K. Pearson 
89 
and hence the differential equation to the curve 
\ dy _ —X 
y dx 13 1 + 13 -iX + /Sa*'- 
, . , \dy -2x{p + \) , ... 
becomes simply - = r-mr^, r., si i (viU- 
y dx \c- {{\ + pf - s~\ + X- 
Whence, integrating, we find 
J 1 , .... 
y ^"[icM(i ^ 
Here x is measured from the modal value which is clearly the same as the 
mean, and corresponds to a length ^{p + s) c, or to the number ^{p + s) of 
protogenic determinants. Further, y^ will be a function of s which remains to be 
determined. 
Let 9?s be the total frequency corresponding to somatic cells of the offspring 
of the hybrids, i.e. to the cells before the reducing division. We must have 
where = ^c- {{1 + p)- — s"-]. Let x = \s tan d, then 
= y„ cos^^ dde/\f"^\ 
Or 2/o = '^sV^+V^'si^. 
here i.}p = i sin^</)C?0 is a well-known integral. 
J 0 
Thus (viii) becomes y = /' ' "f , (ix). 
But X is measured from ^(p + s)c the mean value of protogenic determinants in 
the gametic cells of the offspring of the hybrids. Let us write x = — ^sc + ^, then 
y {\c"-{l+p') + ^''-sc^Y+' ^ 
To find rig, we must note that it is defined by the strip y'hx of a normal curve 
where x'jc = s, for this is the distribution for the somatic cells of the offspring of 
the hybrids. 
Thus the frequency for the germ cells of the offspring of the hybrids of chromo- 
somes with \p + ^jc protogenic determinants arising from offspring of hybrids with 
somatic cells with chromosomes with ]) -\- x'jc protogenic determinants is given by 
the surface 
lic^d ^pi-ix-r^ ^-i^w 
Biometrika vi 12 
W 
