TRANSACTIONS OF THE SECTIONS. 15 



On Fresnel's Formulmfor Reflected and Refracted Light. By the Rev. Baden 

 Powell, M.A., F.R.S. Src, Savilian Professor of Geometry, Oxford. 



The author having recently published in the 'Philosophical Magazine' (July and 

 August 1856) a detailed review of the various questions respecting the rfemores<ra^zo» 

 of these well-known and important formulae, and their applications, is anxious to 

 put before the Section a short summary of the whole case, and to elicit, if possible, 

 a more complete discussion of the questions arising out of it, more especially as some 

 views recently taken would seem calculated to set aside the whole reasoning hitherto 

 adopted on the subject, and to involve the whole application and interpretation of 

 the formulas in doubt. 



The whole of these investigations is founded on the following principles : — 



(1) The law of vis viva (m and m, being the simultaneously vibrating masses of 

 aether without and within the medium ; h, h', h, the amplitudes of the incident, re- 

 flected, and refracted rays), 



>«(A2— A'2)=m;A,^ 



(2) The law of equivalent vibrations as given by Maccullagh (i and r being the 

 angles of incidence and refraction, I the plane of incidence), 



h-\-h'=.hf yWiraiWons perpendicular to I. 



cosr . 



(3) A+ft —ftj • sihraXioas parallel to I. 



(4) Another form of this law, the second case of which was adopted by Fresoel 



h—h'=:h, perpendicular to I. 



cosr 



(5) h — h'z=h — — -. parallel to I. 



(6) MaccuUagh's hypothesis of equal densities, giving 



m sin 2i 

 m, sin 2r 



(7) Fresnel's hypothesis of increased density in the more refractive medium, giving 



(8) MaccuUagh's hypothesis of vibrations parallel to the plane of polarization. 



(9) Fresnel's hypothesis of vibrations perpendicular to the plane of polarization. 

 From these assumptions are directly deduced formulas whose general types are 



h! sin (i — r) 



T~sin(i+r)' (^^ 



k' tan (i — r) 



Ic ~ tan (t+r)' (^^ 



whence h, and k, follow from (1) by (2), (3), or (4), (5). Also V becomes = 0, and 

 changes sign at the incidence of polarization. 



The several hypotheses give these formulas with different signs, and consequently 

 with different values of A, k,. 



(A) Combining Nos. 1, 2, 3, 6, 8, gives 



(«) +A and -\-h' parallel to I i 



(/3) ±fc and +&' perpendicular J ° 



(B) Combining Nos. 1, 2, 3, 7, 9, gives 



(«) +A and — h' perpendicular — Fresnel. 



(^) +i and +it' parallel. 



(C) Combining Nos. 1, 4, 5, 7* 9, gives 



(«) +A and +A' perpendicular — Fresnel. 



(/3) ±i and +A;' parallel. 



