274 



PROFESSOR TAIT ON THE ROTATION OF A 



somewhat less simple form — to which, however, they are easily reduced by 

 putting 



x _ y _ z i 



w = 



X. (ti v x> 



The geometrical interpretation of either set is obvious from the nature of 

 quaternions. For (taking Cayley's notation) if 6 be the angle of rotation : cos/, 

 cos g, cos h, the direction-cosines of the axis, we have 



V . V 



q = w + xi + yj + zk = cos - + sin - (i cos f+j cos g + k cos h) , 



so that 



w = cos - 



x = sin - cos/ 

 2 



y = sin - cos g 



z — sin - cos h 



From these we pass at once to Rodrigues' subsidiary formulae, 



x — — ^ = sec 2 



ID* 



X = — = tan - cos/ 



w 2 J 



&c. = &c 



14. 7/i ^.g system of three angles, corresponding to that usually employed in 

 astronomy — viz., 6 the longitude of node, <p the inclination of orbit, t the angle 

 from node in plane of orbit — to find the quaternion operator. 



Here we relapse into the essential asymmetry of the method of § 10. First, 



