LAGRANGE'S EQUATIONS 339 



The elimination of X from these equations gives 



\~d/dL\ dL 

 I ) --- 



\_dt\ce) de 



dL~] 



6cos0 I ) --- + acos0 ( 1 --- =0, 



a0J 



and on substituting for L from equations (a) and (&), this equation becomes an 

 equation between 0, 0, and their differential coefficients with respect to the time. 

 From this and the geometrical relation (c), 



a sin B = b sin 0, (/) 



we can proceed to determine and in terms of the time. 

 Using equation (/), we can transform equation (6) into 



W = %Cgh sin (0 + e) \ inga sin 0. 



It will be possible to arrange counterpoises on the flywheel in such a way as 

 tomake 



and if this is done, the center of gravity will always be at the same height. 

 This is called balancing the engine. 



If we suppose the engine balanced in this way, we have W = and there- 

 fore L = T. We can, however, determine the motion much more simply than 

 by using Lagrange's equations, for we know that T must remain constant 

 throughout the motion ; and by differentiation of equation (/) we have 



a cos = b cos 0, 

 so that we can replace equation (a) by 



2 T = I0 2 + m(a 2 sin 2 02 + a 2 sin cos tan 2 + 1 a 2 cos 2 sec 2 e 2 ) 



+ M (a sin + a cos tan0 0) 2 

 = 2 [I + wia 2 sin sin (0 + 0) sec + $ ma 2 cos 2 sec 2 

 + Ma 2 sin 2 (0 + 0) sec 2 0]. 



This is constant throughout the motion, but we see that it does not follow 

 that is constant. Thus, although the engine is balanced so as to remain at rest 

 in any position, it will not necessarily run evenly if started into motion. 



Lagrange's Equations for Non- Conservative Systems 



273. For non-conservative systems it has been shown ( 268) 

 that equation (156), namely 



f\L^ 

 J 



, (172) 



h 



must be replaced by 



YSy+ZSzfldt = 0. (173) 



