406 
Multiple Cases of Disease in the Same House 
not need the highest mathematical powers and the most complete mathematical 
training. Now here is a case which absolutely confutes such a suggestion — just 
as it is confuted by almost every medico-biometric problem that arises. None of 
us, who are biometricians or medical men, have yet succeeded in fully solving the 
problem — a most vital one in many respects. We are simply pottering round 
it and nibbling off little corners of it. And why is this ? Solely because we have 
no transcendent mathematical power at our service. Those with such power have 
not the faintest notion of the existence of these statistico-medical problems, and 
those without it do not recognise the essential difficulties of the analysis. Thus 
we go on nibbling off one bit after another of such hard nuts, where the highly 
trained mathematician — if he could be made to grasp our problems — might reach 
the kernels in a few hours. Strange as the confession may seem, when I come 
to these medico-statistical problems, my regret is not for my want of medical 
training, but for the extreme defectiveness of my powers of mathematical analysis. 
This must be my apology for adding still another incomplete solution to those 
of my colleagues. 
2. Mean Values. 
Let it be required to drop n balls into m compartments. Then fixing all 
attention on one compartment, the chance that a ball falls into it is ^ , and that it 
7)1 ■ — 1 
fails to fall into it . Repeating the process n times, we have the binomial 
m 
hn - 1 1 \ n 
\ to to/ 
as giving the theoretically expected frequency or 
(m - 1 1 \ n 
to I — 
V m mj 
gives by its terms the distribution of compartments with 0, 1, 2, ... s ... balls in 
them, i.e. 
n\ fm — l\ n ~ s 1 1' 
Ps = m — — : — r- — (iii). 
1 s ! (n — 8) ! \ to / \mj 
This is the result reached as equation (vii) of Troup and Maynard's memoir. 
Now let us apply Stirling's Theorem 
s!= ^( 1 + il + 2 i + "') 
to (iii) for n\ and (n — s)l and also taking logarithms evaluate 
(n-s)\og e (l-I) 
by the well-known formula 
l0g e (1 -X) = - 
2 3 
