I 



Aherration oj the Concave ij rating. 153 



permissiible with either spherical or concave gratings [see 

 (38), (55), (5(i), and (07)1, — although in this case also the 

 spherical gi*ating has a decided advantage^ — unless again we 

 use very small resolviug-powers. 



We are thus confronted h\ two difficulties in the use of 

 this form of objective spectroscope, and it seems impossible 

 to overcome both simultaneously, except by sacrificing re- 

 solving-power *. This general conclusion is quite in har- 

 mony with the experimental results referred to at the 

 beginning of the paper, with the single exception of those 

 obtained by Mitchell. These require some further con- 

 sideration. As alreadv stated. Mitchell used a orratino: havinoj 

 a linear aperture of 1J:6 mm. and a semi-angular aperture, /5, 

 of '073. He used the B and F types of mounting, and 

 worked with plates about 127 mm. long. The value of / for 

 the B mounting was about 15". and the value of u [from 

 (Od)] would therefore be about 508 mm. The plates (bent 

 to a radius of about 25^ mm.) would cover a field of over 

 14°, or 7° on each side of the axis of the grating. The 

 semi-angular aperture ^ here exceeds the maximum limit 

 imposed by the usual criterion of good definition for a 

 grating of this size by more than 50 per cent, [see (56)]. 

 Hence, judged by this standard, there would be no point in 

 the field for which the definition would be good. Mitchell 

 in his paper, however, states that the plates showed ''excellent 

 definition," presumably over the entire field. Either the 

 preceding conclusions are radically wrong, or Mr. Miteheirs 

 standard of *• excellent definition '"' is decidedly diftereut from 

 that imposed by (37). As 1 shall attempt to show, the latter 

 is not an impossible or even improper explanation of the 

 discrepancy. Definition is, as has already been stated, a 

 function of psychological and physiological phenomena, as 

 well as of strictly physical conditions. As such it is strictly 

 relative, and depends largely for its quantitative estimation 

 on the basis of comparison employed. The latter again 

 differs, or should differ, with the instrument used. If this 

 basis of comparison is less exacting than it should be, then 

 an aberration amounting to very much more than a quarter 



* Certain special solutions of the general equations (42), (4-3), and (48) 

 have been found which theoretically enahle us to greally reduce the value 

 of Z'. No such solution has yet heen found, however, that does not 

 require us to satisfy certain mechanical and optical conditions of con- 

 struction and mounting that are either impossible or highly impracticable. 

 Some of these solutions, and some other points of interest that have been 

 developed in connexion with them, wlQ be given in another paper, 

 " On the most I^tHcient Form of Surface for Concave Gratings and 

 Concave Mirrors for Spectroscopes,'' now in course of preparation. 



