PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 313 
Ci + C 3 — <t~ + (/V — i) er + 
C l9l + C , 2 c h = p* (o- 2 + m - 1) e x 2 + &V) + & 2 (o'* + &V + (A* - 1) ^ 1 2 ). 
e/ and r\p can then be found at once, if the values of the constants are known. 
Remembering that 9l and g 2 will be proper fractions, we can easily find the effect 
of continued panmixia by putting p = co. 
We have 
C) + C 3 - C^ 8 - <r 2 (1 - /3/ 2 ) + ° W 
Mt ~ (ffi + ft) + 1 PW ~ PM' S ~ ~ A' 8 + 1 * 
after some rather lengthy reductions. Similarly 
2 _ <r 2 /3V + ^ (1 - ffi) 
’* w - 0tVt‘ - A* ->V + 1 ' 
If we substitute in these the values of the /3’s, and of c r and cr' given on p. 310, we 
find : 
= °"i> V* = <r'i- 
Thus we see that indefinitely prolonged panmixia carries back not only the means 
of both sexes, but their distributions about the means to the state of affairs when the 
foci of regression were themselves the means of the population .* 
The all-important question concerning panmixia is, as we have seen, that of the 
position and stability of the focus of regression, and it seems to me that this is a 
question which it is only possible to settle by experiments. Nor do the experiments, 
at least from the theoretical standpoint, seem attended by difficulties which are 
insuperable. It is not necessary to select a parthenogenetically reproductive race, it 
is not necessary even to select both parents, it would be sufficient to deal with the 
regression from one selected parent, if this were most convenient.! The simple test 
is this :—If M x be the mean of selected parents, the mean of their offspring, 
and M 2 be the mean of another group of selected parents (e.g., selected out of the 
* In order to ascertain whether the standard deviations would return to tlieir old values, supposing 
natural selection to be suspended, but assortative mating maintained, we should have to solve a series of 
equations of the type : 
v — + /bS-r + Pzhp-\ + zpgk'Yp-i’ip-n 
Vi~ = o’" + ft + ft z'rjp—i + 2ft 2 ft' 3 r Ufa—i, 
and then substitute the.values of oq, oq' and the ft’s from p. 286 in and I have not yet solved these 
equations. In turning the above formulae into numbers, the caution given in the footnote, p. 312, must 
be borne in mind, i.e., the correlation coefficients for inheritance during assortative mating may differ 
somewhat from those holding when it is suspended. 
t Perhaps a common father and series of selected mothers would give the best results. 
MDCCCXCVI.—A. 2 S 
