336 Distribution of Correlation Coefficient in S^nall Samples 
Assuming n odd we must keep the first series in the vahie of -, 7 , and we 
reach 
n-\ 
1^1 + 2 1 , ■nH:n + 2)2 1 3 ^ \ 
^ 2\P n+l + 2p^ 4!' w+ 1 + 2^ ^ + 3 + 2j9^ +^^^-"7 
n -1 
2 (ra - 3) ! (w - 2 + 2^) 
if we use Euler's reduction formula, and note that for n odd, or n — 1 even 
(n-3) (n-5)...2 . 
(w- 2) (n- 4) ... 1 
If we start with n even we reach an absohxtely identical formula by a different 
route. Thus we have 
a' -va' -P-iP^-^ u' +P^P -JllP-^ ^' _ 
A* 2P+1 ~ FH' 2J)-1 2 t ^ 23i-3 g j A* 2P-5 
(xxiv). 
Taking p in succession equal to zero and to unity we find 
V ' w + 1 2 ' 1 . 2 . (n + 1) (n + 3) 4 
1^3^ 52 p« , 
^ 3! («+ 1) (w+ 3) (^^ + 5) 8 •"^ 
^^3' = 1^3 + 3/x,'r - 2r3 = r - p (1 - ^-^2 ^ 
/ 32 p2 32 52^ ^4 32 . 52 . 72 
'^w + 3¥"^ 1 . 2 (n + 3Hn +^5] 4 + ST (n + 3) + 5) (n +~7) 8 
.(xxvi). 
Equations (xxv) and (xxvi) provide the values of the odd moment coefficients 
about zero and this in fairly rapidly converging series. From them we can deduce 
the value about the mean p.^ and thus find the fundamental Table X, p. 377, 
again gives the requisite values of q„ for the range w = 1 to 105. 
