347 



A more rapid solution may, however, be obtained by the ordinary 

 equations of [absolute] vis viva and absolute moments thus : — 



Tracing the absolute motion of the axis in space on the unit-sphere, 

 let {S) be the starting position of the 

 axis, S Q the direction in which from 

 its connexion with the earth, or any 

 other cause, this axis begins to move, 

 (X) any other position of the axis; 

 M, a fixed line in a plane perpen- 

 dicular to SQ; let MX=^, XMS 

 = 6, 7 = starting angular velocity of 

 (X); then, by equation of absolute 

 vis viva. 



and by equation of moments round M, 



de 



dt 



m (cos ^0 - cos ^) + 7 sin ^o- 

 Eliminating [ ^ 



sin2^( ^ = 7' sin - [m (cos - cos ^) + 7 sin ^0 



or, if M be chosen, so that tan MS = tan to = - 



m 



sin 2^ 



which necessitates 



f (m^ + 7') (cos ^0 - cos = 0, 



(^1 = ^5 ? = ^0 = tan -, and^ = const _ ^ 

 \dtj ' m dt sm^o 



+ 7"^ 



If the starting velocity of the axis is solely due to its connexion 

 with the earth before it was set free. 



7 = a; sm ^0 ; 



