172 
On Expansions in Tetrachoric Functions 
The 6's, having been calculated previously by (xxiii), this last formula gives a 
fairly rapid way of calculating the c's, at least the earlier c's. Then 
„ _ 1 I _J p{p+\) ...{p + s-r-l) 
<j\Zq '' {s-r)\{p + q){p-\-q + l) ...{p + q + s-r-\) 
(ao = 1) ...(xxiv). 
What we require generally is the area represented by I —x)i~'^ dx 
<^{-D) 
V'27ro- 
I.e. 
y p-hi- 
<f>(-D)'~dy, 
dx 
0 B(p, 5) 
-7= - ail* 
V27r v27r 
^^-rr \/27r ' ^ 
-00 V 27r 
.\/27r. 
D 
V2 
TT 
= ^(1 +a)-rt,\/l i Ti-a,\/2! To - a, Vs ! T3 
= + a) — a/xi — a2'''"2 — «:/t3 — ... — a^T^ — 
- . . . + a, 
. — ttj Vs ! T, — 
V27r_ 
(«o=l) 
where 
a/ = a^. ! • 
pi:P + l)...{p-Vs-r-l) 
Then a/ = S 6^ , — ^— — ; 7- w t x / i x 
. ..(xxv). 
Now Ci and Co are equal to zero, so that a/, a.,' are zero. Thus there are no 
terms in Tj and To. The argument of the tetrachoric functions and of J(l + a) is y, 
which is equal to - = ^~ — pK^P '^ 9 ) _ applying the above formula for a/, we were 
greatly disappointed to find, that with the h's to 8 decimal places the expression 
under the summation sign in the examples used commenced with 4 or 5 zeros 
after the decimal point. As Vs! and (-^ both increase with s f~ being in our 
case > 1 j accuracy to the seventh place in our a"s could not be obtained. Accord- 
ingly the formula i^ctually used was of a different type. 
Let 
= S C,T,, 
where the argument of the tetrachoric function is again - = — —-^ — — . 
Multiply both sides by t,, weighting by the factor e^^''"^', and integrate from 
— 00 to + 00 , the left-hand side being taken as zero outside x=0 and x=l. 
Th 
en 
~plp+q B {p, q) 
T, e^^-'"^- d^ ■■ 
-00 1 
