385 
355-358] The Period-Luminosity Law 
multiplied by a numerical factor which will depend on the precise model on 
which the star is built, but will always be near to unity. 
This approximate relation P = (y p)~^ may be put in the form 
P z p= 0-0020 (357*1), 
where P is the period measured in days and p is the mean density of the star. 
This enables us to calculate the density which corresponds to a given 
period of pulsation. Calculations for various types of stars are shewn in 
Table XXXI. The third column gives the observed period of the star, which 
is identical, on the pulsation-theory, with the period of the star’s oscillation, 
while the last column gives the mean densities of stars of the type in 
question and of mass ten times that of the sun, as estimated by Seares*. 
The calculated mean densities are seen to agree tolerably well with the 
estimated values. 
TABLE XXXI. Periods and Densities of Variable Stars. 
Star 
Spectral 
Type 
Period 
p (Calc.) 
p (Seares) 
Long Period 
M 
300 
0-00000002 
0 0000006 
Cepheid 
K 
18 
0-000006 
o-ooooi 
G 
4 
0-00012 
0-00002 
F 
0*9 
0-0025 
0-0004 
Cluster 
A 
0*3 
0-022 
0-008 
The Period-luminosity law for Cepheids. 
358. In the case of Cepheids a more precise test of the relation P oc p ^ 
is provided by the period-luminosity law (§ 11). 
This law is usually exhibited in the form of a curve in which log P is 
plotted against absolute visual magnitude m V i 8 . For periods greater than a 
day, the curve shews an approximately linear relation between log P and m V j S) 
which may be expressed by the equation (see Table XXXII below, col. 6) 
log P 4- 0*30 m v i 8 = constant (358*1). 
In a group of Cepheids all of which have the same mass and the same 
effective temperature, the visual luminosity is proportional to the square 
of the radius, and therefore to p~*. In terms of visual magnitude this 
relation becomes 
0*4mv is = | log p + a constant (358*2). 
If P varies as p~^, this equation is seen to be precisely identical with 
equation (358*1), which expresses the period-luminosity law. This law is 
* Astrophysical Journ. lv. (1922), p. 165, or Mount Wilson Contrib. No. 226. Substantially 
better agreement can be obtained by allowing for the differences in mass of Cepheids of different 
types (cf. Table XVII of Seares’ paper).