58 
On the Poisson Law of Small Numhers 
not we a posteriori demonstrate it from the data themselves ? Child suicide may 
be influenced by example, by environmental conditions in different districts, 
possibly even by meteorological conditions in different years. Again, even in 
different army corps the conditions may be far from uniform, the spirit of the 
corps, the teaching with regard to the handling of horses, the experience of past 
life according to whether the corps is raised in town or rural districts may all tell. 
Even Bortkewitsch before he gets his best fit removes four corps or 80 observations 
from his data. We do not criticise this removal, but even uni'emoved he says the 
fit of theory with experience leaves " wie man sieht, nichts zu wiinschen tibrig" 
(p. 25). But the binomial is before removal : 
280 (1-085,714 - -085,7 14)-«-i««''«'^ 
in which q is not very small and is negative, and n is not very large and is not 
positive. It is true that the probable error of q for q insignificant is in this case 
+ -0570, but this only shows that the data were insufficient in quantity to 
determine whether the exponential could be applied or not. 
(9) Mortar as Gases. 
Mortara* in an interesting paper has realised the possibility of repetitions not 
being independent and has discussed a constant Q', by which he proposes to test 
such influence. This quantity Q' should be unity, if the Bortkewitschian hypo- 
thesis can be applied. He then takes 16 or 17 districts with records of 10 years, 
and calculates the mean number of deaths from some special cause per year, say, 
for each district for those years. If this mean number exceeds 10, he casts out 
that district, presumably on the ground either (i) that such a number is no 
longer small, or (ii) that it differentiates the district from those with lower 
numbers. Thus Bologna with 10"9 deaths by murder is excluded and Bergamo 
with 8-4 is included, although Q' = 1 for both. Bologna with 7-1 deaths from 
smallpox is included, but Pavia with 12-3 is excluded although the Q' of the 
former is 2-5 and that of the latter 1-7. What method should be employed in 
dealing with the frequency of the excluded districts which may amount to 50 % 
of all districts is not discussed. Having thus reduced his available districts, 
Mortara proceeds to apply the exponential to each individual district ; he adds up 
the results for each district and compares his totals with the observed totals. It 
will thus be observed that he fits his exponential to ten observations, and then adds 
together five or more districts to get his totals. We can equally well apply this 
process by fitting a binomial to each 10 observations and then adding up such 
results. But it is quite clear that on the basis of ten observations, it is, owing to the 
large probable errors, wholly impossible to assert, whether a binomial of the kind 
required by the Bortkewitsch-Mortara hypothesis,— i.e. one of very small positive q 
and very large positive n — really is justified. We can illustrate this at once from 
Mortara's Tables (see his pp. 42 and 45) for deaths from Chronic Alcoholism. The 
* " Sulle variazioni di frequenza di alcuni fenomeiii deiiiografici raii," Aniiali di Statintica. Serie v. 
Vol. IV. pp. 5 — 81. Koma, 191'2. 
