for Reflected and Refracted Light, 1] 1 



20. In the case of FresnePs and Arago^s result (see above, (9)) 

 of the change in the plane of polarization after reflexion, it should 

 be remarked that the reasoning turns wholly upon the signs of 

 the resulting equation for the tangents of the two related arcs ; 

 and these are derived from the relative signs of the quantities 

 which enter into the numerator and denominator simply as alge- 

 braic quantities, and without any reference whatever to the inter- 

 pretation of those signs as expressive of difference of phase, or any 

 other physical conditions. The resulting relation is a purely tri- 

 gonometrical one, and the direction in which the arcs are to be 

 relatively measured would be just the same whatever might be 

 the physical theory to which they were to apply. This case 

 therefore can in no way be affected by any theory of the change 

 of direction due to the position of the vibrations. Here, then, 

 FresneFs original formulas exclusively apply. 



21. The case then stands thus : — From the experiment of Prof. 

 Stokes, Maccullagh^s formulas are set aside. From the experi- 

 ments of Lloyd and Arago, so far as relates to vibrations perpen- 

 dicular to the plane of incidence, Fresne?s formula (CA') is set 

 aside, since it does not apply directly ; and for such vibrations 

 the subsidiary theory as to the signs is inapplicable ; while Fres- 

 nePs formula (Bh') fully agrees with experiment without any 

 subsidiary explanation. This foi^mula then must necessarily he 

 adopted exclusively. 



22. It is only, then, in the case of vibrations parallel to the 

 plane of incidence that the question remains between FresnePs 

 formula (C^'), which applies directly to the experimental results, 

 but rests on an hypothesis (No. III.) of equivalent vibrations 

 open to question, — and his formula (B^'), which rests on Maccul- 

 lagh's law of equivalent vibrations (No. II.), but does not apply 

 to the experimental facts without a subsidiary interpretation as 

 to the signs. 



23. Now any subsidiary construction, if necessary for the right 

 interpretation of a formula, only shows that that formula is sym- 

 bolically incomplete, and does not include the expression of the 

 whole case it is designed to represent. This, then, would be an 

 additional reason for rejecting that formula and adopting the 

 former in preference. 



24. But this construction (19) is in itself not free from question 

 and difficulty. For let us only consider the case at a point upon the 

 surface : here we have an incident rectilinear vibration in a deter- 

 minate direction ; but at its point of incidence it gives rise to a 

 nascent circular wave, in which it is impossible to say that the 

 vibration is more in one direction than another ; it has no recti- 

 linear direction till we come to take the common tangent (as in 

 the ordinary explanation of reflexion) to two successive waves ; 



