368 Mr MURPHY'S THIRD MEMOIR ON THE 



However, if n be even, and our limits be — ^ and ^, the function 



becomes suddenly infinite at the limits, for the expansion of F„ is 

 then identical with that of (1 -!)-'"+'•. 



38. To express infinite terms the transient function Fn. 

 Put 



i?'„' = cos ra(^ - ^ . ^ cos (« + 2) + ^^i|^-^^ . A'' cos (« + 4) «/) - &c. 



F:'= hcos (n+2)(f> - ^ . A^ cos {n+4>).<l> + <"+!) (^ + ^) ^^ cos (w + 6)0-&c. 



1 1 • ^1 



Then F„ is the limit of F^—F" when h approaches unity. 



Put also 2 cos = a; + - , 



^ X 



hence 2-F„' 



x~^ X 



~ {l+A(a:^+a;-^) + *'}"+' 



W + 1 (W + X^ Tt W + 1 



COSW0+— j— .Acos(w-2)0+^— — ^^.A^cos(«-4)0...— — -.A"cosw0+A"+^cos(«+2)0 



~ fiTaFcosa^TFp^^ * 



the number of terms in the numerator being w + 2. 



In like manner, 

 2F„" X 



+ 



h (aT-' + Aa;)"+i (x + Aa;-')"*' 



_ X (a; + Aa;-')"+' + a;-' (a;"' + Aa;)"+' 



