326 
Ou Random Occurrences m Space and Time 
Thus as the result of our first integration we have 
(T- (71 + 1) ,8) 
f{^) dt,dU... dtn \{T-tn - 2/3) + m (l - . 
We have next to integrate tn from t^-i + /3 up to tn+i — /3, whei-e tn^i is given in 
the first part of the subject of integration the vahie T — ^ and in the second T. 
Accordingly the integration leads to 
(r-(«+i)i3) 
■f^^^ " ■ ■ ■ '^^"-^ r ' ^"2 ! " + ( ^ ~ ^ ^' ~ - ^^^1 ■ 
If we continue this process up to dt,.+i we have 
+1 
(?) -?•-!)! ) 
Now we do not integrate witli regard to d.t,.+i but put = t + t,. and proceed 
to integrate with regard to t,. from . = <,._, + /3 to T — t — {n + r + 1) j3 in the first 
subject of integration and to T—t — (/i — r)/3 in the second subject. We have 
accoi'dingly 
_(r-(»+i))3) 
./(/^) M,aU...M,^,^ {n-r+\)\ 
\ J (n — r)\' 
{n-ry: 
Continuing this process we have finally after writing dt for dt,.^i 
[T-n( + \)^) 
(xxiv) 
for the chance of an interval between t and t + dt. Now if we integrate this from 
t = /3 to t = T — n/3 for the first subject and to t=T—{n — l)/3 for the second 
subject we have 
which agrees with (x), if we remember that n intervals signify ?i + 1 occurrences. 
In precisely the same way we get a form like (xi) (with n changed to ??,+ !) 
by integrating with regard to T from T - ^ to T and replacing f{fi) by 1 ~f{IS). 
The result expresses the chance of any interval t occurring in the series of 
intervals when the period starts somewhere in a closed interval /3. The great 
complexity of the resulting formula compels us again to simplify matters if /3 and 
