433 



i^ denoting here again an imaginary unit, with a direction 

 altogether arbitrary. 



If we make, for abridgment, 



/(Q)=l+5+^^ + _|_ + &C., (u) 



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, 



/ 0« + k r) —f [w] (cos r + i^ sin r) ; (uQ 



which gives, reciprocally, for the inverse function f~^, the 

 expression 



/-Hi«(cos0 + i^smQ)) = \og,x -\- k{e + 2mr), {v") 



u being any whole number, and log ju being the natural, or 

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

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

 the given real and positive modulus |U. And although the 

 ordinary property of exponential functions, namely, 



/(Q)./(QO=/(a + QO, 



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

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

 raise the function/ to any real power by the formula 



{f{w + ^kO)^ =/(?(«' + «R r + 2w7r)), iv'") 



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 



(/(«; + i^ r)y - f {lo' + h, r'), (v') 



will then give 



{ w' + k' {r' + 2«V) \ \ w~ i, (r + 2«7r) \ 

 ^~ z,,2^(r + 2«7rf ' '^ '' 



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



