260 VII. DYNAMICS OF ROTATING BODIES. 



of tangency. Before taking up the discussion of this result, as given 

 by Poinsot, we will consider the analytical method of establishing 

 the result. 



84. Euler's Dynamical Equations. If H x ', HJ, HJ represent 

 the angular momentum about the fixed X', Y', ^'-axes, L', M', N', 

 the moment of the applied couple, the equations of 67, 49) are 



dH f dH' dH' 



<26\ _ _ TJ _JL M 1 N' 



dt *"> dt *->. dt ~ 1V ' 



where (cf. 76) 



H x = 



27) HJ = 



HJ = a, 



Differentiating we have, after making use of 77, 130), 

 dH' dH dH dH 



28 ) - + A + K 



+ H x (ft r- yi q) + S, fop -,) + H, (a t q - ftp). 



If we now choose for fixed axes the instantaneous positions of the 

 moving axes, we have a x = /3 2 = y 3 = 1, all other cosines zero, and 

 the equations 28) become simply 



29) 



dH 



We may obtain the same results by the use of the equations 77, 

 128). Let us take for the point x, y, z the end of the vector H. 

 Its coordinates with respect to the moving axes being H x , H y , H z , 

 substituting them in equations 77, 128) we obtain for their velocities 

 resolved along the X, Y, ^-axes the expression on the left of 29). 

 We must now put for H x , H y , H z the expressions 79, 134). 

 If now the moving axes are taken at random, the moments and 

 products of inertia of the body with respect to them will vary with 

 the time, so that their time -derivatives enter into the dynamical 

 equations, which are thus too complicated to be of any use. It is 

 therefore immediately suggested that we choose for the moving axes 

 a set of axes fixed in the body, and moving with it. The quantities 

 A 9 B 7 C, D, E, F are then constants. If in addition we take as axes 

 the principal axes at the origin of the moving axes, D, E, F vanish, 



