184 DOUBLE REFRACTION AND THE DISPERSION OF 



reducible to this. Then, with the origin of coordinates and the zero 

 of time suitably chosen, the regular part of the displacement may be 

 represented by the equations 



= a cos 2?r r cos 2?r 

 I 



u t 



= 8 COS 27T r COS 27T- , 



I p 



p 



(1) 



p 



r = y COS 27T T COS 27T- 



l p 



where I denotes the wave-length, p the period of vibration, a, /3, y 

 the maximum amplitudes of the displacements 77, f, and u the 

 distance of the point considered from the wave -plane which passes 

 through the origin. Since u is a linear function of x, y, and z, we 

 may regard these equations as giving the values of ;/, f, for a given 

 system of waves, in terms of x, y, z, and . 



4. The components of the irregular displacement, ', ^', ', at any 

 given point, will evidently be simple harmonic functions of the time, 

 having the same period as the regular part of the displacement. That 

 they will also have the same phase is not quite so evident, and would 

 not be the case in a medium in which there were any absorption or 

 dispersion of light. It will however appear from the following con- 

 siderations that in perfectly transparent media the irregular oscil- 

 lations are synchronous with the regular. For if they are not 

 synchronous, we may resolve the irregular oscillations into two parts. 

 of which one shall be synchronous with the regular oscillations, and 

 the other shall have a difference of phase of one-fourth of a complete 

 oscillation. Now if the medium is one in which there is no absorption 

 or dispersion of light, we may assume that the same electrical con- 

 figurations may also be passed through in the inverse order, which 

 would be represented analytically by writing t for t in the equations 

 which give rj, f, (', q ', f, as functions of x, y, z, and t. But this 

 change would not affect the regular oscillations, nor the synchronous 

 part of the irregular oscillations, which depends on the cosine of the 

 time, while the non-synchronous part of the irregular oscillations, 

 which depends on the sine of the time, would simply have its 

 direction reversed. Hence, by taking first one-half the sum, and 

 secondly one-half the difference, of the original motion and that 

 obtained by substitution of t for t, we may separate the non- 

 synchronous part of the irregular oscillations from the rest of the 

 motion. Therefore, the supposed non-synchronous part of the irregular 

 displacement, if capable of existence, is at least wholly independent 

 of the wave-motion and need not be considered by us. 



We may go farther in the determination of the quantities f , r\ t f ' 

 For in view of the very fine-grained structure of the medium, it will 



