106 On the Probable Error of the Correlation Coefficient 
And, again, by squaring (xxxiv) and expanding we obtain in the same way 
f 1 -r 2fdr + (dry 
= (p 2 + 2 P1 + r - 2p«> A - 2«i/8i7 + «i'/8i») 
x (1 - a, + af + a 2 2 - 2a 1 2 a 2 - a.? + a/ + Sa^aS + a 2 4 ) 
x(l-/3 2 + /3 1 2 + /3 2 2 - 2/3, 2 /3 2 - & 3 + ft< + 3& 2 /3 2 2 + & 4 ) 
= P 2 + {2/37 - P' 2 « 2 - P 2 &} 
+ { 7 2 - 2p (a, ft + « 37 + /3., 7 ) + p 2 (ar + ft 2 + *., 2 + /3 2 2 + a 2 /3 2 )} 
+ }- 2a,/S l7 - a 2 rf - /3., 7 2 + 2p (a^fa + Ojftft + 0^7 + & 2 7 + «2 2 7 + &*7 + «*&7) 
+ p 2 (- 2«*a 2 - 2/3r/3 2 - a* - /3 2 3 - a^ft - a 2 ft 2 - « 2 2 & - a 2 /8 2 2 )} 
+ KW + 2a 1 a 2 /3 l7 + 2a 1 /3 1 /3 2 7 + a^y* + &V + «iV + &V + « 2 0 2 r 
+ 2p (- a, 3 /?! - a,& 3 - ai« 2 2 /3i - etjftft 2 - ^fra.fi. 
- 2a 1 2 « 27 - 2/3^/3.37 - « 2 8 7 - /3 2 3 7 - «r/3 27 - a.&'y - a 2 2 /3 27 - a 2 /3 2 2 7 ) 
+ p 2 (e x 4 + ft 4 + S^'-a;- + 3&«& a + a 2 4 + /3./ + 2a 1 2 a,/3 2 + 2a 2/ (3 I 2 /3 2 
+ a//3 3 + a,/3 2 3 + a/ft 1 + a 2 2 /3 2 2 + a^ft 1 + a 2 2 /3 a 2 )) (xlii). 
And taking mean values in samples of n of a normal distribution, 
r 2 + mean (cZr) 2 
= P 2 + ^(l+P 2 -2p (/> + 4p) + p 2 (2 + 4 + 2p 2 )l 
+ - {- 2 - 2p 2 - 4 - 12p 2 + 2p (4/3 + 4p + 16/o + 4p + 4p 3 ) 
+ p 2 (-8-16-4p 2 -16p 2 )} 
+ 1 {1 + 2p 2 + 8p- + 2 + 2p 2 + 4 + 20p 2 + 10p 2 + 2p 4 
+ 2p (- 6/3 - 4p - 2p 3 - 8 P - 24p - 4p - 8p - 16p 3 ) 
+ p 2 (6 + 12 + 24 + 8p 2 + 24p 2 + 1 + 2p 2 + 4 + 8p 4 + 4)} 
= /° 2 + ^{l-3/> 2 + 2p*} 
+ ^ 2 {-6 + 18p 2 -12p 4 } 
+ ^{7-15p 2 +8p«} 
= p 2 + I (1 - p 2 ) (1 - 2p 2 ) + 1 (1 - p 2 ) (1 + 4p 2 - 8p 4 ). 
And by squaring (xxxix), 
P = * _L p 2 (l_p 2 )(l + 5p 2 ). 
