CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
173 
and let p be the correlation of this expression with B, then p will be greatest or 
tbe probable error of the determination of B by means of its correlation with such an 
expression will be least, i.e., ‘67449 cto ^^/(l—p") will be least, when the b’s are pro¬ 
portional to the corresponding c’s of the regression formula. 
Let S be the standard-deviation of the quantity 
Q = bo + biAj -f 62A.2 “b . . . -h 
Then 
= S;'(bVi) -h 2 S (^ 16 . 20 - 10 - 2 ^ 1 . 2 ) 
and 
p = 
The best value of B as determined from Q is 
B = Wq {bi (Ai — mi) -f- h, (Ao — oru) + . . . -j- 6 ,^ (A,,j — n;.,,)} . (iii.), 
with a jDrobable error ‘67449 o-q v^(l — p'). 
This may be taken to be any linear function of the As, since so far h^, h,, . . , b„ 
are n quite arbitrary constants, and the constant bo has to satisfy the condition that 
B takes its mean value when the A’s take their mean values. 
Now select such a value of the b’s as to give the greatest value to p. By 
ditferentiating p with regard to the b’s in succession we find the system of equations 
7‘oiS/'p = bio-i + -j- -h . • . + b„o-.„?‘i„ 
‘^' 02 '^!P — biO-iri2 -f- hoCT'z + hiCriTo^ + b,jO-„7‘2„ 
~ biO-p'ia -j- hiO’oTo^ -{- 630-3 -{-••• “h 
'>'o.X/p — biO-i?‘i,j fi- boO-'p'zti d“ d~ • • • + b,,0-„. 
The solutions of these equations are 
5 
P 
or, the equation to the best value of B, (iii.) above, reduces to the regression 
formula (ii.). In other words, no attempt to reconstruct the organ B from a linear 
relation to the organs Ai, A., . . . A,^ will give such a good result as the ordinary 
regression formula.''' This, of course, excludes all attempts to form type ratios of 
* I note that what i.s here demonstrated is only a special case of Mr. Yule’s general theorem. See 
‘ Roy. Soc. Proc.,’ vol. 60, p. 477. 
biO-i = — 
R 
I^flM P 
b.,cr., = 
It. 
R 
b„o-„ 
uo P 
IL 
E,„ 
