ot ond 
May 7, 1914] 
than at the start; but the average value which their 
product should have, in a large number of cases, and 
the percentage of these cases in which it should lie 
within any given limits, may be computed on the 
principles of geometrical probability. It is thus 
found that the formula z'=sw?/14-6m gives values 
for the hypothetical parallax the average for a large 
number of cases of which will be correct, and that, 
while in individual cases these values will be too large 
or too small, half of them will be within 19 per cent. 
of the true values, and the 
numbers of larger errors will B A 
fall off in very nearly the 
manner corresponding to this 
probable error. If we com- 
pute absolute magnitudes 
from these parallaxes, their 
average for all the stars will 
be a little too bright (since 
the cases in which the com- 
puted parallax comes out too 
small have more _ influence 
than those in which it is too 
large). This may be allowed 
for by adding o-15m. to all 
the hypothetical magnitudes 
so computed—an amount 
almost negligibly small for 
our present purpose. 
We thus obtain a series: of 
hypothetical absolute mag- 
nitudes the average for a 
large number of cases of 
which will be correct. In 
59 per cent. of the individual 
cases the error arising from 
the statistical process—that 
is, from the substitution of a 
mean value of 
sin #, sin 7i,(2—r/a) 
for the true value—will affect 
the deduced magnitude by 
less than +0-5m., and in 
89 per cent. of all cases the 
error will not exceed +1-om. 
The approximation is there- 
fore quite sufficient for our 
purpose. It should, however, 
be noted that, while the error 
of the statistical! process can 
never make the computed 
absolute magnitude of any 
star too faint by more than 
r5m., it may in rare cases 
make it too bright by any 
amount whatever—more than 
20m. in one case in sixty, 
more than 3-0m. once in 250 
cases, and so on. 
We may now proceed to 
compute hypothetical abso- 
lute magnitudes for all 
the physical pairs which 
show even a trace of frela- 
tive motion—including many 
which are ordinarily described as ‘fixed,’ but, on 
careful study of the observations, show very slow 
relative change. With the aid of the splendid collec- 
tion of observational data contained in Burnham’s 
great catalogue and other recent works on double 
stars, and of many observations of spectra made at 
Harvard in generous response to requests for in- 
formation, it has been possible to derive results for 
more than 550 stars. Assuming that the brighter 
NOn2323,, WOl.94| 
NATURE 
257 
component of each of these (which is usually the only 
one of which the spectrum is known) is equal in 
mass to the sun, estimating that of the fainter com- 
ponent on the basis of the difference of brightness 
(with the data for the systems in which the mass- 
ratio is known as a sufficient guide), and proceeding 
as indicated above, we obtain the data plotted in 
Fig. 3. The co-ordinates have here the same mean- 
ing as in the previous diagrams, and the figure shows 
at a glance the relations which would exist between 
the absolute magnitudes and spectra of these 550 
stars if all differences of mass were eliminated, leav- 
ing only those of density and surface-brightness 
operative. Binaries for which orbits have been com- 
puted are shown by solid dots, and physical pairs, to 
which the statistical process has been applied, by 
open circles. 
Our new diagram is strikingly similar in appear- 
ance to the previous ones, even in its minor details. 
