1894.] Reflection and Refraction of Light. 27 



In the first medium are assumed an incident, a reflected, and a 

 pressural wave, and in the second a refracted and a pressural wave, 

 the pressural waves, of course, disappearing on the contractile ether 

 and electromagnetic theories. Taking account both of vibrations in 

 and perpendicular to the plane of incidence, there are 6 amplitudes 

 and 6 retardations of phase, in all, 12 constants, to be determined. 

 From the continuity of the motion at the two boundary planes of the 

 variable layer are obtained 12 pairs of equations, of which 6 pairs 

 determine the motion inside the variable layer, and the remaining 

 ones the 6 pairs of constants. In the actual work, imaginaries are 

 used in the usual way, reducing the equations by one half, the re- 

 quisite number of equations being obtained at the end by changing 

 the sign of the imaginary. 



From the equations of vibration in the variable layer and one half 

 of the boundary conditions, solutions are obtained in ascending 

 powers of the thickness of the layer, which, on substitution in the 

 remaining boundary conditions, give sufficient equations to determine 

 the amplitudes and phases by simple though long algebraic analysis. 



The solutions are found to be convergent, provided 2?rd/X is less 

 than the reciprocal of the greatest value of the refractive index 

 occurring in the variable layer, d being its thickness, and X the wave- 

 length in vacuo of the light employed. This condition gives values 

 for d which are less than the thickness of a soap film producing a 

 red of the first order. Thus the films for which the theory holds at 

 all cannot possibly produce colours of thin plates, which has been 

 stated as a possible objection by Lord Rayleigh. 



The solutions are taken as far as is necessary to give the values of 

 the amplitudes and phases correct to squares of d. The elastic solid 

 theories lead to Green's formulae, but the contractile ether and 

 electromagnetic theories give modifications of Fresnel's formulae, 

 somewhat of Cauchy's type. The corrections to the amplitudes are 

 of order d 2 , whilst the phases are of order d. If yuo> pi, p- are the 

 refractive indices of the media and the layer, then these terms in- 

 volve certain constants which are functions of /to, fi t and of the 

 mean values of /i 2 , l//i* and certain combinations of /t 2 , I//* 1 . 



The effect is considered of supposing the velocities of the pressural 

 waves to be large, but finite and different, the refractive index for the 

 pressural wave from the first to the second medium having any 

 value. The resulting formulae deviate from Green's chiefly in that 



, I*iPo 



M = 2.2 becomes ^-. ? 1 + _, . , ^ -*: : = , where 



sin to sin t! J' 



mo, m l are the large ratios of the pressural to the light velocity in 

 the two media ; and this value of M holds for values of t' , i\ so large 

 that sin i > i/w and sin tj > l/?n t . The effect is always to increase 

 M, and thus Haughton's proposal to substitute for Green's M a 



