332 Distribution of Correlation Coefficient in Small Samples 
Hence = 0 and while ?/2 'will vanish, for all values of r, except r = ± 1 
owing to the factor {n — 2)„=2- Thus (ix) gives us 
^3 = 1, t<3 = pr, 
whence by (xi) 
= 3pr, = 1 + 2p^r^, 
= 4 + llp2,-2, M5 = pr (9 + QpH^), 
v^ = pr {5b + bOph-^), = 9 + 12p^r^ + 24:p^r\ 
and the successive values can be rapidly calculated, much faster than by actually 
differentiating out (iv). It is, however, shortest to insert the numerical values 
of p and r in (xi) and deduce the v„'s and w„'s numerically in succession. (Table A 
was, however, in the present case deduced from (viii). We did this by direct 
calculation of the values of 
Vl — p^ cos"^ (— pr) 
and ys in equation (ix). Equation (viii) then gave us the numerical values of 
l/i) 1/5 > 6tc. in succession.) 
We may write (x) in the form 
yn=^^-=P'VA^^n + ynU) (xii). 
77 
n-i 
where 
1(1 - p2) (1 _ ,2)} 2 C0S^^(-£) 
" (n - 3) ! (1 - pV2)«-2 ^''"^ Vl - 
Here F„, v„ , u„ and U are symmetrical in p and r and accordingly p and r can be 
interchanged. The problem approached this way involves : 
(a) calculating (1 — /3^)^/77 for various values of p ; 
(b) U for various values of pr; 
(c) Vn for various values of p, r and n ; 
{d) determining u„ and in succession from (xi) for various values of pr 
and n. 
Lastly we may use the series for y„ to be given later (see Eqn. (xliii)) which 
develops yn in inverse powers of {n — 1). Actually we have adopted (viii) for 
tabling the ordinates of the first 25 curve-series, and the last expansion for 
verification and higher cases. 
(3) On the Determination of the Moment Coefficients. We shall next determine 
the value of the moment coefficients about r = 0, as origin, and shall deal with 
the even and odd coefficients independently. Let them be /x'jp and jU.'2j)+i- 
Clearly, the total area having been taken as unity ; 
