James Henderson 
165 
Therefore finally 
I „ r (p) f^*' = 1( 1 + - UsTs - (xiv), 
(since Cj = Co = 0) where «s = V^j c, Vs ! . 
Now Cs+i = ^ ^—r {sc,+ from equation (xiii), 
\/ p s + L 
i.e. "'^+' = 2, ^fz^ j 
^(s 4- 1 ) (s + 1) I ! V(s - 2) U' 
therefore = ^ ^^-L ^^^^ ^ ^ ^ ^^^^ ^ >,/ ^sTTH.^jJi^) 
= \/ pT^TT) "^^'""^^ "^-^5 
where (/„ = ] , = a.o = 0. 
The argument of i (1 + a) and of the tetrachoric functions is which equals 
w — p 
— T_- = z, say. 
Since the terms Tj and To do not appear one might hope that only a few terms 
of the expansion (xiv) would be required to obtain a sufficiently accurate result. 
^ (1 + a^) is the ordinary probability integral at z. 
Note that if x is less than /), i.e. z is negative, |(1 - a,) must be used instead 
of 1 (1 + «z) and the tetrachoric functions of even order must be taken of opposite 
sign to those for positive z such as are given in the tables. The odd order functions 
are the same for positive and negative z : 
(2) = - T.,, ( - Z), T2,,+i (Z) = T.^,_^ ,{-z). 
Obviously we could get the area of any portion of the curve between x = a;,and 
X = Xo by subtracting two expressions like (xiv) for 2, and z.,. 
The general expression for , ^ is 
^ r(^) ^ 
^p-1q-x ^ 1 1 \/4! 1 V5I 
1 Ve"! 1 VtT /j9 + 3\ 
lV8l(7jo + 121 ]_ V9! j47p + 60) 
"^p^ 7 1^ 12 r^+^3Vp 8 I 60 
iVlOljy 19 1 ^ 1 Vn!f5 „ 15.3 , 
+ ^ tl8 + 20^^ + V- +^ ^ 1 36 + r40 + 1 ^ 
1 Vi2T f 341 „ 341 I 
+ 11440^' + 280^ + ^r'-^ 
_^ 1 \/l3l f , 493 „ 3349 J 
"12- il62 + 1440 ^" + 2520 ^ + n ^'^ + 
