364 
MR. R. A. FISHER ON THE MATHEMATICAL 
and in consequence the proportion of sterile plates is 
and of fertile plates 
p = e m , 
q = I — e _m . 
In general we may consider a dilution series with dilution factor a so that 
and assume that s plates are poured from each dilution. 
The object of the method being to estimate the number n from a record of the sterile 
and fertile plates, we have 
L = Si (log p) +S, (log q) 
when S x stands for summation over the sterile plates, and S 2 for summation over those 
which are fertile. 
Now 
cp 
d log n 
dq 
0 log n 
= P log p, 
so that the optimum value of n is obtained from the equation, 
0L 
0 log n 
Si (log p) -S 2 1 ] ~ log p ) = 0. 
Differentiating a second time, 
0 2 L 
(log nf 
r, = Si (log p) — So | (log p + 1 + j |; 
q 
now the mean number of sterile plates is^a.9, and of fertile plates qs, so that the mean 
T alue of 
0 2 L 
0 (log nY 
is 
- r, -= -sS ; p log p— p log p (log p + 1 + — log p) 
^ log n L (] 
the summation. S, being extended over all the dilutions. 
It thus appears that each plate observed adds to the weight of the determination 
of log n a quantity 
w — (log pY. 
[ = -sS v-(log p) 2 r, 
■J l q 
