252 VII. DYNAMICS OF ROTATING BODIES. 



and X and X' coincident. Then we have for the transformation of 

 coordinates 



z 1 = y sin a -f z cos a, 



and determining the position of the system by the angle & made by 

 the perpendicular from the center of mass on the axis of rotation 

 with the Y-axis, 



y = r cos #, 



0' = r sin a cos & -\- z cos a. 

 The potential energy is as before 



W = Mgz' = Mgli sin a cos # -f const., 

 thus the equation of energy is 



7) ' Y K \-TZ\ Mgh sin a cos # = const. 



Thus the equation is the same as before, except that the length of 

 the equivalent simple pendulum is increased in the ratio of 1 : sin a. 

 This example includes the case of a swinging gate and of the im- 

 portant physical instrument, the horizontal pendulum of Zollner. 

 The mode of action of the latter depends on the fact that the moment 

 of the force required to produce a given deflection #, 



= --- = Mgh sin a sin #, 



may be made as small as we please by decreasing a, which is 

 observed in practice by making the time of vibration long. 



81. Motion of a Rigid Body about a Fixed Point. 



Kinematics. We shall now consider one of the most important 

 and interesting cases of the motion of a rigid body, namely that of 

 a body one of whose points is fixed, and which thus possesses three 

 degrees of freedom. This case was dealt with very fully by Poinsot, 

 in his celebrated memoir "Theorie nouvelle de la rotation des corps", 

 in the Journal de Liouville, torn. XVI, 1851. On account of the 

 instructive nature of his processes, which are entirely geometrical, 

 we shall present his method first. The treatment of the properties 

 of the moment of inertia, which is contained in the same paper, has 

 already been given in 70 72. 



If one point of the body remains fixed, the instantaneous axis 

 must at all times pass through that point. The motion is completely 

 described if we know at all times the position of the instantaneous 

 axis in the body and in space, and the angular velocity about it. 



