184 UNDULATOIiY THEORY OF LIGHT. 



arc reversed. But to revor.se the molecular movements of a wave is to change 

 its phase half an undiilation. Accordiijg-ly, at the points where 0=^X, and 

 where the dij'crcnce of path is ^A, the difference of phase is ^/^.-f ^/.=A. This 

 thickness should accordingly give a bright ring, and not a dark one; and so it 

 is in fact observed to do. 



If there could be any hesitation about receiving this explanation of the 

 phenomenon, it may be entirely removed by considering the following two 

 experiments. ]\L-. Babinet having produced, by means of Fresnel's mirrors, the 

 fringes of interference already described, received the interfering pencils upon 

 a glass minor, of which one half was transparent and the other half silvered 

 on the back. The reflected pencils, thrown upon a screcn,~ still exhibited the 

 fringes. When both the pencils were reflected from the silvered part of the • 

 mirror, or both from the transparent part, the fringe in the middle continued to 

 be bright, as in Fresnel's original experiment. But when one of the pencils 

 Avas reflected from the transparent glass and the other from the metal, the middle 

 fringe became immediately dark. The other experiment alhided to consists in 

 introducing between the two lenses, in Newton's experiment, a fluid having a 

 refracting power intermediate between that of the upper and that of the lower 

 glass. With a crown glass above, having the index 1.5, and a flint glass beneath, 

 with the index 1.575, the oil of sassafras (index 1.53) or that of cloves (index 

 1.5o9) introduced between will convert the dark rings into bright ones, and vice 

 versa. In this case the rays, at both surfaces alike, arc passing from a rarer to 

 a denser medium. 



When the rings are viewed by oblique light the undulatory theory requires 

 that their apparent magnitudes should be governed by the following law. If 



MM' be the upper of two glass plates, Avith par- 

 allel surfaces, enclosing a lamina of air between 

 them, and if IPQSTVH be the path of a ray 

 incident obliquely at P, and reflected at the lower 

 surface of the lamina at the point S, this will fall 



in at T with another, reflected from the upper 



P,v 4(3 surface at T, Avhose path is NOTVIi. When 



the flrst is at P the other is at N, in P N, drawn 

 from P perpendicularly to NO. These two have the same length of path in 

 the medium MM'. Their difi'ercnce of path will therefore be QS + ST — NO, 

 or 2ST — NO. As the angle NPO is equal to the angle of incidence (which 

 put =;^ :, ) and also KST, we shall have — 



2ST= , and NO=OPsin.'=QTsin.'=2tftan:sin.'=2^'^^. 



cos: cos: 



Hence 2^1 -^0=2o(^~^^\—ZOcQ%i. 

 \ cos: / 



But in order that there may be interference, this difference of path must be a 

 multiple of half an undulation. Hence — 



jl;. 

 2^cos;=/7xy, or 0=7i--^=nx\A.^(iCt. [18.1 



cos: 



In which n is an odd number for the bright rings and an even number for the 

 dark. At oblique incidences, therefore, the thickness at Avliich a given ring 

 appears is greater than at a perpendicular incidence, in the ratio of the secant 

 of incidence to unity, or in the inverse ratio of the cosine of incidence to unity. 

 But this is the law which observation had established before the theory of 

 undulation had indicated its necessity. 



There is still one point to be attended to before the theory of the ])henomenou 

 is complete. The dark rings, as seen by reflection in homogeneous light, are 

 ahsolutdy dark, showiuir that the interference is total. But the amount of light 



