relating to Curved Diffraction-gratings. 49 



Putting r=p 1 and = in (2), we have 



c)p 1 = l—cos(j); (3) 



and, again, putting —r = p 2 and = 17, we have 



c/p 2 =l-t- cos<£ (4) 



Now suppose G S the direction of the star to be kept fixed, 

 and the grating turned so as to vary the value of $ ; then 

 equations (3) and (4) will give the loci of ¥ l and F 2 respec- 

 tively. The loci are the parabolas marked P T and P 2 . They 

 have a common focus G, and common axis GS, and a common 

 latus rectum =2c. 



F 1 is the real focus of reflected light, having a wave-length 



= - sin 6, where a is the distance between the lines of the 

 n 



grating, and n is an integer. 



F 2 is the virtual focus of transmitted light having the same 

 wave-length. 



In fig. 2, P P is part of the parabola (on a larger scale) 



Fier. 2. 



which is the locus of the normal focus of reflected light; 

 G N, G N 7 , G N" are difference positions of the normal to the 

 grating ; and L L, 1/ I/, I/' I/' are respective positions of the 

 spectra, and F, F', F" are respective positions of the normal 

 foci. 



The theorem is, When the sources of light are at an infinite 

 distance the normal foci lie on two parabolas whose common 

 focus is the centre of the grating and common latus rectum 

 is equal to the diameter of curvature of the grating ; the para- 

 bolas for reflected light being convex to the source of light, 

 and those for transmitted light being concave. 



Phil. Mag. S. 5. Yol. 22. No. 134. July 1886. E 



