Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/418

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386
ELECTROMAGNETIC THEORY OF LIGHT.
[784.

Propagation of Undulations in a Non-conducting Medium.

784.] In this case , and the equations become


(9)



The equations in this form are similar to those of the motion of an elastic solid, and when the initial conditions are given, the solution can be expressed in a form given by Poisson [1], and applied by Stokes to the Theory of Diffraction[2].

Let us write


(10)


If the values of , , , and of , , are given at every point of space at the epoch (), then we can determine their values at any subsequent time, , as follows.

Let be the point for which we wish to determine the value of at the time . With as centre, and with radius , describe a sphere. Find the initial value of at every point of the spherical surface, and take the mean, , of all these values. Find also the initial values of at every point of the spherical surface, and let the mean of these values be .

Then the value of at the point , at the time , is


(11)


785.] It appears, therefore, that the condition of things at the point at any instant depends on the condition of things at a distance and at an interval of time previously, so that any disturbance is propagated through the medium with the velocity .

Let us suppose that when is zero the quantities and are

  1. Mem. de l' Acad., tom, iii, p. 130.
  2. Cambridge Transactions, vol. ix, p. 10 (1850).