88 071 a Mathematical Theory of Determinantal Inheritance 
In fact it would often be extremely difficult to distinguish between (ii) for 
/) = 4 and a true \, ^ and \ system of frequency. For example, in any breeding 
experiment yet made we may well doubt whether it has been on a scale 
sufficient to distinguish between 51 and 24 per cent, as compared between 50 
and 25 per cent. We are therefore justified in replacing (ii) by a normal 
curve for practical purposes. If we calculate areas we may take its standard 
deviation V|p; if we calculate ordinates we shall get a somewhat better result 
from 2)1 ^ut for p as large as 12, the results will agree within the errors 
of random sampling for practical series. 
We have now found a fairly simple analytical form for the distribution of 
determinants in the chromosome of a gametic cell of the hybrid. The probability 
that the chromosome of the gametic cell of the hybrid will have a number of 
determinants of protogenic character lying between ^p + x/c and ^p + (x + 8x)/c is 
1 1 
^■^l^iP'^Bx (iv)6^s. 
'\/27r V^pc 
The close resemblance of this result to + shows that the variability of 
the chromosomes in the number of protogenic determinants is practically confined 
to the range of ^p to ^p. More accurately, only two individuals per thousand 
would have more than |j9 + Vp protogenic determinants. 
Now the gametic cells of both hybrid parents will have a distribution of this 
character, and supposing random mating among the hybrids, the frequency of 
protogenic determinants in the zygotes will be given by a normal curve of standard 
deviation S where 
'^- = ^pc + lpc, or 2 = V4-p.c. 
Thus the frequency with which p + s protogenic and p — s allogenic deter- 
minants occur in the somatic cell chromosomes of the offspring of the hybrids 
is given by the normal curve 
y = ^ ^e-^-'"-'^^P'"^ (v) 
where s — xjc, 
(4) We have now to determine the constitution of the gametic cells of the 
offspring of the hybrids or to take away by the reducing division p determinants at 
random from p+s protogenic and^:»— s allogenic determinants. The result is reached 
at once by substituting in (i) of p. 83 the values i(, = p + s, v = p — s, r = p. But 
the complexity of the result would be great (for general theoretical purposes 
apart from special numerical illustration) if we did not use some close-fitting 
continuous curve for the series. Such a curve is provided for in my memoir 
" On Skew Variation in Homogeneous Material *." In the notation of that 
memoir we find 
8(p + l) ' ^^-2(p + l) 
* Phil. Trans. Vol. 18G, A, p. 361. 
