1918]
RANDOM DISTRIBUTION OF LUMINOUS SOURCES
567
or, if we employ the mean value of the 7's instead of the sum, the chance of the mean, viz. (Jj + /2 4- ...)/m, lying between K and K 4- dK is
^ '
nm .ml
We may compare this with the corresponding expression when m = 1, where we have to do with a single /, to which K then reduces. The ratio
o—(m—l^K/n„,,•;»+! Jfm—l
.(5)
When we treat m as very large, we may take
m I = mm V(27rm) . e~m, so that (5) becomes
.(6)
If in (6) K = n absolutely, the second factor is unity, and since the first factor increases indefinitely with m, there is a concentration of probability upon the value n, as compared with what obtains for a single combination.
•* In general we have to consider what becomes of
0» • {as el~x}m-\ .............................. (7)
when m = oo , and oc, written for Kjn, is positive. Here icel~x vanishes when so = 0 and when x = oo , and it has but one maximum when x — 1, ccel~x = 1. We conclude that xel~x is a po'sitive quantity, in general less than unity. The ratio of consecutive values when m in (7) increases to m + 1 is
and thus when m = oo , (7) diminishes without limit, unless x — 1 absolutely. Ultimately there is no probability of any mean value K which is not infinitely near the value n.
Fig. 1 gives a plot of R in (5) as a function of x, or K/n, for m = 2, 4, 6.