242
COMPARISON OF THE ELECTRIC THEORY OF LIGHT
case. Therefore, in finding the differential equation between
and
we may treat
and
in (24) and
and
in (25) as constant, as well as
and
These equations may be written
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Differentiating, we get
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or
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Hence, if we write
for the wave-velocity
,
for the index of refraction, and
for the wave-length in vacuo, we have for the ratio of the two parts into which we have divided the potential energy on the elastic theory,
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(26)
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and for the ratio of the two parts into which we have divided the kinetic energy on the electrical theory,
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(27)
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It is interesting to see that these ratios have the same value. This value may be expressed in another form, which is suggestive of some important relations. If we write
for what Lord Rayleigh has called the velocity of a group of waves,[1]
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(28)
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It appears, therefore, that in the elastic theory that part of the potential energy which depends on the deformation expressed
- ↑ See his "Note on Progressive Waves," Proc. Lond. Math. Soc., vol. ix, No. 125, reprinted in his Theory of Sound, vol. ii, p. 297.