116 CELESTIAL MECHANICS. I. U. 7, B. 



Since, in the case of equilibrium, 0=.lS^s, when there 

 is no motion, and since any uncompensated force must be 

 employed in producing- an increase or diminution of the 

 elocity proportional to its magnitude (230) ; it follows 

 that so much of the force, in the direction x, as is other- 

 wise uncompensated, must be employed in producing a 

 change of the velocity v expressed by du, in the elementary 

 portion of time expressed by d^ : and if the force be called 

 F, or more properly Pdt, because its effect depends on 

 the elementary portion of time in which it is supposed to 

 act, the unemployed portion may be called Pdt — dvzz 



Pdf — d T— ; and the same law will hold good, with res- 

 d^ ° 



pect to the portions of any number of forces thus remaining 

 unemployed, as if the moving point remained at rest. Con- 

 sequently the equation 0=i1aS'^5 (251) will afford us, chang- 

 ing the signs, 0:=:^x (d ■£—Pdi) + hj (d -£— Qdt) -h^z 



dz 

 (d Rdt) ; (f)' We have also, when the body is at 



_ dda: _ ddy , ^ ddz 

 hberty,P=— .<?=^ and K = ^. 



Scholium. We must here carefully distinguish the 

 arbitrary variations ^.r, ^^, cz, from the fluxions dx, dy, d^, 

 the former being subject to no conditions whatever, pro- 

 vided that all the forces concerned be comprehended in the 

 equation, while the latter are confined to the expression 

 of the actual motion of the body M. 



Corollary 1. We are, however, at hberty to as- 

 sume the variations as equivalent to the fluxions, and to 

 substitute dx, dy, and dz, for ^x, cy, and ^z, and in this 



dxddx 

 case the equation will become 0= — r- Pdtdx -f 



