Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/18

 where $$H$$ represents terms containing only the second differential coefficients of $$A$$ with respect to the coordinates, and the first differential coefficients of the coordinates with respect to the time. Therefore, if we conceive of a variation affecting the accelerations of the particles at the time considered, but not their positions or velocities, we have

Comparing these equations with (2), we see that when the accelerations of the particles are regarded as subject to the variation denoted by $$\delta$$, but not their positions or velocities, the possible values of $$\delta \ddot{x}, \delta \ddot{y}, \delta \ddot{z}$$ are subject to precisely the same restrictions as the values of math>\delta x, \delta y, \delta z, when the positions of the particles are regarded as variable. We may, therefore, write for the general equation of motion regarding the positions and velocities of the particles as unaffected by the variation denoted by $$\delta$$,—a condition which may be expressed by the equations

We have so far supposed that the conditions which restrict the possible motions of the systems may be expressed by equations between the coordinates alone or the coordinates and the time. To extend the formula of motion to cases in which the conditions are expressed by the characters $$\leqq$$ or $$\geqq$$, we may write The conditions which determine the possible values of $$\delta \ddot{x}, \delta \ddot{y}, \delta \ddot{z},$$ will not, in such cases, be entirely similar to those which determine the possible values of $$\delta x, \delta y, \delta z,$$ when the coordinates are regarded as variable. Nevertheless, the laws of motion are correctly expressed by the formula (6), while the formula does not, as naturally interpreted, give so complete and accurate an expression of the laws of motion.

This may be illustrated by a simple example.

Let it be required to find the acceleration of a material point, which, at a given instant, is moving with given velocity on the