536 
On the Constants of Index- Distributions 
It will be seen, in accordance with the results drawn from the approximate 
formulae, that the mean value of the opsonic index is greater than unity and that 
there is marked skewness, i.e. no approach even with samples of 100 to a Gaussian 
distribution. Clearly the most probable value of the index is less than unity, and 
is rather farther from unity than the mean. 
Greenwood and White have made an experimental determination by 200 
random drawings. In Diagram I the frequencies of the above curve are reduced 
to 1000 total and plotted against their experimental data increased to 1000. This 
of course exaggerates the apparent deviations, but enables our three diagrams to 
be compared among themselves. 
(5) Further Illustrations. The following frequencies are also deduced from 
Greenwood and White's results for 400 samples of .50 and 800 samples of 25. 
With the same notation as before the constants of the distribution of the 
opsonic indices are as follows : 
II. Distribution of 160,000 indices for 
400 samples of 50 counts. 
For x: v, ^ 3-68140, = 13-82272, 
= 52-92733, v, = 206-69803. 
For z: < = -277213, 1'/ = -078453, 
= •022683, I// = -006705, 
mi = 1-02053, TO, = 1-08443, 
= 1-20055, TO, = 1-38591, 
ytis- -042948, o- = -2072. 
/c = -5415, 
/i3 = -006191, /3i = -4837, 
Mean = 1-0205, 
;Li, = -007567, /S, = 4-1022, 
Mode = -9611. 
The distribution is of Type IV : 
r = 20-86018, = - 22-67044, 
a = -62536, 1/0 = 16-9740, 
and the equation to distribution of the 160,000 indices possible is given by: 
H, / \ - 11-43009 22-67044 tan-i 
F= 16-9740 (l + .^-Q^ j e -62536 (H). 
The origin of the curve is at the opsonic index -3409. 
Diagram II gives this theoretical distribution reduced to 1000 cases and 
compared with Greenwood and White's experimental data for 400 increased to 
1000 also. 
