40 PROCEEDINGS* OF THE AMERICAN ACADEMY. 



relation to one another. The error in ruling which produces them must 

 be of a very complex nature, and we cannot expect the phenomenon to 

 be readily brought under mathematical analysis. In some cases, how- 

 ever, an exact treatment should be possible, for with certain gratings 

 some few of the false spectra are very much stronger than the others. 

 Here it is fair to suppose that we have a comparatively simple error 

 combined with a complicated one. Under the circumstances it seemed 

 profitable to attempt to modify the ordinary treatment as given by 

 Rowland. 



A grating of iV lines may, if N = n m, be considered as a grating of 

 m groups, each consisting of n lines. The spectrum of the first order 

 may be considered as the spectrum of the nth order of a grating of m 

 groups. Each group is in itself a grating of n lines. The intensity I„, 

 any point of a spectrum of a grating of n lines, is known to be propor- 

 sin n a\ 2 , ' . . s tt 



tional to A 2 I — : ) , where a = e (sin i — sin r) - . A' 2 is the inten- 

 se sin a J k 



sity of one line, which may be different in different directions, and e is 



the grating space or distance between two consecutive lines. 



If we consider the grating of m groups, the intensity at any point of 



/sin )7i ci \ 



its spectrum will be proportional to /„ I — : — ) , where a' = na. The 



\ sin a' J 



1st, 2d, 3d n — 1 order will vanish for any value of A, because 



tt 2 7r (n — 1 ) 7T 



l n vanishes for a = -, — 



n n n 



Now let us assume that in the ruling of n consecutive lines there is an 



irregularity, such that the grating of n lines which we take to be repeated 



m times is of itself an imperfect grating. Then I„ will not vanish at all 



tt 2 tt (n — l)ir 



places where a = — , — — and some or the first n — 1 



n n n 



orders of the grating of m groups will be visible. So far this is the 



ordinary theory of false lines. 



Further, let us assume that the grating of n lines has a periodic error. 



To make the matter definite, let us take this error as occurring every 



third line. That is to say, every third line is slightly out of place, while 



the other lines remain in their correct position. Then it is probable that 



7T 2 7T 



I n will have some appreciable value at a = - and at a ■= —. If we 



take the error to be somewhat irregular, the intensity I n will spread to 



both sides of those positions where a = - and a = — . The intensity 



o o 



