MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 
85 
Substituting the values given above we find, after some reductions, 
N??ij \j — nph'j 
~~N~ 
(xxii.) 
This result, which is extremely simple in form, gives the correlation in errors made 
in determining the frequencies in any two classes whatever of any two correlated 
variables. 
I next proceed to find the correlation between errors in u and u, the ratio of the 
ranges occupied by any two classes to their respective standard deviations. 
We have 
Sn 2 + Sn 3 = — NHjS/q ; 
Sn 3 = — NH 3 §/i 3 . 
Hence 
Similarly 
Multiply the first by the second, and summing as usual for all possible errors, we 
have, by using (xxii.) 
1 f Nm a — n 2 n 2 , N?» 3S — n s n/ ( 1 1 \ , Nm 32 — n. 2 'n. A / 1 1 
N 2 H X ' 
5> V TJ _ I -y-i I 
“ N 1 ^ 
K=H, \H/ H,' 1 + 
H, H J 
1 \ / 1 1 
I n z n s \ / 1 
N* AV H 3 / \Hfi HjVJ • 
Collecting the like H’s we find, after very considerable reductions, 
v y h — 
or V V — 
, ^4 U ^ U lX\) UU l - 
1 f Nm n — Nm, 
N [ NSHjH/ ~ 
1 f/*n ~ v\v\ 
( N m 33 — n s n 3 ( Nm 13 — n x n 2 , Nm 31 
AT9.TT TT / \TOTT TT / 
w 
N 2 H 3 H 3 ' 
N [ HjHfi 
+ 
^33 
H S H/ 
+ 
+ 
N 2 H 3 Hfi 
Hdl/ ' H 3 H/ 
. . (xxiii.), 
~ ^ 1 . . (xxiii. 1 "). 
where = m,y/N = proportional frequency. 
A glance at our diagram on the previous page of the correlation table divided into 
nine classes, shows at once the symmetrical formation of this result. By writing at 
the points P, Q, S, and T, the ordinate there of the normal surface, on the supposition 
of no correlation and N = 1, the construction of the result is still more clearly 
brought out. 
We are now in a position to determine the probable errors of p and £. We have 
Sp 
A 3 8/q — 7qSA E 
u~ 
