316 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
of the means of the male and female populations from which the n + 1 th bi-parentage 
is selected. 
Thus we have the finite difference equations : 
Un + 1 = U/i — (A 2 W '» + fig)*) 
V* + 1 — V„ — (j 8 'o U u + fi' ? v u ) 
Assume : 
u n — Ay" -1 , v„ = 
Hence : 
A (x — 1 + A 3 ) = “ B/3 3 , B (y — 1 + /To) = ■— A/3 2 , 
or, 
— 1 )' + {ft -2 + /T 3 ) (x ~ 1) + AsA's — A:. AT = 
or, 
Xl — 1 — 7l and X ‘2 = 1 — 
< 
where y x and y., have the same values as on p. 311. Thus : 
^ = A 1 (i-ri ) J,_1 + A s (l — yi) p ~ 1 , 
v p = - r,)"- 1 + A^(i - y,)'-; 
where 
^ _ (A 3 — y 3 ) m 3 T- A^fis 
1 71 — 73 
A _ _ (A 3 - 7l) + A 3 ? »3 # 
3 7i “ 7s . 
Now this solution^ is the same as that on p. 311, except (i.) that u p and v p , unlike 
jjLp and n’ p) are measured from the selected means, i.e., the mean heights of the male 
and female populations are respectively m 3 — u p and ?n 3 — ty after p-generations; 
(ii.) that in the values u p and v p (1 — yi)'® -1 and (1 — y-:) p ~ l replace Vi p ~ l an( l y/ -1 - 
We conclude, accordingly, since y x , y 2 , and, therefore, 1 — y x , 1 — y 2 are proper 
fractions, that u p and v p grow smaller and smaller, or, if selection be long enough 
continued, the means of the male and female populations will ultimately pass to the 
selection means. 
Of course, if selection be suspended at the n th generation, regression will take place 
as on p. 310, but only to the nearest focus of regression, i.e., m. 2 — u„, m 3 — v„. Thus 
the effect of n selections has been to raise the general means permanently by these 
amounts. 
* The uniparental or parthenogenitic results for progression of the focus follow at once by simply 
putting /3 3 = /to = /3' 3 = 0 in the above formulas. 
