244 THEORY OF STATISTICS. 
correlations of the first order (Table IL. col. 4) are obtained. 
The first-order coefficients are then regrouped in sets of three, 
with the same secondary suffix (Table IIL. col. 1), and these 
are treated precisely in the same way as the coefficients of order 
zero. In this way, it will be seen, the value of each coefficient 
of the second order is arrived at in two ways independently, and 
so the arithmetic is checked: 7, ,, occurs in the first and fourth 
lines, for instance, 7,,, in the second and seventh, and so on. 
Of course slight differences may occur in the last digit if a 
sufficient number of digits is not retained, and for this reason the 
intermediate work should be carried to a greater degree of 
accuracy than is necessary in the final result; thus four places 
of decimals were retained throughout in the intermediate work of 
this example, and three in the final result. If he carries out an 
independent calculation, the student may differ slightly from 
the logarithms given in this and the following work, if more or 
fewer figures are retained. 
Having obtained the correlations, the regressions can be calcu- 
lated from the third-order standard-deviations by equations of the 
form (as in the last example), 
a 
b1g.34="T1234 —y 
2.134 
80 the standard-deviations of lower orders need not be evaluated. 
Using equations of the form 
ores = (1 — r})}(1 - 7159) (1 — 78425)! 
=oy(1 = ri)1 — ri)(1 - Th)! 
we find 
log 0.45, =135740 Gon =228 
log a, .:,=1:50597 i] 
log 03415, =0'65773 Os120=105 
log 0,,3=132914 Oo pon=21'3 
. All the twelve regressions of the second order can be readily 
calculated, given these standard deviations and the correlations, 
but we may confine ourselves to the equation giving the changes 
in pauperism (X,) in terms of other variables as the most impor- 
tant. It will be found to be 
x, =0325x, + 1:383x, — 0-383, 
or, transferring the origins and expressing the equation in terms of 
percentage-ratios, 
X,=-31'14+0'325X, + 1-383.X, - 0383X,,