C. Bar as — Interference of Reversed Spectra. 431 



these interferences is the locus of the corners in the successive 

 parallelograms, B is the distance between two bright bands, 

 while B\ making an angle y' with S, is the direction of these 

 parallel interference bands relative to the vertical. Let the 

 free ends of D and D f be joined by the line E' ; and if D is 

 prolonged to the left and the intercept is D in length, let this 

 be joined with the end of D' by E. Then the triangle BED' 

 and S'BS, DEI)' and S'B'S may be shown to be similar. 

 Omitting the demonstration, the values of B and B' may 

 finally be reduced to 



E V&* + D" + 2Z>X>'cos~0 



sin <f> sin <f> 



, _ E' VD 2 + D n - 2DD' co9 4 , . 



6in cf> ~ sin <£ 



Again the angle y' is on reduction tan y' = ^-. ~^- — r . 

 5 * y y D' - D cos <f> 



113 = D', or S = S' 



b. = *,, b: = D 



sin (<f>/2) ° cos (0/2)' 



tany' = . S1 " ^ x = cos (0/2) /sin (0/2) 

 y sin (0/2) vri ' I vr/ ' 



Thus if = 0, tan y' = cc , y' = 90°, or the fringes are horizon- 

 tal and i? = 2& If y' is nearly zero tan y' = 0/(0 2 /2) = 2/0, 

 changing very rapidly with 0. 



If one grating of a pair, with identical grating spaces D, is 

 moved parallel to itself, in front of the other, the effect to an 

 eye at a finite distance is to make the grating spaces D vir- 

 tually unequal ; or 



B __ B V 1 + (B'/Uy + 2{D'/D) cos0 ? 

 2 cos (0/2) 



so that for an acute angle 0, the fringe breadth is increased. 

 Thus B is a minimum in case of coincident gratings. 



The analogy is thus curiously as follows : The fringes just 

 treated rotate with the rotation of either grating in its own 

 plane and pass through a minimum size with fore and aft 

 motion ; whereas in the above results, the optical grating 

 showed a passage through a maximum of size with the rota- 

 tion of either grating in its own plane and a rotation of fringes 

 with fore and aft motion of the grating. It is in terms of this 

 analogy that an explanation of the optical interferences will 

 have to be attempted, but there are some other complications 



Am. Jour. Sci. — Fourth Series, Vol. XLI, No. 245. — May, 1916. 

 30 



