264 
PROFESSOR KARL PEARSOK, MATHEMATICAL 
We have at once the following results for the total numbers dealt with in each 
case: 
= S (z), N'l = S (Xxz) = XMiNi, 
1\ = S (XxX'xz) = XX'S (xh) = XX' (o-i + M?) Ni . . . . (iii.). 
Tui nine’ to the means : 
o 
M, = S (yz)/N, .(iv.). 
M' 
M 
o = S (Xa2/2)/N\ = [S {(x -M,)(y- M,} z} + M^MoS (z)]/MiNi = Mo + r (v.). 
jMi 
= S (XaX'^2/2)/N", = ^ 
al + Xll 
N,(ai + Mb 
— M j_ - I . Mf — erf S {(« — Mi)’^ (y — M,) z} 
— IVio -j- ? _o , n^2 “I” 
M, 
Ml af + 
Ni (erf + Mf) 
after some reductions. Nov/ make use of (ii.) and we have : 
M'b = 
S {(..: - M,y ^(// - I.L) - - Mo) 2] 
(1 + af/Mf) MfNi 
(vi.). 
But for normal cori elation the equation to the straiglit line of regression is ; 
7 / - Mo = r^(x - Ml). 
Hence for such correlation the mean value of y — Mo for parents x — Mi is equal 
to r — {x — Ml) and the summation term wmuld vanish. For skew correlation, 
O'! 
Mr. Yule has shown that the line just given is the line of closest fit to the curve of 
regression. Hence even in the case of fertility, vdiere the correlation is certainly 
skew, the summation term must he extremely small, or even zero. It follov's, there¬ 
fore, that we may write : 
/o / o 
(j-yai 
MY = M2 + r^(l + 
1 + (rf/Mf 
. . (viL). 
There is still another mean which ought to be found, namely, that of parenis, M''i, 
when all their recorded offspring have been entered on the correlation table. XX"e 
have ; 
M"i = S (Xa:XYx2)/N"i = S (a:-^«)/{Ni (af + Mf)}, 
or, after some reductions 
MY = 
+ Ml + 1 + ff/Mf 
1 
(viii.). 
