160 
On Expansions in Tetrachoric Functions 
The matter is a very important one for Thiele*, Edgeworthf and Charlier| 
have proposed to treat skew frequency distributions by a process, which amounts 
to the same thing as the expansion by tetrachoric functions. 
An attempt made many years ago§ to expand Incomplete V- and B-functions 
by Laplace's method in Incomplete Moment Functions convinced Professor Pearson 
that little was to be gained by a series expansion in the form of a polynomial 
multiplied by the ordinate of a normal curve. A variant of this method, that of 
expressing Incomplete F- and B-functions in a series of tetrachoric functions, was 
tried a year ago and it was found that except for a small distance round the mode 
this method of expressing a frequency distribution was quite ineffectual. The 
matter is of considerable importance because quite recently a Scandinavian actuary 
in America II has been analysing mortality curves by tetrachoric functions and 
asserts not only that they give a good fit but apparently believes that each function 
of the series has some natural physiological meaning ! It is quite possible to re- 
present the survivors of 100,000 persons born in the same year of life by a Fourier's 
series from 0 to 100 years but one would hardly claim any special physiological 
significance for the individual periodic termsH. Such a series however is far 
easier to deal with in later treatment, such as differencing, than a series in tetra- 
choric functions. 
For the numerical calculation of the tetrachoric functions the difference 
equation of these functions is invaluable, i.e. 
where x is the argument of the functions and 
vs Vs(s-l) 
Tables of and 7^ are given in Tables for Statisticians (p. 1 of introduction) 
to five decimal places for a' = 7 to s = 24 (the first six tetrachoric functions being 
given on pp. 42 — 51) and in Biometrika, Vol. xiv. p. 130 to 7 decimal places. - 
For our work /3g and 7,, were required to 7 places (sometimes to 8) to obtain 
the requisite accuracy. The procedure consists in calculating Ti, which is equal to 
^-^=, directly to the required degree of accuracy and then by means of the tables 
\/27r 
referred to above the higher tetrachoric functions are obtained in rapid succession 
on the machine for a given value of the argument. In the testing of our tetra- 
choric series seven-place accuracy was aimed at so that it was necessary to calculate 
Ti to eight places, which was done with the help of Vega's ten-figure logarithms. 
* Vorlaesninrjer over Almiuddig lagUagchedacrc , Kjobenhaveii , 1889. 
t Boyal Soc. Proc. Vol. lvi. p. 271, and in many papers, Journal of li. Statistical Society. 
J Vorlesungen ilber die Grundziige der mathematischeii Statistik (Hamburg, 1920), p. 67. 
§ Biometrika, Vol. vi. p. 68, 1908. 
II Arne Fisher, Casualtij, Actuarial and Statistical Society of America. Proceedings, Vol. iv. Part i. 
No. 9. 
H A normal curve, for example, is quite adequately represented by two or three periodic termg; see 
Phil. Trans. Vol. clxxxvi. A, p. 355, 1895. 
