Karl Pearson and Egon S. Pearson 
133 
Next clearly — ^/f^ stands for 
[ Tixdw = i I — xe dx 
or '£)^,.ro = ^,Ti, 
which is precisely the value given by (viii). 
Thus (viii) is shown to be correct even fur this special case although a form 
like (vi)bis through which it is reached shows difficulties. 
Similarly 2 o' = ^st; * 
The remainder of the t's knowing To and T] come directly from (iv) and the T's 
are always given by (viii). 
Now it is clear that (x) to (xiii) provide a large number of ways of deter- 
mining r. We might find r, i.e. ?\,,.') from the single cell by writing in (x) /i^s' for Tigs'- 
Or we may find 
^S- A-' 
= ~s C'^' {% T„x. T,; + rx. i\% t/ + . . . + rp% T,x. T,; +...;) (xiv), 
where ris^' is given by (x). But hg. is the known centroid of the n^. marginal total, 
and accordingly the above is an equation to find r, i.e. ?v, from a given column of the 
table. 
If we use this value of Vs. in (x) and (xii) to find n^s', and hgs', we obtain the 
theoretical cell frequency and ^/-inean of the cell as found from a column. 
Now sum I'ss' for every value of s and we find k^. the y mean of a column 
depending on the data as found from the column, i.e. 
A;,. = ^ ^'(^^ l^,T„^,To' + r^,T,^,.r/ + ... + /•^'^«t^^,.2V + ...}) (xv), 
where iiss' is the observed cell frequency and Tig^' the frequency found by (x) when 
we insert the value of r as found from (xiv). We are thus in a position theoretically 
to determine on a normal scale the mean of a column from the correlation actually 
determined from that column. This would be the ideal method of determining the 
mean of a row or column ; but it would involve a great deal of hard work, as with 
the two regression curves we should need to find r for every row and colunni by an 
equation of a high order. Hence in most cases we are likely to content ourselves 
by finding r for the whole table and then use this value in (x) to determine n^s' 
and in (xv) to find the mean of the array, kg. plotted to the known h^. on the 
normal scale will give the regression curve. 
* We can thus take Tq = ti and 1\'-=.ti'. 
1 -h.c^ 
~7^^ e - 
