276 
Proceedings of the Royal Society of Edinburgh. [Sess. 
A' = 2/ 0 n/tto- 
A/x'-l = y^sj ttctC 
_ 0- 3 _ 
Ay! 2 ==y 0 Jtt— +y 0 Jircrfi 
A/ 8 -1?o 4+^4 
Adding the corresponding portions of the moments, we have the following 
equations for determining the four unknowns, d, a *, c, and y 0 , 
A + A 1 = 2 Id + y = B X 
• («) 
Afi\ = y 0 Jtt(tc = B 2 .... 
■ • (b) 
r t 2 l z d _ o ’ 3 — 
Ay 2 + A/x-2 = ~Y + y 0 J Try + y 0 vVo-C 2 = Bg 
. . (c) 
A/3 = y 0 + y 0 J- 7 T y = B 4 
• - («*> 
where B 1} B 2 , etc., denote the area and moments obtained from the statistics. 
By using (&) the equations (a), (c), and (d) become 
From a and /3 
or 
+ 2ld = B, 
. . . (a) 
c 
^ 2 + b 2C+ ^ 3 =b 3 . . 
• • • (;8> 
^ 2 b 2 +|b 2 =b 4 . . . 
■ • (y> 
BJ2 
Z 2 - 
which gives c. 
Hence immediately we have 
2 ~c B * '&' +B * e_B * y Bv 
2B 2 c 2 - (3B S - Z 2 B,)c + 3B 4 - B 2 Z 2 = 0, 
0-2 = 2 ?<_?c 2 
Bo 3 
B 2 
\/' 7TO-C 
2 /Jb, - y 
which complete the solution. 
When the theory just described is applied to the elucidation of epidemic 
processes it is found to be fairly satisfactory. It gives a very good fit to 
many epidemics, as will be shown. It is also capable of expressing the 
moment relationships of epidemics satisfactorily. Thus it is generally 
found that /x 4 > S/ul 2 : this is explained, but the form also allows of / ot 4 <3/x 2 
without a change of theory. 
