502 
Proceedings of Royal Society of Edinburgh. [sess. 
the first period of time the infecting power has declined from unity 
to q , say, then at the end of a second period of time it will be q 2 , 
and so on. On this assumption, if a be the number of persons 
infected originally and ap the number infected by these, then the 
succeeding terms representing the course of the epidemic will be 
represented by a, | ap, \ap 2 q., 1 ap 2 q.{p.q 2 }, \ ap 2 q.{p.q. 2 }{p.q 2 }\, 
etc., or the general term will be given by 
(x—2)(x—l) 
a-p -q 2 
which transferring to an exponential form 
(x - 1) log p -f — l°g 9 
is ae. 
As q is by hypothesis less than unity, log q is necessarily 
negative, and in consequence the slope of the epidemic curve is 
seen to be that of the normal curve of frequency of error. The 
normal curve itself, as has been seen, occurs as an epidemic form 
only very rarely. 
This gives an indication how the curve of an epidemic might 
arise, but it can hardly represent the complete solution. All that 
can be said is, that in general one of the curves derived by 
Professor Pearson to represent chance distributions makes a good 
interpolation formula for the ordinary course of an epidemic. 
These curves have been found to fit many classes of statistical 
grouping, and there is nothing in the method by which they are 
derived at all to preclude their application to this class of 
phenomena. These curves are the solution of the equation* 
1 dy _ - x 
y dx a + bx + cx 2 
and the particular one which is found to apply to this case is 
that where the roots of the quadratic expression in the de- 
nominator are imaginary. Its equation is 
y = 
Vo 
e 
- i/tan 
l 
X 
a 
But though this curve expresses somewhat closely the facts of the 
case, yet it does not express the whole truth, as is seen when the 
* See Note at end of paper. 
