77, 78, 79] ENERGY AND MOMENTUM OF ROTATION. 249 



coinciding in direction with the position of the moving axes at the 

 given instant. This theorem, which is by no means self-evident, is of 

 great importance, as is the similar property of the angular velocity, 

 of which we have already made use. 



79. Kinetic Energy and Momentum due to Rotation. 



From equations 119) we find for the part of the kinetic energy of 



the rigid body due to the rotation, supposing -= - = -^ = ~ = 0, 



dv dv ct v 



- qrZmyz rpZmzx pqZmxy 



and for the angular momentum, introducing 119) in 48), 



H x = Zm [y (py - qx) z (rx ##)]= Ap Fq Er, 



134) H y = 2m\2(qz ry} x(py qx)~] = Fp -f Bq Dr, 



H z = 2m [x(rx pz] y (qz ry)] = Ep Dq + Cr, 



the last column being what we obtained in 68, 53). 

 It is evident that 



o^s dT dT rr dT 



Hx= ^ ^V^Jq' Hz= frr' 



so that in this respect p, q, r, H x , H y , H z have the relation of 

 Lagrangian generalized velocities and momenta. 

 Since we have 



the kinetic energy is one -half the geometric product of the angular 

 velocity and angular momentum. 



