514 Distribution of the Correlation Coe^cients of Samples 
is essentially a scries of Legendre functions of the first kind; and may be 
expressed as 
- . tan?' a '-^^ Pp_i (i tan a) ; 
and it is these only wliich occur in the evaluation of the even moments. 
7. It is, however, desirable to obtain general expressions for these integrals 
in terras of 7; and p, and to evaluate them when n is odd. 
For this purpose let us introduce a quantity <f), such that 
cos (f) = cos 0 — k, 
then, when k is sufficiently small, we may expand 0^ by Taylor's theorem, so that 
2 2 + sin ^96' 2 ^|_2 Vsin 6'a6'y 2 
Now let k = p/i Vl — r•^ 
dr , . ^ 8 0' pH>? (1 - / 9 y0\ 
then = ~ + pk\/l — r- ~. — 7^-^ + i^, — o^q] 17 + ---> 
2 2'^ sin 0d0 2 |2 Vsm 6d0) 2 
and differentiating twice with respect to h 
whence, dividing by (1 — r")-, we obtain 
^iZr^Asin (f>d4>J 2 ^ r^^i\sm 0d0/ 2 ^ {sin 030/ 2 
Vsin6'9^/ 2 
r+i 
may be obtained by multiplying by \n — S the coefficient of in 
+1 rP(J,f. 1 - ^ cot 0 
n - 4 
so that I 2" 
p2 , 
' _i Vl _ ^2 sin^^ 
when cos cj) = cos ^ — p/i Vl — = — p (?• + /i Vl — r^). 
Our object might equally be achieved by the evaluation of the integral 
•+i ^,p(-lr / (j) 
-1 1 — r^Vsinc/) sin 0j ' 
Tiie quantity (f) is determined by the equation 
cos cf) = cos 0 - p/i Vl — r^, 
that is cos <^ = — p(r + /iVl — r^). 
