Exponential and Logarithmic Theorems. 43 



a:=~2?v— 1, we shall get e =l-{-v\/ — 1- j-^— j — 



^-^ 1.2.3 



+1:2:3^+1x3^5 -^'•' ' =^-""^-^-T:2 + n-2-x- 



+ 1 ''l'\^ ~ ^ 9*^4^ " ^^' ' •"• ^y addition, subtraction, &.c. we 



^ 2 1.2 1.2.3.4 1.2.3.4.5.6 ' ^^^' 

 -—V— + — &c., {h\ 



2v'':ri 1.2.3 1.2.3.4.5 ' ^ ' 



By adding the squares of H^ and f ~ , we 



2v/31 2 ' 



. e — 24-e , e +2 + e 1 . , 



get 1- + Jl-Jl = 1, .-. since the 



— 4 4 



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 ~ =cos.«2;, fA:), orv — — ^ 4- - 



2 r ,\ J, 1.2.3 1.2.3.4.5 



~-&.c.= sin.pv, 1 — ^"2+ ]^ 2 3 4 ~&c-=cos.j7t?, (A;'), where |) is a 



sin.jov 

 constant which is to be found. By (A:) we get - — ^— =tan.Mv = 



^ ^ ^ ° cos.pt? ^ 



or l+tan.^v-/ — l=e [l-tan.j?v^ - 1], 



^/-l. 6 ^ +1 



22;y-i l+tan.j9Vv/— 1 j ^ , • ,, , , ,• , • , 



• • e ~ ^ =^, and taking the hyperbolic logarithm 



1 — tan.jvyv/ - 1 



we get 2vx^ — l~h. ( an.p?; - > ^ ^^^ ^^^ second member of 



\1 — tan.jDyv - 1/ 

 this by (14), (using tan.pz;\/3I for a,) gives by a slight rednc- 



tan.^pv tan.^py 

 tion v = tm.pv- — ^ — + — - — -&c., (/), and dividing sin.pv 



■y 3 2^5 



by cos.pv as given by (A;'), we get i3in.pv=vi--:^+^ + &c., 



