522 M. Mascart on the Achromatism of Interferences. 



Calling A the angle of the prism, /S the angle made by the 

 entering-face with the surface S, r the angle of refraction of 

 the ray MP on this face, /*' and i' the angles of emergence, we 

 readily find that the equations which determine the order of 

 the achromatic fringe, the angular distances of the neighbour- 

 ing fringes, and the condition for best achromatism, become 

 m\ _ . h 'df _^ 'df\ sin A .^,. 



L \cos"^i^.t; Bi /COS (/t?— z") COS'/' * ' ^ ^ 



d0= ^ "'" ^ .„ (40 



m cos?' COS i' ^ 



T- -^— =sin (/3— i) tanr cos«'— sin A tan i'. . (5') 

 h an ^ ' 



We easily see that the employment of a grating as dispersion- 

 apparatus would not produce an analogous effect. 



We will apply these results to two particular cases. 



Interference-fringes. — Let us consider first the case of ordi- 

 nary interferences, where the fringes are equidistant and sym- 

 metrical with respect to one of them for all colours. The 

 origin being taken at the centre of the phenomenon, the 

 difference in path A is simply proportional to x^ and we 



m\ = ax, -^— =a, ^-. =^0. 

 doc ^i 



There exists only one achromatic fringe, of which the 

 order m and the abscissa a; are given by the equations 



_ m\ _j L sin A 



a cos^ i cos (yS — i) cos r'' 



These values of m and of x are the greater, all other things 



being equal, the farther the prism is from the surface S. 



If the observation is made in a direction normal to the 



surface, that is to say, the screen being discarded, on the path 



of the interfering rays, there remains simply 



m\ , T sin A 

 x= — = AL 



cos /3 cos r' 



J^ewton's Rings. — If the rings are produced by a layer of 

 air between a plane surface and a spherical surface of radius R 

 which do not touch, the thickness of the layer at a distance x 

 from the centre may be expressed by 



^ = ^o + i (^) 



The difference in path at the point considered is then 



.m\=2ecos/=l 2^0 + tt) cost, 



