314 VII. DYNAMICS OF ROTATING BODIES. 



equal to constants. We can not then, as was assumed in 36 in 

 deducing Lagrange's equations, use the equations of constraint to 

 express the position of the system in terms of a less number of 

 coordinates equal to the number of degrees of freedom, three in 

 this case. 



Moreover, we can not even use the equations connecting the 

 velocities, 



x' = acp' sint^, 



y 1 = aqp'cOS^, 



to express the kinetic energy in terms of &', (p 1 , ty' alone, as was 

 explicitly pointed out by Yierkant. On the other hand, we must 

 keep all the coordinates and their velocities in the expression of the 

 kinetic energy, as if there were no constraints, and form the equa- 

 tion of d'Alembert's Principle as in 37, 56), and afterwards introduce 

 the fact that the changes of the coordinates are not all independent, 

 by means of undetermined multipliers, as in 25. 



If , rj are the coordinates of the center of mass of the disk, 

 we have for the kinetic energy, 



218) T = 



where 



| = x -f a cos # cos ^, 



219) r) = y -f a cos # sin ^, 



g = a sin #, 

 and differentiating, 



(' ===#' a(sin#cos4> - & -f cos^sin^ ^'), 



220) / n' = y' - a (sin # sin ^ -9-' cos ^ cos ^ ^'), 

 ' = a cos # #'. 



Squaring and adding, we obtain for the kinetic energy, 



221) T= \ 



+ 2a[ si 



-h cos # ^' ( x' sin ijf -f- y' cos ?/>)]} 



-f |^(#' 2 + sin 2 #- ^' 2 ) -f |C(y + T^'cos^) 2 . 

 J I 



Forming now the equation of d'Alembert, adding equations 216) 

 multiplied by A and ^ respectively, and equating to zero the coeffi- 

 cients of dx, dy, d& 9 Sty, 4^, we obtain the equations of motion 



