H. E. Soper 
105 
We are now in a position to expand (xxxiv) to terms of the fourth degree in 
a, (3 and 7, take mean values of the a, ft, y products as found for samples of normal 
distributions and so obtain f the mean value of the correlation coefficient in 
samples of n of such distributions correct to terms in 1/n 2 . 
r + dr 
= (p + y- 
x (1 - £« 3 + ha* + §a 2 2 - far « 2 - fV« 2 3 + |«i 4 + |f «!* a 2 2 + f^« 2 4 ) 
x (1 - + i/3r + - f m~ + f A 4 + HAW + fVk& 4 ) 
= p-i{p(a 2 + /3 2 )- 2 7 } 
+ l |4p (a, 2 + ft 5 ) + 3/) (a, 2 + A' 2 ) + 2p^ 2 ft - 4 («. 2T + /3 27 ) - 8a, ft) 
+ T V {- 12p (a^ + - hp (a 2 3 + & 3 ) - 4p («i s & + a 2 ft 2 ) - 3/3 (a 2 2 & + e* 2 /3 2 2 ) 
+ 8 (o,» 7 + & 2 7 ) + 6 (a 2 2 7 + /3 2 2 7 ) + 4a 2 /3 27 + 8 (a a a 2 ft + 
+ ^ {48p (a/ + ft*) + 120p (a^a, 2 + ft 2 /3 2 2 ) + 35p («./ + £./) 
+ 48p + a 2 /3^) + 20p (a 2 3 /3 2 + a 2 /3 2 3 ) + 32pa 1 2 /3 1 2 + 24p (a^ft 2 + a 2 2 /3f) 
+ 18p* 2 2 /3 2 2 
- 96 {a*a. 2 y + fc&y) -40 (a 2 3 7 + fty)- 32 (a 1 2 /3 27 + a^y)- 24 (« 3 2 /3 a7 + « 2 /3 2 2 7 ) 
- 64 (a^ft + a,^ 3 ) - 48 (a^A + ai&/3 2 2 ) - 32a 1 /3 1 a 2 /3 2 } (xxxviii). 
Whence taking mean values in samples of n of a normal distribution, 
f = p + -1 {8p + 12/3 + 4p 3 - 16p - 8p} 
+ T^ {_ *8p-80p-16p 3 -48p 3 
+ 32p + 96p + 16p + 16p 3 + 32p} 
+ m~n? {288p + 480/5 + 840p 
+ 192p 3 + 480p 3 + 32p + 64p 3 + 96p + 72p + 144p 5 
- 384p - 960p - 128p - 192p - 384p 3 
- 384p - 192p - 64p 3 } 
= P ~ L P (1 " P ' 2) ~ A P (1 ~ P' ] (1 + 3p2) • .(xxxix). 
811 
Or, expressing the result in terms of n — 1 (by changing n into ri + 1 and 
expanding) we may w r rite to the same degree of approximation 
r = p 
1 
2(n-l) 
from which follows that 
1 -r 2 = (l _ p 3) 
Biometrika ix 
1 + 
M — 1 
4(n-l)J 
1 - 1 ^P-\ 
2(n-l)J 
.(xl), 
.(xli). 
14 
