﻿Exponential and Logarithmic Theorems. 43 



x=~ vv — 1, we shall get e =l-\-v\/ — 1- j^ ^23 



v*\/- I 



_i__!! _ V - -&c.; .". by addition, subtraction, &c. we 



1.2.34 1.2.3.4.5 



get 



2 1.2 1.2.3.4 1.2.3.4.5.6 ^ v * ; ' 





e -e v 3 , v 



%y/~Z\ 12.3 1.2.3.4.5 



c e —e i e +e 



By adding the squares of — ■ ^_u — and r> j we 



J D 2^ - 1 2 



aet e ~ 2 + g + e +2 + g = 1, •*• since the 



B et — j -t 4 j 



sum of the squares of the sine and cosine of any angle =1, 

 (when the radius = unity,) if we put - — ■ ~ — . = sin.pv, 



we must have - ~ =cospv, (&), ovv -—-—-{- 



2 r ..'.* " 1.2.3 1.2.3.4.5 



v 2 v* 



&c.=sin.j)t?, l-I^l^I - ^^ 08 ^' ^' where £ isa 



sin.joy 

 constant which is to be found. By (k) we get c ^ 1 ;=tan.j3i; 



e 



2 "J~ l __i — 2t?y-i 



2ry-l 



, orl+tan.^v/-l=« [1-tanjnV - 1], 



</-!. le +1 



... ^~ l =}± x ^hP^Lzl, and taking the hyperbolic logarithm 



1 — tan._puv/ - 1 



we get 2t>vA— 1=L [ 1 + tan P r ~ j^nd thpsprondmfml^r^f 

 & \l-tan.jwv -1' 



this by (14), (using tan.pw'^T for a,) gives by a slight reduc- 



tan.'pv tan.7w 

 tion v=tan.pv- o + — r~~ -&c., (/), and dividing sin./w 



3^5 



v 3 2y 



by cos.jov as given by (A/), we get tan.pi>=t>+-g- + 3^ + &c > 



