ON DOUBLE REFRACTION. 279 
integration is to be performed is perfectly arbitrary, unless / (=) =i (=) 
at all points. Weare thus led back to the equations (27), which are violated 
in the theory of MacCullagh. 
The form of the equations such as (30) is instructive, as pointing out the 
mode in which, the condition of moments is violated. It is not that the 
resultant of the forces acting on an element of the medium does not produce 
its proper momentum in changing the motion of translation of the element; 
that is secured by the equations (20); but that a couple is supposed to act 
on each element to which there is no corresponding reacting couple. 
The only way of escaping from these conclusions is by denying that the 
mutual action of two adjacent portions of the medium separated by a small 
ideal surface is capable of being represented by a pressure or tension, and 
saying that we must also take into account a couple ; not, itis to be observed, a 
couple depending on variations of the tension (for that would be of a higher 
order and would vanish in the limit), but a couple ultimately proportional to 
the element of surface. But it would require a function ¢ of a totally different 
form to take into account the work of such couples; and indeed the method 
by which the expressions for the components of the tension have been here 
deduced seems to show that in the case of a function » which depends only on 
the differential coefficients of the first order of u, v, w with respect to x,y,z, 
the mutual action of two contiguous portions of a medium is fully repre- 
sented by a tension or pressure. 
Indeed MacCullagh himself expressly disclaimed having given a mechanical 
theory of double refraction*. His methods have been characterized as a sort 
of mathematical induction, and led him to the discovery of the mathematical 
laws of certain highly important optical phenomena. The discovery of such 
laws can hardly fail to be a great assistance towards the future establishment 
of a complete mechanical theory. 
I proceed now to form the function ¢ for Cauchy’s most general equations. 
2, 2 2, 
If we have given the expressions for “a a =e in terms of the differential 
coefficients of u, v, w with respect to a, y, z, they do not suffice for the com- 
plete determination of the function ¢, as appears from the equations (20) or 
(21); but if we have given the expressions for the tensions P,, Ty.» &e., @ is 
completely determinate, as appears from equations (23). In using these 
equations, it must be remembered that the tensions are measured with 
reference to surfaces in the undisturbed state of the medium; and therefore, 
should the expressions be given with reference to surfaces in the actual state, 
they must undergo a preliminary transformation to make them refer to 
surfaces in the undisturbed state. 
Supposing then the tensions expressed as required, in order to find we 
have only to integrate the total differential 
du 
+Ty do 
dw du 
du dv dw 
f pee tat tid atte d as tl ye d dyt Tend dz 
dy dw du 
Hg” gt Tet ot let a) MasB ai iar Dali Ga QSL 
the nine differential coefficients, of which ¢ is a function, being regarded as 
_ * Transactions of the Royal Irish Academy, vol. xxi. p. 50. It would seem, however, 
that he rather felt the want of a mechanical theory from which to deduce his form of the 
function ¢ or V, than doubted the correctness of that form itself. 
