184 Mr Larmor, On the Period of the [May 25, 



2. In order to obtain a clear notion of the action of the internal 

 forcive which causes the free precession, let us examine in a geo- 

 metrical maimer the well-known case of 

 I L a perfectly rigid solid rotating round its 



axis OC of greatest moment of inertia. 

 When this motion is disturbed, let 01 

 represent the instantaneous axis of the 

 rotation co of the solid, and OL the axis 

 of resultant angular momentum which 

 is fixed in direction in space. In the 

 case of a symmetrical solid 00, 01, OL 

 lie in one plane. With reference to the 

 solid itself, 01 and OL will describe right 

 cones round OG with common angular velocity Q, that of the free 

 precession in the solid body of the instantaneous axis of rotation. 

 The line OL being fixed in space, the point L of the body will 

 thus move away from the point L fixed in space with velocity 

 — fl . OL sin 7, where 7 represents the angle COL ; but as the 

 body is rotating at the instant round the axis 01 with angular 

 velocity co, the velocity of its point L must also be co . OL sin (7 — t), 

 where 7 — 1 represents the angle IOL. Thus we have the geo- 

 metrical relation 



— II sin 7 = co sin (7 — 1) (1). 



The angular momentum of the rotating solid with respect to 

 its centre of gravity is at time t made up of 



Goo cos 1 round OG and — Aco sin 1 round OA. 



After an infinitesimal lapse of time St, OC and OA have turned 

 round the axis OL fixed in space through an angle co'ht say ; thus 

 OC has moved through an angle co' sin 7 . Bt, and OA through an 

 angle co' cos 7 . St. These changes in the axes of the constant 

 component angular momenta will introduce two reacting couples, 

 acting round the same axis perpendicular to the plane COI and 

 equal to 



Ceo cos i . co' sin 7 and — A co sin 1 . co' cos 7 : 



and in the steady precessional motion these couples must equi- 

 librate each other, so that 



C cos 1 sin 7 + A sin 1 cos 7 = 0; 



that is, C tan 7 = A tan t (2). 



Combining equations (1) and (2), 



fl = co (cot 7 sin 1 — cos 1) 



= co — j — cos 1 (3). 



