RIGID BODY ABOUT A FIXED POINT. 265 



4. Ifqbe any quaternion, the operator 



q( )q~ 1 



turns the vector, quaternion, or system, to which it is applied, about the axis of q 

 through double the angle of q. 



This was one of Hamilton's early* discoveries in his new calculus, but it was 

 independently obtained by Cayley (only a month or two later)f by the help of 

 the formulae of Rodrigues already referred to. Conversely, when its truth has 

 been established by an independent process, these formulae may be at once 

 derived from it : not only far more simply, but even in a somewhat improved 

 form. 



The quaternion q may obviously be considered as a mere versor, since its 

 tensor does not appear in the operator q ( ) q~\ and a glance at the annexed 



figure proves, by the multiplication of versor arcs, the theorem above stated. 

 (See Tait's Quaternions, § 353, or Hamilton's Lectures, § 282, and Elements, 

 § 308 (9).) 



5. In quaternions we have, of course, whatever be q and r, 



(gf)— 1 = r~ l q — 1 . 



Hence 



q.r( )r~ 1 .q- 1 = qr( )(qr)~ 1 , 



which shows how to combine any two rotations into a single one. 



6. Given the initial and final positions of any two vectors, of a rigid system, 

 drawn from the fixed point; to find the quaternion operator by which the rotation 

 can be effected. Let them be a, 6, a v (3 t , and let q be the required quaternion, 

 then 



qaj- 1 = a x , q^~ l = ft , 



or 



qa. = ufi , qfi = ftj (3). 



Hence 



S(«-« 1 ) ? = 0, S(/3-ft) 2 = 0, 



or 



V 2 i|V(«- ai )(/3-ft) 



* Proc. R. I. A. November 11, 1844. f Phil. Mag. Feb. 1845. 



VOL. XXV. PART II. 3 Y 



