1910-11.] Mathematical Theory of Random Migration. 269 
obtained if n = 0 and <p(x) = c (a constant), so that, in the first instance, the form 
will he considered. 
As is well known 
y — a 
e 
X 2 
< 7 2 
k Mcr 
^ 2 d<j= Jirke V 
the positive sign referring to the negative branch of x and the negative 
sign to the positive branch. 
Hence the equation of the curve is 
y = ak 2 - ake 
cosh 
lx 
from x = 0 to x = c, 
- — 2c 
y = ake k sinh — from x = c to x = oo . 
k 
The curve is symmetrical. Examples of this are given in Diagrams I. 
and II. When these are compared with Diagrams VII. and XXIII. re- 
spectively of my former paper,* it is seen how much better the fit now 
obtained is. 
Mathematical Formulas of Curves which represent possible 
Epidemic Forms. 
Equations and formulse which might describe more or less approxi- 
mately epidemic or migration forms will now be considered in detail. The 
following symbols are used throughout : — 
(1) Let y =f(%) be the curve of distribution 
r 00 
A = I ydx, area of the curve 
J-co 
where the axis of y is taken through the centre 
of gravity 
P'S 
x 3 ydx, etc. * 
When the origin is not in a line through the centre of gravity the moments 
are denoted by 
A*- p 2 » A 1 3 ? ®tc. 
* Proc. Roy. Soc. Edin vol. xxvi. pp. 491 and 507. 
