Page:A Treatise on Electricity and Magnetism - Volume 2.djvu/443

 828.] DYNAMICAL THEORY. 411

ments are functions of z and t only, so that -jj = y -^ and this expression is reduced to


 * 2i-So-

The kinetic energy per unit of volume, so far as it depends on the velocities of displacement, may now be written

��where p is the density of the medium.

827.] The components, X and Y, of the impressed force, referred to unit of volume, may be deduced from this by Lagrange s equa tions, Art. 564.

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These forces arise from the action of the remainder of the medium on the element under consideration, and must in the case of an isotropic medium be of the form indicated by Cauchy,

��828.] If we now take the case of a circularly-polarized ray for which = rcos(ntqz), r? = r sin (nt qz), (14)

we find for the kinetic energy in unit of volume

T = pr 2 n*-Cyr 2 q 2 n; (15)

and for the potential energy in unit of volume 7= /*(4 ) 02-42* + &c.)

= i*Q, (16)

where Q is a function of q 2.

The condition of free propagation of the ray given in Art. 820, equation (6), is dT d y

Tr=^ } (17)

which gives P n 2 -2Cyq 2 n = Q, (18)

whence the value of n may be found in terms of q.

But in the case of a ray of given wave-period, acted on by

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