VARIABLE STARS 
205 
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the mass. The observed progression of spectral type with mass is in the 
opposite direction. 
Our conclusion is that the suggestions in § 136 and § 137 both lead to 
serious difficulties. On the whole, the difficulties of the former seem to be 
of the more fundamental kind; whereas the difficulties of the latter may 
perhaps be set down as numerical misfits natural to an early stage in the 
development of a complex theory. 
Miscellaneous Problems. 
139. In the investigation of § 127 the square of the amplitude has 
been neglected. In typical Cepheids & may amount to T ’ ¥ and P x to nearly 
so that the second order terms are quite considerable; these will give 
rise to terms containing cos 2 nt. 
For a treatment of the theory with retention of terms of the second 
order, reference may be made to Monthly Notices, 79, p. 183. The com 
putations are there carried far enough to show that the complete formula 
for will be of the form 
g x = a x cos nt — a 2 cos 
where a x and a 2 are both positive. Hence the velocity of recession has the 
^ orm V = b x sin nt — b 2 sin 2 nt 
with b x and b 2 positive. This represents a velocity-curve having the general 
characteristics of the observed velocity-curves of Cepheids, viz. a sharp 
decrease from maximum to minimum receding velocity and a slower 
return to maximum with indications of a hump in the curve. The equivalent 
elliptic orbit has its periastron at a> = 90°. 
The close similarity of the light-curve and velocity-curve and the 
relation of phase between them has not as yet received adequate theoretical 
explanation. It is not that any opposition of theory and observation has 
been found; but the difficulty of the mathematics has hitherto proved too 
great an obstacle. We have found that in the adiabatic region of the star 
the outward flow of radiation is greatest at the time of maximum com 
pression; this is true for the whole region (§ 133) or, at any rate, for the 
outer part of it (§ 137). But the greatest outward flow from the surface 
is observed to occur simultaneously with maximum velocity of approach, 
that is to say, a quarter-period later. Presumably the retardation occurs 
in the non-adiabatic region near the surface. The leakage wave discussed 
in § 133 is 90° behind the adiabatic wave in phase, and it grows in import 
ance as we approach the surface. It would, however, be too crude a de 
duction to attribute the 90° retardation of flux to the leakage wave 
supplanting the adiabatic wave. Undoubtedly there will be some re 
tardation in the non-adiabatic region, but no definite prediction of the 
amount of retardation can yet be made.