Astronomy. — “The local starsystem”. By Dr. A. PANNEKOEK. (Com- 
municated by Prof. W. pr Srrrer). 
(Communicated at the meeting of May 28, 1921). 
if 
If A(m) denotes the number of stars of magnitude m, A the space 
density of the stars, which for a given line of sight is a function of 
the distance r, p(M) the luminosity-function, and if for the distance 
r we introduce g=5logr, thus making M=m— eg, then A (o) 
can be found from A(m), if both may be represented by quadratic- 
exponential functions. Thus if we put 
log A(o) = f' 4+- ko —l0?; logp(M) =p + qM —rM’; 
log A (m) = a + bm — cm’ 
we have: 
bg)’ uy 
hap + 3.786—1/, CD shel ogy ee 
r—c r 
Ue cr 
k' = q—0,6+ (b—g) ; ieee 
(p= NE 
By these formulae (in a somewhat different form) Kapreyn and 
Van Ruwn have deduced the distribution of density in the starsystem 
surrounding our sun, representing it by a series of flattened surfaces 
of revolution. *) 
Here the function 4 has been found as a whole from the function 
A. But the observational data determine this function A for a certain 
extent of m only. Now the question arises, whether the value A(m) 
for a given m determines the value A (o) for a certain g. The dif- 
ferential quotients 
0 ; vie ‘ b—q r 
TAR oe eee ae + 0 
rb T—C 
b—q he NG log e 
(Ao loen ze gel. 
BA : °) ey er (r—c) 
show, that for 0, = — a variation of 6 causes not any, a varia- 
We 
tion of c causes only a slight variation of Ay so that A (e‚) depends 
nearly wholly on a= A (0). If we count m and (in order to keep 
1) J. C. Kapreyn and P. J. van Ruin. On the distribution of the stars in space...., 
Astrophysical Journal LII. 289. 
