1910-11.] Mathematical Theory of Random Migration. 281 
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given b y y = ae * (par. 7) and the section of this by the plane of y = 0 gives 
y = ae k. The numbers reduced to the population per unit area are given 
in column ii. of the table. If we take the area and first moment and 
integrate from x = 0 to x = 8, the limit to which a count was made, we get 
X 
y = 23‘04e~ 2-056 ' This gives ^ 2 — fi‘13 or P = *53, a fit as good as could be 
expected, when the uncertainty regarding the numbers in the outer zone is 
remembered. A large part of the value of y 2 is? as before noted, due to one 
zone alone. In all these groupings there seem to be secondary centres 
which interfere with results when the numbers are small. 
The methods of dispersal from a centre were also investigated. For this 
many Daphnise (from 100 to 200) were placed in the centre of the dish 
with a depth of water of about °f an inch. They were contained in a 
cylindrical tube about J inch in diameter. When the level of the water 
was the same on both sides of the tube and when the light was good and 
the camera ready, the tube was removed, the dispersal watched, and at a 
suitable moment instantaneously photographed. The photographs of course 
show no detail, the organisms being simply marked by a paler spot on 
the negative. In all cases a few Daphnise were found greatly more 
energetic than the rest. These were generally above the mean size and 
probably represented an older generation. They were so exceptional that 
they possibly should be rejected from the statistics; all, however, have 
been included. 
The experiments took much time. It was very difficult to manipulate 
the organism without damaging a number and thus introducing a new 
factor. 
In the case of the negative from which the following table is made the 
centre of the group was found by counting the number of organisms in 
each half -inch square of the plate and calculating the mean. This being 
found, circles were drawn round this of diameter \ inch, 1 inch, 1| inches, 
etc., and the organisms in each zone counted. 
The numbers in each zone are given in the table, column i. 
X 
The curve is given by y = 39 ’2e~i^o9 } hence y 2 = 9'7, which gives P = *4. 
In this case, again, one zone gives a large part of the value of y 2 , and 
it is also to be noted that again it is the third zone. Another fact of 
interest in all the experiments of this class is that the centre of the 
migration is not the original centre of dispersal ; the whole mass has 
moved towards the light. 
