ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 7 



The reader will remark the strict analogy between this formula 

 and (6), which would perhaps be more clearly exhibited if we should 



write -jT, -, -T7 for x, y, z in that formula. 



But these formulae may be established in a much more direct 

 manner. For the formula (8), although for many purposes the most 

 convenient expression of the principle of virtual velocities, is by no 

 means the most convenient for our present purpose. As the usual 

 name of the principle implies, it holds true of velocities as well 

 as of displacements, and is perhaps more simple and more evident 

 when thus applied.* 



If we wish to apply the principle, thus understood, to a moving 

 system so as to determine whether certain changes of velocity 

 specified by Asc, Ay, Az are those which the system will really receive 

 at a given instant, the velocities to be multiplied into the forces and 

 reactions in the most simple application of the principle are manifestly 

 such as may be imagined to be compounded with the assumed 

 velocities, and are therefore properly specified by SAx, SAy, SAz. 

 The formula (9) may therefore be regarded as the most direct appli- 

 cation of the principle of virtual velocities to discontinuous changes 

 of velocity in a moving system. 



In the case of a system in which there are no discontinuous changes 

 of velocity, but which is subject to forces tending to produce accelera- 

 tions, when we wish to determine whether certain accelerations, 

 specified by x, y, z, are such as the system will really receive, it is 

 evidently necessary to consider whether any possible variation of 

 these accelerations is favored more than it is opposed by the forces 



*Even in Statics, the principle of virtual velocities, as distinguished from that of 

 virtual displacements, has a certain advantage in respect of its evidence. The demon- 

 stration of the principle in the first section of the Me"canique Analytique, if velocities 

 had been considered instead of displacements, would not have been exposed to an 

 objection, which has been expressed by M. Bertrand in the following words: "On a 

 object^, avec raison, a cette assertion de Lagrange 1'example d'un point pesant en 

 ^quilibre au sommet le plus elev4 d'une courbe ; il est Evident qu'un defacement 

 infiniment petit le ferait descendre, et, pourtant, ce de"plaeement ne se produit pas." 

 (Mdcanique Analytique, troiseme Edition, tome 1, page 22, note de M. Bertrand.) The 

 value of 2 (the height of the point above a horizontal plane) can certainly be diminished 

 by a displacement of the point, but the value of 2 is not affected by any velocity given 

 to the point. 



The real difficulty in the consideration of displacements is that they are only possible 

 at a time subsequent to that in which the system has the configuration to which the 

 question of equilibrium relates. We may make the interval of time infinitely short, 

 but it will always be difficult, in the establishing of fundamental principles, to treat a 

 conception of this kind (relating to what is possible after an infinitesimal interval of 

 time) with the same rigor as the idea of velocities or accelerations, which, in the cases 

 to which (9) and (6) respectively relate, we may regard as communicated immediately to 

 the system. 



