FOUNDATIONS OF THEORETICAL STATISTICS. 
365 
We give below a table of the values of p, and of iv, for the dilution series log p — 2 x 
from x — —4 to x — 11. 
X. 
V■ 
IV. 
S (w;) (per cent.). 
1 
-4 
0-00000014167 
0-000036 
0-002 
-3 
0-0003354626 
0-021477 
0-906 
-2 
0-01831564 
0-298518 
13-485 
-1 
0-1353353 
0-626071 
39-865 
0 
0-3678794 
0-581977 
64-388 
1 
0-6065306 
0-385355 
80-625 
2 
0-7788009 
0-220051 
89-897 
3 
0-8824969 
0-117350 
94-842 
4 
0-9394110 
0-060565 
97-394 
5 
0-9692333 
0-030764 
98-690 
6 
0-9844965 
0-015503 
99-343 
7 
0-9922179 
0-007782 
99-671 
8 
0-9961014 
0-003899 
99-836 
9 
0-9980488 
0-001951 
99-918 
10 
0-9990239 
0-000976 
99-959 
11 
0-9995119 
0-000488 
99-979 • 
Remainder 
0-000488 
Total 
2-373251 
For the same dilution constant the total S (w) is nearly independent of the particular 
2 
series chosen. Its average value being—;-, or in this case 2-373138. The fourth 
6 log a 
column shows the total weight attained at any stage, expressed as a percentage of that 
obtained from an infinite series of dilutions. It will be seen that a set of eight dilutions 
comprise all but about 2 per cent, of the weight. With a loss of efficiency of only 2 to 
2 -| per cent., therefore, the number of dilutions which give information as to a particular 
species may be confined to eight. To this number must be added a number depending 
on the range which it is desired to explore. Thus to explore a range from 100 to 100,000 
per gramme (about 10 octaves) we should require 10 more dilutions, making IS in all, 
while to explore a range of a millionfold, or about 20 octaves, 28 dilutions would be 
needed. 
• In practice it would be exceedingly laborious to calculate the optimum value of n for 
each series observed (of which 38 are made daily). On the advice of the statistical 
department, therefore, Mr. Cutler adopted the plan of counting the total number of 
sterile plates, and taking the value of n which on the average would give that number. 
When a sufficient number of dilutions are made, log n is diminished by - log a for each 
" s 
additional sterile plate, and even near the ends of the series the appropriate values of 
n may easily be tabulated. Since this method of estimation is of wide application, 
and appears at first sight to be a very rough one, it is important to calculate its efficiency. 
VOL. CCXXII.-A. 3 E 
