A STUDY IN VARIATION: 
PUGNAX PUGNUS Martin, and PUGNAX cf. 
SULCATUS J. de C. Sow. 
BRIAN SIMPSON, M.Sc., UNIVERSITY COLLEGE, SWANSEA. 
This paper describes the results of a graphical and mathe- 
matical survey of a group of six hundred specimens of Pugnax. 
These were selected, because of their completeness, from a 
group of two thousand, collected by Mr. A. E. Alkins from 
a small area, and at one locality, in the Carboniferous Lime- 
stone of Castletoh, Derbyshire. The collection is now in the 
Department of Geology, at the University College of Swansea. 
A cursory examination of the group shows that considerable 
variation exists. 
The measurements were all made by a vernier calliper, 
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Ol 
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Fig. 1. Illustrating the terms used in the text. 
reading to the nearest .1 mm., and all the calculations obtained 
by slide rule. 
Four variants have been considered : 
height : from the umbone to the anterior margin of the shell. 
breadth : the greatest measurement parallel to the hinge line. 
thickness : at right angles to these two. 
depth : the greatest measurement at right angles to the hinge 
line and at right angles to the height. This measurement 
was found necessary in order to indicate the height of 
the ridge and the depth of the furrow. 
The percentage ratio of one variant to another has been 
worked out in four cases, namely : — 
JieigM x IO q thickness x IQ0 dej>th_ X I00 thickness x jqq 
breadth height breadth breadth 
and from the results obtained, frequency polygons and scatter 
diagrams have been constructed. 
Homogeneity of the Group. — From a short examination of 
the group it appeared that possibly two forms were represented. 
In order to test this, frequency polygons were constructed. 
In such curves, obvious and marked irregularities have, in 
many cases, been shown to indicate a discontinuous variation 
in the group, caused by the presence of one or more species 
ig33 March i 
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