102 On the Probable Error of the Correlation Coefficient 
When this is expanded, the mean values of a u /3 U a 2 , & 2 , and 7 are of course 
zero. In the following tables the general formulae (xxviii), (xxix) and (xxxi) for 
the mean products two, three and four together of deviations of moments in 
samples of n are given at the head, the suffixes used and their composition in the 
several formulae are shown in the initial columns (omitting repetitions of dp and p 
to abbreviate the printed matter), and the resulting formulae for the mean products 
of the a's, /3's and 7's required in the expansion of (xxxiv) are given in the last 
column, the reductions of the higher moments having been made by (xxxiii) to 
suit the case we are investigating, that of a normal distribution of the two 
characters in the material sampled. Since the first and second moments only of r 
are at present sought it is unnecessary to take products involving higher powers of 
7 than the second. The four additional formulae to y 4 are inserted, however, to 
complete the formulae for the third and fourth moments when required. 
As an illustration, if in equation (xxix) we put 
u = 0, v = 2, u = 1, v = 1, u" = l, v" — l, 
we get mean dp 0 ,dp n dp n = — [p 24 - p r ,p n - p ls p n -p 22 p 02 + 2posPnj»n]. 
Dividing by p 20 p 0 i and using (xxxiii) it follows that 
mean /9 27 2 = \ [(3 + 12p a ) - 3p 2 - 3p 3 - (1 + 2/*) + 2p 2 ] 
= I[2 + 6^] ) 
the penultimate formula in table (xxxvi). It will be seen that the values to be 
attributed to the terms signified by the suffixes between the double rules are the 
right-hand sides of (xxxiii), the composition of the terms being shown in the last 
column of the formula. For 24 put 3 + 12p 2 , for 13 put Sp, for 11 put p, for 22 put 
1 + 2p", for 02 put 1, and the formula for mean /3 2 y 2 may be written down without 
any division being necessary. 
The formulae (xxxvii) are derived from those in (xxxv) and the suffixes between 
the double rules are for reference to the first columns of the latter table. Thus in 
the penultimate formula look up 02 11 in (xxxv) and find 2p. Look up 11 11 and 
find 1 + p 2 . 2p x (1 + p~) = 2p + 2p 3 , the first of the three component terms added 
together in the last column of the formula. 
