in Rectangular Order upon a Medium, 501 



If \, /jl, v are the direction-cosines of the displacement, 

 Z, m, n of the wave-normal, we may take 



g = \0, V = /*#, f = v0, 

 where 



Q __ e i(lx+my+nz— Yt)^ 



Thus 



d8/dx = — W(l\ + m/j, + iiv), &c, 



and the equations become 



o^W 2 = m^KJX + mix + nv), 

 a uN 2 = m 1 m(1\-\-mfL + nv), 

 cvV 2 = m^lX + mjA + nv), 



from which, on elimination of X : //- : v, we get 



V 2 = m/^ + ~+ £)=<tl* + Vm* + <*n*, . (78) 



V x y z' 



if a, bj c denote the velocities in the principal directions 



The wave-surface after unit time is accordingly the ellip- 

 soid whose axes are a, b, e. 



As an example, if the medium, otherwise uniform, be ob- 

 structed by rigid cylinders occupying a moderate fraction (p) 

 of the whole space, the velocity in the direction z, parallel to 

 the cylinders, is unaltered ; so that 



. c 2 = l, a 2 = b 2 = 1/(1 +p). 



In the application of our results to the electric theory of 

 light we contemplate a medium interrupted by spherical, or 

 cylindrical, obstacles, whose inductive capacity is different 

 from that of the undisturbed medium. On the other hand, 

 the magnetic constant is supposed to retain its value unr 

 broken. This being so, the kinetic energy of the electric 

 currents for the same total flux is the same as if there were 

 no obstacles, at least if we regard the wave-length as in- 

 finitely great*. And the potential energy of electric dis- 

 placement is subject to the same mathematical laws as the 

 resistance of our compound electrical conductor, specific 

 inductive capacity in the one question corresponding to 

 electrical conductivity in the other. 



Accordingly, if v denote the inductive capacity of the mate- 

 rial composing the spherical obstacles, that of the undisturbed 

 medium being unity, then the approximate value of fj? is 



* See Prof. "Willard Gibbs's ll Comparison of the Elastic and Electric 

 Theories of Light." Am. Journ. Sci. xxxv. (188S). 



