408 Prof. Henry A. Kowland on Gratings 



ment, a law which I shall show applies to the effect of all 

 errors of the ruling. 



This particular effect was brought to my attention by trying 

 to use a tangent-screw on the head of my dividing-engine to 

 rule a grating with, say, 28,872 lines to the inch, when a 

 single tooth gave only 14,436 to the inch. However care- 

 fully I ground the tangent-screw I never was able to entirely 

 eliminate the intermediate spectra due to 14,436 lines, and 

 make a pure spectrum due to 28,872 lines to the inch, 

 although I could nearly succeed. 



Example 3. — One Groove in m misplaced. 



Let the space a contain m grooves equidistant, except one 

 which is displaced a distance v. The distance is now propor- 

 tional to 



ibu." 2tfyi- ibJp — +v) n ibu^— a 



b/xa 

 I sin 



a 1 

 2m 



tJ/x— « \ S111 « tV 



— +ibpve 



baa 



2 m 



„ 2p-m + l J 

 2m I 



Multiplying this by itself with —i in place of + i, and 

 adding the factors in the intensity, we have the whole ex- 

 pression for the intensity. One of the terms entering the 

 expression will be 



baa . baa 

 sin n -~- sin n —£- 7 _ 



2 2 . ba/ju2p— m + 1 



. baa . baa 2 m 



siu-tt- sm — - f- 

 2m 2 



Now the first two terms have finite values only around the 



points —k- = inNir, where mN is a whole number. But 



2p— m + 1 is also a whole number, and hence the last term 

 is zero at these points. Hence the term vanishes and leaves 

 the intensity, omitting the groove factor, 



. o bap . 9 baa 



sin 2 n —£■ sm 2 n —^~ 



. baa ^ . baa 



sin ^ — sm 2 -£- 



2m 2 



