J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’

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                                3 Propagation of Electromagnetic Disturbances 
in a Crystallised Medium 

[(1]02) Let us now calculate the conditions of propagation of a 
plane wave in a medium for which the values of k and [mu] 
are different in different directions. As we do not propose to 
give a complete investigation of the question in the present 
imperfect state of the theory as extended to disturbances of short 
period, we shall assume that the axes of magnetic induction 
coincide in direction with those of electric elasticity. <s> and that 
(as in the case of an isotropic medium) the terms including [psi] have 
nothing to do with the result.<\s> 

(103) Let the values of the magnetic coefficient for the three axes be [lambda], [mu], [nu], 
then the equations of magnetic force (b) become 
[lambda][alpha] = dH/dy - dG/dz 
[mu][beta] = dF/dz - dH/dx 
[nu][gamma] = dG/dx - dF/dy }(81) 

The equations of electric currents (C) remain as before 
The equations of electric elasticity (E) will be 
P = 4[pi]a<sup>2<\sup>f 
Q = 4[pi]b<sup>2<\sup>y 
R = e[pi]c<sup>2<\sup>h } (82) 
where 4[pi]a<sup>2<\sup>, 4[pi]b<sup>2<\sup>, 4[pi]c<sup>2<\sup> are the values of k for the <s>different<\s> axes of x, y, z 
Combining these equations with (A) and (D) we get equations of the form 
1/[mu][nu]([lambda] d<sup>2<\sup>F/dx<sup>2<\sup> + [mu]d<sup>2<\sup>F/dy<sup>2<\sup> + [nu]d<sup>2<\sup>F/dz<sup>2<\sup>) - 1/[mu][nu] d/dx([lambda] dF/dx + [mu]dG/dy + [nu]dH/dz) = 1/d<sup>2<\sup> (d<sup>2<\sup>F/dt<sup>2<\sup> + d<sup>2<\sup>[psi]/dxdt) (83) 

(104) If l m n are the direction - cosines of the wave and V its velocity 
and if lk + my + nz - Vt = w (84) 
then F G H & [psi] will be functions of w and if we put F' G' H' [psi]' for 
the second differentials of these quantities with respect to w, <s>then<\s> the equations 
will be 

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James Clerk Maxwell
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Cite as

J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7



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