Interference-Bands, 91 



shifted relatively to those of the D-system through half the 

 band-interval. From (1) 



Su = nD{\/b-\ Q /b \ 



= iX D/6 

 by hypothesis ; so that 



\/ b o 



v^-vv (15) 



Thus, in the case supposed, n = 32. After 32 periods the 

 black places of the C-systern will coincide with the bright 

 places of the D-system, and conversely. If no prism had 

 been employed (b constant), a similar condition of things 

 would have arisen when 



2 A 



= 4-2. 



A — Xq 



If (X— X ) or, as we may call it, hX be small, 



X /6 



is of the second order in SX. An analytical expression is 

 readily obtained from (14). We have 



\/b _ 1 + 3SX/X + 3(SX/X ) 2 + ( 8 X/X ) 3 

 X /b - 1 + 3Sa/a + 1(8a;a ) 2 



_, , f(gyxo) 2 +(Sx/Xo) 3 



" M + 3SX/X+3^a/X ) 2 

 = 1 + |(SX/X ) 2 -4(3X/X ) 3 , 

 approximately ; so that, by (15), 



This gives the order of the band at which complete dis- 

 crepance first occurs for X and X + SX, the adjustment being 

 made for X . It is, of course, inversely proportional to the 

 square of SX, when B\ is small. 



The corresponding value of n, if no prism be used, so that 

 b is constant, is 



» = *£ < 17 > 



A 



The effect of the prism is thus to increase the number 

 of bands in the ratio 



2X : 3SX. 



[To be continued.] 



