SOLUTION OF THE EQUATIONS 
125 
dp R = - 
dr 
.(91-3). 
where k x is a constant. We take p. constant throughout the star, the effect 
of varying /x being reserved for discussion in § 94. Hence (91-1) becomes 
k x L p 2 
477-c r 2 T* 
With (91*2) and (91-3) we carry on a quadrature for P and p R . After 
each step we find T and p from P and p R . The mass of the added shell is 
computed from the values of p and added to the current total; hence the 
new value of g is found. We are then ready to determine dp R and dP for 
the next step dr. The usual devices of quadrature are employed, including 
the guessing of half-way values to be subsequently checked. 
The only serious trouble arises with regard to starting values. The 
mass M and radius R of the star are only found at the end of the calcula 
tion; in their place we must have two disposable initial conditions at the 
start. Unfortunately we can only make a beginning by fixing three dis 
posable constants, viz. the constant k x L for the star and the temperature 
T and density p at the starting-point*. We thus over-condition the problem 
and generally fail to reach a solution. The failure betrays itself by p Q 
falling to zero whilst p R is still large or vice versa, thus violating the 
boundary condition that both vanish almost simultaneously. In that 
case we must modify one of the initial quantities, say T, and try again. 
After many trials we contrive to straddle the true solution sufficiently 
closely. Of course a great amount of calculation is wasted on unsuccessful 
trials. 
The following value was taken 
log 10 k x L = 62-6590, 
and the solution began at r = 0-9.10 11 cm. with an adopted density 0-01. 
These values ultimately fix M and R. The temperature at the starting- 
point was varied until two solutions were obtained which kept close 
together until reasonably near the edge of the star, and then failed in 
opposite ways, i.e. p G ran off to zero prematurely in one case and p R in 
the other. The solutions gave respectively 9-687.10 33 and 9-706.10 33 for 
the mass M r up to r = 8-1.10 11 ; and as little mass remained to be added 
the limits were close enough to fix the mass with ample accuracy. 
Finally, as a complete check suitable^ starting values of P, p R and M r 
at r = 8-1.10 11 were interpolated, and the solution carried backwards 
towards the centre with double the number of intermediate steps. Working 
in this direction the mass M r is stripped off in successive shells and the 
test is that M r must just reach zero at the centre. The test was well 
* The density at the centre is zero, the mass being driven away by the intense 
radiation pressure; so we start at some fixed distance from the centre where the 
density is still very low and the interior mass is just beginning to be of account. 
f I.e. suitable for carrying the solution considerably further outwards before 
Pa OT PR ran to zero.