278 
Proceedings of the Royal Society of Edinburgh. [Sess. 
a zero line where the disease is nearly completely absent. The only point 
of difficulty in treating them mathematically is in settling how far the 
calculation of the moments is to be carried into the interepidemic period. 
In general good fits are obtained if the interepidemic cases are neglected. 
Two epidemics are shown in the figures given in the table. 
The fit of the first of these is not good as it stands ; but when it is noted 
that the symmetry with regard to the first moment is produced from an 
unequal distribution with respect to the mean of a kind that the sums of 
the equidistant terms on both sides of the mean are nearly equal, we may 
put the distribution as follows : — 
Actual. 
Theoretical. 
2 + 2 
4 
7*6 
27 + 50 
77 
77*0 
157 + 131 
288 
294-0 
194 
194 
178-7 
This gives = 3*83 or P = ’28. 
Thus the variations in the rise cancel those of the fall. This method is 
subject to criticism, but the fit of any individual epidemic can hardly be 
expected to be good. With regard to the epidemic of 1904 the fit may be 
said to be good. Here = 5*96 or P = *42. 
It is not necessary to give a large number of examples of the way in 
which the present theory fits the facts. In many instances better fits are 
obtained than those shown in my previous paper, where type iv. was 
found to closely represent the epidemic form. Two examples, however, 
may be given, that for the smallpox deaths in Warrington, and that for the 
milk epidemic of scarlet fever in Glasgow. These are shown in Diagrams 
I. and II. In both the correspondence of the facts with theory is very close, 
much closer than with type iv. Before leaving this part of the subject an 
example of the use of the distribution of a- n e ~ for cr may be given. This 
is the only example I have thoroughly worked out, but in a number of 
others I have obtained the medium value. In none is there any evidence 
that the curve obtained has any resemblance to the facts. It is therefore 
very improbable that cr n e~y (r can represent the variation of cr. The example 
illustrated is that of smallpox in London in 1902. As will be seen by 
referring to my previous paper, type iv. gives a quite different representa- 
tion from the curve shown in this case (Diagram III.). 
Examples of the application to random migration will now be given ; 
those with animal forms will be considered, and those with plant forms 
thereafter. With the former it is difficult to secure suitable examples. 
Daphnia pulex was used in many experiments. This crustacean does not 
move so consistently in one direction as others. It moves by jerks, and, 
