RIGID BODY ABOUT A FIXED POINT. 



269 



Thus a rigid body may pass from the position q ( ) q~ l to the position 

 4 ( ) 9 '~ l -> whatever be q and q / , by a rotation about OE'. Also, by q ( ) ^ _1 , Q 

 remains fixed; but by q ( ) ^ /_1 it moves to R, where <QE'R = 2<QEQ' = 

 2<POP'. 



Hence if OE' = — Us = (Us) -1 , the versor arc PP' may be expressed by either 

 of the equal quantities 



2PF 

 (Us) * = q'q-\ 



But the actual rotation about & is 2PP', because Q moves to R. Hence if 

 we put 



q = q + qdt + &c, 



we have 



Ts U = 2PP', 



and thus 



Ts U fa T s U Ts 



1 + qq~ 1 U + &c. = (Us) 7T = cos — — - + Us sin -— - 



= 1 + I U + &c. 

 Hence, as in (6), when St is indefinitely small 



2^ _1 = s- 



10. To express q wz terms of the usual angles ^, 6, (p. 



Here the vectors i, j, k in the original position of the body correspond to 



OA, OB, OC, respectively, at time t. The transposition is effected by— first, a 

 rotation -^ about kj second, a rotation 6 about the new position of the line 



VOL. XXV. PART II. 3 Z 



