110 Prof. Powell on the Demonstration o/FresnePs Formulas 



all incidences greater than that of complete polarization in the 

 formulas for V; that is, at very oblique incidences for rays 

 polarized in one plane equally with those in the other ; in exact 

 agreement with the observed fact. 



16. This is the argument expressly adduced by Dr. Lloyd, 

 derived directly^ without any subsidiary considerations, from the 

 indications of the symbols ; and it has, I believe, been generally 

 received as perfectly clear and conclusive. It should be remem- 

 bered that Dr. Lloyd's paper was communicated and published 

 in 1834, and refers to FresnePs formulas in the form in which he 

 originally gave them, and before any modifications of the theory 

 had been contemplated. Prof. Maccullagh's new views and for- 

 mulas were first communicated to the Royal Irish Academy in 

 1837, and published in its Memoirs in 1838. Thus Dr. Lloyd's 

 reasoning is of necessity wholly independent of any speculations 

 on these newer principles, which, if applied to FresnePs theory, 

 give the formulas with different signs, 



17. Now so far as the mere symbols are concerned, we find in 

 FresneFs formulas (B) this difference of sign, at great incidences, 

 occurring only for A' ; and in the formulas (C) (as is the case 

 also in Maccullagh's formulas) only for k! -, while in (B) for k', 

 and in (C) for h', no such difference occurs. 



This result would therefore seem decisive in favour of FresneFs 

 original formulas (D), to the exclusion not only of MaccuUagh's, 

 but of Fresnel's, in the entire forms (B) and (C). 



18. The reasoning both in this case and the former (9), (10), 

 is indeed of a nature apparently so obvious, that I should not 

 have thought it necessary to state it in detail, were it not that 

 some considerations have been suggested which seem to set it 

 aside, or at any rate to require a closer review of its meaning ; — 

 and which may be stated as follows : — 



19. For vibrations parallel to the plane of incidence, if we 

 make a construction of the course of the rays, then, at an inci- 

 dence very near the perpendicular, the vibrations k and k' respect- 

 ively perpendicular to the incident and reflected rays will lie very 

 nearly in one line ; and if we suppose them in the same direction, 

 then in passing to an incidence extremely oblique, they will come 

 into directions opposite to each other, owing merely to the posi- 

 tion the rays have now assumed. 



Hence it is inferred, whatever relative directions of the vibra- 

 tions we express by the signs + or — at 2 = will be reversed 

 when we pass to i=90°. And thus, for example, in the formula 

 [Bk'), if at z=0° we suppose —A' and -hk to express opposite 

 directions, then at t=90 the same signs would express accordant 

 directions. But in fact at t = 90° we have -\-k! and -\-k, which 

 on this principle therefore now express opposing directions. 



