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Full text of "Scientific Papers - Vi"

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.