340 Distribution of Correlation Coefficient in Small Samples 
This result expresses the mean for samples of n + 2 in terms of the mean for 
samples of n and the third moment of samples oi n. 
Next let 
Now integrate „ by parts twice : 
P J -1 d {pry-^ 
« - 8 + 2;j 
w — 4 -4- '^ ■n r+i 
—-^l {(?i-6 + 2j9)(l-r2) 2 
P J -1 
M - 6 + 2p 
- (». - 6 + 2j9 + 1) (1 - r2)} ' X i-^, dr 
= ~ y {{n-(y + 2p) - (/^ - 6 + 2p + !)/,,„..,}. 
Or returning to the X2p, n notation 
1 — n — 4 + 2p 
X2P,n = -pT^ (^^ _ 3) {(^ - 6 + 2p)X2j,-2,n-2 " - 6 + 2^^ + l)X2^,«-2}- 
As special cases put f = 1 and 2, and change w to w + 2. We have* 
X2,n+2 — ^2 • ,1 _ 1 • jXo.ra _ 2X2,n| ^XXIX;, 
_ 1 - p2 n + 2 ( n TO + 1 ) , 
But Xo,n = 1 and Xi,n = 1 - Ai'2,«» 
-I 9 ' I / 
X4, n — ^ -^/^ 2, « T" A*- 4,n- 
Accordingly 
_ 1 / , 1 \ 
^ 2,ri+2 — ^ ^2 n — 2\^''^ n — IJ ixxxi;, 
which can be verified directly from (xx) or (xx)^'® Again instead of working 
with the series for Xi.n+2. above (xxx), we can replace it by one involving the 
moments about r = 0, directly : 
'^4-™+2-^ p2 l)(n-2)^«'" + (;^l)(n-2)'^-'" n-l 
(xxxii). 
* Tlie process of integrating by parts shows that we must have n > 2. 
