386 
Variable Stars 
[CH. XV 
* Harvard, Circular , No. 314 (1927). 
accordingly seen to represent the relation P oc p~\ the relation which we 
have just seen (§ 357) must connect the period P and the density p of a 
pulsating mass of gas. 
359. Let us examine the form assumed by the law when differences in 
mass and effective temperature are taken into account. 
A star’s bolometric luminosity is proportional to 4mR 3 T e *, so that, if m 
denotes the star’s absolute bolometric magnitude, 
— 0*4 m = 4 log T e + 2 log R + constant (3591), 
and from the approximate relation (§ 118) that the bolometric luminosity is 
proportional to M 3 or to (|7 rpR 3 ) 3 , 
— 0*4m = 3 log p + 9 log R + constant (359*2). 
Eliminating R between these two equations and replacing p from the 
relation that P oc p~^, we obtain 
log P + 0*23m + 3 log T e = constant (359*3). 
The agreement of this formula with observation is almost uncanny. In 
the following table the first four columns express observational data collected 
and averaged by Shapley*, and the fifth column gives the absolute bolometric 
magnitude obtained by applying the bolometric correction from § 48. The 
sixth column gives the quantity which ought to be constant according to 
formula (358*2), while finally the last column gives the value of the left-hand 
member of equation (359*3). 
TABLE XXXII. Observed and Calculated Data for Cepheid Variables. 
Spectral 
Type 
Effect. 
Temp. 
log P 
m Y u 
Wlbol 
log P 
+ 0-3m vU 
log P 
+ 0‘23 m 
+ 3 log T t 
A 0 
10000 
-0*56 
-0*3 
-0-6 
-0*6 
11*3 
Ab 
8500 
-0*31 
-0*3 
-0-4 
-0*4 
11*4 
F 0 
7400 
-0*06 
-0*6 
-0-6 
-0*2 
11*4 
Fb 
6500 
+ 0-23 
-1*0 
-1*0 
-0*1 
11-4 
Fib 
6000 
+ 0-40 
-1-4 
-1*4 
0*0 
11-4 
GO 
5500 
+ 0 59 
-1-8 
-1*9 
0*0 
11-4 
G 2'b 
5050 
+ 0*85 
-2-4 
-2*6 
+0*1 
11-4 
Gb 
4600 
+ 1*22 
-3*9 
-4-2 
0*0 
11-3 
Gib 
4300 
+ 1-62 
-5-4 
-5*9 
0*0 
11*2 
360. The success of the law Pp^ = constant as shewn in this table is so 
striking that one is tempted at first sight to suppose that the Cepheid 
variables, at any rate within the range covered by the table, might unhesi 
tatingly be treated as pulsating spheres, in accordance with Shapley’s sugges 
tion. When, however, the absolute values of the quantities are evaluated