330 Distribution of Correlation Coefficient in Small Samples 
Apply Leibnitz's Theorem and we have 
d^U ^^d^-W ^{n-l){n-2) d"-^U d^U ^^ d^'W 
^> dx^ ^> dx-^ ^ ' 1.2 ' dx-^ " dx^^ + ^'^ > dx-^ ' 
or, (1 - x^) '^-x {2n - 1) - {n - If^^ = 0 (v). 
Put a; = 0 and we have 
(d"U\ . ^.^fd-^U\ 
[d^)r ^'"''^[d-^l' 
but clearly Uq = ^tt and {dU/dx)^ =1. 
Hence by Maclaurin's Theorem 
cos-M-a^) 7T/ 1^ . , 3^P ( 2g-l)M2^-3)^ ...l 
Vl^ 2l' + 2!''+ 4! (2s)! 
/ 22 ^ 42.22 , (2s)2(2s-2)2... 22 \ , 
+ + 3! ^ + + - + - (2.+ 1)! ^ " + - )-(^^)- 
We are now in a position to give the successive differentials of U which may be 
either even or odd. We have for the two cases 
(Z2s jcos~^ (— x) 
^IVr^'2" 
= I i2s - 1)2 (2. - 3)2 ... 12 {1 + + (2^+1)2(2^2 
+ (2.)2 (2. - 2)2 ... 22 {. + (^^%3 + (2^(21+^%. + ^ 
(^2s-l fcOS-l (- X) 
dx^'-'^ [ Vl 
vn 
ibis 
+ (2. - 2)- (-2., ^ir...2'[l + <f + (l»)!(|i±^' ^. ...| 
the development of the several series being clear. 
For calculation of or ?/2s-i the above series are idle, just as they are when 
substituted in the equation for dyjdr = 0 which gives r. They converge far too 
slowly to be of use for numerical evaluations. But as we shortly shall show, they 
are, after certain transformations, most valuable in determining the moment 
coefficients. 
71-1 
AT (I-P')^n 2^^^"-'^^ 
Now w„ = or, (1 — ^2) - 
n+1 « - 2 / 
multiply (v) by (1 - p^f^ (1 - r^)^ / (77 {n - 1) !), 
