Page:Scientific Papers of Josiah Willard Gibbs - Volume 2.djvu/229

Rh If, then, $$\xi, \eta, \zeta$$ denote the components of the actual displacement at the point considered, will represent the average values of these components in the small sphere about that point. These average values we shall treat as functions of the coordinates of the center of the sphere and of the time, and may call them, for brevity, the average values of $$\xi, \eta, \zeta.$$ But however they may be designated, it is essential to remember that it is a space-average for a certain very small space, and never a time-average, that is intended.

The object of this paper will be accomplished when we have expressed (explicitly or implicitly) the relations which subsist between the values of $$[\xi]_{\text{Ave}}, [\eta]_{\text{Ave}}, [\zeta]_{\text{Ave}}$$ at different times and in different parts of the field,—in other words, when we have found the conditions which these quantities must satisfy as functions of the time and the coordinates.

3. Let us suppose that luminous vibrations of any one period are somewhere excited, and that the disturbance is propagated through the medium. The motions which are excited in any part of the medium, and the forces by which they are kept up, will be expressed by harmonic functions of the time, having the same period, as may be proved by the single principle of the superposition of motions quite independently of any theory of the constitution of the medium, or of the nature of the motions, as electrical or otherwise. This is equally true of the actual motions, and of the averages which we are to consider. We may therefore set