328 On Random Occurrences in Space and Time 
(ii) Let us return to our fundamental equation (xxv)'''^ and lot us first find the 
total frequency in an array corresponding to a given n. This will equal (if A- 
represent a constant) 
^ ' (■» + 1 ) 1 
Next, let in be the mean interval fur the array. This will be given by 
(r-(« + i)^) 
(r-(»+i)3) 
n+l (n + l)(n + 2)^ \ii-r^)P) , 
(T-{,i + -i) fi) 
(T-in+l)(3r 
+ 1) ! ' n + 2 
Dividing out by the value first found for N.^ we have 
.(xxvii). 
The nieans of the arrays are therefore points on a rectangular hyperbola of 
which t = 0 and // = — 2 are the asymptotes, and that curve is one of the regression 
lines of the surface. 
Again, if a; „ be the standard deviation of the h. array of intervals 
(h + 1) I ^ "\'{T-t-n/3f 
<^'t,n + in' — 
(7'- (H + l)/3f 
= ^" + . + 2 ^ ^ - ^" + ^ ^ + (n + 2) (. + 87 ' 
whence, substituting for t,i, we find after some reductions 
{T-in + D^Yin + l) ... 
We see accordingly that the system is markedly heteroscedastic, the variability 
(if the array decreasing rapidly \vith 
For the special case of /3 = 0, 
^""=,172' "'•" = « + 2V . + 3 
Tlius the mean and standard deviation of any array tend to equality. 
We have further for the mean interval 
. _ (T-- (n+l) « ( J. _ ^ 1 ) 
t-{l + X ■ ..o) „, (xxx). 
8\e 
(n + 1)! 
