S. JR. Williams — An Achromatoscojje. 103 



total number having passed A, D , or the total number lying 

 between the limits, \ D and \*>. This may be expressed by the 

 equation, 



N' D -N. = N' D . (4) 



In changing the order ]ST D to N' D a certain number of bands 

 have passed \ D . Let this number be m, whence, 



N' D -N D = m. (5) 



At some other point in the spectrum, say \ F , a similar varia- 

 tion has been going on, viz., 



N F - N, = N F . (6) 



Thus while N D was changing to N ; D so N F was changing to 

 N' F , hence, 



W r — TS r = n (7) 



where n is the number of bands passing \ F while m bands 

 were passing A. D , and since X D > \ F , n > m. Moreover, since 



Fig. 2. 



a band either to the right or left of any given band means 

 either an increase or a decrease in order of one, it follows that 

 the number between \ F and \ D is given by the relation 



N F - N D = Pi (8) 



N' F -N' D =^ 2 (9) 



where j?, is the number of bands between X D and X F before the 

 difference in path of the two rays was varied and j? 2 is the 

 number after the change. If equation (5) is subtracted from 

 equation (7) and the difference compared with what follows 

 from subtracting equation (8) from equation (9), it will be seen 

 that 



n — m=p a —p 1 , (10) 



that is, the increase in the number of bands between \ D and \ F 

 is equal to the greater number of bands which pass X F than \ D 



