ROTATIONS IN SPACE OF EVEN DIMENSIONS. 183 



where 



X^ + M' + i'- = 1. (48) 



Expanding and using (48) we get 



<^ = e''^ = / cos g + X sin g- /• {kn + ^'34) + M sin q I- (/.'is — '^•24) + 



V sin q I'{ku + hs). 

 If we take 



i = I-(ki2 + kdi), 



j = I-{ku — koi), 



k= I-{ku + kn), 



then 



4> = I cos 9 + sin g {\i + /uj + vk). 



Also by direct multiphcation it is seen that 



i' = f = k' = - I, 



'ij = f^', jf^' = h ^"^ = i» 



as in quaternions. Two dyadics of this kind can then be multiphed 

 hke quaternions the principal difference being that 1 in quaternions 

 is replaced by /. If M is anti-self-complementary we get similar results 

 with 



i = I- (kr2 — kzi), 



i= I-{kn + k,,), 



/, = I.(k,,-ku). 



Rotations with self-complementary M differ from those with anti- 

 self-complementary M in tlie same w^ay that quaternions referred to 

 a right hand system of axes differ from quaternions referred to a 

 left hand system. 



13. Invariant complexes. If M is self-complementary the 

 rotation 



- r 



leaves all anti-self-complementary two-vectors invariant. For if 

 M' is such a two-vector M and W can be expressed in the form 



M = g (ri + TTo), M' = X (tTi - TTs) 



