524 M. Mascart on the Achromatism of Interferences. 



The value of dO is the same as in the case of locaHzed fringes. 

 I will take one example only. 



HerscheVs Fringes. — By placing the hypothenuse-face of an 

 isosceles prism on a thin plate of glass, in such a manner as to 

 leave between the surfaces a layer of air of sensibly constant 

 thickness <?, W. Herschel* perceived a series of curvilinear 

 fringes parallel to the curve which defines total reflexion. 

 These fringes show themselves clearly when the rays of light 

 are made parallel : they ai'e therefore produced by the inter- 

 ference of plane waves. In this case the angle /5 is zero, and 

 the difference in path equals 2e cos L If we denote by ^(i, n) 

 the deviation i-\-i' —A, which is the same thing as changing 

 the sign of 2 in the preceding equations, the angle of divergence 

 of a fringe of any order m is 



-., 1 cos^ 2 cosr' V cos/ 



di ■= : — : .f = m ■ 



m sin i cos r cos i 4e^ sin i cos r cos i' 



As these fringes are very near to total reflexion, we may 

 replace the angles r, r', and i' by their values R, W, and I', 

 corresponding to this finite direction, and we have obviously 



, ., \^ cos R' 

 rt^' = m -r-o 



4e^ cos E, cos V 



Thus Herschel's fringes present this very singular cha- 

 racter, that their apparent size, at least in the case of the first 

 ones, is proportional to the order of the fringe and to the 

 square of the ratio of the wave-length to the thickness of the 

 layer of air. 



The condition for achromatism is 



cos^ i -^ sin A 



-; r =ij J. 



sm I cos r 



The distance of the achromatic fringe from the limit of 

 total reflexion is therefore independent of the thickness of the 

 layer. The number of fringes visible in the neighbourhood of 

 this region is greater the greater the thickness of the layer 

 itself. 



Talbot t observed this phenomenon in light transmitted 

 through a layer of air compressed between the hypothenuse- 

 faces of two exactly similar prisms, and was able to count as 

 many as 200 fringes with white light. 



The fringes which appear at the margins of the polarized 

 field in NicoFs prisms or in Foucault^s prism are to be 

 explained in the same manner. 



* Phil. Trans, p. 274 (1809^. 



t Phil. Mag. [3] ix. p. 401 (1836). 



