Lucy Whitakbr 
55 
and ■654 respectively. Thus again the single binomial with only two constants 
give a fit slightly better, than the sum of eight exponentials with eight constants. 
Bortkewitsch looking at the observed frequencies and the sum of 8 or 11 
exponentials — without using any satisfactory test for "goodness of fit" — assumes 
that the coincidence is so good as to justify his hypothesis. But a better fit can 
be obtained with two instead of 8 or 11 constants by simply using a negative 
binomial. We must note here that Bortkewitsch is using the final coincidence 
merely as justification of the Poisson-Exponential ; the total frequency is not 
describable in terms of the 8 or 11 constants as it is in terms of the two, fur 
these eight constants are not leally significant for his individual eleven trade 
societies or for the suicides in the individual eight states. If he wants to describe 
the total, he has no constants by which he can do it. If, on the other hand, he 
wishes to describe what has occurred in the individual societies or states, we have 
seen that their binomials differ very widely from Poisson-Exponentials. If, lastly, 
no stress be laid on the individual cases as having too large probable errors, but 
only on the general coincidence with total frequencies, then the same coincidence 
would justify us in using a single binomial with two constants only*. It appears 
to us that to properly test the Poisson-Exponential, we need not 9 or 1-i instances 
in the individual case, but several hundred instances, — more, indeed, than "Student" 
has taken — and that no proof of the " Law of Small Numbers " can be obtained 
on data such as those of Bortkewitsch or Mortara. 
IV. Deaths from the Kick of a Horse in Prussian Army Corps, omitting four 
Corps with Bortkeivitsch. 
Here the results are : 
TABLE X. 
Number of Deaths . . . 
0 
1 
3 
S 
4 
Totals 
Numljer of Corpa 
109 
65 
22 
.3 
1 
200 
Whence : 
m--61, yao = '6079 
and the binomial is : 
200(-99G,.557 + •003,443)1^'™'^. 
This is the first of Bortkewitsch's illustrations for which his hypothesis that q is 
small and n large is really justified by his data. For : 
q = -0034 + -0670, 
/i = 177-1711 + 3449-103. 
The probable error of q for q really zero is + -0674. 
* Of course immensely better general total tits are obtained by using the sums of the actual 8 or 11 
binomials than by the Poisson-Exponential sum or the single binomial, but the results in that case 
involve 16 or 22 non-signiiicant constants. 
