416 Prof. Henry A. Rowland on Gratings 



Hence the sum of all the light is still unity, a general pro- 

 position which applies to any number of errors. 



The positions of the lines when there is any number of 

 periodic errors can always be found by calculating first the 

 ghosts due to each error separately ; then the ghosts due to 

 these primary ghosts for it as if it were the primary line, and 

 so on ad infinitum. 



In case the ghosts fall on top of each other the expression 

 for the intensity fails. Thus when e 2 = 2e v e 3 = 3e l9 &c, the 

 formula will need modification. The positions are in this 

 case only those due to a single periodic error, but the inten- 

 sities are very different. 



/*i 



** - ba ' X + (Ji(1><hP>i) Ji(^«2^i) — Jz{ba lf M } ) J^ba^) +&c.] 2 



Places 



ba 



fju =-£—. [Jo(ba- L fi)J (ba 2 fi)'] 



&c. &c. 



We have hitherto considered cases in which the error could 

 not be corrected by any change of focus in the objective. It 

 is to be noted, however, that for any given angle and focus 

 every error of ruling can be neutralized by a proper error of 

 the surface, and that all the results we have hitherto obtained 

 for errors of ruling can be produced by errors of surface, and 

 many of them by errors in size of groove cut by the diamond. 

 Thus ghosts are produced not only by periodic errors of 

 ruling but by periodic waves in the surface, or even by a 

 periodic variation in the depth of ruling. In general, how- 

 ever, a given solution will apply only to one angle and, 

 consequently, the several results will not be identical ; in 

 some cases, however, they are perfectly so. 



Let us now take up some cases in which change of focus 

 can occur. The term /cr 2 in the original formula must now 

 be retained. 



Let the lines of the grating be parallel to each other. We 

 can then neglect the terms in z and can write r 2 =y 2 very 

 nearly. Hence the general expression becomes 



where tc depends on the focal length. This is supposed to be 

 very large, and hence tc is small. 



The integral can be divided into two parts— an integral over 

 the groove and the intervening space, and a summation for 

 all the grooves. The first integral will slightly vary with 



