20 
I’KOFESSOK K. PP^ARSON OX TllP: IXPLUEXCE OE XATUKAL 
Ill otlier words the regression coefficient of the non-selected organ on the selected 
remains unchanged, wliile that of the selected organ on the non-selected will, as a 
rule, be widely modified. 
Further, let be the mean value of the second organ before selection corresjionding 
to a value of the first; let and Mo he the means of the organs after selection, 
and Yo he the mean value of tlie second organ corresponding to a value Kj of the 
first. Then the equation to tlie regression line before selection is 
^-2 I r I ^2 
A..1 — v’l.-, — Jtii -p ''/a.i — '?’] — aq, 
- 0-1 - 0-1 
and after selection it is 
Y„ = n-o ^ Ki -F M., ~ iq., Ml, 
1 .. s, ~ §1 ‘ 
Ki + nq -F I’jo — /q — /'lo — (nq -|- /q), 
0-1 
_ ^2 Tr I ^2 
— ^*10 — I — ^*12 — ^^^ 1 * 
cr.i 
cr., 
cr, 
But tills is identically the same line as the regression line before selection. Hence 
not only the slope (regression coefficient) of the line, but its position is identical, and 
we have the following result :—■ 
If two local races have been evolved from a single stock by the selection in different 
ways of one organ only, then the regression lines for the two races of any non-directly 
selected organ on the directly selected organ 'ivill be the same in direction and position; 
but the regression lines of the selected organ on any non-selected organ will differ for 
the two races.'^ 
()f course the means, standard deviations and correlations, not only of the selected 
organ but of all the non-selected organs also, will })rohahly have changed. It is only 
certain of the I'egression lines which remain unchanged and serve as a criterion to 
enable us to distinguish between directly and non-directly selected organs. 
Of course the pi'olrlem in Nature will not be as simple as this, for difierentiation of 
the two local races may have arisen from the selection of more than one organ, or 
may have arisen fi'om the selection of two difierent organs, hut the illustration will, 
1 think, indicate the nature of the investigation we are proposing. 
We can easily generalise our tlieorem by considering the form of the selection 
surface given on p. 12. Any result obtained from (xxxv.) whicli does not involve any 
of the c’s will he a result unattected by the selection that has gone on. Now to 
obtain a regression eipiatlon we })ut any num1)er of tlie Fs equal to constants, to A’s 
* The geometrical iiiterjiretatiou in this simple case that the regression line is unchanged is quite 
obvious, and, indeed, may serve as a proof. 
