Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/24

 and reactions of the system. The formula (6) expresses a criterion of this kind in the most simple and direct manner. If we regard a force as a tendency to increase a quantity expressed by $$\ddot{x}$$, the product of the force by $$\delta \ddot{x}$$ is the natural measure of the extent to which this tendency is satisfied by an arbitrary variation of the accelerations. The principle expressed by the formula may not be very accurately designated by the words virtual velocities, but it certainly does not differ from the principle of virtual velocities (in the stricter sense of the term), more than this differs from that of virtual displacements,—a difference so slight that the distinction of the names is rarely insisted upon, and that it is often very difficult to tell which form of the principle is especially intended, even when the principle is enunciated or discussed somewhat at length.

But, although the formulæ (6) and (9) differ so little from the ordinary formulæ, they not only have a marked advantage in respect of precision and accuracy, but also may be more satisfactory to the mind, in that the changes considered (to which $$\delta$$ relates), are not so violently opposed to all the possibilities of the case as are those which are represented by the variations of the coordinates. It may have seemed to some readers of the Mécanique Analytique—a work of which the unity of method is one of the most striking oharaoteristios, and that to whioh its universally recognized artistic merit is in great measure due—that the treatment of dynamical problems in that work is not entirely analogous to the treatment of statical problems. The statical question, whether a system will remain in equilibrium in a given configuration, is determined by Lagrange by considering all possible motions of the system and inquiring whether there is any reason why the system should take any one of them. A similar method in dynamics would be based upon a comparison of a proposed motion with all other motions of which the system is capable without violating its kinematical conditions. Instead of this, Lagrange virtually reduces the dynamical problem to a statical one, and considers, not the possible variations of the proposed motion, but the motions which would be possible if the system were at rest. This reduction of a given problem to a simpler one, which has already been solved, is a method which has its advantages, but it is not the characteristic method of the Mécanique Anaiytique. That which most distinguishes the plan of this treatise from the usual type is the direct application of the general principle to each particular case. The point is perhaps of small moment, and may be differently regarded by others, but it is mentioned here because it was a feeling of this kind (whether justified or not) and the desire to express the formula of motion by means of a maximum or minimum condition, in which the conditions under which the maximum or minimum subsists should be such as the problem naturally affords (Gauss's principle of least constraint being at the time unknown to the present writer, and the conditions under which the minimum subsists in the principle of least action being such that that is hardly satisfactory as a fundamental principle), which led to the formula proposed in this paper. Moreover, as we shall see, they naturally lead to various important laws of motion.

Let us now consider some of the transformations of which our general formula (6) is capable. If we separate the terms containing