R. 0. PUNNBTT 
345 
that the generalized theory here dealt with is not elastic enough to account for 
the numerical values of the constants of heredity hitherto observed" ('04, p. 86). 
These theoretical values do however agree closely with the average values found 
for Spinax, and it must not be forgotten that we are here dealing with obviously 
discontinuous characters capable of simple numerical appreciation, and differing 
in nature, as far as we can at present judge, from any yet investigated. It may 
well be that the heredity of such characters is governed by Mendelian principles, 
though at present these principles are impossible of demonstration, owing to the 
existence of more than one pair of simple, or perhaps also of compound allelo- 
morphs. To account for such families as that comprising embryos Nos. 2G0 — 270 
is difficult except on an hypothesis of dominance, or on Pearson's hypothesis of 
"unit prepotency" ('03, p. 389). Here a family of 11 embryos of both sexes 
exhibits without exception extreme backward homoeosis, whilst the $ parent is 
characterized by marked forward homoeosis. It seems most natural to account for 
such a case by supposing that the unknown parent showed extreme backward 
homoeosis, and that this condition was dominant over that of marked forward 
homoeosis. The data are however too scanty and the possibilities too numerous 
to admit of profitable discussion on these points at present. More is doubtless 
to be learnt by selecting some simpler case in which the breeding can be easily 
controlled. This much new knowledge has at any rate been gained by the 
application of biometric methods, for the magnitude of the parental and fraternal 
correlations cannot but mean that the varying number of units in a primary 
linear meristic series does not depend alone on the individual environment, but 
that it is a character transmissible from one generation to another. 
IV. Summary. 
The chief results obtained from an examination of certain meristic characters 
in 567 specimens of Spinax iiiger may be briefly summarized as follows : — 
(1) A well-marked sexual dimorjjhism permeates the meristic series, the (/s 
showing a greater tendency to backward homoeosis than the $ s. 
(2) Equal numbers of J's and $ s are born but the evidence derived from the 
numbers caught points to a preponderance of $ s in the adult state. 
(3) The variability of the </" embryos (as measured by the standard deviation) 
is markedly higher than that of the cT adults. In the case of the $ s the variability 
is not very different in embryos and adults. This points to more stringent selection 
among the (/s and accords well with the circumstance of their relative scarcity 
when adult. 
(4) A comparison of the variability in different parts of the meristic series tells 
strongly against the hypotheses of vertebral intercalation or excalation ; and this 
holds good either on the view that such processes occur uniformly, or that they 
Biometrika iii 44 
