316 PROCEEDINGS OF THE AMERICAN ACADEMY 



and therefore 



and since 



Sj' = w 9 i, 



, • . q' = nQ\^{Zq-\-k 0)UVy, 

 whence follows the fundamental formula 



log 7 = log T^ + ?i i + (Z 9 + ^- a)UVy ; (11) 

 which can be more accurately written 



log (? = log T(? + n i + [Z (? + {n + 2^-) 9]UVj. (1 2) 

 If q be negative, this becomes 

 log(-(?):=logT<7+«0i + [Z?+Oi + 2Z:+l)0]UVy. (13) 



If (^ be a vector, its angle is — and 



log (± a) = log T« + « i ± ^(" + ^^•) + ^ QU«. (14) 



If n = Z; = 0, and if the real logarithm of Tq be taken, we have 

 what Hamilton calls the principal logarithm, or simply the Logarithm, 

 of a quaternion, and indicates by using 1 instead of log. Tims we have 



\q = \Tq-\-Zq\}yq. (12)' 



The principal logarithm of 7 is a real quaternion complanar with q itself. 

 Its scalar and vector parts are 



Sly = YYq, Y\q = Z qVVq. 



7°. The formulae of 4'* render it possible to find the logarithm 

 of a biquaternion. Let 



log UQ = Q' = q\ + \q'„ 



where Q' is an undetermined biquaternion, and Q = q^^ -\- \q.^ is such 

 that Wq^ = \]Vq.^. Formula (G) gives 



SUQ = 6 SQ' cos {TVq\ + iTVq',), 



TVUQ = T6SQ' sin (TV^'^ + iTVq',), 



.•. T^VUQ + S-UQ = 62SQ' = 1. (15) 



