Prof. Stokes on. Internal Radiation. 475 



In the case of an uncrystallized medium, the following is the 

 equation obtained by Mr. Stewart in the first instance. 



Let R, R' be the external and internal radiations in directions 

 OP, OP', which are connected as being those of an incident and 

 refracted ray, the medium being supposed to be bounded by a plane sur- 

 face passing through O. Let OP describe an elementary conical circuit 

 enclosing the solid angle c0, and let l<p' be the elementary solid angle 

 enclosed by the circuit described by OP'. Let i, i' be the angles of 

 incidence and refraction. Of a radiation proceeding along PO, let the 

 fraction A be reflected and the rest transmitted ; and of a radiation 

 proceeding internally along P'O let the fraction A' be reflected, and 

 the rest transmitted. Then by equating the radiation incident ex- 

 ternally on a unit of surface, in the directions of lines lying within 

 the conical circuit described by OP, with the radiation proceeding in 

 a contrary direction, and made up partly of a refracted and partly of 

 an externally reflected radiation, we obtain 



R cos A Sf =(! — A') R' cos i' ty' + AR cos i ty, 



or (l-A)Rcosity=(l — A')R'cosi'ty' (1) 



In the case of a crystal there are two internal directions of refrac- 

 tion, OP^ OP 2 , corresponding to a given direction PO of incidence, 

 the rays along OP x , OP 2 being each polarized in a particular manner 

 Conversely, there are two directions, P x O, P 2 0, in which a ray may be 

 incident internally so as to furnish a ray refracted along OP, and in 

 each case no second refracted ray will be produced, provided the 

 incident ray be polarized in the same manner as the refracted ray 

 OP x or OP 2 . In the case of a crystal, then, equation (1) must be 

 replaced by 



(1— A)Rcosify=(l — AJR.cose^ + O— A 2 ) R 2 cos z 2 o> 2 . (2) 



In the most general case it does not appear in what manner, if at 

 all, equation (2) would split into two equations, involving respectively 

 R x and R 2 . For if an incident ray PO were so polarized as to furnish 

 only one refracted ray, say OP l3 a ray incident along P x O and polarized 

 in the same manner as OP x would furnish indeed only one refracted 

 ray, hi the direction OP, but that would be polarized differently from 

 PO ; so that the two systems are mixed up together. 



But if the plane of incidence be a principal plane, and if we may 

 assume that such a plane is a plane of symmetry as regards the optical 

 properties of the medium*, the system of rays polarized in and the 



I do not use it as a name for the surface defined analytically by the equation 



As the term wave-surface in its physical signification is much wanted in optics, the 

 surface defined by the above equation should, I think, be called Fresnel's surface, 

 or the wave-surface ofFresnel. 



* According to Sir David Brewster (Report of the British Association for 1836, 

 part ii. p. 13, and for 1842, part ii. p. 13), when light is incident on a plane sur- 

 ace of Iceland spar in a plane parallel to the axis, the plane of incidence, which 

 is a principal plane, is not in general a plane of optical, any more than of crystal- 

 line symmetry as regards the phenomena of reflexion, although, as is well known, 



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