30 

 ON CONCAVE GEATINGS FOE OPTICAL PUEPOSES * 



[American Journal of Science [3], XXVI, 87-98, 1883 ; Philosophical Magazine 

 [5], XVI, 197-210, 1883] 



GENERAL THEORY 



Having recently completed a very successful machine for ruling 

 gratings, my attention was naturally called to the effect of irregularity 

 in the form and position of the lines and the form of the surface on 

 the definition of the grating. Mr. C. S. Peirce has recently shown, in 

 the American Journal of Mathematics, that a periodic error in the 

 ruling produces what have been called ghosts in the spectrum. At first 

 I attempted to calculate the effect of other irregularities by the ordi- 

 nary method of integration, but the results obtained were not commen- 

 surate with the labor. I then sought for a simpler method. Guided by 

 the fact that inverse methods in electrical distribution are simpler 

 than direct methods, I soon found an inverse method for use in this 

 problem. 



In the use of the grating in most ordinary spectroscopes, the tele- 

 scopes are fixed together as nearly parallel as possible, and the grating 

 turned around a vertical axis to bring the different spectra into the 

 field of view. The rays striking on the grating are nearly parallel, 

 but for the sake of generality I shall assume that they radiate from a 

 point in space and shall investigate the proper ruling of the grating 

 to bring the rays back to the point from which they started. The wave 

 fronts will be a series of spherical shells at equal distances apart. If 



J An abstract of this paper with some other matter was given at the Physical 

 Society of London in November last, the paper being in my hand in its present shape 

 at that time. As I wished to make some additions, for which I have not yet had 

 time, I did not then publish it. I was much surprised soon after to see an article 

 on this subject which had been presented to the Physical Society and was published 

 in the Philosophical Magazine. The article contains nothing more than an exten- 

 sion of my remarks at the Physical Society and formula; similar to those in this 

 paper. As I have not before Ihis published anything except a preliminary notice of 

 the concave gratings, I expected a little time to work up the subject, seeing that the 

 practical work of photographing the spectrum has recently absorbed all my time. 

 But probably I have waited too long. 



