Karl Pearson 
147 
'uvw ... l/l 
N ' N "' N N 
as the case may be. 
Now we shall find it convenient to write 
Xs= {m,- ms)lV^s (vi), 
and thus 
X' = SAX,^) (vii). 
Further, if our q linear equations be of type 
htiM^ + htofth + ■■■ + ht,m, + ... = Ht (viii), 
where h and H are constants and t takes every value from 1 to q, we can write 
our conditions in the form 
ktiXj^ + ^(2^2 + ••• + ^tsXs + ■■■ = Kt 
where 
k 
ts 
Vh^tii^i + h^t2'm2+ ... - 
f h^t,m,+ ... 
■ - hf^nis - ... 
.(ix). 
and Kt 
Vh^ti'>ni + ^^2™2 + ••• + h^ts't^s + ■■■ I 
We shall speak of the first of (ix) as the prepared condition. Clearly it corresponds 
to a plane in w-dimensional space in which the constants kti, kt2, ■■■ ^<s. ••• ^^^^ the 
direction-cosines and Kf the perpendicular from the origin on the plane. It is 
convenient to use the notation 
^tikfi + kt^kt'i + ... + kt.kt's + ... = cos («') (x), 
for {tt') is now the angle between the ith and i'th planes. 
Assuming that the frequency surface with which we have to deal may be taken 
as 
z=2;„e-4x^ (xi), 
we may suppose before applying equations of condition (ix) that A'^ , K2, ... K f, ... Kg 
are variates and that we eliminate Xj^, X2, ... Xg, expressing our in terms of 
Ki, K2, ... Kg, Xg+j, -S^?+2' ■■• Xn- We shall have then 
z = Zq expt. — h (quadratic function of these n new variables). 
We now proceed to put K^, /ig, ...Kg constant, but leave the other n — q 
quantities to vary ; we are therefore seeking the value of foi" certain variates 
constant. This is the essence of partial contingency, and the analogy in con- 
tingency to partial correlation. 
(3) As a rule in partial contingency we do not seek to discuss when single 
cells of our multiple contingency solid are constant in frequency, although our 
theory covers that case. What we require usually is the value of x' when we make 
the contents of certain marginal total cells constant. For example, let us consider 
