64 
On the Poisson Laiv of Small Numbers 
the nature of those contributory causes groups. Of course if their dependence 
were of the nature of successive draws from a pack, then the result would be 
a hypergeometrical series and Q'^ would have no physical meaning ftir the series 
at all. 
(11) We will deal with one further illustration out of many considered by 
Mortara which are of like character. In the case of Marriages of Uncle and Niece 
(see Table XIV, p. 65), where the distribution of Q's is the most favourable 
for his theory, the binomials are 
Reggio Marche 
10 ( -7000 
+ 
-3000 )^-» 
Umbria 
10 ( -0000 
+ 
-1000)=° 
Basilicata 
10(1-4000 
•4000)-' ■= 
Sardegna 
10 ( -44545 + 
-55455)''^'^'^ 
Emilia 
10 ( -9818 
+ 
•0182)i-"'"'"' 
Abruzzi 
10 ( -8429 
+ 
.1571)17-8182 
Lazio 
10(1-2548 
•2548)-i2'i«« 
Puglie 
10(1-5111 
Veneto 
10(1-3444 
•3444)-i'^06« 
Toscana 
10(2-2667 
l-2667)-^'2'*3i^ 
Calabria 
10(1-3584 
•3584)-2J™= 
of which only one (Emilia) approaches the conditions for an exponential distribu- 
tion. If we test the totals at the foot of Table XIV, we find the result much to the 
advantage of the binomial, for which P = "902 as against -714 for the exponential. 
(12) On Mortara's own showing nearly all the Q's of his numerous series are 
greater than unity, and very few of the binomials are positive. If we consider the 
distribution of Q's, given in his work omitting Table 13 (Deaths from Malaria) we 
find a range from -5 to 3-6 with a mean Q' at 
1-2565 + -0847, 
while for the distribution of all the ps in the binomials we have determined, we 
find a range from -4 to 15-6 with a mean ^ at 2-5655 + -3817. 
These results are sufficient to show that there is no real distribution of p round 
the value unity but the binomials have a distinct tendency to be negative. 
(13) But the whole theory of Poisson's exponential law in the hands of Bortke- 
witsch and Mortara appeal's essentially vague. The binomial is built up on the 
assumption of the repetition n times of a number of independent events, of which 
the chance of occuiTence is identical and equal to q. The population is n and the 
chance of occurrence q in the case of each individual. The mean frequency of 
occurrence is nq. But if q be very small we have seen that the series is 
/ in" \ 
.-"'(^l + m + - + - + ...j, 
