TRANSACTIONS OF THE SECTIONS. 5 
thematicians, who have endeavoured in vain to explain it; and it was regarded by 
Newton as an insuperable objection to the undulatory theory of light; for, according 
to him, the vibrations of the «ther could not be suddenly stopped at the separating 
surface of the denser and rarer medium: as an elastic fluid communicates motion on 
all sides, the vibrations, on arriving at the surface of the rarer medium, would neces- 
sarily pass into it, and thus there would always be some refracted light; and total 
reflexion, on the hypothesis of undulations, would be impossible. It is curious, there- 
fore, to observe that the phenomenon is now explained, and that for the first time, 
on this very hypothesis, and all its laws deduced with geometrical accuracy. The 
principles, however, on which the explanation is founded, are altogether new, though 
in strict accordance with the theory of waves, and the general laws of dynamics. 
Before the condition of the totally reflected light can be determined, the motion 
which takes place in the rarer medium,—a motion not observable by the senses,—must 
be ascertained. This question is one which it was impossible to solve by the imperfect 
methods and defective principles hitherto employed, though Poisson had caught a 
glimpse of the way in which the solution was likely to be effected. In some investi- 
gations respecting the transmission of motion from one elastic medium into another, 
across their separating surface, he had found that in certain cases the disturbance in 
the second medium would be proportional to a negative exponential, which would di- 
minish very rapidly as the distance from the separating surface increased; so that at 
avery small distance from that surface the disturbance would be quite insensible. This 
result enabled him, in a general way, to remove the objection which Newton had urged 
against the undulatory theory, because it furnished an instance in which the vibration 
did not sensibly spread beyond a certain surface; but, except so far as this, it did not 
give the least assistance in the solution of the question of total reflexion. It showed 
that a solution might possibly be found on the hypothesis of undulations; but this was 
all that it accomplished, for the formulz of Poisson had not the remotest application to 
the case of luminiferous vibrations. The problem next occupied the attention of Fresnel, 
who succeeded completely in discovering the motions of the reflected light, by a most 
ingenious interpretation of the formulz which he had found for the usual case of par- 
tial reflexion, and which give imaginary results when the reflexion is total. Perhaps 
the rare sagacity of Fresnel was never more conspicuous than in this remarkable in- 
stance, in which he applied his formule to a case which they did not originally include, 
and thus succeeded in divining, as it were, the laws that he was in search of. But 
these laws rested on no physical foundation ; nor could they be made to do so without 
previously determining the laws of the insensible refraction with which they are neces- 
sarily connected. These latter laws, however, Fresnel had no means of divining, as 
he did the former. The meaning of the imaginary formule it was in this case im- 
possible to conjecture; and even if it had been possible, still the two sets of laws could 
not have been connected, as the general relations which subsist between them were 
unknown; the relations which Fresnel had found sufficient in the case of partial re- 
flexion being altogether powerless in this. 
To find the laws of insensible refraction, it was necessary to know the general differ- 
ential equations for the propagation of light in a transparent medium; and to deduce 
from these laws the laws of total reflexion, it was necessary to know the general equa- 
tions which subsist at the confines of two transparent media. Both these sets of equa- 
tions were given, for the first time, by Professor MacCullagh, the former having been 
originally published in the Philosophical Magazine for February 1836, the latter in 
the Proceedings of the Royal Irish Academy for December 1839. The equations had 
been first applied to the solution of the problem of reflexion and refraction, when the 
reflexion is partial and the refraction sensible, giving rise to one or two visible re- 
fracted rays, which is the simpler case, and was therefore the first solved, the solution 
having been given in the Proceedings of the Royal Irish Academy at the latter date. 
But it was some months later before Mr. MacCullagh perceived that these same dif- 
ferential equations included also the laws of total reflexion and of the accompanying 
insensible refraction, the only difference between this and the former case being the 
assumption of more general expressions for the integral of the equations of propaga- 
tion. The expression for the displacement parallel to each axis of coordinates now 
contains a negative exponential of which the exponent is a linear function of the co- 
