158-161] 
Gaseous Stars 
175 
the types M, K, G, F, A, B, 0 ,... in this order. The hottest stars would thus 
be the least massive, which is not what is found in nature. 
160. Coming nearer to reality, let us next suppose that a star is formed 
of two kinds of matter. Let it contain a mass M 0 of matter of permanent 
type which does not transform itself into radiation at all, and a mass M — M 0 
of another type which produces radiation spontaneously at a given rate. 
The star’s emission of radiation is no longer proportional to M, but to M — M 0 , 
and the star’s evolutionary path in the diagram will have as its equation 
E 
--W- = cons. 
M -M 0 
(1601). 
This is a straight line such as AP which does not pass through the 
origin but meets the axis of M in the point M = M 0 \ the mass does not 
continually diminish down to M = 0, but only to the limiting mass M = M 0 , 
after which no further decrease of mass occurs. 
Let the curve BGDE in our diagram correspond to the effective tempera 
ture at which a star’s surface is so cool as to be only just visible, so that 
stars are only visible when their representative points are above the curve 
BGDE. The hypothetical star we are now considering would first become 
visible at C, after which its effective temperature would increase until it 
attained a maximum at Q, thereafter decreasing until the star again lapsed 
into invisibility, as a star of very small mass, at E. Its sequence of spectral 
types would be M, K, G, F, ..., F, G, K, M. This is precisely the sequence 
predicted by Russell’s 1913 theory of stellar evolution (§ 154), although it has 
been derived from utterly different premises. It is now clear that a star 
might well be urged, merely by loss of mass consequent on the passage of 
time, to pursue a course along the giant branch and then down the dwarf 
half of the main sequence, the path which Russell regarded as the normal 
evolutionary path for a star. 
161. Passing to higher degrees of generality, let us consider a star con 
sisting of any number of different kinds of matter annihilating themselves at 
different rates. To be specific, let us suppose that a star originally contains 
masses M lt M 2 , M 3 , ... of different kinds of matter which break down at rates 
represented by moduli of decay k 1} k 2> k 3 , — 
After a time t the surviving amounts of these different types will be 
Jfje - *!*, M 2 e~ K d, M 3 e~ K ^, ..., so that the total mass of the star will be 
M = 4- M 2 e~ K ^ + M 3 e~ K ^ + (161*1). 
The star’s rate of emission of energy E is equal to its rate of generation of 
energy — C 2 . Obtaining the value of this by differentiation of equation 
(161*1), we find