ON OPTICAL THEORIES. 187 
The condition of no dilatation holds throughout both media, and the 
stresses over the surface are the same in the two. 
According to this view, a small portion of ether which belongs to one 
of the two media remains of unchanged density, and always forms part 
of the same medium. 
We may, however, consider the question somewhat differently, and look 
upon the ether in the two media as continuous, but of different densities 
on the two sides of the interface. A portion of ether belonging to the 
first medium may cross the interface and become part of the second, and 
in so doing its density is changed. There will thus be a thin sheet of 
ether lying over the interface in which rapid periodic changes of density 
are occurring. 
If, then, we consider the motion on the two sides of the sheet, we 
have for its determination the fact that the quantity of matter within the 
sheet is constant, and therefore that pu=p’u’, while the motion parallel to 
this sheet will ultimately be the same in the two media, and the energy 
in the reflected and refracted waves will be equal to that in the incident. 
But this condition pu=p'u’ does not hold within the sheet where the 
variations of density are taking place, and where the effects of the 
pressural wave are appreciable. The motions denoted by w and wu’ are 
light-motions, exclusive of those which give rise solely to the pressural 
wave. Moreover, it is supposed that this layer of variable density is so 
thin that the phase of the disturbance may be treated as the same over 
its two bounding surfaces. It is further assumed that the above are the 
only conditions which hold at the surface, and these can be satisfied 
without supposing any change of phase to arise from the reflexion, As a 
fact, there are other conditions involved in the equality of the stresses 
over the surface, and to satisfy these it is necessary to suppose that when 
the vibrations are in the plane of incidence the phases of the incident 
reflected and refracted waves differ even at the surface. 
To assume Fresnel’s conditions, as is done by Cornu, without change 
of phase is equivalent to supposing that this sheet of variable density is 
indefinitely thin when compared with the wave length of light. 
Green himself considered the effect of supposing the change in 
refractive index to take place in a gradual manner, replacing the refract- 
ing surface by a regular series of layers, of indices 41, Hy, etc., each of 
thickness 7 ; and proved that the effect of such a series would be to make 
the intensity of the reflected wave more nearly that given by Fresnel’s 
tangent formula. 
The effects of supposing the change of properties from one medium to 
the other to be gradual was discussed by L. Lorenz in the year 1860. 
§ 3. In his first paper! he supposes that Fresnel’s formule express 
the result of a sudden transition, and investigates how they must be 
modified if the transition be gradual. The variable sheet is divided into 
a series of layers, each of constant density. A ray reflected at one of the 
interior layers will on emergence be retarded relatively to the ray 
reflected at the surface. Let 6 be the retardation of the ray reflected at 
a layer on which the angle of incidence is z, and let a, [3 be the angles of 
incidence and emergence, then the disturbances in the reflected ray are 
shown to be :— 
* L. Lorenz, ‘ On the Reflexion of Light at the Bounding Surface of two Isotropic 
Media,’ Pogg. Ann. t. cxi. p. 460. 
