86 0)1 a Mathematical Theory of Determinantal Inheritance 
Here, if the balanced heterogenic group be classed with the protogenic group, 
we should have the ratio 70'5 to 29'5, and most readers will remember worse 
experimental Mendelian quarters than this. The illustration shows that it is 
quite possible to get numbers appi'oaching the Mendelian when there really exists 
a considerable variety in the determinantal constitution of the cells in which the 
protogenic race is dominant. 
Case 4. p = ^. 
Thus we see that there would be 31*5 with dominance of protogenic deter- 
minant, 37 °/„ with balanced heterogenic determinants and 31 '5 7o with dominance 
of allogenic determinant. The distribution is diverging considerably from the simple 
Mendelian quarter and approaching much more nearly a normal distribution (or 
binomial distribution) in the protogenic determinants. 
No special stress is laid on the above examples, I have only taken them to 
indicate, how simple Mendelism passes when we increase the number of deter- 
minants concerned from a pure gamete theory to one in which we have dominance 
with closely Mendelian proportions, but diversity of gametic constitution in the 
dominant quarter, which might not cease to be latent for two or more generations. 
Again, on still further increasing the number of determinants, we pass to what 
is clearly a close approximation to the normal curve of distribution. It seems 
desirable, therefore, to investigate further the general theory of p determinants 
each occurring in two or in 2g " tuned " chromosomes. 
(3) Returning to the fundamental formula (ii), which gives the distribution 
of protogenic determinants in the chromosomes of the germ cell, we note the 
complexity of the analysis required to deal further with it in its present shape. 
But we see that it is formed by the squares of the terms of a symmetrical 
binomial. Now for p even moderately large this binomial is extremely close to 
a normal curve, and accordingly the squares of the ordinates of this curve will 
represent extremely closely our series. But the squares of the ordinates of a 
normal curve are themselves ordinates of a normal curve. Thus the normal 
curve* which fits closely {\->r\y is y= -1- e-i^^V where o- = \/i {p + \)c^, 
and c is the unit of plotting. Very frequently V|j9C is taken for <t, and p being 
TT^ 0-020 7, 
Tr'a 0-653 7„ 
ir^a? 6-694 7„ 
7r=a» 24-163 7, 
ir'a* 36-939 7„ 
7r«a= 24-163 7„ 
7r-a« 6-694 7„ 
iroC 0-653 7„ 
a? 0-020 7, 
Phil. Trans. Vol. 18(i, A, p. 355. 
