ON THE FUNDAMENTAL FORMULA OF DYNAMICS. 9 



the masses of the particles from those which contain the forces, we 

 have V(X8x+ Yoy+ZSz)-2[$mS(x*+y*+z*)]^0, (10) 



or, if we write u for the acceleration of a particle, 



2(X8x+ YSy + ZS^-SZ^mu 2 )^. (11) 



If, instead of terms of the form XSx, or in addition to such terms, 

 equation (1) had contained terms of the form PSp, in which p denotes 

 any quantity determined by the configuration of the system, it is 

 evident that these would give terms of the form PSp in (6), (10) and 

 (11). For the considerations which justified the substitution of Sx, 

 Sy, Sz for Sx, Sy, Sz in the usual formula were in no respect dependent 

 upon the fact that x, y, z denote rectangular coordinates, but would 

 apply equally to any other quantities which are determined by the 

 configuration of the system. 



Hence, if the moments of all the forces of the system are represented 

 by the sum *(Pdp), 



the general formula of motion may be written 



(P<$p)-<52(im 2 )^0 (12) 



If the forces admit of a force-function V, we have 



or (?[F-2(imu 2 )]^0. (13) 



But if the forces are determined in any way whatever by the 

 configuration and velocities of the system, with or without the time, 

 X, F, Z and P will be unaffected by the variation denoted by S, and 

 we may write the formula of motion in the form 



<S2(X+F#+^-Jmu 2 )^0, (14) 



or <$[(P)-2(imo, 2 )]^0. (15) 



If the forces are determined by the configuration alone, or the 

 configuration and the time, SX = 0, SY=0, SZ=0, SP = Q, and the 

 general formula may be written 



S 2(Xx+ Fy+^)-2(|mu 2 ) < 0, (16) 



or 5(P^)-2(Jmt*)^0. (17) 



The quantity affected by S in any one of the last five formulas 

 has not only a maximum value, but absolutely the greatest value 

 consistent with the constraints of the system. This may be shown 

 in reference to (15) by giving to p, x, y, z, contained explicitly 

 or implicitly in the expression affected by 8, any possible finite 



