\:v2\) 



(he limit between the stars that. are visible on A and those that are 

 invisible; this is the limiting magnitnde for ,1,. \n the like manner 

 we find out what magnitnde forms the limit between the stars with 

 2 and with 'A images on A; this is limiting magnitnde for .1,. From 

 (he first follows, with the ditFerence B — A^, the limiting niagnitnde 

 B, from the second follows, in the same manner, the limiting mag- 

 nitnde for .4j, A^, and A^. 



In the application this method proved to involve many difïiciil- 

 ties as yet, as the magnitndes of the stars visible and invisible on 

 A, as well as those of the stai-s with 2 and 3 images, extend fai' 

 the one over the other, and are moreover iri-egnlarly distributed. 

 If //i, is the magnitude measured on B, diflfering from the real 

 magnitnde in by the unequal sensibility of the plate and by errors 

 in the counting, and if the magnitnde on the counting plate, like- 

 wise diverging fron) in, is )n^, then the star will be visible oi* 

 invisible, according to whether ni^ <^ or > m^, the limiting magnitude. 

 If the differences //<i — m and m^ — ni follow the law of errois and 

 if the stars are divided i-egularly over the various magnitudes, there 

 are two criteria foi' (he ascertaining of in^: 



1. for ni^ > in^ (he number of invisible slars is > (he number 

 of visible ones; m^ therefoi-e is thai value of ni^, for which 50'/, 

 of the stars is visible, 50 7o invisible; 



2. foi' //^i > ?/<„ the total numbei- of brighter, invisible stars is > 

 the total number of fainter, visil)le stars; in^ therefore is that value 

 of //i, above which appear a number of visible stars, equal to the 

 number of invisible ones below. 



Now the number of slars for greater ni inui-eases ; the average?//, 

 corresponding with a measured ni^, will consequently be somewhat 

 larger than this latter; the limiting magnitude found according to 

 the first criterion, needs a positive correction, which is somewhat 

 diminished, however, by the differences in^-in. On the other hand by 

 means of the 2'"' criterion the correct limiting magnitude is found 

 if (he number of s(ars is a linear funclion of the magnitnde m '). 



') This can be proved in tlie following manner. The number of stars of real 

 magnitude w that is measured on the one plate in magnitude nix, and likewise 

 the number that ou the other plate show.s the magnitude m.^, is respectively 

 f{in) exp. (_/t^'(,,/^ — m)^)dmdtn^ and /"(m) exp. [~ h^'* (m^ — m^y)dmdm^ 



in which f^m) represents the number of stars of the magnitude m; this f[m) 

 has the form a + bm. 

 If we pose: 



^, ; =z in. = k' 



