‘294 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
daughter generally herself escaping (r u finite and cr 4 very small). The protection of 
the transmitting sex is due, not to smallness of correlation, but to relative smallness 
of variation in that sex. 
(b.) General Formulce for Four Correlated Organs .—Another point—especiallv 
important for the problem of morbid inheritance—is the relative ages at which a 
characteristic appears in parent and offspring. Darwin has noted how a characteristic 
appearing at a given age in the parent will reappear at the same age—sometimes 
indeed earlier—in the offspring. In particular, inherited diseases tend to develop 
themselves at an earlier date in the offspring than in the parent in proportion to 
the intensity of the inheritance. This appears to be especially the case in gout, 
rheumatic fever, diabetes, and phthisis. 4 ' 
Now, the quantities with which we have to deal here are four in number, ages of 
parent and offspring on appearance of disease and intensities of the disease in the 
parent and offspring. We require, accordingly, the formulae for triple correlation. 
Proceeding, as in the earlier discussions, we find, if x x , x. 2 , x z , x± be the deviation of 
the four quantities from their respective means, cr l5 cr 2 , <x ;3 , oq their standard deviations, 
r 12 , r i 3 > r in r 23 > r 24 ) r u> the six correlation coefficients pair and pair of organs or 
characteristic 2 Saq Sx. 2 Sx 3 Saq, the frequency out of a total of n quadruplets of the 
quadruplets with organs or characteristics between aq, x 2i x 3 , oq and oq 4- Saq, 
Xq -f- Saq, aq -f- Saq, aq -j- Saq: 
~ % 
** A O 
l ±TT~(T l (Tc>Crz(T± 
- 9 ( A 1 (—y + ) +U* Y + a 4 ( —) 2 —2^12 AO — 2 v t3 AA - 2 ; 
2 v. \ oq / \ (To / \ cr 3 / \ oq / oqo-o oqa 3 
XxX± o 
v u -I-* -21/03 
0-00-3 
-F04- 
— — 2i’s* 
croaq 
0-30-4 
where 
and 
\ x — 1 
r. 
34 
rj + 2r 23 r 34 r 42 , 
v n — ? 'l2 (1 ■ r 34~) r l3 r 23 r l4 r 24 4" r 34 l r U r 23 W r 13 r 24)> 
/x = 1 - r u - r 
13 
14 
O O O i O0| 00 I 0 0 
r 23~ ~ r 24 — r 34~ + r i2~ r 34~ + ?23~ r 14' + r i3~ r 24 
24 
+ 2 ( r 23 r 24 r 34 + r 34 r 14 r 13 + r l2 r l4 r 24 + r 12 r i3 r 23) 
2 ( r l2 r 14 r 23 r 34 "1“ r l4 r i3 r 23 r 24 "f" r 12 r l3 r 24 r 34) > 
while the remaining X’s and ffs may be written down by symmetry from \ 3 and v t2 . 
Accordingly we have for regression the formula 
K 
h + ^ £l h + hk w h 
0"o ^1 ^"3 A-j (T 
and for the standard deviation of a group of organs x, corresponding to selected 
organs h. 2 , h 3 , Zq (i.e., an array) 
-i = <*ilV x^i- 
* Here, as elsewhere, I have to thank my friend, Dr. R. T. Ryle, for the kindness with which he 
has allowed me to examine the material he has collected with regard to morbid inheritance. 
