Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/27

 any function of the velocity, the terms due to that resistance in the general formula of motion may be expressed in the form where $$v$$ denotes the velocity and $$\phi (v)$$ the resistance. But The terms due to the resistance reduce, therefore, to   where $$f$$ denotes the primitive of the function denoted by $$\phi$$.

Discontinuous Changes of Velocity.—Formula (9), which relates to discontinuous changes of velocity, is capable of similar transformations. the formula reduces to where $$X, Y, Z$$ are to be regarded as constant. If $$\mathfrak{S}(\text{P}dp)$$ represents the sum of the moments of the impulsive forces, and we regard $$\text{P}$$ as constant, we have The expressions affected by $$\delta$$ in these formulæ have a greater value than they would receive from any other changes of velocity consistent with the constraints of the system.

The principles which have been established furnish a convenient point of departure for the demonstration of various properties of motion relating to maxima and minima. We may obtain several such properties by considering how the accelerations of a system, at a given instant, will be modified by changes of the forces or of the constraints to which the system is subject. Let us suppose that the forces $$X, Y, Z$$ of a system receive the increments $$X', Y', Z'$$, in consequence of which, and of certain additional constraints, which do not produce any discontinuity in the velocities, the components of acceleration $$\dot{x}, \dot{y}, \dot{z}$$ receive the increments $$\ddot{x}, \ddot{y}, \ddot{z}$$. The expression will be the greatest possible for any values of $$\ddot{x}, \ddot{y}, \ddot{z}$$ consistent with the constraints. But this expression may be divided into three parts,