H. E. Soper 99 
Precisely the same process evaluates the mean products four together. We 
shall have, putting in representative terms only of each series, and using equations 
(xxiii), (xxiv), (xxv), (xxvi), (xxvii), 
mean dp . dp' . dp" . dp'" = — [a^af a" a"' 2>ff ( 1 —/])'"+ ... 
- a^a^a^'a^" 3/^(1 -fi)f, + ... 
+ mVC/j/i ( 1 -A -/■ +3/i/ 9 ) + - 
- a 1 a 1 'a,"a 3 "'f 1 f. 2 f. s (1 - 3/0 
+ a^a^'al" 'fifif»f t + • ••] 
= ^[(ai«i7i + ...)(«'/. + ••■) 
+ («i«i"/i + --.)(«i'«.'7i + •••) 
+(«i«r/+ •••) (fl/tti"/] + •••) 
- (a, </; + . ..)«/; + . . .) (a/"/ + . . .) 
- (ch <fi +■■■) «/i + • • • ) + • • • ) 
- («!«/"/! + ■ • •) + • • • ) «7 + • ■ ■) 
- (axV/; + . . . ) (c^/i + . . .) «'/, + . . . ) 
- (a/Oi'Vi + • ■ •) + • • • ) «7 + • • •) 
- « ar/i + • • • ) + • • • ) «/; +■"...) 
+ 3(a 1 / 1 + ...)(«i7+ -)«7 + •••)(« "7+ ••■)] 
on collecting terms and rearranging the associations of /'s and a's as before. 
And putting into factors this 
=^JK«i</i+--0-(«7+---)(</i+-^ 
+ {(a 1 a 1 // / 1 + ...)-(a7i+-)«/i+...)} {(a/a "7 +•••)- (ai7i+"-)(«i'7 1 +•••)} 
+ Kai</i + --0-(«7+--0(<7 
And so again if the material is double graded and p uv , Pu'tf, Pu"v" > Pu"'v"' are an y 
four moments involving products of powers of both grades, the general formula 
for the mean products four together of the deviations of such moments in samples 
of n becomes 
mean dp uv . cfyw . dp u v . dpw 
= jTa \xPu+u'v+xf Puv Pu'v') \Pu"+u"' v"+v"' Pu" v" pu"'v"') 
+ ( Pu+u"v+v" PmPu"v") \Pu'+u"' v'+v'" ~ Pu'v'Pu " »"') 
+ {pu+u'" v+v'" puvPu"'v"') (Pu'+u" v'+v" ~ Pu'v' Pu"v")~\ ( XXX )> 
