Karl Pearson 
153 
coordinates and integrate the value of z for the — l)-fold surface beyond the 
value Xd^. Accordingly 
P = (xxxi) 
/ =° 1 2 
is the chance of a sample occurring with as great or greater deviation as the 
Xo sample from the general population. This is the expression from which the 
Tables of " Goodness of Fit " were calculated, the arguments being and I, i.e. the 
value of x^ for the sample and the total number of categories in the sample. 
Thus far there is only difference of method of deduction, not of results. 
(6) We now propose to replace condition (xxx) by a series of g linear equations 
of form (viii). These in the case of sampling will, if the size of the sample be 
fixed, either directly or indirectly involve (xxx). 
The type of these equations in their prepared form is 
^n^i + ^ifi^^ + ■•• + ^u^s + •■■ - Kf 
Each such plane will intersect 
x2=Zi2 + Z22+ ... + A,2+ ... 
in a sphere of lower order. For example, if there be n variates X, the first plane 
gives a sphere of the {n — l)th order, this will be intersected by the second plane 
in a sphere of the (n — 2)th order, so that ultimately we find ourselves reduced to 
a sphere of the {n — q)t\i order, by the intersection of the g'th plane. If K^, 
K2, ... were all zero, the radius of the sphere of the {n — q)ih order would be 
the same, i.e. the radius of the sphere of the wth order. But since these 
quantities are usually not zero we have to determine the radius of this sphere. 
The centre of this sphere must lie in every one of the q planes of the nth order, 
and accordingly on the plane of order w — — 1) in which they intersect. But 
the centre of the sphere of n — g order is where the perpendicular, K, from the 
origin meets this plane of the {n — q — l)th order, and the radius x of the sphere 
of the {n — g')th order is given by x' = X" - To determine x we must find P. 
Now P will be the minimum distance from the origin to the plane of the 
{n — q — l)th order in which the q planes intersect. In other words to find P we 
must make 
2)2 = + Z^^ + ... + Z7 + ... + Z„2 
a minimum subject to the q conditions of type 
^iiZj + A;,2Z2 + ... + A;,,Z, + ... + ^,„Z„ = 
where k^^^ + + ... + + ... + k^^^ = \, 
Using the method of indeterminate multipliers we find n equations of the form 
X, + + Aj^.^s + ••• + Ag/cg, = 0. 
* PUl. Mag. Vol. L. p. 158. 
