K. Pearson 
83 
nuclei may take place in the process of fertilisation, if we suppose the chromosomes 
to retain their individuality, there will be no interchange of chromomeres or their 
determinants between the allogenic and protogenic chromosomes in the zygote. 
Ordinary mitosis will now take place and somatic cells be indefinitely multiplied 
without interchange of determinants. Each hybrid cell will contain 2g chromosomes, 
q with p protogenic and q with p allogenic determinants. 
The nature of the somatic characters of the hybrid will not be determinable 
in any way i;nless we have some theory as to whether the protogenic or allogenic 
determinants are dominant, or blend, or learn by experiment or observation that 
they give rise to certain characters which may differ from those of either pure 
race. We have a hybrid somatic cell in which there is a balance in number of 
the two types of determinants. As long as each ordinary mitosis divides equally 
the number of each kind of determinant in this heterogenic cell, it will not matter 
whether we suppose individual chromosomes to consist solely of determinants of 
one race, or each chromosome to have p protogenic and p allogenic determinants. 
The important point is that this equality of division should be maintained up to 
the reducing division. We shall now suppose that on the reducing division the 
protogenic and allogenic chromosomes fuse and interchange determinants, i.e. there 
is a random selection of p determinants out of the total 2jj, p protogenic and 
p allogenic determinants. This is equivalent to drawing p balls out of a bag 
containing J) black (= protogenic) and p red (= allogenic) balls*. Or, the distribution 
of frequency of the several heterogenic chromosomes which contain p, j) — I, p — 2, 
p — 3,... protogenic determinants is given by the successive terms of: 
(2p) ! 
( 1 . 2 j [ 1.2.3 
.(ii). 
It is obvious that these terms are proportional to the squares of the binomial 
coefficients (I + 1)'*^. 
If we take a second heterogenic cell and give it a reducing division, the 
frequency of the protogenic determinants in the resulting germ cells will be also 
given by (ii). Fertilisation of a heterogenic germ cell of q chromosomes by a 
second will lead to the somatic cell of the offspring of the 'hybrids.' This somatic 
cell contains 2q chromosomes each containing p determinants, and the frequency 
of the number of protogenic determinants in these chromosomes will be given by 
the expression 
{(P}1 
\{2p)l 
p(p-i)Y \p(p-i)(p-2)Y 
.(iii); 
* It is well known that the chances of drawing r, r-1, r-2, r-'i ... black balls out of a bag con- 
taining II black and v red balls are the successive terms of the series : 
m! {u + v-r)\ ^ ^''(''-1) v(v-l) 
(" + (')! (M-)-) ! \ u-r + l^ 1.2 (if- )■+!) ((f-r + 2) 
r{r-l)(r-2) r (- - 1) (r - 2) , | 
1.2.3 («- r-i-l)(!(-/' + 2) (it-r + 3)^ [ ^ 
11—2 
