282 CELESTIAL MECHANICS. T. vii. 32. 



" Now three of these equations are immediately afforded 

 us by the principle of Dalembert. The velocities " lost" 

 during the instant Aty by the particle subjected to the 

 action of the forces X, Y, and Z, are XAt — am, YM — av, 

 and ZAt — aw; for Au, Av, and aw, express the augmenta- 

 tions of velocity which really take place in the given in- 

 stant, and XAt, YAty and ZAt, those which would be pro- 

 duced by the forces X, Y, and Z, if the particle were free 

 and insulated. These supposed velocities, divided by At, 

 will give the measures of the forces capable of producing 

 them ; and calling the quotients X', Y', Z', we have 



r-^- 



'* Now, according to the principle in question, the fluid 

 mass would be in equilibrium, if all the particles were actu- 

 ated by forces capable of communicating to them the 

 velocities lost or gained at each instant; [or in other words 

 the unemployed forces of the whole system must hold each 

 in equilibrium:] we may therefore satisfy the general con- 

 ditions of equilibrium by considering X' Y^ and Z' as the 

 forces, parallel to the coordinates, acting on each particle, 

 instead of X, Y, and Z, which represent the whole forces 

 in those directions. Hence we have 



4^= fX'; ^iz P Y'; and ^=:pZ': or substi- 

 dx dy dz 



tuting for these quantities, and dividing by f ; 



