H. E. Sopbr 
95 
The method adopted in this paper is that of grade groups. It is well known 
that if in an indefinitely large population the fraction / fall into a certain grade 
of a character or combination of grades of two characters, then in taking random 
samples of n the numbers of this group to be found in such samples follow the 
binomial distribution of frequency 
yv, + B y*-i (!_/) + n V"- 2 ( 1 -ff + . . . + (1 -/)", 
and that the mean number is ?i/"and that the deviations from this mean number 
have moments 
mean {dJJ = \f \l - /), 
» wy=^fo--f)(i-V)> 
" = & f ( 1 ~f f + 1 W -A ( 1 ~ V + 6 ^>' 
» WT = ~f (i ~/) 2 (i - V) + ^/O -/) (i - 2/) (1 - 12/+ 12/0, 
etc., the fourth moment being the last which gives terms in l/« 2 [see Pearson, 
Phil. Trans. Vol. 186, p. 347 and Phil. Mag. 1899, pp. 240, 241]. Here df is the 
deviation from mean value /of the frequency of the group in a sample of n. 
Moreover if /, /, f s ... are the totality of frequencies of the various detached 
groups into which the population is divided by the graduation (which in our case 
is a double one) of character the various product moments of the deviations in 
samples of n may be deduced. These and the above, as far as our approximation 
needs, are put in one table as follows : 
mean (df,) 2 = -/(!-/) etc. 
(df)' 
(dfrf 
df, . df 2 . df 
(dfY 
(dfiY.df 2 
„ df . df . df . df, 
n 
1 
n 
1 
ri 
1 
n 
2 
n 2 
3 
ri 
3 
ri 2 
1 
ri 
-f) (i-2/O 
•2/1/2(1 — 2/) 
fff 
U 2 (i-/) 3 
/!/(!-/) 
,//(!-/-/ + 
= -~ 2 /// 3 (l-3/) 
3 
= ^2 ffff* 
..(xviii), 
...(xix), 
....(xx), 
...(xxi), 
. .(xxii), 
, .(xxiii ), 
..(xxiv), 
...(xxv), 
..(xxvi), 
.(xxvii), 
