98 On the Probable Error of the Correlation Coefficient 
It will be observed that nothing in the proof prevents u', v from having the 
same values as u, v and the formula is true for any second order moment whether 
power or product. 
In like manner if p, p, p" are any three moments of the material sampled we 
have the equations of deviations 
dp = a^df x + a 2 df\ + a s df 3 + ... 
dp = a/eZ/i + a 2 'df 2 + a 3 'd/ 3 + . . . 
dp" = al'df x + a"df, + a 3 "df s + ... 
giving 
mean dp . dp' . dp" = mean [a^ha*" (rf/i) 3 + . . . all grades 
+ (ffio/a/' + + a x a"a 2 ) (dff df 2 
+ (e^Os'ag" + a/^a.," + a"a,a 2 ) df\ . (df. 2 f + ... all pairs 
+ («•!«•/ «s" + di«/a 3 ' + <h' a a a s" 
+ « 1 'a 2 "o 3 + a"a,a 3 ' + a"a 2 a 3 ) df Y . df 2 .df 3 + ... all triads], 
and inserting the values from equations (xx), (xxi), (xxii), 
mean dp. dp' .dp" = — [a^'a"/, (1 -/i)(l - 2jQ + ... all grades 
- (a^'a." + a^a"a 2 + a^a"a 2 ) f\f 2 (1 — 2/\) 
-(« a <7 2 'a," + <h' <h^2 + a/^OA/Ul ~ 2 /a) _ ••• a11 P airs 
+ (a^u'a" + a x a 2 '' 'a 3 ' + a{a 2 a" 
+ o/a/'rta + a/'c^a/ + a^'a^a s ) 2/i/ 2 / 3 + . . . all triads], 
and collecting terms of first, second and third degree in f's and suitably 
commuting the f's and as this is seen to be 
= - 2 [(a 1 a 1 'a 1 "f 1 +...) 
- («i </i + ...)(» + ...)- («i «i'7i + • • •) «A +••■)- ( «/ + ■ • ■ ) + • • •) 
+ 2 (a 1 / 1 + ...)(ai'/i+ .. •■•)]» 
the sums being for all grades. 
If then £>, have the double grade values p UV! p u > V ' previously assigned and 
p" stands in the same way for p U " V " where 
Pu"v - (h u "b/'fn + ih u "b/'f u + a.fbff. n +... 
there results the general formula for the mean products three together of deviations 
in moments as follows 
mean dp uv . dp lt ' V < . dp u » v » = — [p u +u'+u" v+v'+v" ~ Pu+u'v+v'Pu"vf' 
Pu+tt" v+v" Pu'v' Pu'+u" v'+o" Puv + 2p u v p»v Pu'V'] (xxix), 
where, as before, the values of the suffixes may be any the same and the formula 
gives power moments equally well with the product moments of the deviations. 
