174 
On Expansions in Tetrachoric Functions 
If we put s = 1, s = 2, 5 = 3 in the above formula (xxvi) for Cg 
1 
1 /m- 
pKp + q)\ -a ;yj-' 
B(p,q) 
dx 
cr- 
= 0, 
o- \/2 ! i 0 
Cs = 
cr VG ■' 0 
3 
1 2 1 
- - 2x;p/(p + 5) + 2)7( /J + g)^} - 
1 
xP-'^ (1 - 
q) 
dx 
B(p + 2,g)-2^-^B(;, + l,5) 
+ 
pi 
Z \ p{p + l) 
a' lip + q)(p + q + l) " {p + qf ^ {p + qf (p+qf (p + q + 1)| 
= 0, 
as obtained before by the other method. 
The terms in Ti and To do not exist, so that the expansion becomes : 
(p + q) 
p- 
:^B(p, q)-a^B(p, q) 
pq 
X 
Bip, q) 
dx = ^{l + a)-a3T3-a,T,- ... -«,t,.- ... , 
where a, = 
Vs + 1 ' 
(s + l) 
Vs + 1 " V(s + 1) 
+ 
s - 1 ) (s - 2) - 3) - + ?) 
xP-^{l-x)i-^ 
2^2! V a J B(p, g) 
J_ r \{^-p/(p+q)y s{s-i) i x-p/{p + q) \'~' 
Vnio [V o- J 2.1! I a- y 
xP-' (1 - 
g(s-l) (g -2) (s- 3) / a;-j;/(jj + ry)Y-^ 
2-^2! V o- 
B ip,q) 
(xxvii), 
dx 
The argument for the tetrachoric functions and for |(1 + a) is ~ . If 
this is negative then we must take \{1 - a). 
From the above expression (xxvii) the coefficients of the expansion can be 
determined both algebraically and numerically, but for the higher coefficients the 
algebraic work becomes exceedingly heavy. It is to be remembered that 
2^ pq 
{p + qfip + q + iy 
