433 



/'b denoting here again an imaginary unit, with a direction 

 altogether arbitrary. 



If we make, for abridgment, 



■^(^) = ^+T+S + tS3 + *^^-' (") 



the series here indicated will be always convergent, whatever 

 quaternion Q may be ; and we can always separate its real 

 and imaginary parts by the formula, 



/ (w + i„ r) -f {w) (cos r + i^ sin ?•) ; (u') 



which gives, reciprocally, for the inverse function /-', the 

 expression 



/-' (/x(cos0 + i^ sin0)) = log|u + ^(6 + 2nvr), (u") 

 M being any whole number, and log/u being the natural, or 

 Napierian, logarithm of ;i, or, in other words, that real quan- 

 tity, positive or negative, of which the function /is equal to 

 the given real and positive modulus fx. And although the 

 ordinary property of exponential functions, namely, 



/(q)./(q')=/(q + q'), 

 does not in general hold good, in the present theory, unless 

 the two quaternions o. and q' be codirectional, yet we may 

 raise the function/ to any real power by the formula 



(f{w + «»»•))« =Aq{w + i^ r + 2«7r)), (v'") 



which it is natural to extend, by definition, to the case where 

 the exponent q becomes itself a quaternion. The general 

 equation, 



Q/" = Q/, (v) 



when put under the form 



(fiw + h r)}' = / {w' + i,, r'), ( vO 



will then give 



\ ^' + »V {r' + g«V) \ \ tc- U ( >• + 2mr)\ 



«'' + ('+2«7r)- ' >^"^ 



and thus the general expression for a quaternion q, which is 



