314 PROCEEDINGS OF THE AMERICAN ACADEMY 



sin ^i = i Sh q, cos q'l = Ch q, 



Sh qi^^ I sin q, Ch q'l =■ cos q ; 



and (2) gives 



09' = cos 9- -f- i sin q. (4) 



Thus, taking TYq for q, and UV^' for i, we can write formula (3) as 

 follows : — 



6' = 6"' (cos TYq + UV^ . sin TV^). (5) 



In the special case of a vector, (3) and (5) become 



6« = Ch « 4- Sh a (3/ 



6« = cos T« + U« sin Ta, (5)' 



where a is any vector whatever. 



4*^. Formulae, analogous to those just given, may be deduced for 

 biquaternions. 



Let Q = «7j -|- \q.„ where i = »/ — i (a scalar), and q^ and q., are 

 complanar. We have VJVq.^ = ± Wqy. Since t is commutative 

 with any quaternion, we have, by (4), — 



G^^'J-2 = (cos Yq., -\- I sin Yq.^), 



which, multiplied by (5) and by 6 '^'2, gives 



QQi + 192= 6S(«7i + 1<72) (cos T V^-j -f VYq^ sin T V7j)(cos Yq.,-\-x sin Yq.^ 



= 6SQ(cos TV^i + VYq, sin TVy,)(Ch TYq, -\- WYq, Sh TYq^) 



_ g r cos TYq, Ch TYq., ± iUV-7, cos TYq, Sh TV^, i 

 ~ ® L =F i sin TYq, Sh TYq, + VYq, sin TV^^ Ch T V^J ' 



6Q = 6SQ[cos {TYq, ± iTYq,) + UYq, sin {TYq, ± \TYq,)]. (6) 

 Observing that 



cos {TYq, ± iTYq.;) = Ch Y{q, + \q,), 

 and that 



sin {TYq, ± iTYq.^ = — VYq, Sh Y{q, + i*/,), 



and substituting these values in (G), we obtain this other important 



formula 



qq^ (3SQ(Ch VQ + ShVQ). (7) 



