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

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ON THE FUNDAMENTAL FORMULÆ 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 ${\frac {d{\dot {x}}}{dt}},{\frac {d{\dot {y}}}{dt}},{\frac {d{\dot {z}}}{dt}}$ 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.^[1]

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

↑ 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 demonstration of the principle in the first section of the Mécanique 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 objecté, avec raison, à cette assertion de Lagrange l'example d'un point pesant en équilibre au sommet le plus élevé d'une courbe; il est évident qu'un déplacement infiniment petit le ferait descendre, et, pourtant, ce déplacement ne se produit pas." (Mécanique Analytique, troiséme édition, tome 1, page 22, note de M. Bertrand.) The value of $z$ (the height of the point above a horizontal plane) can certainly be diminished by a displacement of the point, but the value of i 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.

[1] 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 demonstration of the principle in the first section of the Mécanique 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 objecté, avec raison, à cette assertion de Lagrange l'example d'un point pesant en équilibre au sommet le plus élevé d'une courbe; il est évident qu'un déplacement infiniment petit le ferait descendre, et, pourtant, ce déplacement ne se produit pas." (Mécanique Analytique, troiséme édition, tome 1, page 22, note de M. Bertrand.) The value of $z$ (the height of the point above a horizontal plane) can certainly be diminished by a displacement of the point, but the value of i 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.

[1]