for Reflected and Refracted Light, 8 



the respective rays deduced on the supposition that the vibra- 

 tions are perpendicular to the plane of incidence, another set (K) 

 on the supposition that they are parallel to that plane. Now 

 those of Maccullagh, which correspond closely to FresneFs first 

 set (H), are deduced on the contrary supposition of vibrations 

 parallel to incidence, while those corresponding to FresneFs 

 second set (K) are for vibrations perpendicular to that plane. 



8. In either investigation the formulas (K) are those which 

 represent evanescence of the light at the polarizing angle, while 

 the formulas (H) represent brightness at that incidence. 



But when a ray vanishes at the polarizing angle, we know that 

 its plane, of this second incidence, must be perpendicular to that 

 of its first incidence or original polarization. Hence, according 

 as the vibrations (K) may be parallel or perpendicular to this 

 second plane of incidence, they must be respectively perpendicular 

 ov parallel to the first or plane of polarization. The question 

 thus reduces itself to whether, in polarized light in general, the 

 vibrations qxq parallel ov perpendicular to the plane oi polarization. 



9. M. Cauchy, in an earlier paper {Mem. Instil, vol. x. p. 304), 

 had inferred with Maccullagh, from dynamical views, that the 

 vibrations are parallel to the plane of polarization. But in a 

 later memoir [Bull. Math, July 1830) he deduces formulas cor- 

 responding to FresneFs on the hypothesis of vibrations peipen- 

 dicular to the plane of polarization, and even more formally 

 renounces his earlier opinion and returns to that of Fresnel. He 

 also connects similar equations with higher dynamical principles 



in the Nouv, Exercices Math. (liv. 7). jkuI tp^iXV 



Synopsis of Formulas referred to. 



10. FresneFs formulas for yibvations perpendicular to the plane 

 of incidence {h being the amplitude of the incident, h' of the 

 reflected, and h^ of the refracted rays, and dividing by h, i being /d 

 the angle of incidence, r of refraction), are — 



,,__ — sin (e— r) _ 2 sin r cos i ,ttv 



~" sin(2+r)' '"" sin(i + r)' - ' ' \ ) , 



and for vibrations parallel to the plane of incidence (similarly }. 

 designated by k, k\ and k^j 



/^f^ tan(i-r) /^^^ tan (g~r) \ cosi ^ ^ .^. 



tan(i + r)^ ' \ tan (2 + r)/ cos r* 

 These last may be otherwise expressed thus : 



sin2i— sin2r ■ •• 



.y^-r: 



"> — ~~- — ^^"=~^ — = — ^ — » PB'j.mti 



, / 2sin2r \cos i 4 sin 7 ,,. 



a:,== ( -. — TT- ■• — rr- ) = . ^. . ^^ • ml ic 



' Asm 2i -^ sm 2r /cos r sm m + sm Zr 



B2 



sin 2i + sin 2r ' 



2 sin 2r \cos i 4 sin r cos i 



