312 On Random Occurrences i7i Space and Time 
Then we have 
M, = Ne »' , 
(tr-P) 
M, = Ne {t/ + 2mtr + 2vi'), 
= \ say = — , 
or 2 {m + t^y - (X + t,) {m + U) + t,? = 0 (iii). 
This is a quadratic to find {m + 1^) and therefore m when X is known. 
Now the curve is exponential and we start with a finite ordinate at t = ^, and 
therefore in dealing with Af^ and from the observations we are bound to use 
careful adjustments, i.e. the abruptness coefficients for the moments*. Not knowing 
/S, however, we cannot evaluate the moments of the first group. But we do know an 
approximate value of /3, because we can note what the least interval observed is, 
and we may assume to be something greater than this. In actual observation work 
Equation (iii) did not give good I'esults and the reason for this is fairly obvious. In 
determining the moments we neglect a very important factor, the moments of the 
first group, and throw all the weight of determining jS and m on the later frequencies. 
A start may be made in the following manner. Take approximate values of ^ 
and 7n, say and m„, and calculate the position of the mean and the moments of 
the first group on the assumption that these values are correct, and add them to 
the observed moments of the remainder to obtain the total observed moments for 
equating against our theoretical results for the determination of a new /3 and m. 
■ In doing this we may obtain /So and from the area value and the first moment 
of all the data without the first group (observed frequency «,). Let the limit of the 
first group be t=<y\ then we shall take 7 as close to /3 as is safe in our ignorance 
as to the exact value of /S in order to make the influence of this first group as small 
as possible. For example, if /3 be about "3, we shall take 7 = '5 rather than 1, and 
reckon, say, our groupings from "5 to 1'5, 1"5 to 2'5 etc., rather than from 1 to 2, 
2 to 3 etc. of our units of time. 
Now suppose we find the first moment of N — of the observations after t = <y 
(this being il/i) and also i\r — n-i. These give 
N-n,=Ne , 
ilf J = Ne (7 + mo). 
M 
Then = _ ^ (iv), 
and ^^^^ mo|log,oi\r-log,(i\r-,^)j 
logio e 
Thus our preliminary values /»„ and |3^, are determined. 
* See Biometrikd, \o\. xii, p. 240. 
