144 
Miscellanea 
This gives imaginary values of q-^ and and thus the solution can for the present purpose be 
discarded. 
(iii) n'" = - 3-13390. We deduce 
■5950 + 302614Pi + 15-22557^2 = 0, 
•6000 + 3-23317Pi + 16-44372P2 = 0, 
giving Pi = - 1-21441, P^= -20229 
and g2 + l-21441g + -20229 = 0. 
Hence (/i =- 1-0151, = 2-0151, = - -1993, 232= 1-1993. 
The X equations are 
l = \ + \, - -6845/3-4339 = - l-0151Xi - -1993X2, 
leading to \ = -000043, Xj = -999957, 
or, = -0172, = 399-9828. 
Thus the component series is 
■0172(2-0151 - l-0151)-3-«39 + 399-9828 (1-1993 - ■1993)3-4339^ 
with means at m-^ = 3-4858 and m2 = -6847. 
The first of these components is neghgible, it contains roughly only -02 individuals in 400, 
and the second is sensibly identical with the negative binomial obtained by "Student," i.e. 
400(1-1893 - -1893) -3-6054, 
with slightly modified constants. It provides : 
No. of 
yeast cells 
0 
1 
2 
3 
4 
5 
Calculated 
214 
122 
45 
14 
4 
1 
Observed 
213 
128 
37 
18 
3 
1 
leading to = 3-12 and P = -68, which for all practical purposes is as good as the double Poisson. 
Conclusions. It having been suggested that the appearance of negative binomials as better 
''fits" than Poisson's series for material that is supposed to follow the law of small numbers is due 
to heterogeneity, formulae have been provided for testing whether this heterogeneity is due to 
a second component. If so this component should be small and the first component should 
substantially agree with the primary Poisson's series. The smallness of the second component 
would measure the goodness of the technique in haemacytometer or opsonic index counts. 
Applied to "Student's" first series of counts of yeast cells we obtain (a) two Poisson's series 
neither of which dominates the data or approximates to the primary Poisson's series ; (6) two 
positive binomials, neither of which has any approach to a Poisson series or any agreement with 
the components of (a) ; and lastly (c) two negative, binomials, one of which dominates the series, 
and agrees with the primary negative binomial. This investigation as far as it goes suggests 
either that "Student's" first count is really described homogeneously by a negative binomial, or, 
if it be heterogeneous, then the heterogeneity is manifold, and no weight can be given to the 
results of fitting by the primary Poisson's series. 
The general numerical discussion by the formulae of this paper of a variety of data assumed 
to follow the "law of small numbers" is in hand and will shortly be published. 
