. 
ON OPTICAL THEORIES. 189 
An account of the various theories is also given in papers by Lord 
Rayleigh,! with a careful criticism and comparison of them all. 
In the first part of this paper Lord Rayleigh discusses fully the 
difference between the theories of Green and MacCullagh, and develops 
completely the consequences of the latter, taking into account the full 
effect of the pressural wave. This had been done first by Lorenz in the 
paper already referred to, and he showed that the results to which 
MacCullagh’s theory leads are totally inconsistent with experiment. 
Lord Rayleigh points out that the fundamental assumptions of Green 
and Fresnel amount to assuming an identity between the statical pro- 
perties of the two media, while the dynamical properties depending on 
variation of density are different; while, moreover, as we have seen 
already, Cauchy’s surface conditions, founded on the principle of the 
continuity of the displacements and their differential coefficients with 
reference to the normal, though erroneous if we suppose the rigidity of 
the ether different in the two media, become identical with Green’s if 
we adopt his fundamental hypothesis. The real difference between Green 
and Cauchy lies in their respective treatments of the pressural waves. 
The true surface conditions lead to the following results :— 
Let &, 7, ¢ be the displacements, n the rigidity, m the second 
coefficient, such that m+n is the A of Green’s papers, and D the 
density, while g? = (m + »)/D, y? =n/D for the one medium. 
Let «= 0 be the bounding surface, and let the axis of z be parallel to 
the front of the waves. And suppose f, F, and /, to represent the incident 
reflected and refracted waves, while ¢ and ¢’ are the angles of incidence 
and refraction. 
Then, for vibrations normal to the plane of incidence— 
tan 9’ a’ 
tang 2 
Fi = pad aw nl : ; AS 2 (32) 
tan @ n 
and this becomes :— 
Case I. n= 7’ (Green, Fresnel, Cauchy)— 
FY __ sin (¢’ — 
CET ra aoa Re 
Case II. D =D! (MacCullagh, Neumann)— 
x _, tan (9 — 9) 
ff tan (9 + 9) 
Now, Jamin, Quincke, and others have shown that this latter formula 
is not strictly true, and hence at this point the evidence is already in 
favour of Fresnel’s hypothesis. 
Turning now to the case of the vibrations in the plane of incidence, put: 
(34) 
d®, d¥ 
ee 
_de_ay 
2 dy dx 
1 J. W. Strutt, ‘On the Reflexion of Light from Transparent Matter,’ Phil. Mag. 
August, 1871; ‘On the Reflexion and Refraction of Light by Intensely O ue: 
Matter,’ Phil. May. May, 1872. See as 
