in Theory and Practice. 413 



some. The first error is «i = xo~o?)o"oo * n# an( ^ the second 

 «i = soo^oo ^ n * Hence a periodic displacement of one 

 millionth of an inch will produce visible ghosts and one five- 

 hundredth-thousandth of an inch will produce ghosts which 

 are seen in the second spectrum and are troublesome in the 

 third. With very bright spectra these might even be seen 

 in the first spectrum. Indeed an over-exposed photographic 

 plate would readily bring them out. 



When the error is very great, the primary line may be 

 very faint or disappear altogether, the ghosts to the number 

 of twenty or fifty or more being often more prominent than 

 the original line. Thus, when 



b/i ai = 2-405, 5-52, 8-65, &c. =2ttN- 1 , 



the primary line disappears. When 



V«i = 0, 3-83, 7-02, &c. =2ttN^, 



a 



the ghosts of the first order will disappear. Indeed, we can 

 make any ghosts disappear by the proper amount of error. 

 Of course, in general 



_ 2(n-l) T T 



Thus a table of ghosts can be formed readily and we can 

 always tell when the calculation is complete by taking the 

 sum of the light and finding unity (see p. 414). 



This table shows how the primary line weakens and the 

 ghosts strengthen as the periodic error increases, becoming 



at 27tN — = 2*405. It then strengthens and weakens periodi- 



a 



cally, the greatest strength being transferred to one of the 

 ghosts of higher and higher order as the error increases. 



Thus one may obtain an estimate of the error from the 

 appearance of the ghost. 



Some of these wonderful effects with 20 to 50 ghosts 

 stronger than the primary line I have actually observed in a 

 grating ruled on one of my machines before the bearing end 

 of the screw had been smoothed. The effect was very similar 

 to these calculated results. 



Phil. Mag. S. 5. Vol. 35. No. 216. May 1893. 2 F 



