PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 83 
Equations (14) and (15) clearly give the numbers in the component groups so soon 
as Yi and are found. 
Substituting these values of Zj and Zo in (10) and (H), we have two equations to 
determine and -u/ in terms of y^, y^. Solving them we find 
^3 
• (16)- 
73^/ = J - 3 - 3 (7i + 73 ) + 7i • • • 
72 7i73 
(17). 
These equations clearly give Mj® and and, therefore, the standard-deviations of 
the component groups when and y^ are known. 
For brevity, put 
"^1 = ( 71 '^j)"'’ ^’3 — ( 73 %)^ 
» 
= 7i + 73 . F-2 = 7i73- 
Then 
'^'1 = /^3 - i FJ72 - F ik 7 i +2^2 .( 18 ). 
i ’3 = 1^2 — i H-siYi — hhy 2 + P3 
(l‘J). 
while from (12) and (13) we have 
2 (y^Vi - y^Va) + ^ = (y^ — y^) p. — Ipp — i • • (20)> 
7i 72 
2 {y^\ - yii\) -f 3 {vp - v^) = (y^ - y^) \pp - i iJ^-Jp^} . (21). 
We must now substitute (18) and (19) in (20) and (21). We find 
7i^i - 73% = l7i - 73 ) - 3 ~ +^3 1*. 
yi'^\ - 72^2 
7i 72 
0 0 
Vp — t’.-- 
3 — 
(71 
(71 
(71 
(71 
72 ) “I \^2V\ “ 3 + F P'3 ■“ \Pi + 3lh2^3 f-. 
73 ) “I — ^ + F ^ + ¥ — ^^3 — 2p2 + I P3 ^ 
1:% i'2 
Ps" 
1\ 
\ I 1 P3~Pl I 1 3 _ 2. P 2 P 3 
72 ; i 9 „ 2 9 3 
Vi Vi 
o I o 2?^ 2 'j -1 
t V 2 P 1 ~T t Vs .. zVs~Spll^2 
Vi 
“a _ 
Vi Vi 
"^Px + 6^2 " 
^WVi 
Vi 
20/^3 - 2p^^ + ip^2h - 
^ (w It Vi) _ Q 
2h ~ 
15 (2/Xo/^3 - 4 ^ 5 ) _ ^ 
Vi 
M 2 
whence, 
