282 PROFESSOR TAIT ON THE ROTATION OF A 



qt) = 2q * 



that is, we have to integrate a system of four other differential equations harder 

 than the first. 



Putting, as in § 23, n = iw y + > 2 + Jca a , 



where cb x , <o 2 , o> 3 are supposed to be known functions of t, and 



q = w + ix + jy + kz, 

 this system is 



1 d%D _ dx _ dy _ dz 



2W~X~Y = Z' 

 where 



W = - a,? - u 2 y - u z z , 



X = Ul w + u 3 y - u 2 z , 

 Y = u, 2 w + u^z — « 3 < , 

 Z = w 3 i" + w r c- Ul y. 



or, as suggested by Cayley to bring out the skew symmetry, 



X = . u 3 y — u 2 z + u x io , 

 Y = — u 3 x . + u^ + u 2 xo , 



Z = U 2 X — Wj?/ . + u s w , 



W = - UjX - u 2 y - w 3 2 



Here, of course, one integral is 



w 3 + x 2 + y 2 + z 2 = constant. 



* To get an idea of the nature of this equation, let us integrate it on the supposition that n is a 

 constant vector. By differentiation and substitution, we get 



2q = qq = h^q . 



Hence 



^ Tri _ . Tjj 



q = Qj cos — * + Q 2 sin — * . 



Substituting in the given equation we have 



Trjf-Q 1 sin -~t + Q 2 cos -£t\ = (Q x cos y* + Q 2 sin -£t \ . 



Hence 



Tn . Q 2 = O^jj, 



— Tjj . Qj = Q 2 ^ 



which are virtually the same equation — and thus 



_ / T^ TT . Tjj\ 

 = QJ cos — t + TJjjsin — tj 



ZT.j 



= Q 1 (U,)1T. 



And the interpretation of q( )q~ 1 will obviously then be a rotation about r> through the angle 

 tT?j, together with any other arbitrary rotation whatever. Thus any position whatever may be taken 

 as the initial one of the body — and Q x ( ) Q : —1 brings it to its required position at time t = . 



