DOUBLE REFRACTION, ETC. 183 



by supposing that we have to do with a system of stationary waves. 

 That the relation of the wave-length and the period is the same for 

 stationary as for progressive waves is evident from the consideration 

 that a system of stationary waves may be formed by two systems of 

 progressive waves having opposite directions. 



2. Let x, y, z be the rectangular coordinates of any point in the 

 medium, which with the system of waves we may regard as inde- 

 finitely extended, and let (+(', ij + rf, f+f e the components of 

 electrical displacement at that point at the time t ; r\, f being the 

 average values of the components of electrical displacement at that 

 time in a wave-plane passing through the point. Then rj, f, ', r(, f ' 

 are perfectly defined quantities, of which q, f are connected with 

 x, y, z, and t by the ordinary equations of wave-motion, while each 

 of the quantities ', r\ , f ' has always zero for its average value in any 

 wave-plane. We may call T/, f the components of the regular part 

 of the displacement, and ', if ', f ' the components of the irregular 

 part of the displacement. In like manner, the differential coefficients 

 of these quantities with respect to the time, T/, f, f, if ', f', may be 

 called respectively the components of the regular part of the flux, 

 and the components of the irregular part of the flux. 



Let the whole space be divided into elements of volume Dv, very 

 small in all dimensions in comparison with a wave-length, but 

 enclosing portions of the medium which may be treated as entirely 

 similar to one another, and therefore not infinitely small. Thus a 

 crystal may be divided into elementary parallelepipeds, all the vertices 

 of which are similarly situated with respect to the internal structure 

 of the crystal. Amorphous solids and liquids may not be capable of 

 division into equally small portions of which physical similarity can 

 be predicated with the same rigor. Yet we may suppose them capable 

 of a division substantially satisfying the requirements. 



From these definitions it follows that at any given instant the 

 average value of each of the quantities ', rf ', f ' in an element Dv is 

 zero. For the average value in one such element must be sensibly 

 the same as in any other situated on the same wave-plane. If this 

 average were not zero, the average for the wave-plane would not be 

 zero. Moreover, at any given instant, the values of tj, f may be 

 regarded as constant throughout any element Dv, and as representing 

 the average values of the components of displacement in that element. 

 The same will be true of the quantities ', jj', f ' and , TJ, f 



3. Since we have excluded the case of media which have the pro- 

 perty of circular polarization, we shall not impair the generality of 

 our results if we suppose that we have to do with linearly polarized 

 light, i.e., that the regular part of the displacement is everywhere 

 parallel to the same fixed line, all cases not already excluded being 



