Karl Pearson 407 
and neglect terms of the orders w/m 2 (and therefore of course s/m 2 ) and s 2 /n\ We 
find 
, » s n-s 1 s (s - 1) 
p s = — ( — ) e m 2 n approximately (iv). 
Since in many cases s is a very small number as compared with n or m, this may 
frequently be written 
_ m/nV -™ 
P° = J\{m) e m (V >' 
which is the approximate form given by Troup and Maynard (see p. 402). 
3. Correlations. 
I now turn to the correlation in deviations of p s and p t . I reserve s/x deaths 
out of my total of n deaths ; I intend to put these deaths into fx houses, s apiece, 
so that I reduce my number of available houses to m — /x. I have accordingly to 
distribute n — s/x deaths among m — fx houses. The most probable distribution 
will be 
, (m - a - 1 1 
(m - fi) + 
V m — fj, m — fx,/ 
i.e. by (m) : p s = (m - //.) -r^ — r-: 
J v ' r s ! (n — Sfj, — s) ! V m — /a ) \m - jj, 
, _. . , (n-six)\ (m -ii- l\»-in-t / l \t 
and p t = (m — u) ^, 
Clearly we shall now, introducing the s/x deaths in /x houses, have 
ps + Bp s = /x+ p s ' 
p t + Sp t = p t ' 
Now if we suppose /x small as compared with m or n, we can expand by the 
same theorems as we have used before and determine p s ' and p t '. This will enable 
us to find the ratio of the mean change in p t to an arbitrary change in p s , or if the 
relation turn out linear, we have 
^t = °liR vv (vii), 
B Ps <r pg I* 
where R PtP is the correlation between an arbitrary change in p s and the resulting 
change in p t . 
It will be simplest to work by making the requisite changes in p s as given by 
(iv). We have 
(, SfiV 
I 1 — — j _nfi + Sfi _ls/xt(t-l) 
.(vi). 
mj 
q m- n 2 n n 
