Light from Transparent Matter. 85 



I. The vibrations of polarized light are parallel to the plane of 

 polarization. 



II. The density of the sether is the same in all bodies as 

 in vacuo. 



III. The vis viva is preserved ; from which it follows that 

 the masses of the sether put in motion, multiplied by the squares 

 of the amplitudes of vibration, are the same before and after re- 

 flection. 



IV. The resultant of the vibrations is the same in the two 

 media ; and therefore in singly refracting media the refracted 

 vibration is the resultant of the incident and reflected vi- 

 brations. 



"When the vibrations are normal to the plane of incidence, 

 and therefore parallel in all three waves, the application of these 

 principles gives rigorously Fresnel's tangent expression. If 

 the vibrations are in the plane of incidence, the fourth principle 

 alone leads to FresnePs sine-formula. This only shows that the 

 fourth principle is inconsistent with the others ; for, as we shall 

 see, unexceptionable reasoning founded on I. and II. leads to 

 an altogether different result. The very particular case of IV. 

 required when the vibrations are normal to the plane of inci- 

 dence happens to be correct. In order to prevent misappre- 

 hension, I should say there is a sense in which IV. is perfectly 

 true. If the vibrations belonging to the longitudinal surface- 

 waves be included, it expresses merely the continuity of dis- 

 placement, a condition which must necessarily be fulfilled ac- 

 cording to any view of the subject. But understood in this 

 true sense, it does not carry the consequences deduced from it. 

 It remains then to be seen what the magnitude of the reflected 

 wave would be according to principles I. and II., when the 

 light is polarized in the plane of incidence. Let us take up 

 the question after the method of Green, and inquire what are 

 the consequences of the various suppositions which may be 

 made : and first for light vibrating normally to the plane of 

 incidence. 



The plane of separation of the media being #=0, let the axis 

 of z be parallel to the fronts of the waves, so that z=0 is the 

 plane of incidence. The displacements in the two media are in 

 general denoted by %,r),%', f t , 7] j} f y ; but in this case £, rj, g p r) l all 

 vanish. For the general equation of motion we have 



d^_n_/d^ , d*£\ 



