Karl Pearson 
151 
On the other hand r^^ is not zero, but equals by (xv) 
-VpsPo/Vi - = -Vj)jVi - po 
to the same degree of approximation. Hence by (xx) 
rso^-VpJ (xxii). 
But if we form the partial correlation of deviations in classes and C( for constant 
frequency of number in Cg, we have 
a^st = (rst - '•so'"«)/a/(1 - rso^) (1 - r^o^) 
{o-Vp:Vp;)iV{i-p:)ii-p/), 
or oT.t = - (xxm). 
which agrees with (xv), the classes being now reduced by unity. Further, the 
reduced standard deviation must now be 
= a, V 1 - v j 
= VMp, (1 - p,') 
= VMXp; {I- p/) (xxiv). 
Now take the mean value of the frequency in class C's for a constant Xq or 
for constant frequency in class C^; in our case, if the sample is to be ni this will 
be M — m, we have 
— — ^ . '-^S / y-s 
qCCj [Xq Xq)j 
or 
,x, - Mp, = --^£^Mh. (M - m - M^.„), 
oX, = p, |m - -1-^ [M (1 - p,) - 
o«s = T--^~ = mpj (xxv). 
1 - 
The partial values ^x^, q/^s of the means and correlations of classes for constant 
number in class Cq are given by (xxiii) and (xxv), and are what we might anticipate. 
But (xxiv) should be Vmjo/ (1 — pj). It is accordingly needful to take 
A = mjM, 
or Pf^ = 1 — mjM (xxvi). 
These results have been reached on the assumption that p^ is very large as 
compared with Px, Pi, ■■■ Pn- It follows accordingly that the sample M must be 
large as compared with m, and further the sum of the classes Ci, G^,, ■■■ C i must 
