U2 
Miscellanea 
Further 
Ml 
Ml 
.(xvii) 
which determine Xj^ and Xg. 
It is thus quite easy to resolve a series into the sum of two Poisson's binomial Umits provided 
the roots of the above quadratic are real. As illustration I take "Student's" first count of yeast 
cells on the 400 squares of a haemacytometer. He found: 
No. of yeast 
cells 
0 
1 
2 
3 
4 
5 
Total 
Frequency 
213 
128 
37 
18 
3 
1 
400 
giving: mean = /x/ = -6825, jx^ = -8117, fig = 1-0876. 
From the above values of "Student" I determined 
^2' = 1-2775, y.^' = 3-0675. 
Whence = -6825, = -5950, 0.3 = -6000, 
and the resulting quadratic is 
■129,194m2 - •193,913m + -055,475 = 0. 
The roots are = -3847 and wig = M163, leading to Xj = -59,295 and \^ = -40,705, or, the 
series has for its two components 
vi = 237-18, mi = -3847, 
V2 = 162-82, ?«2 = 1-1163. 
Calculating out the Poisson's. series for these components we have : 
No. of yeast 
cells 
0 
1 
2 
3 
1st Compt. 
2nd Compt. 
161-44 
53-32 
62- 11 
59-52 
11-95 
32-22 
1-53 
12-36 
Round totals 
215 
122 
44 
14 
Observed 
213 
128 
37 
18 
•15 
3-45 
-01 
•77 
-00 
■14 
•00 
-02 
The test for "goodness of fit" for these six groups gives = 2-82 and P= -73, or the fit is 
very good. The negative binomial gave P = -52 and a single Poisson's series only P = -04. But 
the double Poisson's series places of course one constant more at our disposal than the binomial, 
and we can do still better with a double binomial, as we have four constants and only six 
frequencies, while the double Poisson has three constants to six frequencies. It is clear that 
neither of the above components forming 41 % and 59 % of the total number of cells, and having 
their means at -3847 and 1^1163 instead of •6825, gives any idea of a dominant constitution in the 
solution sampled. If in this case heterogeneity accounts for the negative binomial, then the 
difference of the components is not shght, and the heterogeneity being gross would indicate some 
considerable failure in technique. 
If we assume that the counts with a haemacytometer ought to follow the Poisson distribution, 
— and this seems to be theoretically probable, — then the criterion of the binomial might well be 
adopted to ascertain the possibility of some failure in technique. The actual binomial in 
"Student's " first case should be (j + f tf§)^'^ and any binomial with j) very small and 7ip = -6825 
would effectively represent the series ; we could not anticipate getting n = 273 and p = 
closely from the data, but we might certainly anticipate a positive binomial, if the theory of a 
Poisson distribution be correct. If on the other hand we say in this and many similar cases 
that the negative binomial arises from heterogeneity, then it appears to me that we have saved 
our theory at the expense of our technique. I propose now to test this point further by con- 
sidering tiio component binomials. If the theory of heterogeneity be correct, unless it be very 
