J. McD. Troup and G. D. Maynard 
401 
Writing pj, p/, p 2 ', &c., for these values of p 0) p 1} p 2 , &c. 
_ 2 \ 
p 0 ' = me m 
2h ' ne 
n , 
= ~Po 
m 
p2 ~m2! e ~2m Pl 
Pi 
m 2 3! 
dm 
.(xvii). 
The close correspondence between (xi) and (xvii) will be noticed. Referring 
to the example dealt with above, we find the numerical values to the nearest 
whole number for p(, p.,', &c, as found by this method to be : p-[ = 3398, p. 2 ' = 56, 
p s ' = 1, an identical result to that found above for p x , p 2 , p 3 . 
We thus see that the mean and the approximate modal values closely 
correspond when m is large. 
Turning now to the assumption that all houses have the same number of 
inhabitants; there are of course very few towns in which the distribution of the 
population is so homogeneous as to admit of this approximation being used with 
safety. It is clear that where some houses contain only 2 or 3 inhabitants, while 
others have from 20 to perhaps 50, a considerable error might be introduced by 
grouping them together. It is, therefore, necessary to investigate a method of 
obtaining the values of p 0 , p\, &c, when the distribution of the population is not 
homogeneous. 
Keeping to our assumption that all individuals are equally likely to contract 
the disease, it follows that a house containing (say) six people is three times as 
likely to be attacked as a house containing only two. Let us suppose then that 
there are 
m x houses containing an average of a x inhabitants, 
and so on ; where each house in the group contains approximately the same 
number of inhabitants. 
Then the total population N = + m,a 2 + m 3 a 3 + . . ., 
and the number of houses m = + w 2 + m 3 + m 4 + . . . . 
If n is as before the total number of cases of disease, the m x houses will on the 
average contain cases, the m 2 houses »i 2 a 2 cases, and so on. 
Biometrika vin 51 
