410 Midtiiile Cases of Disease in the Same House 
The values of m and n are m = 106,721, n = 3512; the observed values are 
p x — 3350, p 2 — 78, p 3 = 2 ; the higher p's are zero. 
It will be seen that, for such an example as enteric fever (and cancer falls into 
the same category), the approximate formulae are as good as the complete 
formulae, and there is much less danger of error in using them than in dealing 
with high powers of large numbers, such as arise in an equation like (x), which 
is based upon the differences of large numbers*. 
4. Application of Goodness of Fit Formulae. 
I shall now proceed to modify the second and third equations of (xii) or in the 
first place (viii) and (ix). These may be written : 
_ _ (st s + t-l\ 
I \n mj m 
*S^H£z2>l (xiv) , 
H 1+ s)- m i 
since we are neglecting terms of the order n 2 /m? compared with unity ; and 
_ f, _ (s*- s 2 , (s-iy 
I, ^/^^.'-"V (XV). 
Now in the cases with which we are dealing s or t is merely a number of 
the order 1 to 3, and small, while m is very large. Hence such terms as 
(s -l)(t — l)/m and (s - l) 2 /m are negligible as compared with st/N or s 2 /N, where 
N = n (l + £ J . We may therefore to the same degree of approximation as before 
write 
stp s p t 
'Pt"Ps""PtPs i\r 
^ „ T? ivFsft , 
^P^P'i 1 -^) ( Xvii )t, 
where N = ntl + -j (xviii). 
I now propose to find S(s 2 p s ). 
* In the case of a 2 Pi the results from (x) must be worked correct to 10 figures, if we wish to get 
the first decimal place correct in its value. 
t Equation (xvii) is as accurate as (xi) and it is better to use it than (xii). 
