Lucy Whitaker 
39 
from series of observations even of the order 200, much less of order 18, to assert 
that q is or is not really a " small quantity." Thus the observed value of q corre- 
sponding to a population of extremely small q might easily show q = "15 to 'SO !. 
(4) Poisson — Law of Small Numbers. 
A last limitation of the point-binomial is made by supposing the mean //( = nq 
to remain finite, but q to be indefinitely small. We write : 
N{p + qr = N(l-q + qY = N{\- qyi {l + YZT^^^ 
lU ill 
= iV (1 - f/)'? (1 + q)'/ nearly 
= i\V"(^l-fm + ^, ...J. 
Here the successive terms give the frequency of occurrence of 0, 1, 2, 3... 
successes on the basis of each success not being prejudiced by what has previously 
occurred. This is the Law of Small Numbers. It was hrst published by Poisson 
in 1837*. It was adopted later by Bortkevvitsch, who published a small treatise 
expanding by illustrations Poisson's workf. The same series was deduced later 
by " Student " in ignorance of both Poisson and Bortkewitsch's papers, when 
dealing with the counts made with a haemacytometer j. 
The mean is at m from the first group, the other moments as " Student" has 
shewn § are : 
yu-ij = m, /1.3 = 7)1, = 3/?i" + m. 
Hence ^^ — l/m, /?., — 3 = l/m. 
When the mean value is large, /S^ and the higher ^'s approach the values 
given by the Gaussian curve. 
Clearly the Poisson-Exponential formula contains only the single constant 
This will, 
if N be reasonably large and in not too big, be a fsmall or at any rate a finite 
quantity (i.e. not like o-„ for q very small). Hence it might be supposed, although 
erroneously, that the Poisson-Exponential formula was capable of great accuracy 
in addition to its great simplicity. But this is to neglect the fnndamental 
assumptions on which it is based, namely : 
(i) that the data actually correspond to a binomial, 
(ii) that in that binomial q is small and h large. 
Clearly (i) shows us that, if we can find the binomial, it will actually be closer 
to the observed frequency than Poissoa's merely approximate formula. 
* Recherches sur la Prohahilite des Jurjements. Paris, 1837, pp. 205 et seq. 
t Das Gesetz der kleinen Zalden, Leipzig, 1898. 
X "On the Error of Counting with a Haemacytometer," liiomctrika, Vol. v, pp. '6bl — 5, 1907. 
§ They may be deduced at once from (iv). 
