310 
On Random Occurrences in Space and Time 
instant of time. If only one is able to pass, there will be a " closed-time " /3, 
measured by the length of the taxi plus a certain margin of safety, both varying with 
the size of the taxi and the bump of precaution of the second driver. In considering 
experiments likely to exhibit the law, we were repeatedly met by the existence of 
such a " closed-time " /3 intervening between event and event. Some of the most 
important cases to which the theory may be applied, socially or medically, involve 
such a closed time between occurrences. Accordingly it seems desirable to extend 
the theory to cases in which there is of necessity a " closed-time " ^ within which 
no event can follow a given occurrence. As most experimental and many social 
and vital phenomena involve a ^ as well as an m, it becomes necessary to devise 
an experiment which will give some approach to a constant ^ and a constant m. 
If a chronograph be running and marks seconds, and an observer taps on the 
occurrence of random events, he will be unable to tap any number of simultaneous 
events ; they will be separated by at least the reaction time requisite for noting 
the event and tapping the key. But it is by no means certain that the /3 and m 
in such a case remain really constant ; in fact it will be shown that experiments 
suggest that they do not. There may be slight ii regularities due to the apparatus, 
such as in the time of swing of the pendulum or the rate at which the tape is 
moving, but more important than any of these will be the fluctuating " personal 
equation " of the observer. 
A suitable series of random intervals seemed to be those between the successive 
occurrences of the figure " 5 " in the units place of columns of numbers, each con- 
taining at least four figures, in a Census Report. The unit figures were read down 
as far as possible at a constant rate, tapping the key of the chronograph on coming 
to a " 5." Six half-hour readings were taken in this way and it was assumed that 
the values of m and /3 for the six tapes were not significantly different from each 
other, so that the whole might be added to give a homogeneous distribution. It 
will be shown later how far this assumption was justified. 
A constant ^, such as the theory postulates, would be given if the length of the 
interval between two " 5's " were measured by the number of intervening figures. 
But this case would be similar to that of drawing numbered counters from a bag ; 
the intervals could only difi^er by units. By introducing the personal factor we 
obtain a continuous distribution but the constants /S and m are sure to fluctuate 
slightly throughout the experiment. It was hoped that this would not materially 
affect the results. 
II. The Freq uency of Intervals in an indefinitely long Time*. 
The first case to be considered is that of the frequency of intervals which elapse 
between two successive occurrences of the event when the period in which the 
observations are taken is continuous and indefinitely long. We have seen that the 
* We shall use the terminology of " time " throughout, but the intervals may always be interpreted 
as intervals of space and the occurrences as random marks on a line. 
