G. MORANT 
325 
while 
{T-{7l' + l)l3) 
■+1 
(»'+!)! V m ) 
F' ■ = ' — (xxiii)'''^ 
^1 (n' + 1)! \ m / 
If we are only given as a whole, and the several F'n', it is rlifificult to see 
any ready solution of the problem. If we are given the part i^,," of i^,/ which covers 
no occurrences in T we can proceed exactly as in the previi^us problem, wi'iting 
XV = |^^^'}'^ = «-(..' + 1)6, 
and the solution proceeds just as before. If we are given F,l" , the number of cases 
of a single occurrence, but not Fo", the equation seems as intractable as when we 
only know F^ as a whole. Yet this seems a not impossible case in practice, namelj^ 
one in which there would be no record unless at least one occurrence happened. 
We shall now determine the frequency distribution for intervals of size t 
occurring in the period T which we will suppose to contain n intervals. 
We enquire first what is the chance that the ?'th interval of these n intervals 
is of magnitude t. We shall again exclude the possibility of an occurrence having 
happened at an interval less than ^ before the start of T. 
I |JU ^AJ \AA I I /3 I I I 
Then, as before, the probability of the system being as in the diagram above is 
fi/3)e "'-'e "^e "^../^e 
m m m 'in in 
..e 
Here t,.+i ~ t,. = t and is to remain constant while we integrate for all the 
ti ... t,i,+i but t,._f_i. Cleai'ly the above probability can be written 
^{T-nl3) (t„n-T) 
p 
We first integrate ^„+i from ^„_,.^ to T, i.e. we determine 
rT _ "11 +1 ^ I 
e~ '"• 'q(tn+,)dtn+r 
'T-P _('n +l-T) _(2'-f„,+i-P) r-T _{fn +l-T ) 
e e dt,i+, + e "' dt„ 
= e>" {T^tn-2l3)+ m \e"' - Ij. 
Biometrika xiii 21 
