304 Distribution of Correlation Coefjicient in Small Samples 
p (1 - p2)7r 
(3.5...2g + 3)^ (6- + l)(g + 2) 
2 2 f [2.22^(5 + 2)! (s + 2)! 2s + 3 ^_ 
(1 -/)2)7r 
s 
(3 . 5 ... 2s + 3)2 s + 1 p2.+4- 
2/32 2 dp \ [2^' (s + 2) ! (s + 2) ! 2s + 3 22 
1- p2 
P 
2 (Zp 12 
5 
(3 . 5 ... 2s + 3)2 
2*.22»(s + 2)!{s + 2)! 2s + 3) ^ 
1 — p2 (Z (tT „ /I 1 „\ TT „ / 1 1 , 
-7^-5,12^(2' 2' n^2^irr 2' 
2s+4 
1 -p2 (Z 
p2 (Zp 
{F, (p) + A\ (p)} 
p(l-p2) + p 
or 
1 13 ^1 
.(Ixxix), 
where and are as before the complete elUptic integrals. 
In order to obtain the fourth moment coefficient about zero we will return to 
formulae (xx) and (xxi) of pp. 334-5 and write 
, , n-2 
p.2 
n — 1 
n {n — 2) 
^^nd — + 1) _ ]) 
where /a and/4 t.he hypergeometrical series. 
Now the general term oif^ is 
(2 . 4 . 6 . 8 ... 2s + 2)V'-^+' 
{s+l)\(n+ l)(n + 3) ... {n + 2s+ 1)2'+^' 
and the general term of/^ is 
(4.6.8...2s + 2)2p2» 
(s) ! (n + 3) (w + 5) ... (J^ + 2s + 1) 2« 
Hence it follows that '~ — ~ -V- = 
or we have 
w + 1 df, ^ 
4p fZp -^4' 
2 {n- 2) , ■'^(«-2)^_ ,)2#2 
n — 
.(Ixxx). 
Thus if we are able to sum/2 algebraically, we can determine fx^ algebraically* 
* The corresponding formula for ^^3' .and /ij' = r is 
(Ixxxi). 
1 
M3'=P-(i-r)(---2)|^(: 
If we put 11= 'i and f = - (E-^ - (1 - p-) F^) wc find 
, 2 - 2(1 - p"-) 
^- --p^ ^ 
contirniing the result in (Iwix). 
