300 PE OF. K. PEARSON ON THE MATHEMATICAL THEORY OP EVOLUTION. 
We can now transfer the exponential expression to its centre and we find for the 
frequency 
Constant X da e 2 C* 
,-,{s(»•,(■»♦£))«(-...(«,.-»))] dXidXidi 
Where x L , x. 2 , x 3 ... , now denote the coordinates transferred to the new origin. 
The integrations can then be performed without changing the u factor, and finally 
the frequency 
. . , + 
= constant X du e v A/ . 
Hence we notice the following important results : 
(a.) The p + 1 th organ follows a normal distribution, 
(b.) Its standard deviation 2 is given by 
S 2 = o‘ + /3 1 *^ + & 8 X + ' 
+ -fi.fi: q + -fiifi?, q + ■ • • 
(c.) Its mean (since u — v — £) = / 3 ]/q -|- /3. 2 h. 2 + /3Ji 3 + . . . 
We conclude that 
(i.) so long as selection is normal, however complex may be the system of organs 
selected, and however complex their correlation, the distribution of any single organ 
remains normal. This possibly accounts for the persistency with which normal 
grouping reappears in nature. 
(ii.) If we select organs varying about any means whatever, the mean of the 
correlated organ resulting from this selection will be identical with the mean we 
should have obtained by selecting organs actually at the means of selection. 
(hi.) The standard deviation of the organ •which results from the selection is not 
that of an array (cr) arising from selection of the organs actually at the means, but is 
(as we might expect) greater. This greater variability is due to the expression 
2 ^11 
+ fii ^ +... + 2 Aft A q + 
which admits of the following interpretation. 
Consider the selection correlation surface 
z = constant X e - J( «un 2 + «*»■** + • • • + + • • •> 
and give aq and x 2 chosen values y l and y. 2 . 
’Transfer the remaining variables to the “ centre.” The equations to do this are 
