OF THE MOTION OF A SYSTEM. 235 



system, being fixed, the part of the variation, not in- 

 cluded under the sign^^ must vanish for the whole paths, 

 so that we have from the equation (5), ^'£jm(pds:=z^Jmds^p 

 — X/mdt (Pdx+Qdy + Rdz); but the variation of the 

 equation (T), multiplied by dt, affords us ^'2jmvdtd<p-=z'2fin 

 dt (P^x + Q8?/ + 2^gz); or, 'Lfmvdtl(pzz'Zjml(pdszz'Zjmdt 

 (P^ir+QSy+jRSz) [since the variation of any quantity 

 is always the same as its fluxion, with the substitution 

 of the character of a variation for that of a fluxion : the 

 «teps, by which a variation and a fluxion are obtained, 

 being always identical and undistinguishable ; consequently 

 SS^^d^rzO. This equation corresponds to the law of least 

 action, in the natural relation of the force to the velocity, 

 since vi(p is the total force of the body m ; and the principle 

 implies that the sum of the fluents of the finite forces of all 

 the bodies of the system, combined with the elements of 

 the spaces described, is a minimum ; and in this form the 

 principle is applicable to any relation between the force 

 and the velocity, that can be supposed to be mathematically 

 possible. In the state of equilibrium, the sum of the forces, 

 multiplied by the elements of their lines of direction, disap- 

 pears, in consequence of the principle of virtual velocities ; 

 so that, in all cases, the same differential function, which 

 disappears in the state of equilibrium, becomes, after 

 taking the fluent, a minimum in the state of motion. 



