( 628 ) 



used for the deiei'mination (5) and thai of Harvard which served for 

 the derivation of (7). 



In what follows the magnitudes have also be reduced to the 

 Harvard scale. I have adopted the luminosity-curve (2), in which the 

 constants have the values (4), without any change. For the density 

 curve (3), however a new determination was obtained by the aid 

 of the total numbers of the stars of diiferent apparent magnitude. 

 In other words: by the aid of formula (1) 1 derived A as a function 

 of Q from the (jiven values iV,„ (/«, = 2 to 15) and the y/vt^y^ form of if,'. 



The introduction of the analytical functions (2) and (3) has the 

 advantage of greatly facilitating the computations. Of coui-se we have 

 not to forget, however, that they can be relied on only Just to the 

 same extent as that for which we possess observational data. For 

 the luminosity-curve, with the exception only of the stars belonging 

 to the classes of the very greatest apparent brightness, the unlimited 

 use of the formula will not easily give rise to appreciable errors, 

 because extrapolation is only necessai'y for a very small fraction of 

 the total. On the contrary, the density-curve (3) (which, as we already 

 remarked, is not very accurately determined) furnishes values, which, 

 for Q exceeding 60, are to be considered as wholly obtained by 

 extrapolation. It will appear from what follows that up to (> = 60 the 

 values derived from the new materials do not dilFer from those 

 formerly obtained more than seems in accordance with their uncertainty. 

 That on the other" hand, the values for q > 60, which we may extra- 

 polate by means of formula (3), are far too small ; to such an extent 

 that for these greater distances the formula is evidently quite un- 

 satisfactory. 



To begin with, I ascertained how the formula (6), in which the 

 constants have the values (6) and (7), represents the iSf,n of publication 

 18. A table of the integrals entering in the formula (1) has been 

 given in Astronomical Journal N°. 566 for values of m between 

 and 11. (table III) '). 



1) In the calculation of the valuer of 2\ and 2\ a mislake has been discovered : 



T, 2\ 



For m 



