298 PROFESSOR TAIT ON THE ROTATION OF A 



btt a = (a - % ) K, bK, = -(a- pj H 3 



bl 3 = (a - fi^) J 1 6Jj = - (a - /b 2 ) I 3 



bJ 3 = — (a - a*J Ij Mj = {a - fb 3 ) J 3 . 



6K 8 = - (a - ^ H, m x = (a - ^ K 8 . 



These are, again, identical in pairs; for each pair containing the same two 

 constants agrees with the others in giving 



a — 



Pi 



— b a — 



P- 2 



or 



a 2 + V 2 — ((l x + /i 2 ) a + (i^z = 



But, by (43), we have 



Pi + P 2 = U 



1l 2 / ?l\ 



p\p* = -5 - y a - 2) 



2 



- b 2 



and the condition is satisfied identically. 



The final value of the quaternion in the case of the uniform rolling of one 

 right cone on another is therefore 



q — (H a + I x i + 3 x j + KjZ;) cos /t^t 

 — (J-i — Hji + KJ — J-Jc) sin (i^t 



+ "I" 1 ^ Kl + Jli ~ l d ~ H i & ) cos **** 



Putting 

 the ordinary differential equations, corresponding to that just solved, are 



* The tensor of q has been assumed constant. Accordingly we find by this formula 



J H^cos/u^ - I, siii/u/ + j — L fK 1 cos /u 2 < - J, sin ^i 2 n + I, cos ix^t + H, sin ^t H t — ' (J, cos /u 2 £ + K, sin nJL) 



+ ["j, cos /x t < - Ki sin ^t - — g— ' (i, cos ju 2 < - H, sin ii^tj J + I K, cos /u,£ + J! sin ^ t - — g— 1 f H, cos /u 2 « + I, sin n : l J \ 



=(H l > + i l - + j 1 « + K 1 »)[i + (^y] 



= (H I -.+ I l - + J 1 ' + K I -)( 1 -555)- 



