340 Mr. Graves's Explanatioti of a reinarkable Paradox 



limit like the remainder of the series, recedes from as n in- 

 creases. It is demonstrable from the nature of the series (7.), 

 and may, I think, be assumed as an admitted proposition, that 



.,'+^'=.0^./ (8.) 



-1 



y 



Hence e^ ^ 3^+^' -f V - 1 Vl^ "' o+ ^1^ 



— p'^'^y^-V'^' v^^a/-! /Tj- 



cos 



y 



x^o -v^^' 0+ -»/yH^- (9.) 



Now, of the right-hand member of (9.), the first factor, viz. 

 ^^Jvy^+s" is evidently equal to Vj/^ + «% and if it can be 



— ^ _ 1 y 



t I r COS — 



shown that the other factor, viz. e^ V^^ o+ v'^+a^ is 



y-\- \/ 1 ^ 



equal to — , our proposition will be proved, for 



V 'If ^z 



-\ 



it will be proved- that ^„'a/«/-+^=+^-i V"^'*"^,^ l/P^^ 

 = vy + g^x /-g-— 3 = y-\- V— 1« = x; thatistosav. 



vy+ 



-1 y 



it will be proved that I \/y + 2r^+ \/- 1 -7-=! cos — -^ 



^ ^ ■ vg^ 0-1- Vj/^ + s* 



is an e-log of j/ + v^^ ;? or x, according to the most limited 

 definition I can conceive of the term Neperian logarithm that 

 will extend to imaginaries. 



Now e^ a/=T^ = cos d + -/^TT sin 6 (10.) 



if by cos fl be understood the sum of the series 



and by sin fl the sum of the series 



fl3 95 



fl 1— + &c. (12.) 



1.2.3^1.2.3.4.5 ^ ' 



This not only follows immediately from the definition above 

 given of the notation e^^ but the definitions of cos fl and sin 9 

 accord with admitted theorems respecting the sine and cosine 

 when 9 is real. The two series (1 1 .) and ( 1 2.) are convergent for 

 all finite values of 9, and I can see no objection to them as defi- 

 nitions of sine and cosine, even when fl is imaginary. I do ac- 

 cordingly treat them as such in my general exponential theory, 



