36 THE INTERFEROMETRY OF 



non vanishes with the spectrum lines as the slit is widened, but, on the other 

 hand, persists as far as the interference of light for a narrow slit. Finally, 

 the apparent occurrence of more than one line is referable to the presence of 

 more than one nearly horizontal wide band in the field of the telescope. Thus, 

 for instance, cases between b and c near c and between c and d near c, figure 24, 

 are the ones most liable to occur when both diffractions take place at a single 

 grating. This result will be used in paragraph 15. 



15. Tentative equations. In the first place, the actual paths (apart from 

 the theory of diffraction) of the two component rays, on the right and left 

 sides of the line of symmetry, II'Z, figure 23, will be of interest. The compu- 

 tation may be made for the method of two gratings at once, as the result 

 (if the distance apart of the gratings is C = o) includes the method with one 

 grating; i.e., the more complicated figure 23, where G is the transmitting and 

 G' the reflecting grating, resolves itself into a case of figure 24, with but one 

 grating, G. M and M' are the two opaque mirrors, I the normally incident 

 homogeneous ray. Supposing, for simplicity, that the grating planes G and 

 G' are parallel and symmetrically placed relatively to the mirrors M and M', 

 as in the figure, the ray Y diffracted at the angle 0i is reflected into X at an 

 angle 2 -0i and diffracted into Z normally, at an angle 2 , on both sides. 

 Under the condition of symmetry assumed X+Y (X'+Y'} =o, or without 

 path-difference. Let N be the normal from / to M, and n the normal from 

 I' to M, with a similar notation on the other side. Hence if / be given an 

 inclination, di, 0i is incremented by ddi, Y-\-X passes into y \-\-y-\-x, Y f -\-X' 

 into y' \+y' +x' , decremented at an angle dQ\, while both are diffracted into 

 Z'. Since generally 



sn 6i sm = \ cos 



for homogeneous light and the same di. Hence ddi = dd'i = dQ, say. 

 If 5 = 02 0i, and = 0a+0i, the auxiliary equations 



02 N n 



sin 8 cos (V 2 ) 



are useful. From a consideration of the yC and yx triangles, moreover, 

 the relations follow : 



N Nn cos 



y~T~y^ T fs/~ JQ\ y^ ~, /. /,,\ _ // i j/i\ ^ y 







cos (5/2 -dO) ~cos(o-/ 2 )cos(0 1 +c?0) cos(9 2 -dO) 



and from the y\C and y'x triangles, similarly, 



, , N , = N-n ,_ .cos (di- 



" " 1- - " 



COS (5/2+d0) COS ((T/2) COS (0i-d0) COS (0 2 +d0) 



Hence, after some reduction, the path on one side is 



2Ncos(ff/2)_ Nn 



cos (0 2 d6) cos (cr/2) cos (02 



