130 



THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. 



of a constant called the moment of inertia of the body multiplied 

 by the square of its angular velocity, and also that if we find the 



FIG. 27. 



angular velocities about three mutually perpendicular axes of 

 symmetry the energy may be found by adding the three parts 

 obtained for the energy of rotation about the three axes. We 

 will resolve the motions of the gyrostat into three angular velo- 

 cities, about the axis of the top, the axis of the inner ring, and 

 an axis perpendicular to both. About the axis of the gyrostat 



the angular velocity is -^- = </>', but there is also the angular 



cut 



velocity -J = -v/r' about the vertical axis, which has the com- 



Cut 



ponent ty' cos 6 about the axis of the gyrostat. The velocity 



7/J 



about the second axis is -7- = 6', and about the third is the other 



dt 



component of the velocity about the vertical, ijr sin 0. If A is the 

 moment of inertia of the gyrostat about its own axis, B that 

 about either of the other two, we have for the kinetic energy 



T = J [A (<p + ^ cos 0) 2 + B (<9' 2 + f 2 sin 2 0)], 



so that </> and ty are cyclic coordinates. For the components of the 

 forces tending to increase T/T, 0, cf>, 



