460 
ASTRONOMY: H. SHAPLEY 
this class of variable stars have given results for different spectral types 
with considerable certainty. The average density of first type stars 
(B, A) is about 0.15 on the solar standard, while none exceeds the sun 
in density and none is less than one-hundredth as dense. Also a number 
of second type stars are found to be much denser than the average first 
type star, even exceeding the sun in rare cases. This condition fits in 
with either theory of the order of stellar evolution. Hence for the 
purpose of differentiating between the rival hypotheses it is of impor- 
tance to know how definitely we can say that there are also stellar densi- 
ties much lower than any known for A and B type stars. The present 
communication outlines the data now in hand relative to low-density 
eclipsing stars of the second spectral class. 
The values of the densities already derived for the stars considered in 
this note refer to the individual components of each system, and there- 
fore, since for such purpose knowledge of the relative dimensions and 
masses is necessary, the results are involved with rather complicated 
computations of the orbital elements. It is possible, however, to derive 
upper and lower limits of the mean density of each system without the 
necessity of any data relative to the stars or their Hght curves other 
than the period of revolution and the duration of eclipse, which are the 
two quantities most commonly and most easily obtained for eclipsing 
variables. In the more detailed discussion of this subject the following 
simple relations are obtained, which give in a circular orbit the mean 
density, Jq, in terms of the sun: 
The quantities P and t are the period of revolution and the semi-duration 
of eclipse. In very exceptional cases the second relation may give too 
high a lower limit, but never a value as much as twice too large. 
The above formulae are especially suitable for the easy computation 
of the order of the density in systems of long period, which are usually 
also the cases where very low densities are found. With their aid 
results for five low density binaries of the second spectral type were 
readily determined and the values found to correspond closely to those 
computed rigorously from the orbital elements. The light curves on 
which the results are based are from various sources, and, while not 
of the first order of accuracy, are probably sufficient in all cases to leave 
no doubt that the low densities really exist. Three independent classi- 
fications of the spectra were made at the Harvard College Observatory. 
0.054 
0.216 
P 
