H. E. SoPKR, A. W. YouNCx, B. M. Cave, A. Lee, K. Pearson 36.5 
Putting w = 3 and writing /j = — \ log^ (1 — p^), we have 
1 + + (3 + p2) log^ (1 _ p2)} (ixxxii). 
1 - 
This completes the moment coefficients for samples of three. 
As p may be determined by considering the ratio of negative to positive cor- 
relations in samples of two, so it may be determined by considering the ratio of 
positive to negative correlations in samples of three. Let nip be the number of 
positive and the number of negative correlations, then since 
^,l-p^ I IdU 
l-p^ d j+i Udr 
TT dp I 0 f Vl — 
1 — p'^ d [+1 cos~'^ {— pr) dr 
and m„ = 
7r dp} jV(1 -r2)(l -r2p2) 
\ — p'^ d f" cos~^ (— pr) dr 
Now put in the latter integral r = — r, then 
1 — p2 1"+^ — cos~^ (pr) dr 
m„ 
TT dp] Q f ^(1 - r2) (1 _ y2p2j ■ 
But cos~i (— pr) + cos~^ (pr) = tt, hence 
+ ^ dp rV(l -r^)(l-r^p^) 
'1 r^/jtZr 
orifr = sin^, = (1 — p^) 
= -(l-p2) 
Or again if p cos 0 = Vl — p^ tan ^, 
oVl - r2(l - rV)- ' 
TT 
3 psmcjfdcf) 
0 (1 -p2sin2(/,)3 
TT 
(Z (p cos ^) 
0 (1 -p2 + p2cOs2</.)^ 
»»j, + m„ '^Vlo (l-p2)^ ^^^^^ 
= I COS = p, 
0 
