324 
On Random Occurrences in Space and Time 
TABLE II. 
Frequency of Periods of 10 seconds with n occurrences. 
Frequency 
11 = 0 
1 
3 
3 
4 
5 
6 
7 
8 
9 and 
over 
Totals 
Observed 
Calculated 
22-0 
19-5 
86- 0 
87- 7 
187-0 
189-3 
214-0 
229-6 
221-0 
197-8 
119-0 
123-1 
57-0 
57-2 
19- 0 
20- 2 
6-0 
5-5 
0- 0 
1- 4 
931-0 
931 -3 
! 
laborious than the second and it will generally be quite accurate enough. Using 
the values of and m found from the frequency of intervals in an indefinitely 
long time with the ^ to -5 grouping (i.e. /3 = -370,552, m = 2-488,007) to give the 
frequency of periods of 10 sees, with n occurrences, gave for goodness of fit to the 
observed distribution P = -67, which shows that it may not be always necessary in 
practice to recalculate the constants when they have been found for one phase of 
the work. But it will always be necessary to do so unless, as in the case considei'ed, 
/S is small relative to the length of the period taken, and there are thus few periods 
having an occurrence within /3 of their beginning or end. 
IV. Lengths of Intervals in limited Periods of Time. 
- We now pass to the consideration of the number of intervals which will occur 
in a limited period of time T, and also to the distribution of the lengths of intervals 
within that period when there are, say, n intervals in the period. In this case one 
occurrence and no occurrences will not give an interval. If n' = number of intervals, 
then they involve n' + 1 occurrences, and if Fn' be the frequency of n' intervals, we 
shall have Fn' proportional to 
(T-(n' + l)^) 
(n' + iy. 
or for the frequency of intervals N', 
N' 
{«' + ])! 
[T~(n+l)Si) 
.(xxii). 
1 {n +1)1 \ m 
s ' 
T 
where n^' = integer part of — — 1 . 
If N" = total frequency when we count the periods T in which there are no 
intervals, then since no intervals may arise from 0 or 1 occurrence, we shall have 
