279 
1910-11.] Mathematical Theory of Random Migration. 
moving more or less vertically, fulfils more nearly than any other Professor 
Pearson’s criterion that motion in any one direction is as likely as in any 
other. It has also the advantages of being not specially attracted by light 
and of being more or less opaque, so that it is easily photographed. Cyclops, 
Littorina rudis, etc., were also used as is described. Of plants only two 
species were investigated, an Oscillatorian and Aspargia hispida. 
Daphnia pulex. 
Experiments were made in two ways with Daphnia pulex. In some 
experiments a large number of the crustaceans were placed inside a 
cylindrical tube in a large flat dish of white porcelain and then liberated ; 
in others the water flea was allowed to take up its position as it liked in 
the dish. It usually chose to distribute itself from one corner along one 
side of the dish. Examples of the manner in which this happened are 
given below. The corner being an impenetrable boundary may be taken as 
representing a centre of diffusion, so that the simple fundamental integral 
should apply and the grouping should conform to the exponential. This 
is what takes place (Diagram IV.). 
Table showing the Number of Daphnia per Unit of Length along the 
Margin of the Plate from one Corner. 
Unit of Length. 
Actual (a). 
Theoretical. 
Actual ( b ). 
Theoretical. 
0-1 
18 
18T 
23 
20-7 
1-2 
11 
10T 
11 
13*6 
2-3 
3 
5-6 
7 
9T 
3-4 
4 
3-9 
6 
6-0 
4-5 
2 
2-5 
5 
39 
5-6 
1 
1-5 
1 
2-6 
6-7 
IT 
4 
1-7 
7-8 
1 
•7 
2 
IT 
8-9 
1 
•75 
The curve is given, 
X 
for [a) by y — 18‘e 2 ‘ 225 
for (6)by?/ = 25’e 2 ' 4 . 
In the first example the fit is excellent (x 2 = 2’8 P = ’9), in the second 
not so good (x 2 = 6'34 P = -6). The want of fit, however, is largely due to 
the group of four near the tail end of the distribution, which contributes half 
of the divergence. Except for this group a further divergence might be 
expected nine times in each ten trials. 
