68 Generalised Probable Error in Multiple Normal Correlation 
TABLE II. 
Values of the Functions yfr^ mid -^.^ required to determine the Constants of a 
Normal Frequency Distribution from the Moments of its Truncated Tail. 
h' 
Yl 
7/ 
It 
Yl 
i// 
0-1 
•588 
1-311 
1-6 
-787 
2^358 
0-2 
■605 
r371 
1-7 
-796 
2^437 
0-3 
•622 
1-432 
1-8 
-804 
2-517 
0-4 
•638 
1 -495 
1-9 
-813 
2^598 
0-5 
■653 
1-560 
2-0 
-820 
2^679 
0-6 
•668 
1-626 
2-1 
-828 
2^762 
0-7 
•682 
1-693 
2'2 
-835 
2^845 
0-8 
•696 
1-762 
2-3 
-842 
2^929 
0-9 
•709 
1 -833 
2-Jf 
-848 
3^013 
1-0 
•722 
1-904 
2-5 
•854 
3^098 
1-1 
•734 
1-977 
2-6 
•860 
3-184 
1-2 
•746 
2-051 
2-7 
•866 
3-270 
1-3 
•757 
2-126 
2-8 
•871 
3-357 
1-4 
•767 
2-202 
2-9 
•876 
3-445 
1-5 
•777 
2-280 
3-0 
•880 
3-532 
This is quite sufficient for practical purposes, for the difficulties of practice do 
not arise from the paucity of figures in this table, but from quite other considera- 
tions, now to be described. In the first place, when we have a truncated distribu- 
tion like that indicated in the above figure, the group of maximum frequency is at 
one end of the distribution, and any variation in this frequency widely modifies 
not only 2 but Vi'. Thus the probable error of random sampling in the "stump" 
group is very influential. Next, the frequency is almost certain to be given in 
definite ranges, and the correction of the raw moments for such a case with the 
maximum frequency group at the terminal is tedious even if it can yet be 
considered satisfactorily determined. Thus in actual statistics, I have found that 
truncating at different frequency groups much modifies the values of the unselected 
frequency constants. For these practical reasons no very great number of decimal 
places is needful in \}ri and yjr.. I think, however, the tables may be of service 
in determining the constants of the untruncated distribution to a first approxi- 
mation, especially if the final values of those constants be based on truncating 
at two or three points and averaging the results. Further they may serve to 
give the fi^'st idea of the root of the nonic required when we break up a distri- 
bution into two components, the constants of one component being indicated by 
the tail of the frequency on that side. 
I have to thank Dr A. Lee for the laborious arithmetic involved in the 
preparation of the tables accompanying this paper. 
A further use of the incomplete normal moment tables in evaluating the 
incomplete B- and F-functions will be considered in another paper. 
