322 
On Random Occurrences in Space and Time 
We have 
f R 
-(n-l)log,,(r-(«-l)yS). 
/")( /"n- 
Proceeding as before, the weight Wn of •v/r,i will be " "• " ^ - and for the best 
value of m, 
u" = S Wn" \- logio e - logio n - logi„ «i + n logio - "/3) 
-(»-l)logi„(r-(«-l)^)-^' 
has to be made a minimum. Thus we reach 
Mo f 
0 = S ]wn" - logio e - logio n - logi„ in + n log,o {T~nfi) 
1 t \m 
-(»-l)logio(2^-0« - 
from which m can be found since j3 is known. 
((^) We have not succeeded in finding the mean number of occurrences or the 
standard deviation of occurrences, when they are limited terminally as above 
described. 
Corollary. If we take 13 = 0,/(/3) = 1, then our formula reduces to 
fn = N~ - (XXI), 
the well-known Poisson expansion, which gives the frequency of n occurrences in 
time T. 
(e) We come now to the application of the above to the experimental data. The 
tapes were divided into 10-second periods and all those were rejected which were 
either preceded by an occurrence in time less than fi, or in which there was an 
occurrence within /3 of the end of the period. But at this stage the value of j3 
which the sample being dealt with would give was, of course, not known. That 
found from the distribution of lengths of intervals in an indefinitely long time was 
used, since the difference between the two would certainly be small, but usually in 
practice such a close approximation would not be available. 
The constants /3 and m have first to be found. Of a total of 931 periods there 
were 22 in which there was no occurrence of the event, so the Xn = ( -^4^) "Tf^Gthod 
could be safely used. The values of ;y„ calculated for all possible values of n and 
weighted by (xiv) lead to the two equations (xv) and hence 
a = 4-444,441, & = -157,272. 
