408 Multiple Cases of Disease in the Same House 
But fx being small compared with n and m, and remembering that we have 
neglected terms of order s 2 /n 2 and w/m 2 , we have on expanding the exponential 
Pt -, (st s + 1 — 1 
Pt \ n in 
For the special case of t = s: 
p s \n m , 
Accordingly from (vi) we find, — showing a linear relation- 
s f, _ (s 2 2s - IN) 
-> _ /st s + t — 1 
opt = - fipt - - 
Whence from (vii) 
m 
si s + t-1 
D v» m , . .... 
CT <T_ i£_ = ; - — (VIU . 
Pt Ps PtPs fs 2 2s — 1 
P- , 
It remains to find a 2 p to the same order of approximation. 
Clearly <r p <r p R p p must be a function perfectly symmetrical in s and t. 
Hence we must have 
, _ (st s + t - b 
a 'p t Ps 
(T Pt (T p s R PtP s r r- -> t . | 
1 -p 
n m 
whence it follows that 
a . _ L _ /s 2 2s - 1 
m 
( \n m j I 
where X is independent of s or £, because and <7y cannot respectively involve 
a particular if or s ; i.e. X can only depend on m and n, or be an absolute number. 
If s = 0, we have 
^, = ^o(l-|), 
which is the correct value only if X = 1. 
Hence generally we have 
^p=pA i -ps(i- 2 ^)\ ( ix >- 
To test this value take Troup and Maynard's more general result (see p. 398) : 
, _ m (m - 1) (m - 2) n - 2s n\ _ ' _ a / x 
Call this a* = e + p s - p 2 . 
