XIL—PARTIAL CORRELATION, orl 
lation giving the correlation between X; and X, or other pair of 
variables when the remaining variables X, . . . . X, are kept 
constant, or when changes in these variables are corrected or allowed 
for, so far as this may be done with a linear equation. For examples 
of such generalised regression-equations the student may turn to 
the illustrations worked out below (pp. 239-247). 
3. With this explanatory introduction, we may now proceed to 
the algebraic theory of such generalised regression-equations and 
of multiple correlation in general. It will first, however, be as 
well to revert briefly to the case of two variables. In Chapter IX, 
to obtain the greatest possible simplicity of treatment, the value 
of the coefficient »=p/o 0, was deduced on the special assump- 
tion that the means of all arrays were strictly collinear, and the 
meaning of the coefficient in the more general case was sub- 
sequently investigated. Such a process is not conveniently 
applicable when a number of variables are to be taken into 
account, and the problem has to be faced directly: i.e. required, 
to determine the coefficients and constant term, if any, in a 
regression-equation, so as to make the sum of the squares of the 
errors of estimate a minimum. We will take this problem first 
for the case of two variables, introducing a notation that can be 
conveniently adapted to more. Let us take the arithmetic 
means of the variables as origins of measurement, and let z;, , 
denote deviations of the two variables from their respective 
means. Then it is required to determine a, and &,, in the re- 
gression-equation 
z=, +b, kL 58) 
so as to make Z(z, -a,+b,,.,)% for all associated pairs of 
deviations x, and a, the least possible. Put more briefly, if 
we write 
N.S o=3(x, ~ a. + b;5.2,)% . . (9) 
so that s,, is the root-mean-square value of the errors of estimate 
in using regression-equation (a) (¢f. Chap. IX. § 14), it is required 
to make s;, a minimum. Suppose any value whatever to be 
assigned to b,,, and a series of values of a, to be tried, s,, being 
calculated for each. Evidently s,, would be very large for 
values of a; that erred greatly either in excess or defect of the 
best value (for the given value of &,,), and would continuously 
decrease as this best value was approached ; the value of s, , could 
never become negative, though possibly, but exceptionally, zero. 
If therefore the values of s,, were plotted to the values of a; on 
a diagram, a curve would be obtained more or less like that 
of fiz. 44. The best value of a, for which s,, attained its 
aq