92 Oil a Mathematical Theory of Determmantal Inheritance 
Accordingly the chance from parental somatic cells lying between and 
a-i + Sa-i and and x<^->r^x^ of getting gametic cells respectively between |i and 
?i + and ^3 and |o + Sfs is measured by 
-4-^{8(f;- + .V^)-8(xi|, + a;,y + 6(Xj2 + a;/)} 
c-ij dx^dxo^d^id^2, 
where Zo' is a constant depending on p and c only. Write for 77 - and the 
chance becomes 
-i^ {8(2e/-2^?, + V')-8(^x, + (.r,-.r,)y + 6K2 + ,T,2)} , , 
^^'e -f-jj' dx^dxndrjd^^. 
Now integrate this expression throughout the range of values of ^2 and we 
shall have the triple correlation surface connecting ^1, x^ and In order to do 
this we must make the result a perfect square in fa. We thus find 
Integrating throughout the range of ^2, we only modify the constant Z^' 
without changing the terms in 7;, x^, or x^. Thus the frequency surface is finally 
of the form : 
( t]^ .rj'^ x.^ '2i7.rj 2t).C2 2.ria;2) 
z = z,e~ Wp^ h'^^ ^ i^p~ Wp~ ^ '<^hn (xiv). 
Let 7- be the correlation between chromosome in the parents' and in the 
offspring's somatic cells, then : 
The correlation of r) with either and the correlation of x^ and 
^2=0. 
Let o-j and be the equal standard deviations of Xi and x^ ; 0-3 of t]. 
Then the bracket term ought to be equal to 
1 \r]- x^{l — r-) xi{\ — r'^) 2r]Xir 2r)X27' 2r'^XiX. 
1 - 2r- (0-3'- o-i" 0-1^ o-go-i o-gCTi O-J- 
Hence we deduce : 
I 1 1 - 7- _ 5 1 r _ 2 r- 1 _ 1 
^7 1 - 2}-"- ~c^' of 1 - 2r^ ~ c^' 1 - 2r' ~ c^' 0-1^ 1 - 2r- ~ cy' 
The second and fourth equations give us r^=l/6, agreeing with the value 
obtained from the first, second and third. The first and second then give 
= fc^i^ and o-j^ = ^c^p. 
Thus finally we have : 
Variability of the somatic cells in or in the offspring of hybrids = ^c\/p- 
Variability of the somatic cells in Ho or offspring of Hi=^^ ^Cfjp. 
Correlation of the somatic cells in parents and offspring i/2 = = "4082. 
Regression of the filial on the parental somatic cells = "5000. 
These results seem to be of not a little interest. They indicate that if two 
pure races be crossed, the variability in protogenic determinants of the chromosomes 
of the somatic cells will be still increasing to the third generation, i.e. it is 1"225 
times as great in the H.2 generation as in the Hi generation. The correlation, 
