28G PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
(7.) Special Case of Three Correlated Organs. 
We need not stay long over the general theory as it has been fully treated by 
Bra. vais. We indicate its general outline in a modified form. By p. 263 we have, if 
x, y, z be the deviations from the means of the three organs, and cr 1 , oy, <x 3 their 
standard deviations, 
x /. x 2 . . V 2, . \ 22 2ys % zx 2 xy \ 
a ( ^1 r + ^ 2 -^ + ^3 2“—"— Vl ~ v ' 2 ~ 111 
= (J (3 V ° 1 2 A- <T 3 * 0 - 20-3 0 * 3 o', <r,<r 3 / (Jx ay dz. 
This may be written in either the form, 
or, 
- ! A / J«_ _ v 3 y _V2±V _ A 2 *i - V& (JL - JL y iAl + M:< y 
p (Je 1 ' O'! A i 0' 2 A, <r 3 / ^ g ‘^A| V o'! o - 3 A 2 A t - v 3 2 / 
_A^A^A^ — 2i/|i' 2 t'3 — Aji'^ 2 A 2 V2“-*“ A 3 v 3 2 £*• 
X e 2 (a x a 2 - V3 2 > o? dx dy dz 
-4A 1 (JL-_pj'-A 2 _i ) 2 
Jr = (JC V °1 A 1 °2 *1 0 - 3 7 
_ i f / ,V \ 2 A 2 Aj — / £ \ 2 — Vo 2 2 yz VjA] 4- Vat's 7 
X e 2 l V cx 2 ' Aj. V o' 3 / Aj O 2 O 3 Aj ' dx dy dz 
• (A), 
(B). 
Integrating A for x, y, 2 successively between i 00 , we have, if n be the number of 
correlated triplets, and 
X = ^1^3 ~ Zvi v 2 v 3 ~ Vp — ~ h v s 2 > 
n — C. ( 2 tt -) 3 - cr iCTjCTg/ v y, 
or, 
C = nv 7 x/((27t) 3/2 oyo-gOg). 
Integrating B for x between dz 00 , we have 
_ 1 f /J/_\ 2 AjA, —V3 2 , / 2 \ 2 A^A, - v 2 2 _ 2i/s vQ! + ) 
P' — Q'g " d crd A! hj/ n 0 ' 20 's Ai 1 (]\ 
dy dz. 
But this must be the correlation distribution for y and 2 treated independently of x, 
or, comparing with p. 264, if i\, r. 2 , r 3 be the three correlation coeflicients for the pairs 
yz, zx, xy respectively, we have 
A-i/(A-iA 3 — vp) = 1 rp =■ Ai/^AgAx — vp). 
(^i x i + ^ 3 )/(\A 1 - v-p) = i\. 
Integrating A for x and y from rh o°, we must have the distribution for 2 treated 
independently, or a normal distribution <x 3 ; this gives at once 
X i X 2 - v i = X- 
