M. Lorenz on the Theory of Light. 85 



p is here the above-mentioned undetermined exponent, which 

 in Fresnel's expressions is equal to nothing. The second source 

 of uncertainty is the double sign, which, however, it must be 

 observed, is reversed in the two cases. The positive direction of 

 vibrations which take place in the plane of incidence may ; for 

 instance, be chosen so that, the plane of incidence being supposed 

 horizontal, it is to the left of the observer when he is turned 

 towards the ray, whether incident or reflected. Now it is a fact 

 that when light is reflected at an angle of incidence of nearly 

 90°, for example, the azimuth of the plane of vibration does not 

 change its sign, calculated in the manner already indicated, 



whence it follows that an y*~ igj anc i sm ; a ~^; are in this case 



tan (a + /3) sm(a + £) 

 (a + /3;>90°) positive or negative at the same time, which can 

 occur only if the two expressions have opposite signs. 



Nevertheless we use the formulas that have been given only 

 for the case in which the angle of incidence differs infinitely little 

 from the angle of refraction. If/3 = a-f^a, the ratios for the 

 vibrations in the plane of incidence become as 



l:l + ( 1 _JL_W ± *L_. . . (3) 

 \sm2« tana/ tan Zu 



and for those which take place perpendicularly to the plane of 



incidence, as / i n \ j„ 



\..\^{~--^-)du:+^ r . . . (4) 

 \sm2a tan a/ sin 2a 



We will now, in the first place, suppose the heterogeneous 

 body divided into layers in some given direction, in such wise 

 that each layer perpendicular to the axis of x may be regarded 

 as homogeneous for an infinitely small thickness, and as differ- 

 ing to an infinitely small amount from the next following layer. 

 Within the limits of such a layer the ray moves as in a homo- 

 geneous body ; at the limiting surface it is partially reflected ; 

 and both in the incident and in the reflected ray the vibrations are 

 perpendicular to the direction of the ray. We here take into con- 

 sideration the elementary rays or the virtual motion of the light ; 

 the actual motion thence resulting may perhaps exhibit a differ- 

 ent direction of vibration, since, for example, all the rays reflected 

 from the different layers may neutralize each other, although they 

 may still exert an influence on the direction of the vibrations in 

 the transmitted ray. 



If we assume that the plane of coordinates ocs coincides with 

 the plane of incidence, the excursion at the entrance of the light 

 into the substance, at the surface x=0, may be expressed by 

 terms of the form 



AeCtt-^W^i^ ( 5 ) 



