AT THE SURFACE OF AN UNCRYSTALLIZED BODY. 



17 



(i) s s' 



If we take s = s, — - gives - = — , which is in ffeneral inconsistent with (6) ; we must there- 

 ' ' {3) ^ V V ^ 



fore take s^= — s. We may suppose s' to have either sign. Hence by (3) and (4), we have 

 V+ V=V', s(V- V) = lixs' r, which give 



s — /xs' , . .„ 2s 



(7), 



r=. 



(8). 



S + /xS ' ' S + /xS 



(15.) We shall now interpret these results. 



Supposing p and s positive, the normal propagation of the wave V tends to increase x and x 

 (see Article 6) ; and since p, = p, «, = — s, that of the wave F tends to increase a> and diminish z ; 



/ ? 

 and since p = ixp, s' = ± \/ l , that of the wave V tends to increase x, and to increase or 



M 

 diminish z. Hence we have two cases according as we take the upper or lower sign of s'. 



Fig. (l) represents the first case; X'AX and Z' AZ z 



are the co-ordinate axes; NA, N^A, N' A are the normals /-j\ 

 to the waves V, V^, V' respectively, the arrows representing 

 the direction of normal propagation, N and N' tending to 

 increase x and z, and N^ to increase x and diminish «. 

 Since p,= p, and p' = ixp, we have Z NAZ'= L NAZ', , 

 and wa N' AZ = fx sin JV^Z'. This is the ordinary case 

 of reflexion and refraction. If a, a^, a be the maximum 

 values of F, F^, and F' respectively, the intensities of the 

 three rays N N ^ and N' will be proportional to a-, af, a'. 

 Now by (7) and (8), we have 



s — ixs 



2s 



s + fxs s + fxs 



These are Fresnel's formulae for the intensities of the reflected and refracted rays of a ray 

 polarized in the plane of incidence. 



Fig. (2) represents the second case, in which s' is negative ; 

 and therefore N' tends to diminish z. This case may occur in 

 the following manner. An incident ray along NA will produce 

 a reflected ray along AN^, and a refracted ray along AN", 

 /. N"AZ being equal to zN'AZ; and another incident ray 

 along N'A will produce a reflected ray along AN", and a re- 

 fracted ray along AN^. Now let the intensities of the two rays 

 along AN" be equal, and let one of these rays be half a wave 

 behind the other ; then they will interfere and destroy each 

 other, and we shall have remaining only a ray along NA, one 

 along N'A, and one along AN^ (namely, the sum of the two 

 along AN^. This is exactly the second case. 



(16.) It is evident, that, in the ordinary case where the rays N^ and N' are the effects 

 produced by the ray N, the normal propagation of N' will be from and not towards the plane 

 of separation : therefore s' must have its positive value, and consequently the second of the above 

 cases cannot occur. 



(17.) If we suppose either F^ or V equal to zero, the equations (7) and (8) give us 

 either a — /xs' = 0, or s = 0, neither of which equations can be generally true. Hence the 

 incident ray must, in general, be accompanied by a refracted and a reflected ray, or the equations 

 of connection cannot be satisfied. 



Vol. VIII. Pabt I. C 



