Opacity of Tourmaline Crystals. 121 



propagation of electric displacements in the medium are repre- 

 sented. So the equations (3) become 



. n d¥ d?F 



A n da d 2 G A 



4^0,-^-^=0, 



47r^ 2 C 3 -^- =0, 



(8) 



which are not the equations of any kind of wave-propagation, 

 but represent as going on in the plane (xy) the diffusion of 

 electricity by conduction through the medium at logarithmic 

 rates whose values along the x and y axes are proportional to 

 —fjbfii and —fi 2 ^2 respectively; that is to say, the disturbances 

 will be propagated just in the same fashion as heat would be 

 in a medium whose thermal conductivities and specific heats 

 had different values in different directions. The spread of 

 electricity in this case resembles therefore that of the spread 

 of heat in a thin film when the heat starts at a point and dif- 

 fuses around. In its generality the case is therefore compa- 

 rable, as Maxwell points out, to that of which Fourier gave 

 the complete solution (art. 384, i Analytical Theory of Heat,' 

 p. 382, Freeman's translation), for the diffusion of heat in all 

 directions, where the temperature is represented by the triple 

 integral 



in which, if we write r = \/{u — x) 1 + (/3 — yf + (7 — e) 2 , the ex- 

 ponential e- r2 / 4kt will represent the value contributed to the 

 mean temperature at the point (oc, y, z) whose distance from 

 the origin is r. In the present case of diffusion of electric 

 currents in the plane (xy), a similarly constructed double inte- 

 gral may be employed to represent the distribution of vector- 

 potential, the quantity symbolized by h (the coefficient of the 



intrinsic rate of diffusion) being replaced by j- ~ and 



-. rr in the exponential. 



4:7T/Jb 2 Kj 2 r 



Now, where the whole energy of the electromagnetic dis- 

 turbances is thus converted in the conducting medium into 

 the energy of currents diffused through its substance, no wave 

 will be propagated ; or, as Maxwell has shown, the substance 

 will be opaque to light, the energy of the diffused currents 



