M. Greenwood and J. D. C. White 
521 
index is beyond these limits, the chances are less than ten to one against its being 
drawn from a normal population. We now come to a most important qualification 
of the value of these tables. 
In the table as given, the measure is of the probable deviation of a sample 
from the real mean of the samples as determined from a long count. If the control 
mean, i.e. the denominator of the fraction which we call the opsonic index, has 
been determined from a large number of cells, say 1000, and the tested serum 
is measured on the basis of a small sample, then our results are directly applicable ; 
but this supposition does not represent the usual state of affairs. In ordinary 
practice, two small samples, each of not more than 100 cells, serve as control and 
test, there is therefore, so to speak, a double source of variation ; odds based 
on the results obtained above would over-estimate the reliability of any given 
determination. This is a point of considerable importance which has, we think, 
escaped the attention of writers on the subject ; its elucidation would involve 
a careful examination of the whole theory of index distributions, and would 
TABLE XL 
Constants of the Experimental Index Frequencies. 
[Mean, Mode and a are expressed in terms of the Index, i.e. in physical units.] 
Material 
Number of 
Samples 
Mean 
Mode 
/3i 
ft 
r 
25's 
50's* 
lOO's 
800 
400 
200 
1-0362 
1-0205 
1-0120 
•9432 
•9536 
•9689 
•2585 
■2204 
-1553 
-7099 
-6952 
-3654 
4-3914 
4-8945 
3-7138 
•9619 
•3635 
-9032 
11-2690 
Material 
m 
a 
ft- 3 
Skewness 
y 
P 
P (Fit) 
25's 
50's 
lOO's 
6-6345 
-8-5166 
5-6343 
1-3914 
1-8945 
-7138 
-3600 
-2912 
-2780 
326-1439 
491-3142 
27-4967 
48-7621 
•0853 
■8953 
•9355 
326-l'439\ 
Equations : ?/=antilog 43-6116 {x' 
A change of -1 in the index is taken as the statistical unit of grouping and index ^25 as zero. 
/ ^2 \ -6-6345 8-5166 tan-i-_5_ 
3/ = 6-2092( 1 +— — \ xe ^'^343 
V 5-6343V 
A change of •! in the index is taken as the statistical unit of grouijing and index -45 as zero. 
The origin is at 1*4468. 
/ 491-3142 \ 
y=antilog 71-8335 [x^'^^"'^'^^ xe ^ J. 
A change of -1 in the index is taken as the statistical unit of grouping and index -45 as zero. 
* A slight slip has been found in the values of /Sj and /Sj for this series, they should be -6836 and 
4-9855. This error does not, however, visibly alter the form of the curve in Graph 7. 
