196 THE THEORY OF SCREWS. [197 



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 instan 

 taneous 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 con 

 jugate diameters of the projected ellipse, it follows that the harmonic axes 

 must project 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. 



