536 HENRY A. ROWLAND 



subsidiary spectra varies as the square of the order of the spectrum and 

 the square of the deviation of the line, or lines from their true position. 



So sensitive is a dividing engine to periodic disturbances that all the 

 belts driving the machine must never revolve in periods containing an 

 aliquot number of lines of the grating; otherwise they are sure to make 

 spectra due to their period. 



As a particular case of this section we have also to consider 



PERIODIC ERRORS OF RULING. THEORY OF " GHOSTS " 



In all dividing engines the errors are apt to be periodic due to 

 " drunken " screws, eccentric heads, imperfect bearings, or other causes. 

 We can then write 



y = a t n + a^ sin (e^ri) + a^ sin (e?n), + etc. 



The quantities e 1? e z) etc., give the periods, and a 1} a 2 , etc., the ampli- 

 tudes of the errors. We can then divide the integral into two parts as 

 before, an integral over the groove and spaces and a summation with 

 respect to the numbers. 



ds . 



I I 'e~ ib < Ax + w) ds = le-tbw j " 



Vy 1 */0 



It is possible to perform these operations exactly, but it is less com- 

 plicated to make an approximation, and take y"- y r = a, a constant 

 as it is very nearly in all gratings. Indeed the error introduced is 

 vanishingly small. The integral which depends on tho shape of the 

 groove, will then go outside the summation sign and we have to per- 

 form the summation 



sine^ + o s sine 2 



Let J n be a Bessel's function. Then 



cos (u sin ?) = 7 () + 2 [J" 2 (w) cos 2 y + J t (w) cos* ^ + etc.] 

 sin (w sin </>) = 2 [Ji (w) sin ^ + /, (w) sin 3 ^ + etc.] 



But e~ iu sin<f> = cos (u sin ^ ) i sin (w sin ^>) . 



Hence the summation becomes 



X [Jo (^0 + 2 ( iJi (S/iOi) sin e^n + J t (ft/taj cos 'Ze^n etc.)] 

 X [J (bvctt*) + 2 ( iJ t (b/jifty) sin e z n + J t (J//a 2 ) cos Zeji etc.)] 

 X [/ (bra,*) + etc.] 

 X [etc.] 



