340 



If the connexions of the moving system expressed in relative co- 

 ordinates do not involve the time, we deduce the equation of relative 

 vis viva precisely in the same way as that of absolute vis viva is obtained 

 when the co-ordinate axes are fixed, — i. e., 



the J 2 {mP"dp"), the work done by the second set of forces fictives" 

 vanishes, inasmuch as these forces are perpendicular to the displacements 

 of the particles to which they are applied. 



When the motion of the moving axes is one of uniform rotation 

 round a fixed line, {P') is evidently a force (wV) along the shortest dis- 

 tance from the molecule to the fixed line, and directed outwards from 

 this line, P'dp' = w^rdr, 



and the equation of relative vis viva assumes the very simple form 



where / and /q the moments of inertia of the moving system round 

 the fixed line at the time (t) and at the origin of time (to). 



The problem to be solved may be stated as follows : — 



A solid of revolution turns round its axes of figure with an angular 

 velocity (n). Its centre of figure being fixed relatively to the earth, and 

 the resultant of the earth's attraction being supposed to pass through 

 this fixed centre, it is required to determine the motion of the axis, 



1°. "When the axis is restricted to a plane ; 



2°. When the axis is restricted to a right circular cone; 



3°. When the axis is unrestricted. 



If we choose for co-ordinate axes three lines at right angles through 

 the centre of the gja^oscope moving with the earth, the motion of these 

 axes may evidently be resolved into two — a motion of translation of the 

 origin in a complicated curve in space, and a uniform angular rotation 

 (w) round an axis* drawn through the origin parallel to the earth's axis. 

 The former evidently does not affect the relative motion of the gyroscope, 

 and may be (as far as the present purpose is concerned) considered as 

 non-existent. 



For the complete determination of the motion of a solid body round 

 a fixed point, three equations must be deduced from the dynamical con- 

 ditions of the problem. In the present instance, the simplest that pre- 

 sent themselves are the following : — 



* This axis shall call, for shortness, the polar line. 



