4 ON THE ASTIGMATISM OF ROWLAND'S CONCAVE GRATINGS. 



In order to demonstrate the truth of this assertion let us consider 

 the pencil of monochromatic rays that will, after the reflection at 

 the grating, concur in the focus at C. It may be divided into what 

 I may be allowed to call vertical „ fans" of rays, each of them 

 being limited by two vertical planes passing through the slit and 

 including an infinitesimally narrow strip of the grating. Now all 

 the rays contained in such a „ fan", in order to concur at C without 

 any difference of path, must issue from an apex situated in the 

 line C Q , being the axis of the spherical surface, part of whose 

 equatorial region is occupied by the grating. 



On the other hand the horizontal fans of rays into which the 

 pencil may be divided, by theory of diffraction have their apices 

 in the slit. So all the rays that concur at C must have passed 

 successively through two caustics : the one realised by the slit, 

 the other only virtual, lying along the line GH, where it may be 

 realised by another slit, if the source of light be placed at a suf- 

 ficient distance. It will be easily seen that the length of the first 

 caustic, the available part of the slit, is #XQA./QB = è sin 2 1>, 

 that of the second GH = aXQA/BA = atg 2 v, a and b being 

 the horizontal and vertical dimensions of the grating. 



The existence of the second caustic, that is of great importance 

 for the complete theory of the instrument, may be very simply 

 demonstrated ad oculos by stretching a thin wire in Q along G H 

 across an incident beam of sunlight: the result is a perfectly defined 

 narrow black band passing horizontally across the field of the eyepiece. 

 Any other horizontal strip of the field has its own conjugate 

 horizontal strip, of a somewhat greater width, in the proportion 

 9 sec v/q, a little above or below Q in a vertical plane passing 

 through G H. Yet every single point in the strip of the field, be- 

 longing to one single A, derives from the conjugate horizontal caustic 

 //GH in its full length; conversely every point of a horizontal slit 

 above or below G H has its horizontal linear image in the field 

 depicted by rays of different A's. 



If the horizontal band, seen iu the field, is required to have a 

 width />, the horizontal slit in G H must be replaced by a rectan- 

 gular diaphragm height h X Q B / B C = h sec v, length as before 

 GH = ö tg 2 v. At the same time the vertical slit ought to be 

 lengthened by the quantity h cos v, until it gives passage to all the 

 rays issuing from the diaphragm that can reach the grating; so 

 the full length becomes h cos v -\- b sin 2 v. All the rays that are ob- 

 structed by the diaphragm, if admitted would only tend to increase 

 the disadvantageous illumination of the field by scattered light. 



