158 On Expansions in Tetrachoric Functions 
The real reason for adopting 
Co't,, + c/ti + C./to + c/to + . . . , 
instead of the above expression, is that the calculation of the constants Cq', c,', c.^ ... 
is more direct than that of Co, Cj, c.... because the tetrachoric functions are semi- 
orthogonal functions*. It will be seen that the problem of expansion in tetra- 
choric functions is closely related to a theorem of Laplace. If U he a. unimodal 
function of x within the range under discussion and the integral I = judx be 
required, Laplace transfers to the mode 7n as origin so that x = m, + ^ and writes U 
in the following form : 
U = U„, e - ^"'^^ ( 1 + p + a, t + ...). 
He extends the limits to oo in both directions by supposing U—0 outside the given 
range and in the integration applies the well-known values of I c'^^-'^'d^, i.e. 
zero if s be odd, and again if s be even (= 2?-), 
[ e-i^>^prf^=:(2r-l)(2r-3)...3.1.\/2^a-'-+i. 
J — CO , 
It will be seen that Laplace is really proceeding by expansion in tetrachoric 
functions as the process is precisely the same whatever be the limits of the integral 
of U. Following Laplace we develop our function in " incomplete normal moment 
functions," i.e. I — dx-f; it is better to use tetrachoric functions. The series in 
-'-00 v27r 
tetrachoric functions seems to converge slightly better than that in incomplete 
normal moment functions. 
If we have 
F{x) = a„Ti + (tiTo + aoT;j . . . 4- a^-iTg + 
then, assuming we may integrate the right-hand side of this equation term by 
term (i.e. assuming uniform convergence) between x and oo , 
smce 
jF(x)dx=a,n + -^- + -^ + ..., 
T 
T^dx:^^ (iii). 
Jx Vs 
* A series of functions /i (a;) , /2 («) .../s (x) ...f^,(x) is orthogonal if J/^ (.r) /,/ (.r) rf.r = 0 when s and 
s' are not equal, the integration being throughout the range. They are semi -orthogonal if 
/ 
(p (x) being a function of x peculiar to the series. In other words a system is orthogonal if the sums of 
the products of different order functions vanish without weighting for x. A system is semi-orthogonal 
if we require to weight the values of x to obtain the vanishing of the product sum. This weighting is 
the great disadvantage of semi-orthogonal functions. In our case of the tetrachoric functions the weighting 
factor is f^'"" or the tails of series are excessively weighted. 
t Discussed Biometrika, Vol. vi. p. 59. Tables of these functions up to s = 10 are given in Tables 
for Statisticia7is, pp. 22 — 3. 
