14 ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 



of which we can perfectly define any infinitesimal change in the 

 configuration of the system ; and let 



da)-, dwn 



^w * = -dt' etc - 



where dw lt du> z are to be determined by the change in the con- 

 figuration in the interval of time dt ; and let 



etc - 



Also let i7 



It is evident that U can be expressed in terms of cb 1 , o> 2 , etc., 

 t^, oig, etc., and the quantities which express the configuration of the 

 system, and that (since S is used to denote a variation which does 

 not affect the configuration or the velocities), 



Moreover, since the quantities p in the general formula are entirely 

 determined by the configuration of the system 



dp dp , 

 p = -f- a)! + -f- ft> 2 + etc., 



where -*- denotes the ratio of simultaneous values of dp and dto,, 



OCOj -. 



when dco 2 , etc. are equal to zero, and 3-, etc. are to be interpreted 



<Xft) 2 

 on the same principle. Multiplying by P, and taking the sum with 



respect to the several forces, we have 



(Pp) O^ 



where Q = lP Q=lP etc. 



If we differentiate with respect to t, and take the variation denoted 

 by S, we obtain 



.. 



w 2 + etc. 



The general formula (12) is thus reduced to the form 



2 + etc. ^ 0. (26) 



If the forces have a potential F, we may write 

 dV dU\. dV 



/<m 



etc., (27) 



2 



where -, denotes the ratio of dV and dot, when dw*, etc. have the 



value zero, and the analogous expressions are to be interpreted on 

 the same principle. 



