QUANTUM THEORY 
65 
no bound electron. Then the number with no bound electron but with a 
free electron in the range dxdydzdudvdw is by (46-1) 
and this must be continuous with the number given by ( 45 ‘ 5 ) as having 
only a single bound electron in this range. Accordingly 
The number of systems with just one bound electron N (1 — x) is 
obtained by summing (44-2) for all possible orbits of that electron. Thus 
N (1 — x) — B {2e~*d lxl -f- 6e~ x d RT + ... -j- r (r + 1 ) e~ Xr l RT + ...} 
which determines the ionisation x when the temperature T and free 
electron density cr 0 are given. 
47. Although we have considered only one peculiarly simple system 
the formula (46-2) derived from it is valid always. There may be no such 
system in the assemblage considered—no atom ionised down to the last 
electron or none sufficiently free from disturbance by free electrons or 
neighbouring atoms. But there is always a chance of such a system, and 
the chance however infinitesimal is sufficient to justify the formula. 
There can be only one equation determining B in terms of cr 0 however 
many different kinds of systems may be involved, so that all systems must 
give the same result as the simple system which we have been able to work 
out fully. This requires in particular that in any kind of atom the weight 
of a space-velocity range for bound electrons approaches the limit (45-6) 
when the negative energy approaches zero. 
To make the argument specific, define a system of class A to be one 
in which there is a nucleus and a bound electron with coordinates x, y, z 
to x 4 - dx, y + dy, z + dz relative to the nucleus and velocity in a range 
dudvdw corresponding to small negative energy, provided that there is 
no other nucleus or electron within a distance 8 of the nucleus. Let a 
system of class B be one in which there is a nucleus and a free electron 
with coordinates similarly specified and with velocity in an equal range 
but corresponding to small positive energy, provided that there is no other 
nucleus or electron within a distance 8 . The postulate is that the numbers 
of systems of classes A and B must be continuous. The factor representing 
the proportion of systems spoiled by the intrusion of other matter within 
(46-2). 
(46-3). 
From (46-2) and (46-3)