184 
On Expansions in Tetraclioric Functions 
ri^ gpiSg-x x — p 49 — 49 
Table VIII gives the results for L, . . dx and, since z = — ; - = --^ = 0 
" 'o r(49) v> 7 
for the expansion from the mean, all the tetrachoric functions of even order vanish. 
It will be observed that the values of the series vary in a similar fashion to the 
others and not one of these gives the result correct to more than four decimal places. 
TABLE VIII. 
Jo f(49)^^"' ' = ^ 
(Expansion with regard to the Mean.) 
Tetrachoric 
Terms in 
Value of Series 
s 
Functions 
Series - OgTs 
up to term 
0 
1 -00000000 
•5000000 
-5000000 
-5000000 
3 
•11664237 
- -1628675 
+ -0189973 
-5189973 
5' 
•00638743 
+ -1092549 
- -0006979 
-5182991 
7 
•01785148 
- -0842920 
+ -0015047 
-5198041 
9 
•01470566 
+ -0695373 
- -0010226 
-5187815 
11 
•00895618 
- -0596711 
+ -0005345 
•5193160 
13 
•01079260 
+ -0525526 
- -0005672 
•5187488 
15 
•01015854 
- -0471442 
+ -0004789 
•5192277 
True value -5189993. 
After a careful study of the tables and graphs we are forced to the conclusion 
that a tetrachoric series is of no practical utility as a representation of skew 
frequency curves such as y = y„xP~^ e"'^ and y = yaX™~^(l —x)"^ ~\ and although it 
may be rash to generalise from our results on these two types it would seem 
that such a series cannot be generally suitable to represent skew frequency dis- 
tributions. Moreover, the types, which have been discussed, are of common occur- 
rence and for these the expansion is certainly futile. 
The true values of the incomplete F-function were taken from Tables of the 
Incomplete r-fiinction which will be shortly issued by H.M. Stationery Office. The 
values of the incomplete B-function were determined by direct calculation ; the 
power of (1 — x) was expanded and the result readily obtained with the help of the 
relation 
In his Vorlesimgen ilher die OrundziXge der mathematischen Statistik (Hamburg, 
1920) Charlier, when dealing with skew frequency curves, gives as the general 
equation for the skew frequency curves of his Type A 
1 7 = 4- /S3 (/)„"' <^„i^ + (/>o^ + . . . 
