84 M. Lorenz on the Theory of Light, 



any doubt that these formulae correspond exactly with experi- 

 ment, and that the slight discrepancies which Jamin has detected 

 receive a natural explanation from the fact that the passage from 

 one body to another takes place gradually. (Cf. Poggendorff's 

 Annalen, vol. cxi. p. 460 ; also vol. cxiv. p. 244.) These slight 

 discrepancies only serve as a further confirmation of the exact 

 applicability of the formulae when the refractive index of the two 

 bodies undergoes a sudden or infinitely small alteration; of which 

 cases the latter only will be here taken as the basis of the calcu- 

 lation. On the other hand, it is to be borne in mind that, although 

 these formulae are taken as the starting-point of the theory, they 

 are to be regarded as deduced solely from experiment, so that 

 equal weight must be given to any other formulas which lead to 

 equally concordant results. 



If we accordingly endeavour to give to the formulae their most 

 general and most fully developed form, it becomes evident, in the 

 first place, that we may add as a factor to the expression for the 

 amplitude of vibration of the refracted ray an arbitrarily chosen 

 power of the refractive index, since, in the refracting body, neither 

 the intensity nor the direction of polarization are determined by 

 experiment, and, besides this, the factor in question will disap- 

 pear again. Even if we could determine the intensity experi- 

 mentally in the refracting body, this factor would still be unde- 

 termined ; for we may consider it as experimentally demonstrated 

 that in one and the same substance the intensity is proportional 

 to the square of the amplitude of vibration; but it is unknown 

 what power of the index of refraction enters here as a factor. 

 We shall nevertheless immediately show that this factor is the 

 same for vibrations taking place in the plane of incidence and 

 for those perpendicular thereto, an assumption which is neces- 

 sary for the attainment of homogeneity in the following results. 

 On this supposition, the factor in question has no influence on 

 the rotation of the plane of polarization in the refracting body, 

 and hence it cannot be determined by means of experiments on 

 this rotation. 



Again, the sign of the excursion of the reflected ray is to some 

 extent undetermined. If the angle of incidence be denoted by 

 a, and the angle of refraction by /3, the ratio of the excursion of 

 the incident, refracted, and reflected light will be, according to 

 EresneVs formula thus expanded, for vibrations in the plane of 

 incidence, as 



, 2 cos cc sin /3 /sinaV tan (a— /3) ^ 



' sin(a+/3)cos(a— /3) \sin#/ : - tan(a+£) ' 

 and for vibrations perpendicular to the plane of incidence, as 

 t 2 cos a sin {3 /sin a V . — sm ( a— P) 

 sin(a+/3) \sm#/ : + sin(a-F/3) 



