542 HENRY A. ROWLAND 



Places. Intensities. 



AI _ fJL A. [/! (K,"i) /o (fas.i) J 3 (ba^J 7 2 (ba&J + etc.] 2 



*" + [</i C^iAti) t/x (ifl^x) <7 3 (>iAO /i (far,*0 + etc.] 8 , 

 etc. etc. 



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 dia- 

 mond. 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, however, 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 *r* 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 * depends on the focal length. This is supposed to be very 

 large, and hence K is small. 



This 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 change in the distance of the 

 grooves apart, but this effect is vanishingly small compared with the 

 effect on the summation, and can thus be neglected. The displace- 

 ment is thus proportional to 



K y*\ 



CASE I. LINES AT VARIABLE DISTANCES 

 In this case we can write in general 



y = an + atf + a^n 3 + etc. 



As K, a u a 2 , etc., are small, we have for the displacement, neglecting 

 the products of small quantities, 



(an + Oin a + a a n 8 + etc.) a 3 2] . 



