• OF ARTS AND SCIENCES. " 317 



Let us agree to define the angle of Q as such that 



SUQ = cos Z Q, and TVUQ = sin Z Q. 



This definition will be admissible if it be not found inconsistent with 

 other established definitions. Then (15) gives 



SQ' = nQ\, 

 and 



ZQ = TVg\ + ITVq', + kQ, 



n and k being any integers, which differ by an even number. Multi- 

 plying Z Q into Wq^, we have 



Z Q.UVy, = Y{q\ + i^g + IcdVVq,, 



(Z Q + kQ)\]Yq, = Yiq\ + iq',) = VQ', 



or Q' = nQ i + [Z Q + (w -f 2 k)Q]l]Yq, = log UQ. 



Whence, for the general logarithm of a biquaternion, of which the 

 real components are complanar, we find 



log Q = log TQ + log UQ 



= logTQ + »9l + [ZQ + (n + 2iOa]UV5',; (16) 



and, for the principal logarithm of such a biquaternion, 



1Q = 1TQ+(ZQ)UV^,. (17) 



8°. For the product of any number of complanar quaternions (when 

 the sum of their angles does not exceed 180"^) we have by (9) 



log {q. q'. q" ) = log q -\- log q> -{-... , 



or 



log nq =y log q; (18) 



and since JJYq == IIYq' = UV^" etc., we therefore obtain at once 

 from (12) the general formula 



log nq = i: log T^ + 7iQ[ -f [JZq + (« -f 2k)e^\]Yq 



= log nTq 4- nQi -{-[Znq-ir (n + 2^-)9]UV^. (19) 



This formula gives the principal logarithm only when (2^Zq) <180° 

 (9"^). The two expressions of this formula show that in general 

 (^Z? being <180°), 



Znq = ZZq; 



which is otherwise easily seen to be true. 



