334 Distribution of Correlation Coefficient in Small Samples 
and accordingly 
Now 7o = 2 f ' sin"+23'-3 f7<^ = 2g„+2^_2, say, 
and = 2 ( sin"-i0 c?^ 
0 
.(xvi), 
IS known to be = ) =-f^ ~ ^tt 
(n — 1) (w — 3) ... 2 
if w be odd as supposed above. Thus finally we have 
X2. = (1 - ^~P=-^ F (p, p,'^ + p, pA (xvii). 
n 
A Table of q,^ = 1 sin"~^<^f7^ from w = 1 to w = 105 is given on p. 377 below. 
■' 0 
Now (xvii) has only been proved for n odd. If n be even we must take the 
first series of (vii) and this gives 
n - 1 
(n + 1)2 (H - 1)2 1 3 
P — T • , , V + 
4 ! ^ n + 2^ - 1 ■ n + 2 4- 1 
where V = 2 sin^+^P-^ c/,dcf, = 2q.„+.^ 
In +2P-2 
0 
(u-2) (M-4)...2 ... 
^'?" = (^iT(n^r3y7:3-^ 
since « is even. Thus 
X.. = (1 - p2)'^^i^ ( ^ + 
= (1 _ ?«±2.32^ /^^ ^ + p^] (xix), 
or (xvii) holds whether n be even or odd. 
As particular cases we have for p = 1 
P-2 = + CT." = 1 - X2 
22 . 42 . 62 p6 
^ 1 . 2 . 3 . (w +'l) (n + 3) (w + 5) 8 
.(XX), 
