: THEORY OF STATISTICS. 
minimum value, say a, 4, could be approximately estimated from 
such a diagram ; but it can be calculated with much more exact- 
ness from the condition that ¢f a’; a”; be two values close above 
and below the best, the corresponding values of s,, are equal. Let 
a, and (a, + 8) be two such values. Then if 
(wy = ay + brg.2g)2 = 2(2; — ay + 8 + by)? 
when 6 is very small, the value of a, is the best for the assigned 
value of b,,. But, evidently, the equation gives, neglecting 
the term in 82, . 
3(@) = ay + byp.2) = 0, 
that is, 
a,=0 
whatever the value of by, This is the direct proof of the 
a, ; i . 
Fic. 44. 
result that no constant term need be introduced on the right 
of a regression-equation when written in terms of deviations 
from the arithmetic mean, or that the two lines of regression 
must pass through the mean (Chap. IX. § 10). We may 
therefore omit any constant term. If, now, b,, is to be assigned 
the best value, we must have, by similar reasoning, for slightly 
differing values, 4,,, b;, +8, 
S(@) = byg.p)? = 3(w; — [bg + 8],)2 
That is, again neglecting terms in 82 
Sy(@) = bg.) = 0 
or, breaking up the sum, 
3 _ 2m) Lik] 
12 (x2) == Ann a, 
232 
(c)