ON THE POISSON LAW OF SMALL NUMBERS. 
By LUCY WHITAKER, B.Sc. 
PART I. THEORY AND APPLICATION TO CELL-FREQUENCIES. 
(1) Introductory. 
Let p denote the probability of the happening of a certain event A, and 
<! = ^ — p, the probability of its failure in one trial. Then it is well known that 
the distribution of the frequencies of occurrence n, n — 1, n — 2, ... times in a series 
N of » ti'ials is given by the terms of the point binomial 
N{p + qr (i). 
The fitting of point-binomials plotted on an elementary base c to observed 
frequency distributions has been discussed by Pearson*, and he has indicated that, 
if c be unknown, the problem can be solved in terms of the three moment coefficients 
//._,, /X:,, yu.4 required to find c, p and n. In actual practice but few cases of frequency 
can be found which are describable in terms of a point-binomial, and of these few 
a considerable section have n negative, p greater than unity and q negative; thus 
defying at pi'esent interpretation, however well they may serve as an analytical 
expression of the frequency. 
The hypothesis made in deducing the binomial {p + g)" as a description of 
frequency is clearly that each trial shall be absolutely independent of those which 
precede it. In this respect it may be said that binomial frequencies belong to the 
teetotum class of chances, and not to those of card-drawings, when each drawing 
is unreplaced. In the latter case the " contributory cause groups are not inde- 
pendent," and our series corresponds to the hypergeometrical rather than to the 
binomial type of progression "f*. 
Using the customary notation /3i = /"•a'V/^/, /Ss = iJi-iln-2, tbe binomial is determined 
from : 
n = 2/{3 - /S., + A}, c = (7 V6 - + 3/3i 
pq = h - /3. + A)/(6 - 2/3, + 3/3,) 
* "Skew Variation in Homogeneous Material," Phil. Trans. Vol. 186, A, p. 347, 1895. 
t Phil. Tram. Vol. 186, A, p. 381, 1895. 
