92 On the Probable Error of the Correlation Coefficient 
it a posteriori by means of experimental samples for given p and given n. Now 
the only type among the skew curves mentioned applicable in the present case 
is of the form : 
^n'-sJ <A 
where, if the origin be at the mode, we must have 
w 1 /a 1 = m 2 /a 2 (ii) 
Now if we suppose p to be positive, we clearly have 
Oj = 1 — r, a 2 = 1 + r. 
Hence from (i) and (ii) 
1 + 
.(iii), 
1 + f) 
r = (m 2 - m^j^m, + v^) (iv) 
Now let a r denote the standard-deviation of the distribution. Then we easily 
find (Phil. Trans., loc. cit. p. 368) 
Thus 
and 
It follows that 
A(J= 2^+1) _ 
+ m 2 + I 
a 2 = 4(m 1 + l)(w 2 + l) . 
r = (?/i 2 — rn^Kyri! + ??i 2 + 2) (vii), 
1 - r 2 = 4 + 1) (m 2 -f l)/(mj + m 2 + 2) 2 , 
o> 2 = (1 - f 2 )/(m 1 + m, + 3) (viii). 
o 1 - , 
+ m 2 + 3 = — = X, say, 
■ ii - hi , = r — „ r — 1 !- = r (X — 1 ). 
Accordingly 
m, = \(\-\)(\-r)-\ 
m, = i(X-l)(l + f)-l 
Substituting in (iv) we have 
and d = r-r = 2r/(X - 3) 
r = r (X -l)/(X-3) 
.(ix). 
..(x). 
