264 Proceedings of tlie Royal Society of Edinburgh. [Sess. 
infectious disease a : if such element infect pa persons, and if q be the rate 
of loss of infectivity per unit time, as has already been shown to be probable, 
the number of each group of infected persons (supposing the supply of 
susceptible persons large) is a : ap : apxpq : ap\xpq 2 : ap s q 3 xpq s , etc., 
x-l (x-\)(x- 2 ) 
or, in general, if x denote the unit time, y = ap q ~2 , which, as q is 
necessarily less than unity, is the normal curve of error. If, instead of 
finite, infinitesimal differences are employed, the result is expressed by the 
formula 
y = ay x cp x2 . 
That the normal curve is to be taken as that from which variation is to be 
expected both when the space and time distributions of epidemics are 
examined then seems clear, and it remains to discover in what manner the 
natural process differs from that so far developed theoretically. 
It is, in the first place, to be noted that if the distribution is from a 
central area instead of a point, a disturbing influence on the shape of the 
curve comes into play. This can be allowed for at once. For purposes of 
convenience a two-dimensional solution is given, such a solution being easily 
extended to three dimensions in any particular case. If we consider the 
modification of the normal curve produced when the mosquitoes start from 
an area and not from a point, the moments of the resulting distribution will 
not be those of the normal curve, that is, not 
but 
f 
- a 
x n e 2<j2 dx, 
fa 
/ da (x + a) n i 
J - a J - co 
x 2 
2a2 dx ; 
or, for the even moments, since the odd moments are zero, 
r . a 2 
P 2 = ^2 + ~cr 
^ 4 = Aq + 2 a 2 /x 2 + — , 
so that 
JH = 
u , 
/x 4 + 2a 2 //, 2 + -g 
, 2\2 
/ X 2 
3/x 2 2 + 2 ck 2 /i. 2 + ^ 
~ j _ 2«V 2 a 4 
/V + 
3 ' 9 
since for the normal curve yu 4 = S^ 2 2 . 
