6 REPORT—1843. 
ordinates, this exponential being multiplied by a quantity which contains, in general, 
both the sine and cosine of the phase. By reason of this exponential, the vibrations in 
the rarer medium rapidly decrease in magnitude as the distance from the surface of 
separation increases, becoming insensible at a very small distance from that surface ; 
and when the aforesaid expressions for the displacements are substituted in the equa- 
tions of propagation, we get certain relations among the constants, which relations are 
in fact the laws of the insensible refraction. The laws of the totally reflected light are 
then easily deduced by means of the equations which subsist at the separating surface. 
It is to be observed that the reflexion is not assumed to be total, but is proved to be 
so, from the equations last mentioned ; and the laws of this reflexion come out the very 
same (for ordinary media) as those discovered by Fresnel in the singular way before 
alluded to. 
On account of the novel character of the laws of insensible refraction, Mr. Mac- 
Cullagh entered into some details respecting them. The vibrations are in this case 
elliptical, every particle of the rarer medium describing an ellipse, which has the di- 
rections and the proportion of its axes everywhere the same; but the magnitudes of 
its axes rapidly diminish as the distance of the particle from the common surface of 
the media increases. It would, however, take up too much space to go into these 
laws, which will be published in the Transactions of the Royal Irish Academy. 
The problem of total reflexion was considered by Fresnel only with reference to two 
ordinary media. The preceding method, however, is general, and solves the problem 
in its widest extent. The most complicated case is that in which the rarer medium 
is supposed to be a doubly refracting crystal, the crystal being covered with a fluid of 
greater refractive power than itself, so that total reflexion may take place at the com- 
mon surface. We then have insensible refraction within the crystal, and it is found 
that this refraction is double, giving rise to two insensible waves, in each of which the 
vibrations are elliptical. The laws of this insensible double refraction are very general, 
and include, as a particular case, the laws of double refraction discovered by Fresnel. 

Attempt to explain theoretically the Phenomena of Metallic Reflexion. 
By Professor Luoyp. 
The physical hypothesis from which the author sets out is, that the elasticity of the 
zther (which is usually assumed to change abruptly at the confines of transparent 
media) varies gradually at the surface of a metal, so as to constitute, in fact, an infi- 
nite series of thin plates of infinitesimal thickness. In such a medium it is natural 
to suppose that there will be an infinite series of infinitesimal vibrations, reflected at 
every point in the course of the ray, the sum of which will constitute the resultant 
vibration. The magnitude (V) and phase (A) of this resultant vibration will be given 
by the formule 
VoosA.=,f v cos a, VsnA=/f osin a; 
where v denotes the magnitude of the infinitesimal vibration reflected at any point 
of the varying medium, and « its phase, and where the integrals are taken between 
limits porreapansing to those of the varying medium. In these expressions the quan- 
tity v is readily de uced, in terms of the angle which the direction of the ray makes 
with the normal to the bounding surface, by setting out from Fresnel’s expressions 
for the reflected vibrations in the case of a finite change of elasticity. The general 
value of « is also readily expressed in terms of the same angle, when the relation be- 
tween this angle and the distance from the first surface of the varying medium is 
known. The latter relation cannot be certainly known a priori; but the author 
showed that there was reason to believe that it was expressed by the simple formula 
sin @= sin © e {*; 
© being the angle of incidence on the exterior surface of the medium, and q a con- 
stant. This being assumed, the two components of the resulting reflected vibration 
are expressed by single integrals, and the problem is therefore reduced to quadratures ; 
