STUDIES IN STELLAR STATISTICS, II. 29 
or, if R is introduced instead of r, 
ga = TRE : y" = ne gil = TL å 
where 
ESL c 102 BÖN COS 
gl =S NOS 10227 cos d Sim a 
t" — 10°: rM = 102" ain Ó : 
= 
The quantities 3", 7'', C" may be called the reduced spherical 
eoordinates of a star!. Their values are independent of the value 
adopted for À and can be calculated directly for each star. Certain 
parameters of the frequency surfaces may be calculated directly from 
these reduced spherical coordinates without knowing the value of A 
and it is hence advisable to perform the computations as far as possible 
without introducing the value of À before necessary. 
The moments of the first order of the frequency distribution 
give the coordinates of the centre of the stellar system. Let X”, Y", Z" 
be these coordinates, then we have 
5 rl = M (x) = RM(§") 
Y" = M(y") = RM(q") 
M(z") = RM(C") 
| 
pA 
If D denotes the distance of the centre from the sun and b and 
a the declination and the right ascension of the centre, so that 
= c DicoSuD EOS airs 
%. =D» GOS 0 SIT 4% 
AN — D=sin br, 
we find that a and b are independent of À, if only such stars are 
used as have the same value of À, h. e. stars of the same subclass. 
Even if stars of different subclasses are taken together, we conclude 
1 The factor 10°:™, which may be suitably called the photometric factor of reduc- 
tion, is tabulated in the table at the end of this memoir. 
