G. MORANT 
319 
have to be found from the observations. The expression will be still more com- 
plicated if we introduce the frequency of n intervals in T, when an event has 
occurred in the interval /8 before T starts. For this case we shall have to replace 
dT 
f(/3) by 1 - f(/3), uniltiply by — and integrate from T— ^ to T, or this chance 
will be 
These integrals could be integrated out by parts, but the process wo'uld be 
lengthy. 
Now in a long time the mean interval between events is /3 + m, and the mean 
occupied space is /3. Hence the chance that we start our epoch at an occupied 
space is ■ and at an unoccupied space is --r r = f(B). 
^ /3 + m r r (/3 + m.) •' ^'^ ' 
(b) The mathematical theory is thus complete but is clearly of too complicated a 
character for much serviceable application. To test its accuracy experimentally it 
is desirable to throw out from our observations all sets of T in which an event 
occurs within /3 before the start of T or within /3 before the end of T. If /3 be 
_^ 
relatively small both 1 — and 1 — e will be small and this will not involve 
the rejection of a large number of T-periods. 
When this is done, our formula reduces to 
$ + 771 nl \ m ) ' 
The curve of frequency of periods T with n occurrences will accordingly be of 
the form 
/T-n/Sy 
0 n ! V m. J 
T 
where N = total number of periods of length T and Ho = largest integer in 
H 
To simplify this further we ought to find a finite value for the series 
(a - nhy' a 
o — —^^ , where ?«„ = integer part oi ^ • 
There appears to be no discussion of such a series, nor any likelihood of its being 
expressible in a simple form. 
