ON CONCAVE GRATINGS FOR OPTICAL PURPOSES 493 



these waves strike on a reflecting surface, they will be reflected back 

 provided they can do so all in the same phase. A sphere around the 

 radiant point satisfies the condition for waves of all lengths and thus 

 gives the case of ordinary reflection. Let any surface cut the wave 

 surfaces in any manner and let us remove those portions of the surface 

 which are cut by the wave surfaces; the light of that particular wave- 

 length can then be reflected back along the same path in the same 

 phase and thus, by the above principle, a portion will be sent back. 

 But the solution holds for only one wave-length and so white light will 

 be drawn out into a spectrum. Hence we have the important conclu- 

 sion that a theoretically perfect grating for one position of the slit and 

 eye-piece can be ruled on any surface, flat or otherwise. This is an 

 extremely important practical conclusion and explains many facts which 

 have been observed in the use of gratings. For we see that errors of 

 the dividing engine can be counterbalanced by errors in the flatness of 

 the plate, so that a bad dividing engine may now and then make a 

 grating which is good in one spectrum but not in all. And so we often 

 find that one spectrum is better than another. Furthermore Professor 

 Young has observed that he could often improve the definition of a 

 grating by slightly bending the plate on which it was ruled. 



From the above theorem we see that if a plate is ruled in circles 

 whose radius is r sin [JL and whose distance apart is dr / sin //, where Ar 

 is constant, then the ruling will be appropriate to bring the spectrum 

 to a focus at a distance, r, and angle of incidence, //. Thus we should 

 need no telescopes to view the spectrum in that particular position of 

 the grating. Had the wave surfaces been cylindrical instead of spher- 

 ical the lines would have been straight instead of circular, but at the 

 above distances apart. In this case the spectrum would have been 

 brought to a focus, but would have been diffused in the direction of 

 the lines. In the same way we can conclude that in flat gratings any 

 departure from a straight line has the effect of causing the dust in the 

 slit and the spectrum to have different foci, a fact sometimes observed. 



We also see that, if the departure from equal spaces is small, or, in 

 other words, the distance r is great, the lines must be ruled at distances 

 apart represented by 



r sin n 



in order to bring the light to a focus at the angle p. and distance r, c 

 being a constant and x the distance from some point on the plate. If 

 f* changes sign, then r must change in sign. Hence we see that the 



