Prof. Stokes on Internal Radiation. 477 



In the case of an isotropic medium, K and R' are alike in all 

 directions, and therefore the ratio of cos ify to cos i co ought to be 

 independent of i, as it is very easily proved to be. The same applies 

 to a uniaxal crystal, so far as regards the ordinary rav. But as re- 

 cards the extraordinary, it is by no means obvious that the ratio should 

 be expressible in the form indicated — as a quantity depending only 

 on the direction OP'. Mr. Stewart has, however, proved that this is 

 the case, independently of any restriction as to the plane of incidence 

 being a principal plane, on the assumption that the wave-surface has 

 the form assigned to it bv Huvgens. 



It might seem at first sight that this verification was fairlv adducible 

 in confirmation of the truth of the whole theory, including the 

 assumed form of the wave-surface. But a little consideration will 

 show that such a view cannot be maintained. Huvgens's construction 

 links together the law of refraction and the form of the wave-surface, 

 in a manner depending for its validity only on the most fundamental 

 principles of the theory of undulations. The construction which 

 Huygens applied to the ellipsoid is equally applicable to any other 

 surface ; it was a mere guess on his part that the extraordinary wave- 

 surface in Iceland spar was an ellipsoid ; and although the ellipsoidal 

 form results from the imperfect dynamical theory of Fresnel, it is 

 certain that rigorous dvnamical theories lead to different forms of 

 the wave-surface, according to the suppositions made as to the 

 existing state of things. For every such possible form the ratio 

 expressed by the right-hand member of equation (3) ought to come 

 cut in the form indicated by the left-hand member, and not to involve 

 explicitly the direction of the refracting plane : and as it seemed evi- 

 dent that it could not be possible, merely by such general considera- 

 tions as those adduced bv Mr. Stewart, to distinguish between those 

 surfaces which were and these which were not dynamically possible 

 forms of the wave-surface, I was led to anticipate that the possibility 

 of expressing the ratio in question under the form indicated was a 

 general property of surfaces. The object of the present Note is to 

 give a demonstration of the truth of this anticipation, and thereby 

 remove from the verification the really irrelevant consideration of a 

 particular form of wave-surface ; but it was necessary in the first 

 instance to supply some steps of Mr. Stewart's investigation which 

 are omitted in the published abstract. 



The proposition to be proved may be somewhat generalized, in a 

 manner suggested bv the consideration of internal reflexion within a 

 crvstal, or refraction out of one crystallized medium into another in 

 optical contact with it. Thus generalized it stands as follows : — 



Imagine any two surfaces whatsoever, and also a fixed point O ; 

 imagine likewise a plane n passing through 0. Let two points P, P , 

 situated on the two surfaces respectively, and so related that the tan- 

 gent planes at those points intersect each other in the plane IT, be called 

 corresponding points with resjject to the plane IT. Let P describe, on 

 the surface on which it lies, an infinitesimal closed circuit, and P' the 

 " corresponding" circuit ; let co, co' be the solid angles subtended at 

 by these circuits respectively, and /, i the inclinations of OP, OP 



