By Student 
359 
Conclusions. 
We have seen that the distribution of small particles in a liquid follows the law 
,„ Ci , ni- 'Hi'' 
e-'" U + m + + ... + --+ ... 
( 2 ! r\ 
where m is the mean number of particles per unit volume * and the various terms 
in the series give the chances that a given unit volume contains 0, 1, 2, ... r, ... 
particles. We have also seen that this series represents the limit to which 
any point binomial (p + g')" approaches when q is small, insomuch that even 
(M + ioY"" X 1000 is represented by {I + o + + ... + + ...^ x 1000 with 
a maximum error of about 45 in 180. 
For the rough calculation of odds with n small compared to ^ the exponential 
series may be used instead of the binomial as being less laborious. 
'iv^ Finally, we have found that the standard deviation of the mean number of 
particles per unit volume is a/-^ where m is the mean number and if the number 
of unit volumes counted, so that the criterion of whether two solutions contain 
different numbers of cells is whether vii — m^ is significant compared with 
TABLE I. 
Distribution of Yeast Cells over 1 sq. mm. divided into 400 squares. 
2 
2 
4 
4 
4 
5 
2 
4 
7 
7 
4 
7 
5 
2 
8 
6 
7 
4 
3 
4 
3 
3 
2 
4 
2 
5 
4 
2 
8 
6 
3 
6 
6 
10 
8 
3 
5 
6 
4 
4 
7 
9 
5 
2 
7 
4 
4 
2 
4 
4 
4 
3 
5 
6 
5 
4 
1 
4 
2 
6 
4 
1 
4 
7 
3 
2 
3 
5 
8 
2 
9 
5 
3 
9 
5 
5 
2 
4 
3 
4 
4 
1 
5 
9 
3 
4 
4 
6 
6 
5 
4 
6 
5 
5 
4 
3 
5 
9 
6 
4 
4 
4 
5 
10 
4 
4 
3 
8 
3 
2 
1 
4 
1 
5 
6 
4 
2 
3 
3 
3 
3 
7 
4 
5 
1 
8 
5 
7 
9 
5 
8 
9 
5 
6 
6 
4 
3 
7 
4 
4 
7 
5 
6 
3 
6 
7 
4 
5 
8 
6 
3 
3 
4 
3 
7 
4 
4 
4 
5 
3 
8 
10 
6 
3 
3 
6 
5 
2 
5 
3 
11 
3 
7 
4 
7 
3 
5 
5 
3 
4 
1 
3 
7 
2 
5 
5 
5 
3 
3 
4 
6 
5 
6 
1 
6 
4 
4 
4 
6 
4 
4 
2 
5 
4 
8 
6 
3 
4 
6 
5 
2 
6 
6 
1 
2 
2 
2 
5 
2 
2 
5 
9 
3 
5 
6 
4 
6 
5 
1 
3 
6 
5 
4 
2 
8 
9 
5 
4 
3 
2 
2 
11 
4 
6 
6 
4 
6 
I 
5 
3 
5 
7 
2 
6 
5 
5 
1 
2 
7 
5 
12 
5 
8 
2 
4 
2 
1 
6 
4 
5 
1 
2 
9 
3 
4 
7 
3 
6 
5 
6 
5 
4 
4 
5 
2 
7 
6 
2 
7 
3 
5 
4 
i 
5 
4 
7 
5 
4 
8 
4 
6 
6 
5 
3 
3 
5 
7 
4 
5 
5 
5 
6 
2 
3 
8 
3 
5 
6 
6 
4 
2 
6 
6 
7 
5 
4 
5 
8 
6 
7 
6 
4 
2 
6 
1 
1 
4 
7 
2 
5 
7 
4 
6 
4 
5 
1 
5 
10 
8 
7 
.5 
4 
6 
4 
4 
7 
5 
4 
3 
1 
6 
5 
3 
3 
3 
7 
4 
3 
7 
8 
4 
7 
3 
1 
4 
4 
7 
6 
7 
2 
4 
5 
1 
3 
12 
4 
2 
2 
8 
7 
6 
7 
6 
3 
5 
4 
The prism standing on unit area. 
