275 
1910-11.] Mathematical Theory of Random Migration. 
This can be fitted by radial moments, or, in general, more easily by first 
reducing the statistics so that the numbers per unit zone is taken as a 
basis of calculation. 
The distribution may also be summed parallel to one axis and compared 
with a table of the integral 
,00 a/ ag+'j/ 2 
ale k dx. 
Jo 
In some cases, especially when an epidemic occurs in a locality where a 
disease is more or less uniformly endemic, it is useful to have the means of 
separating the epidemic portion of the disease from that which is endemic. 
Such a combination is very common in the case of such diseases as scarlet 
fever, enteric fever, diarrhoea, etc. Successive approximations are necessary 
to obtain a solution, but a first approximation can be obtained by using the 
normal curve to represent the statistics of the course of the epidemic. 
Let the whole amount of the disease be as represented in the diagram 
The endemic prevalence of the disease is represented by the rectangular 
base and the epidemic portion by the curve above. In general the parts 
of the curve beyond the limits may be neglected as only affecting the 
result to a small extent. Taking the middle of the rectangle as origin, 
denoting its height by d and its length 21, where l is known and d un- 
known, we have for the area and for the moments of the rectangle round 
an axis through the middle, 
A=2 Id 
A/x 2 = 
2 IH 
3 
Tor the normal curve the equation is 
y = y^ > 
and its area and moments become 
