194 THE THEORY OF SCREWS. [197 



197. 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, is 

 in stable equilibrium under the influence of gravity. The body is slightly 

 disturbed : it is required to determine its small oscillations. 



Since three co-ordinates are required to specify the position of a body 

 when rotating about a point, it follows that the body has freedom of the 

 third order. The screw system, however, assumes a very extreme type, 

 for the pitch quadric has become illusory, and the screw system reduces to 

 a pencil of screws of zero pitch radiating in all directions from the fixed 

 point. 



The quantity u e 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 present case to the well-known momental ellipsoid. 



The quadric of the potential ( 193) assumes a remarkable form in the 

 present case. The work done in giving the body a small twist is propor 

 tional to the vertical distance through which the centre of inertia is 

 elevated. In the position of equilibrium the centre of inertia is vertically 

 beneath the point of suspension, it is therefore obvious from symmetry that 

 the ellipsoid of the potential must be a surface of revolution about a vertical 

 axis. It is further evident that the vertical radius vector of the cylinder 

 must be infinite, because no work is done in rotating the body around a 

 vertical axis. 



Let be the centre of suspension, and 1 the centre of inertia, and let 

 OP be a radius vector of the quadric of the potential. Let fall 1Q per 

 pendicular on OP, and PT perpendicular upon 01, It is extremely easy 

 to show that the vertical height through which / is raised is proportional 

 to IQ 2 x OP 2 ; whence the area of the triangle OPI is constant, and there 

 fore the locus of P must be a right circular cylinder of which 01 is the 

 axis. 



We have now to find the triad of conjugate diameters common to the 

 momental ellipsoid, and the circular cylinder just described. A group of 

 three conjugate diameters of the cylinder must consist of the vertical axis, 

 and any two other lines through the origin, which are conjugate diameters 

 of the ellipse in which their plane cuts the cylinder. It follows that the 

 triad required will consist of the vertical axis, and of the pair of conjugate 



* Trans. Roy. Irish Acad., Vol. xxiv, Science, p. 593 (1870). 



