Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/148

independently; so the components of the force of restitution are

${\frac {\cos \alpha }{\epsilon _{1}}},{\frac {\cos \beta }{\epsilon _{2}}},{\frac {\cos \gamma }{\epsilon _{3}}}$.

This resultant force has not in general the same direction as the displacement which produced it; but it may always be decomposed into two other forces, one parallel and the other perpendicular to the direction of the displacement; and the former of these is evidently

${\frac {\cos ^{2}\alpha }{\epsilon _{1}}},{\frac {\cos ^{2}\beta }{\epsilon _{2}}},{\frac {\cos ^{2}\gamma }{\epsilon _{3}}}$.

The surface

$\gamma ^{4}={\frac {x^{2}}{\epsilon _{1}}}+{\frac {y^{2}}{\epsilon _{2}}}+{\frac {z^{2}}{\epsilon _{3}}}$

will therefore have the property that the square of its radius vector in any direction is proportional to the component in that direction of the elastic force due to a unit displacement in that direction: it is called the surface of elasticity.

Consider now a displacement along one of the axes of the section on which the surface of elasticity is intersected by the plane of the wave. It is easily seen that in this case the component of the elastic force at right angles to the displacement acts along the normal to the wave-front; and Fresnel assumes. that it will be without influence on the propagation of the vibrations, on the ground of his fundamental hypothesis that the vibrations of light are performed solely in the wave-front. This step is evidently open to criticism; for in a dynamical theory everything should be deduced from the laws of motion without special assumptions. But granting his contention, it follows that such a displacement will retain its direction, and will be propagated as a plane-polarized wave with a definite velocity.

Now, in order that a stretched cord may vibrate with unchanged period, when its tension is varied, its length must be increased proportionally to the square root of its tension; and similarly the wave-length of a luminous vibration of given period is proportional to the square root of the elastic force (per unit