265 
1910-11.] Mathematical Theory of Random Migration. 
The latter is always less than 3, so that if the centre of distribution is 
extended, that is, if the original mosquitoes are uniformly dispersed over a 
space bounded by two parallel lines, the subsequent distribution will 
resemble a curve of type ii. rather than the normal curve. But, as has been 
said, type iv. almost uniformly occurs. Some other modification is therefore 
necessary. This may be found in the fact that <j is not constant. In the 
simplified problem of Professor Pearson 2 cr 2 = nl 2 where n is the number of 
the flights and l the length of the mean flight. The curve is thus given by 
N -% 
y = - n . If l, however, vary, we do not get the normal curve at all, 
but a derivative. The frequency of any value of l or cr may be taken as 
given by f(cr) when the limits of cr are a and /3. The surface of distribution 
T\^ f Pf( ) -- 
derived from this y = x —\ ^-^-e 2(r2 dcr. Further, if the distribution take place 
7 r Ja. cr z 
from a definite area as above described, the final form of the distribution of 
the organism becomes 
*2 
-^Zbdcr 
fx+c rp 
y = y 0 <f>xdx /(cry 
Jx-c Ja 
( 1 ) 
when cpx denotes the mode of distribution in the area. This, if the forms of 
<p(x) and (/ cr) are known, I take to be the fundamental epidemic or random 
migration equation. 
To return for a moment to the form found for the time distribution of 
an epidemic 
y = • 
This may be put in form 
x logp+*a;2 w q 
y = ae 
ilOgqlx + '^lY 
— ae x lo g v Jo &2 
_(X+C) 2 + _C 
= ae 0-2 0-2 
_|2 c 2 
ae 0-2 0-2 
changing the origin of x 
( 2 ) 
if log q = - — and 
cr 2 log q 
From the symmetry of the epidemic c must in general be a constant, so that 
we have the relation 
^ log $ = log y 
or 
p = q c 
as the relationship between the infectivity and the rate of loss of infectivity. 
As q is less than unity c will be negative in sign, so that we have as the 
