, q„ possess a common property, viz. 



except when m = n, which exception does not apply to the last integral 

 when m = ?i = 0. 



From the same results the following identities are obtained : 



, ff'^*!T\. 1. ■ («')^ = cos {n cos-' (1-20} 

 1.3.5....(2tt-l)</^" ^ ' ' \ Ji 



(» + l)2''+'rf"(«T^ • J/ , -,. w, o*\\ 



i-3-^i-^^^-^,, =sm U« + 1) cos ' (1 -20}. 



