12 
PROFESSOR K. PEARSON ON MATHEMATICAL CONTRIBUTIONS 
But 8% — (8 b -f- 8 d) (8c -j- 8 d), 
U’z/j 0”ff 2 ^ nyn. z — O'/jO'c'l be "4~ &b&d^ bed I &d * cd “f" O',/ , 
cid — be 
= “IT - .(xxxvi.), 
therefore 
ad — be . . 
o-k^k^hk — .(xxxvn.). 
ad — be ... 
\/ (b H- d) (a + c)(c + d) (a + b) v ’ 
This is an important result; it expresses the correlation between errors in the 
position of the means of the two characters under consideration. But if the prob¬ 
abilities were independent there could be no such correlation. Thus r !lk might be 
taken as a measure of divergence from independent variation. We shall return to 
this point later. 
Since S?q = — HNS/q we have SrqSd — — HNSc/S/q whence we easily deduce 
r d>h = — r dh .(xxxix.). 
Similarly r d , h = — r dt .(xl.). 
Now d is a function of r, h, and k. Hence if d — f(r, h, k), 
Sd = % Sr + % Bh + % Sk 
dr dh dfc 
= y 0 Sr + y x 8/i + y 2 Sk .(xli.). 
Whence transposing, squaring, summing, and dividing by the total number of 
observations, we find 
y 0 2 err = cr/ + y x 2 c r/~ + yv cr* 2 — 2y i cr d cr h r dh — 2y*cr d ar k r dk 
+ 2y 1 y. 2 o- /i o- / ;.r M 
= ^ + (mr)” < + {ex ) 1 + 2 {w) 0V0 '"' r "- 
“1“ -I |£^r ) <J d o'.. ^ |- <r., .(xlii.). 
Substituting the values of the standard deviations and correlations as found above, 
we have 
°> 2 — | c K a + ^ + c ) + (hn ) ( a + &) + c ) + (^) ( 01 + c ) (d + &) 
“+* ( a d ~ ^ c ) + H§ + a ) +KN C ^ C + a ) } • • • ( xlm 0- 
