182 CELESTIAL MECHANICS. I. iii. 14. 



law of virtual velocities is observed, the sys- 

 tem must remain in equilibrium. 



For if it were otherwise, and the points ?w, m', . . . , ac- 

 quired the increments of velocity v, v\ . . . , while ^mS^s 

 remain edmO, the system wonld be held in equilibrium by 

 the forces mS, 771 S\ diminished by the forces expended on 

 the velocities, which may be called viv, m'v\ . . . [making 

 the increment of time unity] ; and if we call the variations 

 in the directions of these forces ^v, ^v, . . . , we shall have, 

 by the proposition, Ozz^mS^s — ^mv^v: and since ^mS^s 

 =zO, we have also Ozz^mv^v. But as the variations ^v, 

 Su', must be subject to the conditions of the system, we 

 may suppose them equal to vdt, or to v, and we have then 

 0=:Smt;", which can only happen when v=:0, v'~0,. .. 

 since all squares are positive : it follows, therefore, that the 

 system must remain at rest in consequence of the forces 

 mS, m'S'y . . . , alone. 



Scholium. The conditions of the connexion of the 

 different parts of a system with each other may always be 

 reduced to equations between the coordinates of the diffe- 

 rent bodies concerned. Suppose these equations to be 

 M=0, u'zuO, w"=:0, we may always add to the equation 

 {izz^mS^s (I) the quantity S^Sm, the functions aSm, a'§m',... 

 of which it is the sum, being* dependent on the coordinates, 

 [and of such a nature as to substitute an expression de- 

 rived from them for the variations of the perpendiculars to 

 the surfaces and for those of the distances of the bodies (245, 

 Sch. 3)]; the equation will then become 0:=z^mS^s -\-^y^du. 

 In this case the variations of all the coordinates will be 

 arbitrary,and their coefficients maybe separately made equal 

 to nothing, which will give as many different equations for 

 the determination of a and x'. If we compare this equa- 



