RIGID BODY ABOUT A FIXED POINT. 277 



as given by Hamilton ; and we shall omit for the present the consideration of 

 whether 7 is a constant vector or not. 



19. Let a be the initial position of ©•, q the quaternion by which the body can 

 be at one step transferred from its initial position to its position at time t. 

 Then 



zr = qa.q~ l 



and Hamilton's equation (14) becomes 



2 . mqaq~ x V . sqaff 1 = 7 , 



or 



2 . mq {aS . af 1 sq — q~ l sqa 2 } q~ x =y. 



Let 



<p^— 2 • m(aSag — a 2 g) (16), 



where (f> is a self-conjugate linear and vector function, whose constituent vectors 

 are fixed in the body in its initial position. Then the previous equation may be 

 written 



or 



For simplicity let us write 



q<p(q~ 



1 eq)q~ 1 = 



- 7> 



<p{q~' 



tq) = q~ l 



yq. 



T 



- 1 ei = V 



1 



<r 



l 71 = l. 



1 



(17). 



Then Hamilton's dynamical equation becomes simply 



<Pn = l (18). 



20. It is easy to see what the new vectors r\ and £ represent. For we may 



write (17) in the form 



£ = qnq~ l 



7 = ?& -1 



(17)', 



from which it is obvious that n is that vector in the initial position of the body 

 which, at time t, becomes the instantaneous axis in the moving body. When no 

 forces act, 7 is constant, and £ is the initial position of the vector which, at time t, 

 is perpendicular to the invariable plane. 



21. The complete solution of the problem is contained inequations (7), (17), 

 (18).* Writing them again we have, attending to (17), while introducing v\ instead 

 I of g into (7), 



* To these it is unnecessary to add 



T(j = constant , 



as this constancy of Tq is proved by the form of (7). For, had Tq been variable, there must have 



been a quaternion in place of the vector jj. In fact, — (Tq) 2 = 2S . qKq = (Tq) 1 S»j = . 



VOL. XXV. PART II. 4 B 



