RIGID BODY ABOUT A FIXED POINT. 295 



of which the integral is obviously 



a = y — 1 Say + % COS -^ t + Xsia-~t, 



where k and X are vector constants of integration. 

 The two last terms must be, together, equal to 



and, as they vanish alternately, the tensors of k and X must be equal. Also 

 unless 



SxX= 



the tensor of this part of a will vary. Hence 



« = - Uy SaU 7 + TVaTJy . (Vx COS ^ t + UX sin ^ A . 



Let us, for simplicity, take the usual i,j, k of quaternions as coinciding with 



U7, U/c, UX, and let 



— SaUy = COS j8 . 



Then 



TVaUy = sin |8 . 

 Also let 



T 7 



B = *• 

 Thus we have 



« = i cos jS + (y cos %£ + A; sin %£) sin /3 

 whence 



2^ -1 = — (=r — — WB cos ]8 [* cos /3 + (/ cos nt + h sin nt) sin /3] + ni 



= 2ai + 2h(j cos nt + k sin ??i) , 



where 



2b = - nB (^ - -^ cos /3 sin |8 



2a = — nB(^ — -^ J cos 2 /3 + n = n( sin 2 /3 + -^ cos 2 /3 j . 



51. For the complete solution of the problem, it remains that we integrate 

 the equation above, which we may write as 



q = Vai + b(j cos nt + k sin nt)] q 



= (ai + lis) q (41), 



if we put 



w =j cos nt + k sin nt . 



