6 Fraternal and Parental Correlation Coe^cients 
We find 
^nq[nq + ^ + 2{q-l)r-}s' 
X^{y.y-2)f = hnq (9-1) (1 + 2 (g - 2) r + [\nq - 1) + g^- 3^+ 3] r''] 
2 (2/1-) S (y^) = 1 (9 - 1) [»r/ + 4 + 2 (9 - 2) r} rs' \ (11). 
(^(3/12/2))^ =ln(n-l)g^{l+(9-l)?f 
S(2/i=)^(3/,2/.) =0 and 2"(y^)^C^) - 0 
The calculation of a* may now be continued. We find, by substituting the 
above mean values in (5), 
n-g^o^ = \n-(f - 1 - 2 {i}q + 1) (9 - 1) r + (g - 1) [2nq -{q- 1)] r=}. 
Fi "om (3) is found 
v"q" {a-)- = s* [n-q- - ^»q +1-2 (nq -l){q- l)r + (q- If ?-^|, 
and accordingly 
_ 2s'' 
a\,. = cr^ - (cr"^y = ~ {uq - 1 _ 2 (9 - 1) r + (9 - 1) {iiq - g + l)r% 
or arranged according to powers of nq 
= 11^ + - - 4 " 
This formula for the s.D. of the squared standard deviations is thus exact^ 
supposing that the correlation be normal. 
??(/ 
For great values of » or rather of — , - , , - we may consider the S.D. of a 
" l+(9-l)r- 
differential, so that 
(7- = tr" + 8(t'' = cr" + 2o"8cr. 
From 8a- = 2aSa we find by squaring and taking mean value for a great number 
of samples, 
cr'v = 4o-'-o-o.'-, 
and by substituting the value of av, omitting the last term, 
or, as with the accurac}' obtainable we have 
,s" = cr-, 
it follows that : o-^- = 1 1 + (<? — 1) '"'l- 
zuq • '-^ 
We notice when comparmg this formula with (1) that only for ?• = 1 and r = 0 
does the rule 
hold good. 
