266 
Proceedings of the Royal Society of Edinburgh. [Sess. 
general rule that as p increases q decreases, or the greater the infectivity 
the more rapidly it is lost. 
In the case of the epidemic the frequency of each value of <x is quite 
2 
unknown. As log q = — — and as q must from the initial assumption lie 
cr 2 
between 0 and 1, the limits of or are evidently — oo and 0. The situation of 
the commonest value of a is therefore between these limits, and must in 
general lie nearer zero than infinity. In choosing an arbitrary form for cr 
so as to get an approximation, it must be of such a form that (2) will be 
integrable and expressible in a form suitable to calculation. 
f° 
The equation obtained by integrating (2) is y = I e 0-2 0-2 fordo-. It is 
(j2 
obvious that if this be finite when cr = 0 that the term e +a2 must disappear, 
c2 cr 2 
so that part of f{cr) must be e 0-2 : the other part may be taken as e k 2 
c2 cr 2 
The function e 0-2 k * has a maximum when cr — J ck. The constant c is 
quite unknown, nor does it seem ascertainable from the method of analysis, 
but it disappears, and as k is at our disposal it is evident that the maximum 
value of cr can be placed where the statistics to be examined demand. We 
thus have as a possible form of the epidemic equation 
r 00 _ 
y= e a ' 2 k2 dx. 
Jo 
_ of 
To increase the variety of the distribution we might take <j n e 0-2 * 2 as 
representing the variation of cr : in this case the final integral becomes 
roo 
y= / (r n e 0-2 k *d(r. 
Jo 
The epidemic due to an organism instantaneously becoming infective 
and thereafter losing its infectivity at a rate corresponding to the geo- 
metrical progression should at least approximately fit the above curve if 
the distribution of or has been at all closely guessed. When we turn to 
Professor Pearson’s approximate form of solution of the random migration 
problem we find that it also has a term with cr in the denominator. The 
distribution is given by 
y = 
X 2 
2^2 
where N is the total number of mosquitoes starting from the point of origin 
and cr — J nl 2 , n being equal to the number of flights and l to the length of 
