270 
Proceedings of the Royal Society of Edinburgh. [Sess. 
To complete the symbols 
The fundamental forms which have been chosen for investigation are 
a ; 2 c r 2 x 2 
P°*~&dcr 
m= a 
ay. 
cr'*e 
ro o ^ Z " 00 
/ cr 2n e 0-2 * 2 c?<x and ?/ = a / < 
;o Jo 
These are symmetrical round the origin. On the hypothesis before 
mentioned they are integrated for each value of a? from x — c to x + c, 
multiplying each term by a suitable function so that for the working 
equations we have, 
[x+c [ z 
y = a <J>xdx 
Jx-c Jo 
[x+c [* 
y = a (jixdx / cr n e ^ 
Jx-C Jo 
(M) 
W 
(2) The first of these forms is much easier to evaluate, and in addition 
gives the closest representation of the facts. Considering it in the first 
instance and assuming that <fi(x) — a (a constant), we have the curve of 
distribution in time or space given by 
= a f X+ °dx I ° 
Jx-c Jo 
a ; 2 <r 2 
This form implies either that the rate at which infective organisms are 
given off is constant or that the distribution of organisms before migration 
begins is uniform. 
In the 
application required 
n 
is in general 
zero. The areas and 
moments of the first three curves, 
however, are 
given in the following 
table : — 
n = 0. 
n= 1. 
n = 2. 
A 
ck 2 7T 
ck 4 n/V 
2 ck G Jir 
k 
U 
15 k 
V 1 
2 
T 
16 
A: 2 c 2 
/r 2 + ^- 
3 
3 79 c 2 
/*2 
t + it 
t* 2+ tt 
-fc 4 + llll--- 
i 5 
li 
2 
‘ + ‘2cW + ~ 
5 
9 & 4 + 6c 2 F+ — 
5 
/c 2 
From the value of obtained from the statistics can be easily calcu- 
lated, and thence k. The latter value may be compared with value of k 
deduced from the value of v v and the nearest value of n ascertained. The 
equation of the curve corresponding to n = 0 is as follows : — 
- 2c 2x 
y = ak — ake k cosh k from x = 0 to x — c 
z 2 * . 2c 
y — ake k sinh from x = c to x = oo . 
