142 DYNAMICS OF A RIGID BODY. 



equilibrium has been effected by a small twist about a 

 screw on the cylindroid which contains two of the har- 

 monic screws, then the twist can be decomposed into 

 components on the harmonic screws, and the instanta- 

 neous screw about which the body is twisting at any 

 epoch will oscillate backwards and forwards upon the 

 cylindroid, from which it will never depart. 



If the periods of the twist oscillations on two of 

 the harmonic screws coincided, then every screw on 

 the cylindroid which contains those harmonic screws 

 would also be a harmonic screw. 



If the periods of the three harmonic screws were 

 equal, then every screw of the complex would be a har- 

 monic screw. 



130. Oscillations of a Rigid Body about a Fixed Point. 

 We shall conclude the present Chapter by applying 

 the principles which it contains to the development 

 of a geometrical solution of the following important 

 problem : 



A rigid body, free to rotate in every direction around 

 a fixed point y is at rest under the influence of gravity. The 

 body is slightly disturbed : it is required to determine the 

 nature of its small oscillations. 



Since three co-ordinates are required to specify the 

 position of a body when rotating about a point, it fol- 

 lows that the body has freedom of the third order. The 

 screw complex, however, assumes a very extreme type, 

 for the pitch quadric has become illusory, and the 

 screw complex reduces to a pencil of screws of zero pitch 

 radiating in all directions from the fixed point. 



The quantity U Q appropriate to a screw reduces to 

 the radius of gyration when the pitch of the screw is 

 zero ; hence the ellipsoid of inertia reduces in the pre- 

 sent case to the well-known momental ellipsoid. 



The ellipsoid of the potential ( 126) assumes a 



