100 Oil the Probable Error of the Correlation Coefficient 
and it is to be recalled that this formula omits terms in — , not wanted to the 
degree of approximation laid down. 
Comparing (xxx) and (xxviii) it appears that the mean values four together 
are equal, within our degree of approximation, to the sum of the products of the 
mean values two together of the complementary pairs making the four, the 
division being possible in three ways. 
mean dp uv dp u >tfdp u » V '>dp u »w = mean dp uv dp U ' V > x mean dpwdpw 
+ mean dp uv dp u ",.y x mean dp u > v >dp u >» v "' 
+ mean dp uv dp u >" v »> x mean dp u ' v 'dp u %" ...(xxxi). 
It is unnecessary to find the general formula for the mean products of 
deviations five together, which by p. 96 will contribute nothing within our 
approximation, and formulae (xxviii), (xxix) and (xxx) applied to the expansions 
of (xvii) and its powers are sufficient to evaluate the general formulae for the 
mean and moments of deviations of r as far as terms in Ijn". 
It is not proposed to exhibit these general formulae for moments of devia- 
tions of r in terms of the higher moments of the given distribution at length, 
but to proceed at once to the simpler case of a normal distribution in the two 
correlated characters and reduce the higher moments to second moments and the 
coefficient of correlation, p, as such distributions, it is well known, admit. In 
order to reduce in this way the values of the various mean products at the same 
time that they are evaluated by the formulae (xxviii), (xxix), (xxx), the necessary 
formulae of reduction are next obtained. 
The expression for dr involves dp l0 , dp 20 , dp 01 , dp 0i and dp u , and (xxix) shows 
that we shall require to reduce |), ;o , pn...p r>0 ...p i0 ...p 30 ... in the above way. 
Now it is well known that in normal distributions, following the Gaussian 
law of frequency, the odd moments from the mean in either character vanish 
and the even moments are derived from the second moment by a simple 
formula of reduction, from which there results that 
/'so = P-m = Pio = 0, Pob = Pos = Pm = 0, 
p i0 = ZpJ, p m = 5 . 3p., 0 s , p 0i = 3jt>02 2 , p w = 5 . 3p 0 .f. 
And, utilising these results, the higher product moments of normal distributions 
in two characters may be derived from the first product moment and the second 
moments by two well-known properties of the Gaussian surface. 
If x, y are deviations from their mean value of two normally correlated 
characters the mean value of y for a given x is — ■ x, and if y is the deviation 
Put 
