Miscellanea 
143 
manifold, we might anticipate two binomial components, i^i (Pi + g'l)" + (Fs + 9'2)") with 
n positive and large, both and being small, while n^^q^ and n^q^ would be approximately -3847 
and 1-1163, the frequencies and being roughly in the ratio of 3 to 2. 
Returning to "Student's" data we find fi^' = 8-9275, whence 
= -6825, 02 = -5950, a.^ = -6000, = -4800. 
Substituting in (xii)''''^ we obtain for the cubic: 
- •021,3269»3 + •028,2321»^ + -363,3020ft + -051,0825 = 0. 
This has three real roots, approximately. 
n' = 4-89997, n" = - -14234, and n'" = - 3-43390. 
We will consider in succession these cases: (i) n' = 4-89997. The first two equations of 
(xi) provide 
-5950 - 2-66173^1 + 19-10974P2 = 0, 
-6000 - l-72548Pi + 7-71894^2 = 0, 
leading to Pj = -55304, P, = -04590, and the quadratic 
q^ - •55304r? + -04590 = 0, 
whence we deduce the binomial factors 
?i = -4514, pi = -5486, and q^ = -1017, = -8983. 
The first two equations of (viii) give 
1 = Xi + Xj, -6825/4-89997 = -451,355Xi + -101,685X2, 
leading to 
\ = -107,535, X2 = -892,465, 
or, in a population of 400, 
i/j = 43-014, v.^ = 356-986. 
Accordingly the compound series is given by 
43-014 (-5486 + .4514)«-8999' + 356-986 (-8983 + •1017)i-8999', 
with means of the components at 
Wj = 2-2118 and = -4983. 
We see that neither the sizes of the component populations nor their ineans have any relation 
to the previously discussed component Poisson series ; further the present series * diverge widely 
from Poisson series, n is not large nor q^ or ^2 very small. Calculated to the nearest whole numbers 
we obtain: 
No. of 
yeast cells 
0 
1 
2 
3 
4 
5 
1st Compt. 
2 
9 
15 
12 
4 
1 
2nd Compt. 
211 
117 
26 
3 
0 
0 
Combination 
213 
126 
41 
15 
4 
1 
Observed 
213 
128 
37 
18 
3 
1 
which leads to = 1-27 and P = -93. 
Thus the fit is excellent, but it does not correspond to the heterogeneity of a double Poisson 
series. 
(ii) n" = - -14234. Here the first two equations of (xi) provide 
-5950 + -77965Pi + -I626OP2 = 0, 
-6000 + l-27469Pi + 1-75727P2 = 0, 
and give -Pi = - -81529, P. = -24996, 
with g2 + .81529? + -24996 = 0. 
* It should be noted that such fractional binomial series tend ultimately to become negative, 
although with negligibly small frequencies . 
