538 



internal radiation, if the law of exchanges be true, is clearly indepen- 

 dent of the position of this surface, which is indeed merely employed 

 as an expedient. This is equivalent to saying that the constants which 

 define the position of the hounding surface must ultimately disappear 

 from the expression for the internal radiation." This anticipation he 

 shows is verified in the case of the expression deduced, according to 

 his principles, for the internal radiation within a uniaxal crystal, on 

 the assumption that the wave-surface * is the sphere and spheroid of 

 Huygens. 



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 cty, and let cty' 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 j 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 i fy = ( 1 A') R' cos i' ty' -f- AR cos i ty, 

 or (l-A)Rc<m'fy=(l A')B/cosi'30' (1) 



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

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

 the rays along OP^ OP 2 being each polarized in a particular manner. 



* To prevent possible misapprehension, it may be well to state that I use this term 

 to denote the surface, whatever it may be, which is the locus of the points reached 

 in a given time by a disturbance propagated in all directions from a given point j 

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



2 +y 2 +2 2)( a 2^2 + 2 y 2 + c 222)_ a 2( J2 + c 2^ 



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. 



