48 THE INTERFEROMETRY OF 



Now if B' is referred to the original normal it becomes B" '= Q'+i 

 or 



If the sodium lines are to coincide, 6=9", or approximately 



sin (Q(i i')} sin i' = sin 9 cos 9 . (i i'} sin i' = 



or on eliminating sin 9 



i f -i\ n X X 



sim sin 2 (i i ) cos 9 = ^ jy 



which is nearly 



i-i' X 



In case of the Wallace grating below 



=1.75X10-* X=58. 93 Xio- 6 i-i f = 10^X3-29 (D-D'} 



Thus if the inclosed angle i i' between the plates is i degree, or 0.0175 

 radian, D D' = S-^Xio'" 7 , about 0.3 per cent of D and equivalent to about 

 43 lines to the inch. With adequate facilities for measuring i, this method 

 may be useful for comparing gratings, not too different, in terms of a normal 

 or standard, practically, since the finite equations may also be expanded. 

 In a similar way the slight adjustments of the longitudinal axes of the two 

 spectra may be made by rotating one grating around a horizontal axis ; but 

 this correction is less easily specified. Finally, one should bear in mind that 

 with film gratings there is liable to be an angle i-i' between the adjusted 

 plates. Fortunately this has very little bearing on the method below. 



The range of displacement of grating within which the fringes may be 

 used with an ordinary small telescope extends from contact of the two gratings 

 to a distance of e = 2 to 3 cm. beyond. 



In figure 31, which is a plan of the essential planes of the apparatus, G, G' 

 being the ruled faces of the gratings in parallel, 7, I', I", three impinging 

 rays of white light diffracted into D, D f , the points a, b, c, a , a" , b are in the 

 same phase, so that the path-difference of the rays from b at g and / is easily 

 computed. If the single ray I is diffracted into D and D' or I and I' into D, 

 I and I" into D', I' and /" into D and D', the equations for- these fringes 

 should be (if AP is the path-difference), 



= <?(i-cos 0)=*(i-Vi- 



where D is the grating space, e the distance apart, and X the wave-length. 

 Thus the micrometer value of a fringe for a color X should be, under normal 

 incidence, 



For two colors X and X' 



= e(i-cos 6}=eM (+w')X' = e(i cos 6'}=eM' 



