388 Supplementary Tables for Tetrachoric Groupings 
We accordingly proceed to find d/N for these values of r by interpolating in 
these two tables for the above values of h and k; in accordance with the general 
rule previously given, the interpolation is first carried out for k and afterwards 
for h. 
From the tables we have 
r 
= 9 
k = 
h = 
1-4 
1-5 
h-l-l 
•0686 
•0591 
7i = l-2 
•0645 
•0562 
which gives 
r 
= -9 
k = 
1-4032 
fc=l-l 
•0683 
h=l-2 
•0642 
and finally 
r 
= •9, 
k = 
1-4032, 
h= T1218, 
d/N = 
Similarly 
we 
obtain 
r 
= •95, 
k = 
1-4032, 
h = 1-1218, 
d/N = 
Interpolating for r, bearing in mind that the interval of r from one table to 
the next is - 05, and using d/N = "0676 as argument, we obtain the result r = "903. 
Accuracy of the Result. 
In order to test the accuracy of the result, the value of r in the above example 
was calculated by the usual method, using the Tables of Tetrachoric Functions and 
including terms up to the twelfth power of r. 
The equation obtained was 
•057090 = -010516?- + -024948?- 2 + -01322?- 3 + -003733r 4 + -003868r 5 - -000225^ 
+ -002923r 7 - -000121 ?- 8 + -001597/- 9 + -000442r 10 + -000630?- 11 + -000017r 12 , 
whence solving by Newton's rule the value of r is found to be "904. 
It is not suggested that this remarkably close agreement between the two 
values - 903 and - 904 is always to be expected ; but the difference between the two 
values will always, I think, be found very much less than the probable error of r 
and will therefore be without any significance. 
Further Note on the Tables. 
If D hlc be the tabulated value of d/N for arguments h and k for any one of the 
values of r for which the Tables are constructed, then the volume of the frequency 
solid on the area bounded by the lines corresponding to the values /V, h 2 , k lt k 2 , is 
given by D hlkl + D h2k2 - D hlk2 — D/^,. Consequently within the limits r = S to 
r = l the distribution of the frequency within the one quadrant, for which the 
Tables are constructed, may be readily found. 
Gases where the Method of the Fourfold Table fails. 
A careful examination of the Tables shows that, when the values of h and k 
differ widely and r is large, then the corresponding values of d/N differ very little 
from one value of r to the next ; in such a case the probable error of r will be 
