400 Chance Distribution of Cases of Disease in Houses 
where it is to be observed that the conditions (iii) and (iv) are satisfied by the 
new values. The expression (xiii) is a little less than each of the following : 
(p 0 - 1) ! ( Pl + 2) ! (p 2 - 1) ! p., ! . .. (1 !)^ + 2 (2 !)*>*" 1 (3 f)*» (xiv), 
(p 0 - 2) ! ( Pl + 3) ! p 2 ! (p 3 - 1) ! ... (1 !)^ + 3 (2 |)A<3 
and so on. 
Dividing each of these by the original expression (xiii), we get 
(^ + 2)0^+1) 
Po P-2 2 ! 
= 1, nearly, 
whence 
(^+3^+2X^+1) Qearl 
and so on ; 
(Pi + 2) (p 1 + 1 ) , 
p 0 2! 
+ 3) ( Pl + 2)(p 1 + l) 
Poipo- 1)3! 
and so on. 
, nearly, 
Now wbere m is considerably greater than n, as in most of the problems of 
this nature with which we have to deal, p 0 will obviously be considerably greater 
than p lt and the values of p^, p s , &c, found above will rapidly diminish so that we 
may put for a first approximation, 
Po =Po, 
Pl=pi 
Pl 
Po 
P*=Po '- 
Ps=P> 
PoJ 2!' 
and Pl =pi, 
a Pi 
Po 
Q fPlV 1 
PoJ 2! : 
therefore adding, 
and 
m =p 0 
n =p 1 
4 2 t 
Po \p, 
Dividing (xv) by (xvi), we get 
_Po 
n ~p x 
■> ! 
+ 
+ 
Pl 
PoJ 3! 
+ ... 
Pi 
=p 0 e Po 
-p 1 e 
■ •(xv), 
.(xvi). 
therefore 
=p 0 e 
n = p x e 
