324 GENERALIZED COORDINATES 



Let us compare this motion with a slightly different motion in 

 which Newton's laws are not obeyed. In this second motion let the 

 coordinates of m lf at the instant at which they are x lt y^ z 1 in the 

 actual motion, be supposed to be x[, y{, z[, and let the components 

 of velocity at this instant be u(, v[, w(, so that 



, dx[ 



<*5*a 



Let us agree that the modified motion is to differ so slightly 

 from the actual, that any quantity such as x[ x l} u[ u lt which 

 measures part of this difference, may be treated as a small quan- 

 tity. Let us denote x{ x l by Bx v and use a similar notation for 

 the other differences. 



Multiply equations (148), (149), (150), which are true at every 

 instant, by Sx v Sy l} Sz l} and add. We obtain 



du^ ~ dv l r, dw l 



1 dt 1 l dt l l dt 1 



= X 1 8x l -}-Y l Si/ 1 -}- Z-fz^. (1^1) 



Now - Sx* = ( 



= - 

 ~~ dt 



d . 



= dt (U ^~ 

 Hence 



du, ~ dv. ^ dw, 



m^ * ox l -\- m l oy^ -\- TYI^ - 



dt dt dt 



= l \Tt ( U ^ X * + V 



= JT 1 &B 1 +r i 8y 1 + Z 1 & 1 , ( 152 ) 



by equation (151). 



