196 DOUBLE REFRACTION AND CIRCULAR POLARIZATION 



where I denotes the wave-length, p the period of oscillation, u the 

 distance of the point considered from the wave-plane passing through 

 the origin, a t , fa, y l the amplitudes of the displacements rj, f in 

 the wave-plane passing through the origin, and a 2 , /3 2 , y 2 their 

 amplitudes in a wave-plane one-quarter of a wave-length distant 

 and on the side toward which u increases. If we also write L, M, 

 for the direction-cosines of the wave-normal drawn in the direction 

 in which u increases, we shall have the following necessary relations : 



^ = 0, La 2 +M/3 2 +Ny 2 =0. 



4. That the irregular part of the displacement (, rf t f ) at ani 

 given point is a simple harmonic function of the time, having the 

 same period and phase as the regular part of the displacement ( rj, 

 may be proved by the single principle of superposition of motioi 

 and is therefore to be regarded as exact in a discussion of this kin< 

 But the further conclusion of the preceding paper ( 4), "that the 

 values of ', r[ t f at any given point in the medium are capable 

 expression as linear functions of q, f in a manner which shall 

 independent of the time and of the orientation of the wave-planes and 

 the distance of a nodal plane from the point considered, so long as the 

 period of oscillation remains the same," is evidently only approxima- 

 tive, although a very close approximation. A very much closer 

 approximation may be obtained, if we regard ', rf , f ', at any given 

 point of the medium and for light of a given period, as linear functions 

 f n> f an d th e n i ne differential coefficients 



dx' dx y dx' 



s- pir 



-i , CIA^. 



dy' 



We shall write r\, f and diff. coeff. to denote these twelve quantiti( 



From this it follows immediately that with the same degree of 

 approximation ', /)', f ' may be regarded, for a given point of the 

 medium and light of a given period, as linear functions of 7), f an( 

 the differential coefficients of r\, f with respect to the coordinal 

 For these twelve quantities we shall write , r\, f and diff. coeff. 



5. Let us now proceed to equate the statical energy of the medium 

 at an instant of no velocity with its kinetic energy at an instant of 

 no displacement. It will be convenient to estimate each of these 

 quantities for a unit of volume. 



6. The statical energy of an infinitesimal element of volume may be 

 represented by cr dv, where <r is a quadratic function of the components 

 of displacement + ', r\ + */', f + f '. Since for that element of volume 

 ', rf, f ' may be regarded as linear functions of 77, f and diff. coeff., 



