1911-12.] Point Binomials and Mendelian Distributions. 401 
The first of these leads to groupings represented by the binomial or 
multinomial (l + l) p or (l-fO-fl)*, etc. The second leads to groupings 
represented by (3 + lp or (3 + 0 + 1)^, etc. The third to groupings repre- 
sented by such multinomials as (1 +0 + 2 + 0 + 0 + 1)^, where the number 
of zeros is unequal on the two sides of the middle significant term. 
The fourth are represented by the groupings 1 + 2(n — 1) + 1 and 
^_ 1 _(- 24 - 7 i— 1 , either alone or in combination, when n is a positive 
power of 2. 
1. Blending.— Blending is known in many instances, as when a red 
cow mating with a white gives a roan ; but of quantitative blending, to which 
the present analysis applies, I have not found many examples. One is 
given by H. M. Leake in the paper already referred to.* In this case he 
shows that when plants with a different leaf factor are mated the leaf 
factor of the cross is the arithmetical mean. When his diagram is ex- 
amined it is found that he mated plants of a certain range of leaf factor 
with others of a like range of higher value, with a distance between the two 
approximately equal to the range of each type. For a first approximation 
each type can be represented by the distribution 14- 2 4-1, so that the F 2 
generation will be given by 
( 1+2 + 1+0 + 0 + 1 + 2 + 1 ) 2 , 
or 
1 + 4 + 6 + 4 + 1 + 2 + 8 + 12 + 84-2 + 1 + 44-6 + 4 + 1 . 
This is approximately what is obtained on experiment, but there is more 
compression, so that, although in fact the units of each type fell inside the 
limits chosen, many probably contained elements which, when pure, ranged 
beyond these limits. If we write, instead of the above, 
(1 + 2 + 1 + 0+1 + 2 + 1 ) 2 , 
or 
1+4 + 6 + 4 + 3 + 8+12 + 8 + 3 + 4 + 6 + 4 + 1 , 
the result will be nearly that shown by the diagram. 
2. Dominance . — If dominance exist, the proportions in which the 
numbers will turn up is as 3 to 1, so that the general formula for a mixed 
dominance will be (3-}-l) ?71 (l + 3) n . This in general approximates to 
Type IV. Data on which to make analysis are very scarce, so scarce that 
I have not thought it worth while to carry out the arithmetical work 
necessary to test individual cases. There is, however, one case given by 
Mr Leake in the Journal of Genetics ( loc . cit.), concerning the length of the 
vegetative period in hybrid cottons, which is sufficiently interesting to 
require an approximate solution. Here dominance obviously comes specially 
* Journal of Genetics , vol. i. p. 227. 
VOL. XXXII. 
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