Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/19

 frictionless surface of a body (which it cannot penetrate, but which it may leave), and is acted on by given forces. For simplicity, we may suppose that the normal to the surface, drawn outward from the moving point at the moment considered, is parallel to the axis of $$X$$ and in the positive direction. The only restriction on the values of $$\delta x, \delta y, \delta z$$ is that Formula (7) will therefore give  The condition that the point shall not penetrate the body gives another condition for the value of $$\ddot{x}$$. If the point remains upon the surface, $$\ddot{x}$$ must have a certain value $$N$$, determined by the form of the surface and the velocity of the point. If the value of $$\ddot{x}$$ is less than this, the point must penetrate the body. Therefore, But this does not suffice to determine the acceleration of the point.

Let us now apply formula (6) to the same problem. Since $$\ddot{x}$$ cannot be less than $$N$$, This is the only restriction on the value of $$\delta \ddot{x}$$, for if $$\ddot{x} > N$$, the value of $$\delta \ddot{x}$$ is entirely arbitrary. Formula (6), therefore, requires that

—that is (since $$\ddot{x}$$ cannot be less than $$N$$), that $$\ddot{x}$$ shall be equal to the greater of the quantities $$N$$ and $$\frac{X}{m}$$, or to both, if they are equal,—and that The values of $$\ddot{x}, \ddot{y}, \ddot{z}$$ are therefore entirely determined by this formula in connection with the conditions afforded by the constraints of the system. The following considerations will show that what is true in this case is also true in general, when the conditions to which the system