Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/21

That is, if ${\dot {x}}=0,{\ddot {x}}$ has the greater of the values ${\frac {X}{m}}$ and 0; otherwise, ${\ddot {x}}={\frac {X}{m}}$.

In cases of this kind also, in which the function which cannot exceed a certain value involves the velocities (with or without the coordinates), one may easily convince himself that formula (6) is always valid, and always sufficient to determine the accelerations with the aid of the conditions afforded by the constraints of the system.

But instead of examining such cases in detail, we shall proceed to consider the subject from a more general point of view.

Comparison of the New Formula with the Statical Principle of Virtual Velocities.—Case of Discontinuous Changes of Velocity.

Formula (1) has so far served as a point of departure. The general validity of this, the received form of the indeterminate equation of motion, being assumed, it has been shown that formula (6) will be valid and sufficient, even in cases in which both (1) and (7) fail. We now proceed to show that the statical principle of virtual velocities, when its real signification is carefully considered, leads directly to formula (6), or to an analogous formula for the determination of the discontinuous changes of velocity, when such occur. This will be the case even if we start with the usual analytical expression of the principle

$\textstyle \sum \displaystyle (X\delta x+Y\delta y+Z\delta z)\geqq 0,$

(8)

to which, at first sight, formula (6) appears less closely related than (7). For the variations of the coordinates in this formula must be regarded as relating to differences between the configuration which the system has at a certain time, and which it will continue to have in case of equilibrium, and some other configuration which the system might be supposed to have at some subsequent time. These temporal relations are not indicated explicitly in the notation, and should not be, since the statical problem does not involve the time in any quantitative manner. But in a dynamical problem, in which we take account of the time, it is hardly natural to use $\delta x,\delta y,\delta z$ in the same sense. In any problem in which $x,y,z$ are regarded as functions of the time, $\delta x,\delta y,\delta z$ are naturally understood to relate to differences between the configuration which the system has at a certain time, and some other configuration which it might (conceivably) have had at that time instead of that which it actually had.

Now when we suppose a point to have a certain position, specified by $x,y,z$, at a certain time, its position at that time is no longer a subject of hypothesis or of question. It is its future positions which