Q. MoRANT 
333 
Clearly the regression coefficient 
a\_ pu_ 
(Tt at III 
which agrees with (xxxiv). 
It is further easy from (xxxviii) and (xxxvi) to verify that 
_ t T 
n+ — = - + \ (xliii) 
111 VI 
which completes the checking of (xxxiv). 
As the mean «('s for arrays of t are not hard to find*, it would be fairly easy t' 
deduce m from a correlation table by the slope of this regressiori line. I have not 
so far succeeded even in this simple case in hnding either correlation ratios in 
a simple form ; but as In and t as well as Nn and a,, are known theoretically, it would 
be simple to calculate their theoretical values from the given values of in and T 
arithmetically, using 
Vt,n=0 ^Tj—, 
and to compare this value with that found from the observed value of these quan- 
tities. Proceeding in the same way we, of course, find 
■o 
1 =^ ^ 
III 
or the other ?; is equal to the correlation coefficient, owing to the linearity of the 
regression. 
(iv) We will now apply the process of finding the frequency of intervals in 
periods T supposing the periods to contain n intervals. The fundamental formula 
for this is (xxv)'''' which is of the form = C'„ {T -t- ii/3)"', where On is a constant 
dependent on n. The frequencies of lengths of intervals for different values of ii 
are shown in Table III, which is thus a correlation table for the number of in- 
tervals in periods of 10 sees, and the lengths of the intervals. No satisfactory 
method of finding the constants m and /3 from such a table of observed frequencies 
has been discovered so we are obliged to use the values found from the frequency 
of occurrence of the event in period T, i.e. the ratio of frequency method. The P 
for goodness of fit for the whole table is "233, but, as in the case of intervals in an 
indefinitely long time, this low value is due to the bad fit of the first two groups 
(/3 to 1 and 1 to 2) for the different values of n. Taking the first group in each case 
to be /8 to 2 gives the much improved value P = "707. The P for the right marginal 
total of Table III (Fig. IV), that being the distribution of lengths of intervals in 
10 sees, irrespective of the number of intervals in that period, when the first 
group is from /3 to 2, is "882. 
* The intervals of time, subranges in time, must be small. Tbeie is, however, none of the 
dilliculty due to the abruptness and unknown value of ji (in the general case) which accompanies the 
tindintr of the i,/s. 
