222 MOTION OF A VARIABLE SYSTEM. [393. 



, d T ^ ( . dx , dv , dz\ . . 



and =2 m \x+y-^+z . (15) 



dq \ dq dq dqj 



From (15) and (14) we find 



at dq \ dq dq dqj dq 



393. Thus prepared we can introduce the new co-ordinates 

 q into the equations of motion (4) by multiplying these equa- 

 tions by dx/dq, dy/dq, dz/dq, and adding them throughout the 

 whole system ; this gives : 



_, f-.dx , dv , dz\ v> / v dx , -, r dy , ^dz\ / * 



^m (x +y^+z ] = S(^r 4- F-^ + Z ; (17) 

 \ dq ' dq dqj \ dq dq dqj 



the coefficients of X, //,, all disappear in the summation, since, 

 by hypothesis, the new co-ordinates satisfy the conditional 

 equations (i) identically. 



The right-hand member of (17) we shall denote by Q (comp. 

 Arts. 1 80, 211) ; the left-hand member can be put into a more 

 convenient form by means of (16) and (12). Thus we find 

 finally the equations of motion in the second Lagrangian form : 



at dq dq 



As there is one such equation for every one of the Lagrangian 

 co-ordinates g v q 2 , q m , the number of such equations is 

 m = $n k. They are obtained from the type (18) by attaching 

 successively the subscripts i, 2, m to each of the symbols 



q, q, Q- 



394. In the particular case of a conservative system, i.e. when 

 there exists a force function U such that 



, , , , 



vx ay az 



the quantity Q in (i 8) is evidently =dU/dq ) so that the equa- 

 tions of motion assume the form 



dL s Jl = ^ ( T + U). (19) 



dt di d v 



