G. MORANT 311 
chance of non -occurrence during an interval t, when the occurrence measured in a 
t_ 
very long period averages once in time m, is e . If there are N intervals, it follows 
that the frequency of intervals of length lying between t and t + dt will be given by 
N -- N -- 
fn dt = —e '" dt, or the frequency curve is ?/, = — e . Now we have to consider the 
case in which an occurrence is always followed by a "closed-interval" /3, so the form 
N 
of the curve will be v, = — e ™ . 
m 
Thus, having observed the frequency of intervals in an indefinitely long time, it 
remains to find the values of ^ and ni which will fit this curve best. The beginning 
of the range, being at the point t = /3, is not known a priori, so that the moments 
of the observed distribution cannot be found in the usual way. The following- 
method is used to determine suitable values of j3 and m, applying for fitting, as now 
usual, the method of moments. 
The curve starts at /8 and runs theoretically to ^ = go . Let 
/ . = — e "' dt; 
in } 13 
or, integrating by parts, 
, = N + sinl3'-' + s(s-l) /3^-- + ...] 
- Ne [tr' + smt,'-'' + s{s- 1) rnHj."--] . . .(i). 
Accordingly : 
I,j^ = N[l-e j = nt^ = Ni\, say, 
= X {& + m-e ™ (tr+m)] 
= N\l3-tr + i\{tr + m)], 
I.,j^ = N\l3-+2ml3+2vi'-e "' {t,- + -Ind, + 2'm-)] 
= N {13' - t,' + 2m. (/3 - tr) + nt^{t; + 2mt, + 2m')\ (ii). 
Now /] — /] = 1st moment of all observations about t = 0, 
excluding the frequency up to intervals t,-. 
/o 00 — ^2,n, = 2nd moment of all observations about i = 0, 
excluding the frequency up to intervals t,.. 
Let these be respectively ilfj and M.,. 
20—2 
