266. REPORT—1862. 
eyen conveying when pointed out the expression of any simple physical 
relation. 
The year 1839 was fertile in theories of double refraction, and on the 9th 
of December Prof. MacCullagh presented his theory to the Royal Irish Academy. 
It is contained in “ An Essay towards a Dynamical Theory of Crystalline 
Reflexion and Refraction”*. As indicated by the title, the determination of 
the intensities of the light reflected and refracted at the surface of a crystal is 
what the author had chiefly in view, but his previous researches had led him 
to observe that this determination was intimately connected with the laws of 
double refraction, and to seek to link together these laws as parts of the same 
system. He was led to apply to the problem the general equation of dynamics 
under the form (5), to seek to determine the form of the function ¢ (V in his 
notation), and then to form the partial differential equations of motion, and the 
conditions to be satisfied at the boundaries of the medium, by the method of 
Lagrange. He does not appear to have been aware at the time that this method 
had previously been adopted by Green. Like his predecessors, he treats the 
ether within a crystallized body as a single medium unequally elastic in dif- 
ferent directions, thus ignoring any direct influence of the ponderable mole- 
cules in the vibrations. He assumes that the density of the ether is a constant 
quantity, that is, both unchanged during vibration, and the same within all 
bodies as in free space. We are not concerned with the latter of these 
suppositions in deducing the laws of internal vibrations, but only in inyesti- 
gating those which regulate the intensity of reflected and refracted light. 
He assumes further that the vibrations in plane waves, propagated within a 
crystal, are rectilinear, and that while the plane of the wave moves parallel to 
itself the vibrations continue parallel to a fixed right line, the direction of this 
right line and the direction of a normal to the wave being functions of each 
other,—a supposition which doubtless applies to all crystals except quartz, and 
those which possess a similar property. ‘ 
’ In this method everything depends on the correct determination of the form 
of the function V. From the assumption that the density of the ether is 
unchanged by vibration, it is readily shown that the vibrations are entirely 
transversal. Imagine a system of plane waves, in which the vibrations are 
parallel to a fixed line in the plane of a wave, to be propagated im the crystal, 
and refer the crystal for a moment to the rectangular axes of a", y', z', the 
plane of ay’ being parallel to the planes of the waves, and the axis of y' to 
the direction of vibration ; and let « be the angle whose tangent is 2, With 
respect to the form of V, MacCullagh reasons thus :— The function V can only 
depend upon the directions of the axes of a’, y', z' with respect to fixed lines 
in the crystal, and upon the angle which measures the change of form produced 
in the parallelepiped by vibration. This is the most general supposition which 
can be made concerning it. Since, however, by our second supposition, any 
one of these directions, suppose that of w', determines the other two, we may 
regard V as depending on the angle « and the direction of the axis of a! 
alone,” from whence he shows that V must be a function of the quantities 
X, Y, Z, defined by the equations 
dn dé dz dé di dn 
Se dy nae ey 
This reasoning, which is somewhat obscure, seems to me to involve a fallacy. 
* Memoirs of the Royal Irish Academy, vol. xxi. p. 17. 
