274 REPORT—1862. 
of any given refrangibility, in those cases in which they are not determined 
by being at the same time planes of crystallographic symmetry, is a matter 
needing experimental verification. However, as no anomaly, so far as I am 
aware, has been discovered in the systems of rings seen with homogeneous 
light around the optic axes of crystals of the oblique or anorthic system, 
there is no reason for supposing that such planes do not exist. 
APPENDIX. 
Further Comparison of the Theories of Green, MacCullagh, and Cauchy. 
In a paper “On a Classification of Elastic Media and the Laws of Plane 
Waves propagated through them,” read before the Royal Irish Academy on the 
8th of January, 1849*, Professor Haughton has made a comparative examina- 
tion of different theories which have been advanced for determining the motion 
of elastic media, more especially those which have been applied to the expla- 
nation of the phenomena of light. Some of the results contained in this 
Appendix have already been given by Professor Haughton ; in other instances 
I have arrived at different conclusions. In such cases I have been careful to 
give my reasons in detail. 
Consider a homogeneous elastic medium, the parts of which act on one 
another only with forces which are insensible at sensible distances, and which 
in its undisturbed state is either free from pressure, or else subject to a 
pressure or tension which is the same at all points, though varying with 
the direction of the plane surface with reference to which it is estimated, 
Let w, y, z be the coordinates of any particle in the undisturbed state, «+, 
y+v, z+w the coordinates in the disturbed state, and for simplicity take the 
density in the undisturbed state as the unit of density. Then, according to 
the method followed both by Green and MacCullagh, the motion of the 
medium will be determined by the equation ge 
du d*u dw 
(Wc but ae uta iw) de dy dz= (\\e du dy dz, . (19) 
where @ is the function due to the elastic forces, To this equation must be 
added, in case the medium be not unlimited, the terms relative to its boundaries. 
The function ¢ multiplied by dw dy dz expresses the work given out by the 
element dx dy dz in passing from the initial to the actual state if we assume, 
as we may, the initial state for that in which ¢=0. According to the sup- 
position with which we started, that the internal forces are insensible at 
sensible distances, the value of ¢ at any point must depend on the relative 
displacements in the immediate neighbourhood of that point, as expressed by 
the differential coefficients of uw, v, w with respect to #, y,z. For the present 
let us make no other supposition concerning ¢ than this, that it is some 
function (—f) of those nine differential coefficients; and let us apply the 
equation (19) to a limited portion of the medium bounded initially by the 
closed surface 8. We must previously add the terms due to the action of the 
surrounding portion of the medium, which will evidently be of the form of a 
double integral haying reference to the surface 8, an element of which we 
may denote by dS. Hence we must add to the right-hand side of equa- 
tion (19) 
Ed§, 
the expression for E having yet to be found. 
* Transactions of the Royal Irish Academy, vol. xxii. p. 97. 
