RIGID BODY ABOUT A FIXED POINT. 267 



where 



a = 1 -t- ik + j[L + ki> . 



8. The result of § 7 may be stated in even a simpler form than (5), for we 

 have always, whatever quaternion q may be, 



V ^" 1= ^ 2(U?)_1 



and, therefore, if we suppose the tensor of q, which may have any value what- 

 ever, to be a constant (unity, for instance), we may write (5) in the form 



■£ = 2j (6). 



An immediate consequence, which will be of use to us later, is 



q.q~Uq = 2q . . . . (7). 



9. It may appear to some that the demonstration of § 7, founded on the 

 differentiation of quaternions, is not very convincing. For such it is easy to put 

 it in an expanded form in which no process of differentiation of a function of a 

 quaternion is alluded to — though in principle it is the same proof. 



Let q become q + r in the indefinitely short interval t. Then the change of 

 position of the extremity of 



— = qa,q~ x 



may be expressed either as 



Vsar • r or as (q + r) a (q+ r)— 1 — quq -1 . 



Hence 



rV.eqa.q~ 1 = (q + r) a (q + r)~ x — qa.q~ x , 



— 1 [(1 + J -1 r ) "C 1 + 9T l T )~ 1 — "l^ 1 j 



= T'(l + g -ir) ( (l4g " lr)a ^ 1 + K - g " lr) -( 1 + g-^a + g-JT 1 ^)")^ 1 



= t»(i + r v) ( (1 + sr^CVr 1 '.-))^ 



But r is the change of q in time r, and we may therefore write 



r — q r 



Substituting, expanding, and neglecting small quantities of the orders r 2 and 

 upwards, we have 



