James Henderson 
159 
Let Ts- - -7= e~**', where is the polynomial in x of degree (s — 1) 
V6'I v27r 
in (ii). 
Let Tg- be another tetrachoric function and suppose s' is greater than s. Then 
r - 2 7 1 1 1 r / dY'-'e-i"^' 
J = ^7^] VT"! Vl^ i-. dj WJ""- 
Now since Ts(x))and Ts (— oo ) will always be zero owing to the exponential 
factor (s > 0) we can integrate by parts transferring the ^ from the exponential 
to the polynomial, therefore 
-J■r2^ 
77^ - » 
The integrated part at every step vanishes at the limits and ultimately 
1 If" d/- 
— A.l- 
Vs ! V*' ! V27r i -* dx"'-'^^' V27r '' " 
Since p^-i is a polynomial of degree (« — 1) and s is > s the differential of the 
polynomial vanishes, i.e. 
f oo 
r^T^'e^'^'' dx = 0, s^ s' (iv). 
J — oo 
If s = s then the differential of reduces to — 1 ) ! so that 
r . ' 7 1 1)! r e-^-^' 
J -00 \/27r s ! J - ^ V27r 
= 7^^ ^•>0 (V). 
These equations (iv) and (v), which give the fundamental properties of the 
tetrachoric functions, enable us to expand any function F{x) in terms of tetra- 
choric functions if we can find the value of the integral 
, „ 1 /"^ 1 
F {x) Tg e^-'^'' dx = ,-=^ -j= jJs-i F{x)dx ( vi ). 
J -co v27ri Vs ! 
Since ps-i is an integral function of x, this amounts to saying that we can 
expand any function of which we are able to determine the successive moment- 
coefficients. 
The practical value of the functional expansion when obtained is, however, 
a very different matter. That depends on the convergency of the series and our 
experience has shown us that in the most common cases the convergency is so 
slight or non-existent as to render the expansion idle. 
