182 - PHILLIPS AND MOORE. 



If now 



M = ^ Mc, <t> = e'-^ . (44) 



we can write in the form 



where ^*i is an arbitrarily chosen vector and g is a constant of the 

 motion. The motion rotates A'l through the angle q in the fixed plane 

 ku. Hence every vector moves in a fixed plane and all are rotated through 

 the same angle q. For this reason we shall call the rotation equiangu- 

 lar when 



M= ^ Mc (45) 



Conversely, if all vectors are rotated through the same angle q, in 

 particular the rotation in the fixed plane must be through the angle 



qi = qi = q 



and so (44) is satisfied. 



In the case just discussed, if x is any vector, since x, <j)X, (f)\x lie in a 

 plane and the angle from x to '(})X is q and that from </)a; to (f)\v is also q, 



X -\- (fr X — 2 cos q 4> X. 



Since this is true for all values of .r 



(^2 = 2 cos g - /. (46) 



This is the characteristic equation of 0. Conversely if any rotation 

 has this for characteristic equation, x, 4)X, (fy^x, lie in the same plane. 

 This plane will then be invariant and the rotation in the plane will be 

 through the angle q. The rotation will therefore be equiangular. 

 These rotations resemble very much the rotations that result from 

 quaternion multiplication. If the angle of rotation is q, by (44) 



M-M = 2ql (47) 



In terms of a given set of orthogonal vectors ki, k^, kz, ki, any self- 

 complementary two-vector of this kind be written, 



M = 9 [X (A-i, + A-34) + M [kn - k,i) + V (A-14 + ^-23)] 



