H. E. Soper 
97 
instance, give terms in 1/n 2 , then mean (df\ + df. 2 + df 3 + df 4 + df ; ) 5 would have such 
terms, which is contrary to the formula arrived at above for mean (df) s . 
Having obtained the various mean products of deviations of group frequencies 
shown in equations (xviii) — (xxvii) the mean products of deviations of moments, 
formed by associating with such group frequencies their grade values, follow. 
Let ftj, a 2 ... be the values to be assigned to the grades 1, 2 ... in the 
formation of the moment p (these values will in the present case be the product 
of one power of one grade of one character with another or the same power of 
one grade of the second character). Then 
P = <hfi + Cts/a + a,f, + ... 
and if a/, a,/ ... are the values proper to a second moment,^/, in like manner 
V = ((if i + ct 2 '/ 2 + a 3 'f s +■..., 
and if in random samples of n deviations df lt df\ ... in the frequencies lead to 
deviations dp, dp', in the moments, all deviations being taken from the above 
universal values which are also the mean values in samples, then 
dp = (h.dfi + (hdf, + a a df 3 + ... 
dp = (h'df + a.!df. 2 + a 3 'df 3 + . . . 
and so 
mean dp . dp' = mean [c^a/ (df])- + a 2 a 2 ' (df. 2 )- + ... + a^a.!df\df, + a l 'a,df 1 df,+ ...] 
= - (1 -/) + a 2 a.// 2 (1 -/ 2 ) + . . . - a^ff, - a/cu/J, -...] 
= \ [«i</i + (t-Sh'f* +■■■- (ai/. + K-2.fl + ■■■) («.'./! + (h'f* +•••)]• 
If then p is the u, v moment defined by 
Puv = «i" W v fn + <bJ% + a./' &,»/:„ + . . . 
obtained by summing the products of the group frequencies f by the uth power of 
the grade value a of the first character and the vth power of the grade value b of 
the second character in that group ; and p is the u , v moment defined by 
p«„> = af'b/fn + af'b/fu + <h u 'h/U + •••» 
it follows that the first term in the above square brackets is 
a 1 " +v 'b 1 0+v 'f n + a 1 u+u 'b 2 v+v 'f n + ... 
or p u+U 'v+v', and the general formula for the mean products two together of 
deviations of grade moments is* 
mean dp m . dp u:v - = - [p„ +U 'v+v -puvPu'tf] (xxviii). 
* See W. F. Sheppard, " On the application of the theory of error to oases of normal distribution 
and correlation," Phil. Trans. 1899 (192 A), in which paper (p. 127) are given formulae for the mean 
products, two together, of errors of moments calculated from the means of samples. In the present 
paper, it should be noted, the moments of the samples are crude, being calculated, not from the means 
of the samples, but from the mean values of the measured characters in the whole population ; and dp 
is the deviation in the value of the crude moment in any particular sample from its mean value in all 
samples, which is mean a x (/ x + dfi) + a-i{f-z + df-i) + ■•• =«i/i + 02/2+ ••• =P or the moment in the whole 
population. This latter is a true moment, the general means having been taken as the origin of 
measurement. 
Biometrika ix 13 
