57 
the same p{M)) also o from the zero point m,, this means, that in 
the formula 
log A (m) = a + b (mm) — e (mm) 
the first term a —= log Alm) ane A (9) for 
zl 
‘or 
Assuming that the observations determine the values for A(m) of 
Van Ran from m= 4 to in —= 16 we find, by applying this formula, 
that they determine the 4 computed from them for 9, =10 to 17 
in the Milky Way (i.e. for r—100 to 2500 parsec) and for 9, = 9.5 
to 15 in the polar regions (i.e. for »— 80 to 1000 parsec). 
As m, and g, are conjugate values, it is only rational to take o, 
as the zero point in the formula for 4. If we put 
log A (9) =h + k(e—e@,) — / (o—e@,)’: 
9, == ™, ra: 
we have 
b—q 1 1 1 
Ee =m, + ; Bee 0 k==6— 0.6; 
2r l 6 at 
— 0,6)? b—0,6 r 
h=a— 0,6m, —p + 3,786 — Cm) Ga oe + tlo En 
Ar en = 
If we insert now the values p = — 2,394, ¢q = ai 0,186, 
r= + 0,0845'), we get: 
eht b —0,186 Lecce A ey Mange 
= ™ +) 69° LER ’ == hk) 
2 
k 
h=a-+ 4,937 — 0,6m, + —— Daan 1 log (l + 0,0345). 
If over a limited extent of m the number of stars A(m) may be 
represented by a quadratic-exponential formula it determines A (©) 
over a limited extent of @ also. An adjacent extent of m affording 
a formula for A with other constants determines another part of 
the curve for Aig). In case of an irregularly fluctuating course of 
A(m) and A(@) we may divide them into separate parts and represent 
each of them by such formulae, thus using the quadratic-exponential 
formula in an interpolatory manner. It may be noticed that in this 
case the coefficients c and / (which become zero together) may be 
negative, if only /-+7>0. Of course this solution of the problem 
to find A from A is not rigid, but only a practical and approximate 
one. If c approaches r very nearly, small errors in c cause enormous 
deviations in /, making A wholly uncertain; if c has a great negative 
value the result has no real meaning. If c surpasses the value '/,, 
1) Kapreyn and Van Rawn, l.c. p. 297. 
