376 Proceedings of Royal Society of Edinburgh. [sess. 
which is Fresnel’s “sine-formula”; a very surprising and interest- 
ing result. It has long been known that for vibrations perpen- 
dicular to the plane of the incident, reflected, and refracted rays, 
unequal densities with equal rigidities of the two mediums, 
whether compressible or incompressible, gives Fresnel’s sine-law : 
and unequal rigidities, with equal densities, gives his tangent-law. 
But for vibrations in the plane of the three rays, and both 
mediums incompressible, unequal rigidities with equal densities 
give, as was shown by Rayleigh in 1871,* * * § a complicated formula 
for the reflected ray, vanishing for two different angles of 
incidence, if the motive forces in the waves are according to 
the law of the elasticity of an ordinary solid. How we find for 
vibrations in the plane of the rays, Fresnel’s sine-law, with its 
continual increase of reflected ray with increasing angles of 
incidence up to 90°, if the restitutional forces follow the law of 
dependence on rotation which I have suggested f for ether, and 
if the waves of condensation and rarefaction travel at velocities 
small in comparison with those of waves of distortion. 
§ 15. Interesting, however, as this may be in respect to an ideal 
problem of dynamics, it seems quite unimportant in the wave- 
theory of light ; because Stokes J has. given, as I believe, irre- 
fragable proof that in light polarized by reflexion the vibrations 
are perpendicular to the plane of the incident and reflected rays, 
and therefore, that it is for vibrations in this plane that Fresnel’s 
tangent-law is fulfilled. 
§ 16. Of our present results, it is (35) of § 14 which is really 
important ; inasmuch as it shows that Fresnel’s tangent-law is 
fulfilled for vibrations in the plane of the rays, with the rotational 
law of force; as I had found it in 1888 § with the elastic-solid-law 
of force ; provided only that the propagational velocities of condensa- 
tional waves are small in comparison with those of the waves of 
transverse vibration which constitute light. 
§ 17. By (28) we see that when a~ l , the velocity of the wave- 
trace on the interface of the two mediums, is greater than the 
* Phil. Mag., 1871, 2nd half year. 
t See * of § 1. 
X Dynamical Theory of Diffraction. See footnote § 5. 
§ See footnote § 14. 
