128 The Luminiferous Medium ^ 



independently ; so the components of the force of restitution are 



COS a COS )3 COS y 

 l ft 3 



This resultant force has not in general the same direction 

 as the displacement which produced it ; but it may always 

 he decomposed into two other forces, one parallel and the other 

 perpendicular to the direction of the displacement ; and the 

 former of these is evidently 



The surface 



COS 2 a COS 2 )3 COS 2 7 



I {_ I _ 



fl 2 3 



X 2 V* 



i 2 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 com- 

 ponent 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 



