1 380 



And if tills function and llie niodtde of accuracN for magnitudes 

 diverging 1'" may be considered as equal, the correction for both 

 limiting magnitudes is equal with the 1"'' criterion, so that the dif- 

 ference in magnitude A^-A^ is correctly found also in this way. 



In table II the 2"^^ and 3'^^ columns contain the numbers of stars 

 with 0, with 2, with 3 images (?ij n, Ug). In order to smooth the 

 very considerable, accidental irregularities of these numbers, the 

 total of every 3 consecutive numbers have been plaged in the fol- 

 lowing columns (?i/ ?i/ ?i/). Column /> shows how many percentages 

 n^" is of the sum total; where in the increase this amounts to 

 50 Vo' t''6 limit lies between invisibility and two images; where in 

 the decrease it amounts to 50 ' ,, the limit lies between two and three 

 images, according to the ist criterion. Next to that stand the sums- 

 total .s' of the fainter, visible, and the brighter invisible stars; the 

 limiting magnitude, according to the 2nd criterion lies where these 

 become equal. 



From the values />.^ we find as limiting magnitude 13,67 and 

 12,54; to this must be added the corrections of page 1327, so that they 

 become 13,65 and 12,62. From the 2"^' criterion we likewise find 



the number of stars having on the one plate the magnitude Wj, on the other 

 wig becomes 



/'(///j) exp- ( — A' (m, —»',)') (^m^ dni^. 

 If m,, is the limiting magnitude, so that mo ^nij means invisibility or visibility, 

 theil the number of invisible and the number of visible stars of the magnitude 

 W] is given by : 



dm^ l/(m,)exp.(— A'(»M,-Ai,)')rfm, en </?/r, |/'(?w,) exp. (— A'(w^ m,)')(/m,. 



For mi = niQ these two are not equal, in consequence of the factor 



nm,)=a+b ' y \ \ 



Tlie number of bright invisible start;, that have therefore ni^ < m^. m^ > w^o 

 and I he number of faint visible stars that have w, > in^ and «Jg < "^o '^ 



I dm^ I dm^/(m^) exp. ( — A* (;/;,— w,)') 



— CO win 



and 



I d7ii^ 1 dm.^ / (w'i,) exp. ( — A' (Wj — mj). 



'"0 ^ 



These two double-integrals are equal, m^ and ni., being completely interchange- 

 able here. 



