738 
MR. J. LARMOR OX A DYNAMICAL THEORY OF 
the incidence on a rarer medium is so oblique that one or both the refracted waves 
disappear; if we simply treat these waves as non-existent, the four surface-conditions 
cannot all be satisfied. The natural inference is that the solution of the problem now 
depends on the particular form of the wave ; the fundamental simple-harmonic form is 
the obvious one to choose, so let the vibration be represented by 
A exj) t 27rX“^ ijx -f- my -j- nz — vt), 
real parts only being in the end retained. The satisfaction of the interfacial condi¬ 
tions,—which must now be chosen all linear as we are runnino- a real and an imao-iuarv 
O Cj 
part concurrently, and they must not get mixed up,—leads to a complex value of n for 
one or both of the refracted waves and of A for both of them. The interpretation is 
of coarse, in the first case purely surface waves, in the second a change of phase in the 
act of reflexion or refraction. With this modification the celebrated interpretation of 
the imaginary expression in his formulie, by Fresnel, becomes quite explicit, and the 
general problem of total or partial crystalline reflexion is solved for the type of medium 
virtually assumed by him, without any detailed consideration of the nature of the 
elasticity. The hypothesis is implied, and may be verified, that the .surface waves 
penetrate into the medium to a depth either great, or else small, compared with the 
thickness of the layer of tran.sition between the media,—a point which has not always 
been sufficiently noticed. 
Reflexion at the Surfaces of Absorbing Media. 
27. The fact that homogeneous light in passing through a film of metal does not 
come out a mixture of various colours, or more crucially the fact that the use of a 
metallic speculum in a telescope does not interfere with spectrum observations, shows 
that the equation of vibration of light in a metallic medium is linear, and therefore 
that to represent the motion of the light in the metal requires simply the introduction 
of an ordinary exponential coefficient of absorption. The interface being the plane of 
xy, the light propagated in the absorbing medium will be represented by the real 
part of an expression of the form A exp t 27rX“^ {lx + + uz — vt), wdiere n is now 
complex with its real part negative if the axis of 2 is towards the direction of 
propagation. If the opacity of the medium is so slight that the light gets down some 
way beyond the interfacial layer of transition without very sensible weakening, 
we may therefore solve the problem of reflexion by an application of the ordinary 
surface-conditions stated in a linear form, but with a complex coefficient of elasticity ; 
for we may treat the layer of transition as practically indefinitely thin. This comes 
to the same thing as the method used first by Cauchy, of simply treating the index 
of refraction as a complex quantity in the ordinary formulae for transparent media ; 
and it should give a satisfactory solution of the problem, provided the opacity is not 
excessive. 
