1214 



in the mean density from tlie centre towards tlie margin was only 

 due to spherical aberration. F'nrthermore it was assumed that, for 

 each plate, the mean density at equal distances from the centre 

 had the same value; this means that the photographic plate was 

 considered to have been focussed on its centre in a position perpen- 

 dicular to the optical axis of the telescope. 



Each plate of the Harvard Map, the film-area of which is 19 X 21 

 centi-metres, was divided into 7 concentric zones by circles having 

 their centres at the centre of the plate and radii of 2, 4, 6, 8, 10 

 and 12 centi-metres respectively; each of the three outer zones 

 consisted of four disconnected parts. Now, on each j)late and for 

 each of these seven zones the mean density was computed from 

 Heme's counts ') and so, for each plate, the density was found as 

 a function of the distance from the centre. In order to eliminate 

 as far as |)ossible the individual peculiarities of the plates these 

 mean densities of the zones have been dixided by the mean density 

 of the whole plate, thus giving the relative densities in the seven 

 zones. These occnr in Table VIII of the investigation mentioned 

 above. *) 



In order to derive from this xariation of the relative density the 

 variation of the limiting magniliuie, a relation between star-density 

 and magnitude was to be assumed. Following the method used by 

 HE^tE 1 expressed this relation by the following Ibi'inida, which has 

 been given by Charlier in his "Studies in Stellar Statistics" ') 



'" (m — m„)' 

 {7>i)z=— I e 



N 

 A (v,) = -— \e 2 k' dm 



Here .4(7??) is the number of stars covering a certain area of the 

 sky and brighter than the magnitude m, while ^V, k, and ?/i, are 

 constants, which Charlier has determined by means of star-counts 

 on the Carte du Ciel. For the computations, necessary to deduce 

 with the above formula the distance-correction from the changes 

 in the density. I wish to refer to my first paper. ■•) 



In determining the distance-correction 1 had to consider a com- 

 plication which had not a priori been expected. There seemed no 

 reason why the 55 plates taken with two equivalent instruments 

 under similar conditions, should not. apart from local irregularities 



1) 1. c. table VII, p. 34. 



2) 1. c. table Vni, p. 35. 



8) Lunds Universitets Arsskrift. N.F. Afd. 2. Bd. 8, Nr. 2, p. 32. 

 *) 1- c. p. 37 et seq, 



