420 Prof. Carl Barus on Interferometry 



Here #>©, so that a — Q — © is the deviation of the oblique 

 ray from the normal ray in air. Finally, the second 

 reflexion of the oblique ray necessarily introduces the angle 

 of refraction within the glass, such that 



sin (© — a) =yu<0 sin j$ x (3) 



If D is the grating space, moreover, 



sin i — s'm0 = \ e /D and sin i — sin © = X /D . (4) 



6. Equations for the Present Case. — From a solution of 

 the triangles preferably in the case for single incidence, as 

 in fig. 8, the air-path of the upper oblique component ray at 

 a deviation ; a, is 



2^ COS©/ COS (©-a). 



The air-path of the lower component ray is 



2 cos ®(y n —e sin ©(tan 6\— tan ©0)/ cos (©-a). 



The optic path of this (lower) ray in glass as far as the final 

 wave-front in glass at fii is 



fi e e(l/ cos X + 1/ cos ft). 



The optic path of the upper ray as far as the same wave-front 



in glass is 



N sin fr-^2 _^|^(y w -y n + e sin ©(tan ^ - tan © 2 )) 



-^(tan^-tanft)} . (5) 



Hence the path difference between the lower and the upper 

 ray as far as the final wave-front is, on collecting similar 

 terms, if the coefficients of y m ~y n =y (where y is positive) 

 and of e be brought together, as far as possible, and if the 

 path difference corresponds to n wave-lengths X , 



n\~ — 2y cos u + 2e/x Q cos«(l/ cos © x — cos (6 X — ©J/ cos 6 X ) 

 + ^ fl (l+cos(0 1 -A))/cos0 1 , ... (6) 



which is the full equation in question for a dark fringe. It 

 is unfortunately very cumbersome and for this reason fails 

 to answer many questions perspicuously. It will be used 

 below in another form. 



The equation refers primarily to the horizontal axes of the 

 ellipses only, as e increases vertically above and below this 

 line. The equation may be abbreviated, and in case of a 

 parallel compensator of thickness e (y being the path difference 

 in air) may be written 



«>u=-2ycos« + ( e -/)(Ze-Z,) ... (7) 



