158 REPORT—1885. 
amplitude of the vibration multiplied by the density of the medium. 
Then Fresnel’s results may be expressed by the statement that the trans- 
versal of the incident ray is the resultant, in the mechanical sense of the 
word, of those of the reflected and refracted rays. 
This first suggestion of MacCullagh’s was modified by reading some 
of Cauchy’s work on double refraction, from which it appeared possible 
that the vibrations of polarised light might lie in the plane of polarisa- 
tion instead of at right angles to it. Adopting, then, this hypothesis, 
a transversal represents in addition the direction of vibration; and if the 
further supposition is made that the ether is of the same density in all 
media, so that reflexion and refraction arise from variations in its 
rigidity and not in its density, expressions very nearly identical with 
Fresnel’s can be found for the intensities of the reflected and refracted 
rays, while at the same time the principle of the continuity of the 
displacement normal to the surface is satisfied. 
§ 3. These three principles— 
(1) The ether is of the same density in all media, 
(2) The displacement is the same on both sides of the surface of 
separation of the two media, 
(3) Theenergy of the incident wave is equal to that of the reflected 
and refracted waves 
—were applied by MacCullagh to the problem of reflexion and refrac- 
tion at the surface of a crystal, and the results of a first investigation 
were communicated to the meeting of the Association in 1834, 
The theory as there given was somewhat modified in consequence of a. 
paper by Seebeck in Poggendorff’s ‘ Annalen,’ and took its final form in 
a memoir read before the Irish Academy! in January 1837. MacCullagh 
in this paper states his fundamental principles, not as based on mechanics, 
but merely as those which had led him to a solution, the results of 
which agree closely with the experiments of Seebeck and Brewster. 
The analysis of the problem is greatly simplified by the introduction 
of the idea of ‘ uniradial directions.’ 
In a crystal, for any given direction of incidence, there are two posi- 
tions for the incident transversals, which give rise each to only one 
refracted ray—there are corresponding positions for the reflected trans- 
versals. These directions of the incident transversals are the wniradial 
directions. 
For a uniradial direction the incident, reflected, and refracted trans- 
versals lie in one plane, and the refracted transversal is the resultant of 
the other two. : 
The transversal is normal to the plane containing the ray and the 
wave normal. The polar plane is defined as a plane through the trans- 
versal and parallel to the line joining the extremity of the ray to the 
point in which the wave normal meets the surface of wave slowness, here 
designated the ‘index surface.’ j 
It is hence proved that for a uniradial direction the incident and 
reflected transversals lie in the polar plane of the refracted ray, and then 
the principles of equivalence of vibrations and of vis viva lead to 
equations to determine the relation between the azimuths of the trans- 
versals referred to the plane of incidence. , 
' MacCullagh, ‘On the Laws of Crystalline Reflexion and Refraction,’ Januaryyl 
1837, Trans. of Royal Irish Academy, vol. xviii. 
