544 



the summation is 



HENRY A. ROWLAND 



( 



+ etc. 



sn 



dq 



rf* sin <y __ 

 ~dq* q 



_ <7 cos <y sin q 



S * D 



2q cos <y + (2 g 2 ) sin g 

 q* ' 



g (6 g 2 ) cos q (6 3g 2 ) sin 

 ~~ 



< 



etc. 



etc. 



These equations serve to calculate the distribution of light intensity 

 in a grating with any error of line distribution suitable to this method 

 of expansion and at any focal length. For this purpose the above 

 summation must be multiplied by itself with -f- i in place of i. 



The result is for the light intensity 



j^ sin q 

 dq q 



+ 2c - + etc. 



16 

 n* 



Uq 3 



9 



sin q 



16 



As might have been anticipated, the effect of the additional terms is 

 to broaden out the line and convert it into a rather complicated group 

 of lines, as can sometimes be observed with a bad grating. At any 

 given angle the same effect can be produced by variation o'f the plate 

 from a perfect plane. Likewise the effect of errors in the ruling may 

 be neutralized for a given angle by errors of the ruled surface, as noted 

 in the earlier portions of the paper. 



