366 Mr MURPHY'S THIRD MEMOIR ON THE 



T -u - ^±1 u + (^ + i)(^+^) r/ _ &c 



1 1.2 



which latter series is also produced by reverting to that which expresses 

 C7„ in terms of 71 in Art. 30. 



35. To find a function reciprocal to t" when the limits of t are 0, 

 and GO . 



Let M„ be the required function, and put t = e"', 



then ^lUj'^ = 0, from # = to ^ = x ; 



«» /V- 1 N™ ^ i>_ = to T = 1, 



therefore /"- (h. 1. t)-" = 0, from t = 



^» " 



h_ 

 1-* 



hence m„ = t C/„ = t ; — ^ttt when h = \ 



1 . ^...ndh'' 





36. Tb ^«c? « function F„ ^t>A^cA *Aa?/ ie reciprocal to cos" ^, ^A^ 



- awa -. 



2 2 



ZmeV* o/*^ Je^?^ — - and - 



Following similar steps to those adopted in the preceding Articles 

 we shall obtain, 



w + 2 

 in cosines F„ = cos n<p — . cos {n + 2)(f> 



(« + l)(w + 4) , ,,,^ (w+l)(w + 2)(« + 6) , ,-,-, . 

 + —^-12 ^•cos(« + 4)0 -^ ^ g g ^cos(w + 6)<^, &c. 



