180 CELESTIAL MECHANICS. I. iii. 14. 



omit the terms R^r, R^r. . . , if we limit the variations to 

 the surfaces in which the bodies are compelled to move. 

 The equation (k) will then become 



0=zi:mS^s. (/) 



Hence it follows that, in the case of equilibrium, the 

 Bum of the products of the forces, into the elementary va- 

 riations of their directions, will be equal to nothing, pro- 

 vided that the conditions of the connexion of the system be 

 observed in those variations. 



It may be further shown that this theorem, which is here 

 demonstrated upon the supposition that the bodies are 

 united at invariable distances, is true in general, for all 

 conditions of the connexion of the different parts of the 

 system. In order to prove this, it will be sufficient to 

 show, that, observing these conditions, we have, in the 

 equation (^), Ozi2p8/'+Si?3r, since it will then follow 

 that S??J*S^«=:0 also. But it is clear that ^r, Sr. . . will 

 necessarily vanish when the variations are confined to the 

 given surfaces, and we have only to show that SpS/'z^O 

 under the same circumstances. 



Let us, therefore, conceive the system to be subjected 

 only to the forces p, p', p'\ . . . , and suppose the bodies to 

 be at liberty to move in obedience to them upon the given 

 surfaces : these forces may be resolved into others, some 

 of which ^, q\q"y,..t will act in the directions of/,/', 

 /",. . . ; which will destroy each other [as the forces p in 

 the former supposition, in virtue of the equality of action 

 and reaction], without producing any motion in the curves 

 in question ; others T, T', T", . . ., will be perpendicular 

 to the curves described ; and others again will be in the 

 directions of the tangents of those curves, and capable sepa- 

 rately of giving motion to the system : but it is easy to see 



