Karl Pearson 
409 
Then 
,_„ 7n—l m n (m — 2)"~- s (n — s) \ (n — s)\ 
e IP* = - -x — .... x 
m (m-l) m - 2s 
We may evaluate the factors separately, 
_ (n — s)\(n —s)\ 
»!(n-2s)! ' 
U-2 = 
and by Stirling's Theorem = — ^ 
2k -2s + 1 
log,e s = -(2n-2, + l)(i + ^ + ...) + ( W -2 8 + i)(| + g + ...) 
s 2 , 
= etc., 
or 
ii„ = e «= 1 nearly, 
^ _ (m - 1) m n (m - 2)"~ 2g _ / _ 2_\"^" A _ 1 \-< 2 ' l - 25+1 > 
Ml _ m (m- 1) 2 ' 1 " 28 V m/ \ W 
, 2 (re -2s) 2n,-2s-l - ; /« 
= 1 - — — ' H — - + terms ot order — 
which we have agreed to neglect. 
Hence 
and accordingly 
e p * = «! u 2 = l H 
, - f* 2 2s - 1 
.(xi), 
which agrees with our previous value in (ix). 
Clearly in the enteric fever case, to which Troup and Maynard apply their 
results, the term (2s— l)/m is quite negligible, and we may always write when m 
is large compared with n : 
_ »i I n ' 
e tn 
Stpspt 
y 
.(xii). 
To test the accuracy of (xii) the case of enteric fever has been worked out by 
the long formulae, (iii) and (x), and by the short formulae of (xii). We have 
,2 ! 
Pi 
P3 
ff \ ! 
Long Formulae ... 
3398-34 
214-49 
55-88 
52-10 
•613 
•612 
Short Formulae ... 
3398 33 
214-60* 
55-89 
52-33 
■613 
•612 
* Equation (xi) used to find a{ 2 , but only this first value for which (xii) is not close enough. 
Biometrika viii 52 
