96 On the Probable Error of the Correlation Coefficient 
the last five values being approximate and wanting terms in lju? to render them 
exact. 
The method of derivation of the product from the power moments is illustrated 
in the following example. 
Mean (df^ 2 .df 2 .df 3 = mean {{dffi x mean df 2 df 3 for constant cZ/i}. 
Now in samples of constant df x the number of l's is n^ + ndfi and of not l's 
n — n^ — nd^, amongst which latter restricted population in the whole community 
f f 
the frequency of 2's will be and of 3's - , . Hence the mean number 
f 
of 2's in such samples will be (n — nf x - ndf x ) » , and of 3's 
f 
differing from the mean numbers in all samples, nf 2 , nf 3 , by — ndf x — . and 
1 — Ji 
— ndfi 1 , , and the mean product of the deviations from such means in the 
restricted samples will be 
_ (n _ w/i _ nrf/i) ._^_._A_ 
by (xix). It follows that the mean product of the deviations ndf 2 , ndf s , which 
are measured from the means of all samples, will be 
- (n - nf\ - n df,) . ^ . ^ + (- nd/, j (- n df Y , 
i.e. - /i 
in samples of constant df^. Dividing by we get the value of mean df 2 .df 3 for 
constant dfi and so obtain finally 
mean (dfr df 2 df 3 = mean - + J^ + WJ.J^L-, 
which by (xviii), (xx) and (xxiii) 
1 ff f , 1 /^(1-2/;) 3 
= - ? >A/ 2 /3(i-3A) ) 
to our approximation. 
The other formulae were arrived at in like manner but the process is lengthy 
and these formulae and the general formulae that follow have been verified by a 
shorter process, which however being less direct in method is not introduced in 
this paper. 
There is no necessity to take the products of the deviations more than four 
together, for these do not give terms in \\v?. Did any products, five together for 
