190 KINETICS OF A RIGID BODY. [344, 



344. To prove that q' is constant, it should be remembered 

 that, by (7), Art. 321, we have 



'-"?;& \ 



As H is constant in our case, it only remains to show that o>/p r 

 is constant. This follows from the expression (9) for the kinetic 

 energy T t given in Art. 322, viz. 



for, as there are no external forces, no work is done, and the 

 kinetic energy must remain constant ; hence co/p' is constant, 

 and co is directly proportional to p f . 



Moreover, the expression (14) of Art. 322 shows that 



a) cos <f) ~= const., (i) 



H 



that is, the projection a> cos< of the angular velocity co on the 

 invariable direction remains the same throughout the motion. 



345. It has been pointed out in Part I., Art. 35, that the 

 motion of a rigid body with a fixed point can always be 

 regarded as produced by the rolling of the cone of the body 

 axes over the cone of the space axes, these cones having their 

 common vertex at the fixed point O. The body axes, i.e. the 

 lines /' of the body that become instantaneous axes of rotation 

 in the course of the motion, form a cone, invariably connected 

 with the momental ellipsoid at <9, and intersecting this ellip- 

 soid in a curve fixed in the body. This curve has been called 

 by Poinsot the polhode (or path of the instantaneous pole P\ 



Fig. 43). 



The cone of the space axes /, which is fixed in space, inter- 

 sects the invariable plane in a curve called herpolhode (or creep- 

 ing path of the pole). During the motion of the body, the 

 polhode rolls over the herpolhode. 



