VARIABLE STARS 
199 
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Since T x and dQ are known correct to the first power of the amplitude, 
we can now find W correct to the square of the amplitude. 
We shall take the values at = 3 as representative of the average 
conditions in a star. Less than 20 per cent, of the mass is outside = 3; 
but in view of the rapidly increasing dissipation per unit mass in the outer 
parts, this seems a fair representation. Denoting by [fj,. the central 
amplitude of , so that roughly 
By a rough quadrature the mean value of this at time of greatest velocity 
is found to be about 2 »2 m 2 
■f±n lx LfelJe 5 
so that the whole mechanical energy of pulsation of the star is 
The following numerical results are obtained for 8 Cephei*. We have 
e - LjM = 160 ergs per gm. per sec. 
Negative potential energy, Q. = 8-65.10 14 ergs per gm. 
* The values of a and j8 on which (134-3) depends only roughly fit 8 Cephei; 
but the formula is only intended to give the order of magnitude; probably the most 
[£i]o = 0-1SR/R, 
we have by (133-2) at = 3 
-¡ft = 2-2e [£j] c cos nt 
T x = (/ - 1) Pl = - 0-355 x 3-54 [^] c cos nt, 
(134-25), 
so that T x = - l-38e [f x ] c 2 (1 + cos 2 nt). 
Hence W (per gram per sec.) is 
W = l-38 e &] c 2 
and the rate of dissipation of energy by the whole star is 
(134-3), 
1-38 &VL 
The kinetic energy of the pulsation per gram is 
(134-4). 
By (134-4) and (134-5) the time of decay is 
(134-5). 
(134-6). 
KiL - -05. 
M = 1-75.10 34 gm. 
L = 2-80.10 36 ergs per sec. 
serious inaccuracy is the use of a mean value for e.