OP ARTS AND SCIENCES. 315 



For bivectors, we have, if a = a^ -\- \a^, provided that «^ and a^ 

 are parallel, 



6<^ ^ cos (T«j -f- iTa^) -f- U« sin (T«j -)- iTaj), 

 = Ch a -(- Sh a. 



5°. The usual algebraic definition of the natural logarithm, applied 

 to a quaternion, gives 



e"'^' = q. (8) 



Let log q = p, log q' = p'. 



,\ 6^ = q, e^' = q', 



and by (1) eP+P' = qq', 



which gives, for complanar (but not for diplanar) quaternions, the gen- 

 eral formula 



log qq' = log 5' + log q'. (9) 



This formula, however, is subject to certain limitations as will 

 be shown later, in 9". But in particular and always, we have 



log q = log TqVq = log Tq + log \Jq. (10) 



6°. Let logUg = q , where q' is an undetermined quaternion. 



By (5) 



ei' = JJq= 6^5' (cos TV^' 4- UV^' sin TV?'), 

 whence, 



SU5' = 6^^' cos TVq' = cos Z. q, 



TYUq = T. 6^3' sin TYq' = sin Zq; 



. • . T'YJJq -\- S-Vq' = 6'Sg/ = 1, 



,'. 6^5' = (— )" 1, and Sq' = n Q t; 



where i is the scalar J — 1, and n any integer. Comparing these results, 

 we find 



cos Zq =^ ( — )" cos TVq', sin z^ 5- = T sin TVq', 



± TVq' = Zq-\-kQ, Yq'=±{Zq^k c))\JYq', 



where k is any integer which differs from n by an even number. But 



± UV5' = UVU?, and \q \\ VUy, 



