263 
1910-11.] Mathematical Theory of Random Migration. 
will be spoken of as a ‘ flight,’ and there may be n such ‘ flights ’ 
from locus of origin to breeding ground, or again from breeding 
ground to breeding ground, if the species reproduces more than 
once. A flight is to be distinguished from a ‘ flitter,’ a mere to- 
and-fro motion associated with the quest for food or mate in the 
neighbourhood of the habitat. 
“(3) Now, taking a centre, reduced in the idealised system to a point, 
what would be the distribution after random flights of N indi- 
viduals departing from this centre ? This is the first problem. 
I will call it the ‘ Fundamental Problem of Random Migration.’ 
“ (4) Supposing the first problem solved, we have now to distribute 
such points over an area bounded by any contour, and mark the 
distribution on both sides of the contour after any number of 
breeding seasons. The shape of the contour and the number of 
seasons dealt with will provide a series of problems which may 
be spoken of as ‘ Secondary Problems of Migration.’ ” 
The proof of the theory given by Professor Pearson contains also im- 
plicitly the proof that if the normal surface of error describes the distribution 
at any moment, it will at all subsequent times. This can be seen, however, 
_ a;2+2/2 
quite easily otherwise. Thus if y = y^e 20-2 be the distribution of disease 
where y 0 is the number of cases per unit area at origin and Jx 2 + y 2 is the 
distance of any point from the origin, then the amount of disease at any 
_ x' 2 +y' 2 
point x', y' is y 0 e 20-2 . This element gives rise to a new normal surface 
_ (x-x') 2 +(y-y') 2 _ x' 2 +y’' 2 
y^e 2<t2 2<r2 due to the infection at x'y\ If these several surfaces 
be integrated from — oo to + co with respect to x and y' successively, we 
get the new distribution 
ym o 
-(x-x’)2+(y-y')2 
2 a 2 
x’2+y'2 
20-2 dx'dy' 
x 2 +y 2 
= y^e 4 * 2 , 
so that the standard deviation cr is multiplied by J 2 — that is, that the slope 
of the surface is flattened. In other words, the longer the disease is present 
in a town the more uniformly, other things being equal, it will tend to 
distribute itself. 
At the present point it will perhaps be well to recapitulate the method 
by which it was considered likely that the normal curve of error should 
represent the course of an epidemic in time. If there be an amount of 
