6 ON THE ASTIGMATISM OF ROWLAND'S CONCAVE GRATINGS. 



the last or fourth spectrum with v = 68°, siu v = 0.928. A knitting- 

 needle held in the horizontal caustic, that now lay at 714 cm. from 

 the slit, was accurately represented by a narrow black line across 

 the solar spectrum. This proves that the definition in the images 

 of horizontal lines, produced by the vertical fans holds good even 

 at this great angle of incidence. 



I still may remark that the whole action of the hollow grating 

 with a radius q, may for these fans be regarded as the result of 

 three successive operations : one being that of a first concave mirror, 

 with a radius 2ç>, but reduced by astigmatism to a radius 2(>secr, 

 that brings the incident rays to parallelism ; the second that of a 

 plane grating, which occasions the diffraction at an angle v; the third 

 that of an other concave mirror 2 ç>, which makes the diffracted 

 parallel rays converge into a focus. The distances and dimensions 

 of two conjugate images may be simply calculated by the formulae 

 for oue mirror with /=q/(\ -|-cosi>), as may be proved in the fol- 

 lowing manner. 



Let B K (Fig. 2) be part of a very narrow vertical strip, and B 

 the centre of the mirror, C the centre of curvature; D and E two 

 conjugate foci determined by their height *, = D M, £ 2 = FL over the 

 horizontal plane LB M, by B M = R, BL = r and /_ M B L = v ; 

 q being the radius B C of the sphere, K I = I. 



Now with a sufficient degree of approximation we successively 

 find 



l 2 I 2 COS V 



BI = — , IM— . R , 



2q 2q 



R I 2 cos v 

 K D» = 1 M* + (/ - m) 2 = R 2 - - -+(L — *]) a 



Q 



l 2 cos v l % — 2 I z-, -\- z{ 2 



consequently 



I 2 l 2 —2lz. 2 + z 2 



For the point B, 1 = 0, we have 

 — B D = - R 



2R' 



-BE=-r 



2r' 



