DYNAMICS OF A RIGID BODY. 145 



velocity about any other axis. This axis must be in the 

 plane S, if small oscillations are to exist at all, for 

 the initial angular velocity, if not capable of being 

 resolved completely on the two harmonic axes, will have 

 component around the vertical axis OL The effect of 

 an initial rotation about OI will be to give the body 

 a continuous slow rotation around the vertical axis, which 

 is, of course, not admissible when small oscillations only 

 are considered. 



If, therefore, the body performs small oscillations only, 

 we may regard the initial axis of displacement as lying 

 in the plane S, while we must have the initial instan- 

 taneous axis in that plane. The initial displacement 

 may be resolved into two displacements, one on each of 

 the harmonic axes, and the initial angular velocity may 

 also be resolved into two angular velocities on the two 

 harmonic axes. The entire motion will, therefore, be 

 found by compounding the vibrations about the two 

 harmonic axes. Also the instantaneous axis will at 

 every instant be found in the plane of the harmonic 

 axes, and will oscillate to and fro in their plane. 



Since conjugate diameters of an ellipse are always 

 projected into conjugate diameters of the projected 

 ellipse, it follows that the harmonic axes must pro- 

 ject into two conjugate diameters of a circle on any 

 horizontal plane. Hence we see that two vertical planes, 

 each containing one of the harmonic axes, are at right 

 angles to each other. 



We have thus obtained a complete solution of the 

 problem of the small oscillations of a body about a 

 fixed point under the influence of gravity. 



