ig^ ANALYTICAL TORUVLM. 



1 2 A r 1 . 1 . 1 



1. Cotan. A = -~ - — I -^-,--, + ^—^ + ^r::^, 



-{- &c. >, where it zz. 180% and a — — * 

 For, by the common trigonometrical formula. 



Sin. A = A (I- ~) {1-|^) (1 ~^)X&c. .-.hyp.log. 



sitj. A = hyp. log. A -I- hyp. log. (1 — -* ) + hyp.log. (1 — ^) 



+ &c.: therefore taking the differentials, 



cos. A ^ , 1 Ik, dec 



sin. 



_ j_ 2 A 



~A ,r 



In the same way we «iay deduce the second theorein I propose to offer, 



- = cotan.A = -_-_x | -,_,+ -,_,+ &c. ^ 



where j3 =: rr 2 «, For, since cos. A rr (l — -y) (l — ;^ ) 



(I — gi) » &c. hyp. log. COS. A — hyp. log. (l -^ -^) + hyp.lpg. 



R'- fl» 



{\ — ~{) ■\- hyp. log. (1 — — ) +&C. .'.difFerenciugagainasbefore, 

 — sin.A 2/3rf^C 11 , 1 , c 



and therefore tan. A =: ^, J __1^, + -J^, + -^^ &c.J (2) 



V 2 ^ 



By the combination of the two formulae, for sin. A, and cos. A, we 

 may obtain an elegant analytical expression for tan. A. 



