162 
On Expansions In TetraGliorlc Functions 
concerned. It may be expanded with regard : (i) to the mean and the standard 
deviation, or (ii) to the mode in the manner of Laplace*. 
(i) The mean of the function (viii) is easily found to be at x = p, the mode is 
Sit X = p — 1 and the standard deviation is "J p. 
Referring to the mean as origin the function becomes 
y= — v(^) — 
Let y = (f) (- D) - , where D = -j- and z = (x). 
V27r dz 
Except for a numerical factor the right-hand side is a series of tetrachoric 
functions. 
Let (j)(-D) = c,-c,D + c.,D' ...{-iyc,D'+ .... 
The function 0 (— D) has to be determined, i.e. we require to find the succes- 
sive c's: 
= c„T,4-Ci\/2! T.,4-c,,\/3!T,-t-... + c,_iVs! Ts + ... (xi). 
To determine the c's. With the origin at the mean the function y must be 
taken as zero from — oo to — p, while from — p to + cc it is given by (ix). The 
c's will be obtained most easily by multiplying both sides of (x) by e*^ and equating 
the coefficients of powers of 6 on both sides of the equation, i.e. we make all the 
moments of the two expressions for the curve the same, for the coefficient of 6^ on 
either side is the sth moment f. Thus 
J —CO J - cc V ZTT 
but ^ = 0 from x = — <x> to - p. 
Accordingly /__^ ^^-^ = m ^= dl. 
. Now x = p i- ^ and z = ^j*^ p. 
r'^ pe{x-p) ^,p-\o-x Q-hzi 
Thus - — , dx = \/« e^'^" 0 (- D) -= dz. 
The left-hand side is equal to 
e-^* ' - p^- — dx 
, f " uP-' e-" du T i. /-. n^ 
= e-''^(l - d)-P. 
* Laplace's method is really an expansion in incomplete normal moment functions but as we have 
seen (p. 158) these may be replaced by tetrachoric functions. 
f We owe this elegant method of determining the c's to Mr H. E. Soper. Originally the c's were 
determined by use of the fundamental property of the tetrachoric functions but that method, while 
leading to the same result, is more laborious. 
