Miscellanea 
177 
Hence we see that So-,, is small, if the frequencies of interchanged subgroups are small as com- 
pared with iV and accordingly ; 
^-yl-y= ^ - (")• 
We now turn to the change in the product-moment. 
P + dP= 6' ( ir.ri/d.rSi/) - V,. Jv (i/s- - {y, - y,-) - + ^y\ 
where wbxbt/ is the total frequency of individuals, with characters l)etween ,»• and ./ and 
y and y + by and Xg and are the means of the arrays corresponding to y^ and But 
P=S {u\vy8x8y) — iVxy, hence : 
bf=iys-ys') {(^'8'-^) i's-bys--{^s-^') >-sbya)- 
Thus 8P/pJ^-^-^' ^ - ^^-^^ ^'Jf (iii). 
Now if r be the correlation before and r+Sr after a change is made, wo have, since 
r=PI{Nax<Ty), 
8r 8P 8ax So 
.(iv). 
Now we have suj^posed at present no change to be made in the .r's ; thus we may treat So-j; as 
zero, and using (ii) and (iii) we have, rearranging : 
(ys-ya'Y i'a'8ys'+Vs8y, 
2o- 2 JV 
.(V). 
Xow suppose the regression to be originally linear, then we have Xg-.r= — '- ('A - //) not only 
for s and s' but for all values of s whatever. In other words the whole series of terms in square 
brackets vanishes and summing for all pairs of interchanges : 
iys-ys')^ (I'a- fi.'A' + Vs^ys) 
V- 27v^7 
If we make similar interchanges of and x^y we can show that* : 
_^iysZj^a')^(.Va'^ya' + Vebya) _ (■Vp-Xp.f (U p^Sx^. + Up^X p) 
S'" -ys'){xp- Xp.) ( wy hx„ 8 j/a - ?t>2 bxp. Bys - tvs 8xp + 8 x„, S//.,0 
Here S denotes a summation or integration for all possible interchanges of the y arrays, i.e. say, 
columns of the correlation table ; and S' denotes a like summation for all possible interchanges 
of the .r-arrays, say the rows of the table. S'" is a summation involving the frequency at all 
points where interchanged rows and columns cross. Of course this result assumes that the units 
of grouping of both characters are so "fine" that the squares of the ratios of the array frequen- 
cies to the total frequency are negligible. 
We may now draw some interesting conclusions from (vi). Suppose the material to be such 
that the correlation is linear under some arrangement. Then for slight interchanges the squares 
and products of the interchanges are negligible and §/• will be zero. Thus, r being positive, we 
* The reader will find a verification of this formula arising fiom writing (i) the coirelation table 
with its columns inverted, then drjr= -2, and (ii) again in addition with its rows written backwards, 
in this case 5)-/r = 0. In (i) the fir.st term only remains and its numerator =4iVcry'^. In the second case 
the numerators of the three terms are respectively 4Ncr/, -IxW^- and iNra-^a-y. 
Biometrika v 23 
