168 On Expansions in Tetrachoric Functions 
To find the area up to abscissa x we have 
r (i''±i)L!^'"^f/^ = _L[^ [c,T, + V2!c,T,+ ... + \/(7Tiy!c,T,+,+ ...}cZ^ 
; _y I {p + i) ^/p' J -00 
= J" {c,T, +\/2]ciTo+...+V(s+l)!e,T,+i + ...](^^ 
= I (1+ a,) -CiTi - V2 ! c.,T2 - Va ! C3 T;j - ... - Vs ! c,t,- ... as c„ = 1, 
i.e. / - , -, .dx = l{l +a,)- a/xi - a/r^ - cl/t; - ... - a/x^ - . . . , 
Jo T{p +1) 
where a/ = ! . 
Substituting in (xviii) to obtain the difference equation for the a"s we have 
Vs ! V/ (V(s - 1) ! s V(s _ 3) ij ' 
therefore a,' = ^ ^ {sa',_, 4- V(s - 1) (s - 2) a',.,] (xix), 
and a.o = 1 , a/ 
V2 
(to 
By this formula the a's are i-eadily obtained numerically. It is to be noted 
that in this case the terms in Tj and T2 do imt vanish, as they did in the expansion 
from the mean. The argument of J (1 + 0(2) and of the teti'achoric functions is 7^, 
VjO' 
and the remarks with regard to sign made above must be again observed. 
Coefficients in the expansion from the mode : 
1 , \/2"i 
a-o = 1, ch = — , (u = — ^ , 
V/ P 
' "pV?? ("3"}' ' V' \^^' r I " 60 
, V6l(p'- . 19 , , V7~! [5 , 153 
, VsTfSM 341 , ^ , V9"! fp'^' 493 „ 3349 , 
=7^il440?^'" + 280?^ +^}' |l62 + 1440^^ + 2520^ 
We note that the coefficients of powers of 6 in the functions 
ff! e-^ 
' o 4 
and (f)'(d) = e 
(in the expansion from the mode we had p for |) in cj)' (0)) are closely related. 
