G. MORANT 
327 
m are not a priori known by selecting those intervals only in which no event 
occurs in the period /S before T starts or in the period /3 before the end of T. 
In this case the chance of a definite interval in an n interval period being 
between t and t + dt '\» 
This chance is clearly the same whether the given interval occurs between the 
rth and r + 1th events or between any other pair of events. It therefore i-epresents 
the cha,nce of an interval- of the required length being in any position. But if we 
are going to tabulate the frequency of intervals there will be n cases in which such 
a length of interval could occur, and we shall have for our frequency surface 
varies as. '« i ), • 
To obtain the constant of variation, C, we must find 
\ h \ [ 1)! ) 
or, C = 
\X 1»"+'(7? + l)! 
where N' is the total number of intervals. 
Thus finally = ^ ^ ,.Mn.M., ,^ - ' . (-^^^^^ 
i\ m-+>Oi+'l)T"^"'j 
We proceed now to deduce some consequences from this result. 
(i) If /3 = 0, f{/3) = 1, and we no longer have difficulties about terminal 
conditions, and the chance that a given interval shall be of length between 
t and t + dt in an h -interval period is 
Gt = e~>''^^^Klt (xxvi). 
Integrating t from 0 to T 
rr _1 fn 
G,dt = e ■ 
7?i"+' («+ 1) I 
This is the total chance of n intervals regardless of their size, and summing 
from 9i = 1 to 00 , 
,S' I G, dt = e "' S \ , I = e"- f e"' - 1 - ^ 
' ^ "'+'(7; + l)!j \ m 
T - 
= l-e "' e "'■ 
m 
= certainty — chance of no event — chance of one event 
which it clearly should be. 
21—2 
