518 Distributio?i of the Correlation Coefficients of Samples 
now becomes 
0^ dt 
and the range of the curve is extended from — oo to + oo . 
It is interesting that in the important case, r = 0, the frequency reduces to 
dt 
and the curves are identical with those found by " Student " for z, 
the probability integral of which he has tabulated in his first paper, 
11. Tlie moments of these curves are obtained by the evaluation of the 
expressions 
p / d Y^-' 0- dt p / d Y'-' tdt 
l_A^m 2"^~^^.^yi^>' I^Asinedd) 2^Y7^" 
and so on; of these the first is known already (HI) to have the value 
TT \n - 3 
(1-P-) - 
and the others may be obtained in succession, for 
3"-! 6"' tPdt 8"-^ r°° 1 (9^ tPdt 
so that the first moment 
a Y'~' ^f^* 9 7rb-4 TT |n-4(n-2)/9 
, - n — 'i p n — 2 
hence t = ^ , = j: t. 
/t — o V 1 — p'^ n — o 
The mean, therefore, is greater than the true value r by a constant fraction 
of its value. And this fraction decreases in the simplest possible manner as n 
increases. 
In the same way, we may evaluate the second moment, 
and = , J 1 -f + ) T' 
the third moment 
(n-3)(n-4)(7i-5) r (?i-3)- } 
and the fourth moment 
