1881.] the Electromagnetic Theory of Light. 159 



,, c d 2 a . f 1 dv 1 diu 



therefore ^=^| :g ,____ 



for v = — , etc. ; 



, f cPa _ 1 d /da _ dc\ 1 ^ /(#> c£a\ 



dtf /j,K 2 dr \dz dxj fiK 3 dy \dx dy) ' 

 Putting a, b, c for the principal coefficients, 



d 2 a _ j- 2 d z a -„ tf « ^- 2 rf 2 c - 2 cf 6 

 d£ 2 cfo 1 ' c/y cfcc&c dxdy' 



and two similar equations. These equations have been given by 

 J. J. Thomson for an isotropic medium in his paper already re- 

 ferred to. From these we can shew that the magnetic induction 

 is propagated with a velocity given by the same construction while 

 its direction also satisfies Fresnel's formula. It is interesting to 

 notice that the equations for the magnetic induction and the elec- 

 tric displacement are different in form, though they lead both to 

 Fresnel's construction for the velocity of propagation. 



It is frequently necessary to determine the magnetic force in 

 terms of the electric displacement. 



If a, /3, 7 be the components of magnetic force, 



— = 4tt Ip^-c 2 — I etc 



dt 4?r T dz ° dy] ' etCl ' 



f=\S, etc., 

 S being a function of 



Ix + my + nz — Vt= w, 

 and X, fju, v direction cosines of electric displacement, 



dg _ dS _ n dS 

 dz diu V dt 



4?r f#> -i 1 o 



-rp- -io fxn — cvm)- o, 



and two similar equations. 



Multiply by I, m, n respectively, and add, the sum is zero. 

 Thus the direction of magnetic force is in the wave front. 

 Multiply by \, /x, v respectively, and add, then since 



1 ( 6* _ c -) + 5 (c * _ #j + » (J _ p) = o, 



A, jjb v 



the sum is again zero. Thus the direction of the magnetic force is 

 perpendicular to the electric displacement. 



