A. Rhind 
129 
and IBs are calculated from /3i and /Sg, on the assumption that the values of 
them obtained from the finite difference momental formula are sufficiently accurate 
to use in Pearson's formulae for S^^j, S^^, jB^.^j and 2^^, etc. 
The following are the finite difference ;S-formulae used : 
(even) = (. + 1) |^ + (l + |) ^ /(l - a) , 
/3n (odd) = (n. + 1) || A.-, + (l + I) /3™-.} /(l - ^ «) ' 
where a - (2/3^ - 3 A - 6)/(/S, - 3). 
The process of calculation adopted was as follows : 
Fundamental values of /3i, /3o were adopted; these are indicated in Table VI., 
and the resulting values of /Sg, /Sj, /Sj, calculated by the above formula were 
then found to seven figures. These are tabulated to six figures as they may be of 
service for the determination of other constants as occasion arises. 
The values of for the different values of fi^ and ySa being known, it was seen 
that a very simple diagram would permit of a statistician ascertaining at once 
from his values of and the type of his frequency distribution. In fact this 
diagram brings out very suggestively the normal curve "point" (G), the Types II., 
III., V, and VII.* "lines" and the Types I., IV. and VI. "areas" of occurrence. 
By aid of this diagram and a reasonable consideration of the probable errors of 
his /3i and /Sg the statistician can readily determine within what limits he is justified 
in using any special type of frequency curve for given data. 
The following values of analytic constants for the fundamental values were then 
found (Table I.), VFS^,^ (Table II.), Rp^p,^ (Table III.), and intermediate 
values deduced by interpolation. It is believed that these interpolated values 
recorded to two places of decimals only are amply sufficient for the practical uses 
to which these tables will be put. 
From these tables were found, again using interpolation, the probable errors of 
the two chief desired physical constants : (i) d, the distance from mean to mode or 
the modal divergence, and (ii) Sk, the skewness of the distribution. In Table IV. 
we have the values of '^N'Zd/o- and in Table V. the values of ViVSgjfc provided. 
The actual quantities tabled in Tables I., II., IV. and V. are such that when 
multiplied by ■67449/\/F we obtain immediately the required probable erroi's. 
The value of this numerical factor can at once, however, be extracted from 
Winifred Gibson's Tables for computing probable errorsf. 
* It is convenient to call ?/=?/„ ^1 + j Type VII., see Biomelrika, Vol. iv., p. 174. 
t Biomctrika, Vol. iv., p. ,385. 
Biometrika vii 17 
