XXXIV. On the Demonstration o/Fresnel's Formulas for Reflected 

 and Refracted Light. — No. III. By the Rev. Baden Powell, 

 M,A,y F.R.S. ^c, Saviiian Professor of Geometry in the Uni- 

 versity of Oxford*. 



1. TN two former papers (Phil. Mag. July and August 1856), 



Jl especially in the last paper, I have shown that, on received 

 principles, the original formulas of Fresnel are apparently neces- 

 sary for the application of theory to certain experimental results, 

 to the exclusion of some newer modifications, though deduced on 

 more systematic theoretical groundji. 



In opposition to this, however, another view has been sug- 

 gested (as there mentioned), which, if true, would set aside all 

 the reasoning hitherto adopted on the subject, but which to me 

 seems open to great doubt in itself. 



It is, however, clear that FresnePs original formulas cannot 

 both be deduced on any common principle hitherto proposed, it 

 being, as far as yet appears, necessary to assume a separate 

 hypothesis for each of the two cases, and these not apparently 

 reconcileable with each other. 



2.^ If the considerations I have adduced in my second paper 

 (§§ 24, 25) be regarded as well founded, it becomes highly im- 

 portant to find some mode of deducing both FresnePs original 

 formulas on a common principle. 



But whether the arguments I have advanced be thought valid 

 or not, it must still be allowed, on all hands, to be a matter of 

 some interest if possible to suggest a proof free from the objec- 

 tions mentioned. 



Since writing those papers, it has appeared to me that this 

 may be effected, provided the following considerations be ad- 

 mitted relative to the law of equivalent vibrations, which (as 

 before hinted) appears to be the doubtful element in the former 

 investigations. 



3. In the case (a) of vibrations perpendicular to the plane of 

 incidence, and where the incident, reflected and refracted, vibra- 

 tions are all parallel to each other and to the surface, there is no 

 difficulty. Here there is no geometrical construction from which 

 to find the relation of the amplitudes. In this case the proof of 

 equivalence depends directly on mechanical considerations alone, 

 agreeably to the reasoning referred to before (first paper, § 26). 

 Here hy h', and h^ being the mechanical values of the amplitudes, 

 we have simply for the law of equivalence, 



h,=:h-^h'. 



4. In the case (13) of vibrations parallel to the plane of inci- 



* Communicated by the Author. 



