342 Mr MURPHY'S THIRD MEMOIR ON THE 



Hence -^^ = - 2. (« + !)(» + 2)... (2»- 1), 



^^" = (m + 1)(« + 2)...(2m-1), &c. 

 and the value of j9„ is the rational function 



^ 1.2...{n-l) ^^ +^'^ +A,f ...+A„^,], 



in which the coefficients are successively formed from the equation 



{n-m-lf.A„ + {m + 2){2n-m-l).A„^i 



+ 2(-ir "i^-'^)-("^ + ^) n{n-l)...(n-m-l) _ 

 ' '1.2...{n-m-l)' 2n{2n-l)...{2n-m) 



and the omitted part in the integral of the proposed equation is 



6|p„h.l. (I) + (-l)».^. 



18. When m = —^, the general equation of Art. 16. becomes 



and putting ^ = cos"^(l — 2#), we have Q„ = cosn<p, §'_,„+,) = sin ncp, the 

 complete solution is therefore M = a cos«^ + 6 sin «^. 



Though the trigonometrical functions were the first used in analysis 

 as reciprocals, for the purposes of expressing functions by means of 

 definite integrals and of expanding them, in the former instance of 

 their application there remain a few remarkable cases which do not 

 seem to have been noticed, with which we shall conclude this Section. 



19. The two functions which possess the remarkable properties al- 

 luded to, are 



e = e^'<«» . COS {x sin &), and 6' = e^ ""^^ sin {x sin B). 



