84 On a Mathematical Theory of Determinantal Inheritance 
this is raised to the power of '2q, since the two hybrid or heterogenic germ cells 
have the same constitution of chromosomes. 
Now several possibilities arise at this stage of the development of the theory : 
Suppose we have only a single chromosome in the germ cell, or two in the 
somatic cell, then p = 1 gives us the following condition of affairs : \ of the zygotes 
would consist of two chromosomes both with protogenic determinants, one-half of 
two chromosomes one with a protogenic and one with an allogenic determinant, and 
one-quarter with allogenic determinants alone. This is I take it pure Mendelism. 
The single determinant is the unit character and the protogenic and allogenic 
determinants are the allelomorphs. But a difficulty occurs in the fact that we 
must content ourselves with two chromosomes for most cases of inheritance. If we 
accept De Vries' view of an interchange of chromomeres bearing the determinants 
occurring at random between paternal and maternal chromosomes at the reducing 
division, then to reach pure Mendelism we must assert that only two chromosomes 
in the somatic cell, one in the germ cell, contain the inheritance factor for any 
character, and that this factor is built up of a single determinant. If we suppose 
for example that all the chromosomes, say 2^ of them carry the determinant, then 
our distribution would be {\ + \ and it is clear that a homozygote would only 
result among the crossings of hybrids in a percentage far below the simple 
Mendelian 25. Thus De Vries' explanation appears to involve the narrow differen- 
tiation of the chromosomes, two only being concerned with any single character ; 
or if more or all of them are to contain the determinant of any one character, 
then there must be some special mechanism to insure that all chromosomes at the 
random interchange of determinants before the reducing division interchange in 
precisely the same manner. We are not compelled to assume such a mechanism, 
which there is at present no cytological evidence in favour of, until it has been 
demonstrated that the experimental destruction, or observed absence of one or 
more chromosomes in a cell involves no loss of any series of characters. If a perfect 
zygote should develop from two germ cells, when half the nucleus of one has been 
removed, the problem of how De Vries' suggestion could lead to Mendelism would 
have to be further considered. Meanwhile let us proceed on the assumption, that 
there exist not a single Mendelian unit but p determinants for the character 
either in a single chromosome of the germ cell, or in q "tuned" chromosomes. 
The distribution of protogenic determinants in the two, or the " tuned," 
chromosomes of the somatic cells of the zygote will be given by the system of 
chances involved in the series 
The symbols tt and a here refer to protogenic and allogenic determinants ; 
the chance of s protogenic and s allogenic determinants occurring in the two 
chromosomes (s -f- s = 2^) or in the " tuned " q pairs of chromosomes, being given 
by picking out the numerical coefficient in the above expansion of 7r*a*'. 
I illustrate this on the first four cases. 
