322 UNIVERSITY OF VIRGINIA PUBLICATIONS 



If two constructions C and S can be found, different from zero, then 

 e"'' will have been constructed as a complex number, if on the other hand 

 we find C = and ;S =f= then e''' will construct an imaginary number, 

 if C =t^ and S = Q then e^^ is real. If no such functions C and S exist 

 and caimot be constructed then e''' is not a complex number as previously 

 designed but is some new number which must be designed and the complex 

 system extended to include these new numbers. 



5. We investigate the construction of the functions C{x) and Six) under 

 the hypothesis that there exist the two real one-valued continuous func- 

 tions of X which satisfy (8) subject to the definition of the integral number. 

 Since 



gix giy = gi(x+y) 



there results 



C{x + y) + iS{x + y) = C{x) Siy) - S{x) S(y) +i{C{x) S{y) + S{x) C{y) ] 

 Equating real and imaginary components, 



C{x + y) = C{x) C{y) - S{x) S{y), 

 S{x + y) = C{x) S(y) + Six) C(y). 



Also e'*-''~y' = e'Ve"'' gives in like manner 



C (x) dy) + Six) Siy) 



Cix-y) = 

 Six~y) = 



CKy)+s-^iy) 



Six)Ciy)-C(x)Siy) 

 C'iy) + S%y) 



(8) 



(9) 



In (8) put 1/ = and solve, whence C(0) = 1 and S (0) 

 In the same equations put y = — x, whence* 



CHx) + SKx) = ^^^) - '^^^) 



Ci-x) S-ix) 



IC'ix) + S\x)}{C'i-x) + SK- x)} = 1. (10) 



Square and add equations (8). Also square and add equations (9), using 

 the second of (10). Whence 



C\x^y) -f S'^ix^y) = {C%r) + S%x)] {C-i^y) + S\^ y)] (11) 



upper signs together and lower signs together. 



Consider now the two functions tpix) and ;/'(.r) defined by 



* For brevity we write [C{x)Y^ = C"(x), etc. 



