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

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                                (V<sup>2<\sup> - a<sup>2<\sup>(m<sup>2<\sup>/[nu] + n<sup>2<\sup>/[mu]))F' + a<sup>2<\sup>lm/[nu] G' + a<sup>2<\sup>ln/[mu] H' - lV[psi]' = 0 
(V<sup>2<\sup> - b<sup>2<\sup>(n<sup>2<\sup>/[lambda] + l<sup>2<\sup>/[nu]))G' + b<sup>2<\sup>mn/[lambda] H' + b<sup>2<\sup>ml/[nu] F' - mV[psi]' = 0 
(V<sup>2<\sup> - c<sup>2<\sup>(l<sup>2<\sup>/[mu] + m<sup>2<\sup>/[lambda]))H' + c<sup>2<\sup>nl/[mu] F' + c<sup>2<\sup>nm/[lambda] G' - nV[psi]' = 0 } (85) 
If we now put
V<sup>4<\sup>  V<sup>2<\sup> 1/[lambda][mu][nu]{l<sup>2<\sup>[lambda](b<sup>2<\sup>[mu] + c<sup>2<\sup>[nu]) + m<sup>2<\sup>[mu](c<sup>2<\sup>[nu] + a<sup>2<\sup>[lambda]) + n<sup>2<\sup>[nu](a<sup>2<\sup>[lambda] + b<sup>2<\sup>)} + 
+ a<sup>2<\sup>b<sup>2<\sup>c<sup>2<\sup>/[lambda][mu][nu] (l<sup>2<\sup>/a<sup>2<\sup> + m<sup>2<\sup>/b<sup>2<\sup> + n<sup>2<\sup>/c
a<sup>2<\sup>)(l<sup>2<\sup>[lambda] + m<sup>2<\sup>[mu] + n<sup>2<\sup>[nu]) = U (86) 
we shall find 
F'v<sup>2<\sup>U - l[psi]'VU = 0 (87) 
with two similar equations for G' * H' 
V = 0 (88) 
Hence either <s>V = 0<\s>, U = 0 or (89) 
VF' = l<s>[text?]<\s>[psi]', VG' = m<s>[text?]<\s>[psi' and VH' = n<s>[text?]<\s>[psi]' (90) 

The third supposition indicates that the resultant of F' G' H' 
is in the direction normal to the plane of the wave but 
the equations do not indicate that such a disturbance if possible 
could be propagated, as we have no other relation between [psi]' and 
F' G' H'. 
The solution V = 0 is unimportant 
The solution U = 0 gives two values for V<sup>2<\sup> corresponding to 
values of F' G' H' which are given by the equations 
l/a<sup>2<\sup> F' + m/b<sup>2<\sup> g' + n/c<sup>2<\sup> H' = 0 (91) 
a<sup>2<\sup>l[lambda]/F' (b<sup>2<\sup>[mu] - c<sup>2<\sup>[nu]) + b<sup>2<\sup>m[mu]/G' (c<sup>2<\sup>[nu] - a<sup>2<\sup>[lambda]) + c<sup>2<\sup>n[nu]/H' (a<sup>2<\sup>[lambda] - b<sup>2<\sup>[mu]) = 0 (92) 

(105) The velocities along the axes are as follows 

Direction of Propagation x y z 
Direction of the <s>disturbances<\s> x  a<sup>2<\sup>/[nu] a<sup>2<\sup>/[mu] 
<s>F G H or<\s> y b<sup>2<\sup>/[nu]  b<sup>2<\sup>/[lambda] 
electric displacements z c<sup>2<\sup>/[mu] c<sup>2<\sup>/[lambda] 

                            
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Manuscript details

Author
James Clerk Maxwell
Reference
PT/72/7
Series
PT
Date
1864
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J. C. Maxwell’s, ‘Dynamical theory of the electromagnetic field’, 1864. From The Royal Society, PT/72/7

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