PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 309 
those taller sons averaging 6', their sons will still only average 70 /,- 45 , and, however 
long we persist in this process of selection, we shall produce no secular change; the 
population will remain after the y> th selection just where it was both as to mean and 
variation after the first. The only way to produce a secular change is to continually 
increase the standard of the selected (or to alter the focus of regression). No steady 
selection would appear to produce “truer breeding.” 
(ii.) Panmixia arid Uni-parental Regression .—Continual selection of the same 
magnitude for p generations, merely giving us the same mean and variation, we may 
now ask wdiat would be the effect of suspending natural selection for q generations. 
Take first the case of parthenogenetic reproduction, or that of uni-parental regres¬ 
sion. The first parentage after suspension of natural selection will have m 2 r s cr L /cr. 2 
for its mean, and \/{crp (l — r 3 2 ) -f- ,sq 2 r 3 2 oq 2 /oq 2 } for its standard-deviation. 
Successive parentages can be found by substituting these values successively in 
themselves for the quantities rn 2 and s 2 . We find at once that after q generations 
of suspended selection the mean of the population will differ from the focus of 
regression by 
m . 2 (r 3 oq/oq) 7 , 
and the standard-deviation will be given by 
V 2 _ ^2 
*9 
°Y 0 - r z) 
— P’s o~i/q- a ) ag 2 
1 — P's a \! a dr 
+ i r 3 ofi/on) 2 * 2 s i- 
Now if the population simply repeat itself without any natural selection (if there 
be no reproductive selection at work) oq = oq, and in most cases I have come 
across r 3 oq/oq is a fraction. Hence, as q is indefinitely increased m . 2 (r 3 oq/oq) 7 becomes 
1 — r. 2 
indefinitely small, and 2/ = oq 2 , or = oq 2 -—— — —qp if cr 1 be not equal to err. 
We see, therefore, that both as to mean and variation the population with 
suspended natural selection tends to rapidly regress to the general population from 
which it was selected. This is still true if there has been a continuous secular, 
as distinguished from a periodic natural selection, for we have only to suppose m 2 
and s . 2 to be the final result of such selection. If then the focus of regression 
does not progress with continuous selection, all that has been asserted as to the 
effect of suspended natural selection holds, at least so far as concerns a return to the 
condition of things which prevailed when the focus of regression was the mean of the 
general population. But unfortunately the advocates of panmixia want more than 
this, namely, either an indefinite regression of the focus of regression itself, or to 
place it, if steady, at an indefinitely distant point. The first result would be 
perfectly parallel with our second hypothesis—a progression of the focus of regres¬ 
sion,—but would demand rather a reversal than a suspension of natural selection. 
The second result seems quite inconsistent with any statistics of successive genera- 
