M. Mascart on the Achromatism of Interferences. 521 



A — mX=f(^x, i),^ 



a + x = h tan i,> (1) 



'D = d-i=(})(i,n).y 



The order m of the achromatic fringe is defined by the con- 

 dition that, for a constant value of in, the angle 6 which de- 

 termines the direction of the emergent rajs be the same for 

 the neighbouring colours. If we diiferentiate these equations, 

 putting dm = 0, and dO^O, and replacing the expression 



— XjT, which depends only on the nature of the prism, by L, 

 ^"^^^" ^-^fl + I^VL(^.|^ + |A|^. . . (2) 



The law of the phenomenon of interference being known, 

 as well as the nature of the prism and its direction, equations 

 (1) and (2) determine all the quantities mx, ic, i, and d which 

 correspond to the achromatic fringe. 



As the angle of divergence of a fringe is in general very 

 small, we shall obtain it by making dm = l in the differential 

 equations and considering X and n as constants. The angle of 

 divergence dd of a fringe is then determined by the equation 



x(i + ^)=(^9/+|C)rf«. . . (3) 



V o* / \cos'' I c-v 01/ 



For a fringe of any order this angle depends on the wave- 

 length, but in the neighbourhood of the achromatic fringe we 

 may take into account equation (2), which gives 



^^=^l|^ (4) 



m on ^ ^ 



The factors L and ~ vary very slowly with the colour. 



We see then that, whatever be the law of interferences, the 

 value of dQ in the neighbourhood of the achromatic fringe is 

 in inverse ratio to the order m and is almost independent of 

 the wave-length, so that a very great number of fringes will 

 appear there. 



It is even possible, by making choice of a suitable inclina- 

 tion for the prism, to render the coincidence still more perfect, 



if the differential coefficient of the product L ^ with respect 



to the index of refraction is zero, which is given by the condition 



l|!| + ^|^=o (5) 



O'i dn 0'^ 



