66 
On the Poisson Law of Small Numbers 
from which n has disappeared, and iu this exponential we have seen that 
Bortkewitsch and Mortara suppose m small, i.e. 10 or under. We have seen 
that there is no reason why m should be absolutely small, and that the name 
given by Bortkewitsch to the Poisson-Exponential — i.e. the " Law of Small 
Numbers " — -is misleading. But supposing the mean occurrence m to be small, 
it by no means follows that q need be small and n finite. For if q = '2 and n = 4, 
7)1 would be "small" — and the sort of small number with which our authors deal, 
but the mere fact that the mean frequency of occurrence Avas 2 would not justify 
our using the Poisson-Exponential for 
(-8 + -2)^. 
The fact is that when our authors speak, of the deaths in a Prussian Army 
corps from tlie kick of a horse, or the suicides of schoolgirls, or the deaths from 
chronic alcoholism as being "small," they I'eally mean small as compared with the 
number of persons exposed to risk. They had probably in mind all the men in 
the army corps, all school-girls or all individuals liable to death in the towns 
considered. But are all men in the army corps, — or only the cavalry, the artillery, 
etc., — equally liable to death from the kick of a horse ? Is every school-girl equally 
liable to commit suicide or only a very few morbid and unhealthy minded girls? 
Is every individual equally liable to die of chronic alcoholism, or only perhaps the 
10 or 12 confirmed and aged drunkards in a town ? The moment we realise these 
doubts, what is the population n to be considered ? It is not vi being small, but 
the smallness of vijn that leads us to believe that the binomial may have passed into 
an exponential. But if only six school-girls per year in a community are in the 
least likely to commit suicide, what is the justification for the "law of small 
numbers," if the average number of suicides be "65 ? Further, if we pass to even 
a large community in which the tendency to commit suicide is graded — a very 
probable state of affairs — m might be small and n large, and yet since q is not 
constant, the binomial and its exponential limit would not be applicable ; and this 
non-applicability would not depend on "lumpiness" — i.e. contagion or example in 
occurrence. Thus the probability might be : 
with all the ps independent (as in spinning differently divided teetotums) and not 
correlated (as they would be in drawing successive non-returned cards from a pack). 
It would seem therefore that a priori we should not expect the conditions for the 
exponential to be fulfilled in most of the cases selected by Bortkewitsch and 
Mortara, although with perfect mixing we might expect it in the cases cited 
by " Student." 
(14) In order to test this point on adequate numbers, the ages at death of all 
persons dying over 70 years of age were extracted for a period of three complete 
years from the notices of death in the Times newspaper for the years 1910 — 1912 : 
see Table XV. These announcements of death are those of individuals in a fairly 
limited class, which may be considered stable in numbers for these three years. 
