Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/23

 The reader will remark the strict analogy between this formula and (6), which would perhaps be more clearly exhibited if we should write $$\frac{d\dot{x}}{dt}, \frac{d\dot{y}}{dt}, \frac{d\dot{z}}{dt}$$ for $$\ddot{x}, \ddot{y}, \ddot{z}$$ that formula.

But these formulæ 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 $$\Delta \dot{x}, \Delta \dot{y}, \Delta \dot{z}$$ 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 $$\delta \Delta \dot{x}, \delta \Delta \dot{y}, \delta \Delta \dot{z}.$$ The formula (9) may therefore be regarded as the most direct application 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 accelerations, when we wish to determine whether certain accelerations, specified by $$\ddot{x}, \ddot{y}, \ddot{z},$$ 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