240 
Miscellanea 
Or, supposing the ancestry to ascend indefinitely by the same law, 
aP=y [r) + 
-r-yivfi+f+f^) ,vih 
= y (r,ap + fa/3 + r) 3 + 
Multiply each equation by 77 and subtract it from the one below it. We find 
a/3 2 (/3-7 ? ) = y^a/32 
■(«) 
or |8— r)=yq in each case/ 
Substitute this value of yrj in the first equation and we have 
*_„-,) 
(/3-, )(l-,/3) 
a/ * = ^-2^+1 J 
Hence 
which determines the variability of the array with given ancestry. 
Returning to Equation (x) we have 
2 _ 7 (l-2a^ + /3 2 ) 
.(xii), 
or J7 2 — + 1 = 0, 
*~ 2 
where it is necessary to take the root less than unity. 
Application I. Galton's Rule. Sir Francis Gal ton took the "contributions" of each 
generation of ancestry to be ^, j, \, Thus each parent contributed on an average 
each grandparent T ' 5 , each great-grandparent ^j, etc. 
Our equation, if we neglect assortative mating, is 
x 0 - a? 0 =$ ( JT, - Xi)+H*2 ~ ^2) + ... 
=S |c p ~(A'„-X„)| 
=£{y (^(Xp-X,,)}, 
if we put o- p =o- 0 , i.e. suppose population stable. 
.-. y=l, s>2r,=i, 
or by (ix) ,8=217 = 1/^2. 
-„ (1-^) _1 
Therefore by (x) "=V?^W = n' 6 
Thus the mid-parental correlation = n/i''= 'G 1 
s/2) " 
