ON DOUBLE REFRACTION. 267 
If the form of V were known, the rectilinearity of vibration and the constancy 
in the direction of vibration for a system of plane waves travelling in any given 
direction would follow as a reswlt of the solution of the problem. But in using 
équation (5) we are not at liberty to substitute for V (or g) an expression 
which represents that function only on the condition that the motion be what it 
actually is, for we have occasion to take the variation 6V of V, and this varia- 
tion must be the most general that is geometrically possible though it be 
dynamically impossible. That the form of V, arrived at by MacCullagh, is 
inadmissible, is, I conceive, proved by its incompatibility with the form 
deduced by Green from the very same supposition of the perfect transversality 
of the transversal vibrations ; for Green’s reasoning is perfectly straightforward 
and irreproachable. Besides, MacCullagh’s form leads to consequences abso- 
lutely at variance with dynamical principles*, 
But waiving for the present the objection to the conclusion that V is a 
function of the quantities X, Y, Z, let us follow the consequences of the theory. 
The disturbance being supposed small, the quantities X, Y, Z will also be small, 
and VY may be expanded in a series according to powers of these quantities ; 
and, as before, we need only proceed to the second order if we regard the 
disturbance as indefinitely small. The first term, being merely a constant, 
may be omitted. The terms of the first order MacCullagh concludes must 
vanish. This, however, it must be observed, is only true on the supposition 
that the medium in its undisturbed state is free from pressure. The terms of 
the second order are six in number, involving squares and products of X, Y, Z. 
The terms involving YZ, ZX, XY may be got rid of by a transformation of 
coordinates, when VY will be reduced to the form 
ny Sits: Ves e(O RP REO! vl oe V0) 
the constant term being omitted, and the arbitrary constants being denoted by 
— 2@, —3b°, —2¢°. Thus on this theory the existence of principal axes is 
proved, not assumed. If MacCullagh’s expression for V (10) be compared with 
Green’s expression for ¢ (8) for the case of no pressure in equilibrium, so that 
A=0, B=0, C=0, it will be seen that the two will become identical, provided 
f 2 
first we omit the term p = ed = in Green’s expression, and secondly, 
we treat the symbols of differentiation as literal coefficients, so as to confound, 
; du dw dv dw : ‘ , 
= instance, du da a dake, The term involving p does not appear in the 
du dv, dw 
dosages ee 
therefore does not affect the laws of the propagation of such vibrations, although 
it would appear in the problem of calculating the intensity of reflected and 
refracted light ; and be that as it may, it follows from Green’s rule for forming 
the equation of the ellipsoid of elasticity, that the laws of the propagation of 
transversal vibrations will be precisely the same whether we adopt his form of 
g or V (for the case of no pressure in equilibrium) or MacCullagh’s. Indeed, 
if we omit the term p ae +74E)> the partial differential equations of 
expressions for transversal vibrations, since for these 
da dz 
motion, on which alone depend the laws of internal propagation, would be 
Just the same as the two theoriest. Accordingly MacCullagh obtained, though 
* See Appendix. 
« + See Appendix. MacCullagh’s reasoning appears to be so far correct as to have led to 
correct equations, although through a form of V which may, I conceive, be shown to be 
inadmissible. 
