ON EXPANSIONS IN TETRACHORIC FUNCTIONS. 
By JAMES HENDERSON, M.A., B.So. 
(1) We define the tetrachoric function of order s to be t, (x), where 
''^''^ = ^Ti\~dx) 
Other writers have adopted various other vahxes for the external numerical 
factor but this is immaterial. The factor was chosen because it gives an ex- 
Vs ! 
tremely simple expression for the volume of a quadrant of the normal bivariate 
frequency surface, and because for tabulating the numerical values of the functions 
it is necessary to have some reduction factor of this kind to keep them of manage- 
able size. We can usually drop the argument x and speak of Tg. The values of t, 
for s = 1 up to s = 6 are tabled to five decimal places in the book, Tables for 
Statisticians and Biometricians*, for values of ^ (1 — a) (which is really To, when 
the argument is negative) from "000 to '500 at intervals of •001. With a different 
multiplier they have been tabled by Charlierf to four decimal places only for 
s = 1, 4 and 5 {x = "00 to 3). ' 
The general form of the tetrachoric function of order s is 
1 f , (6--l)(s-2) , (6--l)(s-2)(s-3)(s-4) . ^ ] 
Ts = ^ 1^-' - ^ 2711 * 2kY\ ^ " * ~ ''^^•J 
X 7^ e (ii), 
V27r 
that is, the ordinate of the normal curve of errors multiplied by a polynomial of 
degree (s — 1). Tj is simply the ordinate of the nonnal curve, while T(, is the area 
of the tail of the normal curve up to a given abscissa x, with the addition of an 
r.r 
arbitrary constant. This constant may be so selected that = I dx, and 
will be found from the tables of the probability integral. It will be equal to 
1(1 + a), if X is positive and ^(l — a), if x be negative in the usual notation. 
Accordingly the expansion of a function of x, f{x) in a series of tetrachoric 
functions, is really the expansion of the difference of the function and a multiple 
of the probability integral in terms of 
' X X- 
C„ + C'l - + C.2 — + 
k (T a- 
where a and Co, Cj, ... are at our choice. 
* Cambridge University Press, p. 1, and pp. 42 — 51. 
f Vorlesungen ilber die Grundzilge der tnathematischen Statistik, 1920. 
