7 
NOTICES AND ABSTRACTS OF MISCELLANEOUS COM- 
MUNICATIONS TO THE SECTIONS. 
MATHEMATICS AND PHYSICS. 
Professor Hamilton gave a sketch of his new theory of Iogo- 
logues and other numbers of higher orders ; (see Transactions of the 
Royal Irish Academy;) also a similar account of his new theory of 
varying orbits. 
He likewise explained to the Section the method of investigation 
pursued by Mr. G. B. Jerrard, for accomplishing the solution of 
equations of the fifth or of higher degrees. 
A short Account of some recent Investigations concerning the Laws of 
Reflexion and Refraction at the surface of Crystals. By Mr. 
M’Cullagh. 
To understand the nature of the general problem which a com¬ 
plete theory of double refraction requires to be solved, let it be sup¬ 
posed that a ray of light is reflected and refracted at the separating 
surface of an ordinary medium and a doubly refracting crystal, 
the light passing out of the former medium into the latter. This 
limited view of the subject is taken merely for the sake of clearness 
of conception; since we might suppose that both media are crystal¬ 
lized, without increasing the difficulty of the problem. The question, 
it is obvious, naturally divides itself into two distinct heads. The 
first relates to the laws of the propagation of light in the interior of 
either of the two media, before or after it has passed their separating 
surface ; and this part of the subject has been fully treated, accord¬ 
ing to their different methods, by MM. FrCsnel and Cauchy. The 
second division of the subject had been left completely untouched. 
It relates to the more complex consideration of what takes place at 
the separating surface of the media, the laws according to which 
the light is there divided between the reflected and refracted rays, 
including a determination of the attendant circumstances indicated 
by the wave theory, with regard to the vibrations in the reflected 
and refracted rays. In the case above mentioned, when the incident 
light is polarized, there are four things to be determined, namely, 
the magnitude and direction of the reflected vibration, with the mag¬ 
nitudes of the two refracted vibrations. The four conditions neces¬ 
sary for this determination are furnished by two new laws, which 
could not be easily stated without entering too much into detail. 
The results, applied to determine the polarizing angle of a crystal 
in different azimuths of the plane of reflection, agree very closely 
with the admirable experiments of Sir David Brewster on Iceland 
spar. In the course of these experiments it was observed that the 
polarizing angle remained the same when the crystal was turned half 
