362 Distribution of Correlation Coefficient in Small Samples 
Now let X = p sill then 
■z 
„ . , dV dU 
JN ow sm 0 = , 
dx dp 
hence jUj,' = - — - | " sin^-^ </> rf^ 
But C/ 
0 cosh z — p sincf)' 
thus u„' = ^ ^ f- [ dz ( ^ r-'— — 7 d6.... 
n dp } J _w cosh z — p sin </) 
2 
Write ifj = ~\- (f) and cosh z = r}, 
, , l-p2fZ,-", j-( - l)f-icos2'-ii/.# , 
we have u„ = - f/2 ^ ^ (Ixxv). 
7T dp ' 0 .'0 V ''f P r 
Let p = 1, then we find 
7^ dp.K Jo + + (r7-p)tan2ii/f' 
But 
fZ (tan ^i/r) 
tan" 
^ / tan 10 \ ] " _ 77 1 
Thus 
o(^+p)+ (^-p)tan2l!/r Vri^-p' ( /r] + p]\ Vrj^ - p^' 
L VV ^-p^ ■ 
d ["^ dz 
0 
Now take 77 = . ^ ,, = cosh 2, 
' sin 0 
hence — ^-?^J^, dd)' = sinh zdz, 
sm^ 0 ■ 
= Vri^~~ldz, 
thus dz = — cosec (p'dcf)'. 
It follows that 
