Prof. Guthrie on V— 1. 285 



\±x — ±x 



(\±x) 3 =+x 



(\±wy=+x 



&c. &c. 



Or, since all ambiguities of sign may be considered as referring 

 to a supposititious factor unity, and omitting such factors as 

 always understood, we have 



|± = ±, |± 2 =-, |± 3 =+,and|± 4 =+. 



It will be easily seen that in this way the sign j + answers to the 

 sign v — 1 as ordinarily used. 



With these definitions, the fundamental laws of algebraical 

 powers will be found to be true, viz. 



a ,m xx n = x m+n f (1) 



(x m ) n =#»*■», (2) 



x m .y m —{xy) m i ...... (3) 



certain conditions being observed, — and likewise the theorem that 

 if a + \±b = c + \±d, then a=c and b = d; for a + \±b = c + \±d 

 must be taken as meaning a + b = c + d, and a— b-=c — d when 

 a=b and c=d. Whence every result ordinarily deduced by 

 means of the symbol V — 1 can readily be demonstrated. 



The other principal symbol to which we require to give a mean- 

 ing is e* when x is an ambiguity. The meaning we give to it is 



x 2 

 the expansion 1 + x + ==-§ + . . . , which, when x is ambiguous 



(say y\±), will be seen to be 



A x\ X 4 \ . ( X 3 X 5 \ — 



The convergent expansions, 



1 a? 2 x 4 , x 3 oc b 



l j^+ L 4 ~~ " ' ^~" "L 8 -J? "~ * * * * 



arc called cos a? and sin# respectively; and the other properties 

 of these functions are easily deduced from purely algebraical 

 considerations. 



But how, if this be the conventional meaning we attach to e*'** 

 do we know that the laws of exponents are true of this function ? 



