120 A First Study of Natural Selection in Clausilia laminata 
peripheral spiral at corresponding points of young and of adult shells. The proper 
measure of this variability is the standard deviation of the peripheral radii, in a 
number of individuals, measured at the same angular distance from the standard 
columellar radius, or the standard deviation of all the measures falling within a 
very narrow vertical strip of the diagram, Fig. 2. It would be impossible to find 
this standard deviation directly, without measuring many thousands of individuals ; 
and the labour involved in preparing even a small series of measures is so great 
that a sufficiently large series for the purpose seems unattainable. Some indirect 
way of comparing the variability of young and of adults must therefore be 
found. 
Before describing the method adopted in comparing the variability in young 
and in mature shells, a remark must be made. The variability of different parts 
of the peripheral spiral, as it appears from the distribution of the dots on Fig. 2, 
differs greatly in different regions. It is smallest in the immediate neighbourhood 
of the standard columellar radius, and increases as we pass away from this in 
either direction. The variability here indicated is not the variability of radii 
measured in different individuals at constant angular distatjce from the origin of 
the peripheral spiral : on the contrary, it is largely determined by the correlation 
between the different parts of the peripheral spiral and those radii of the 
columellar spiral which are nearly 5 mm. in length. The relative variability would 
be largely altered if we chose another columellar radius by which to determine 
the plane of reference. The artificial character of the variability which is here 
exhibited does not pi-event us from using it to determine the probable error of 
the values obtained, or to establish a comparison between two sets of individuals 
which have been treated in exactly the same way. 
If we consider any group of measures, such as those lying between two of the 
vertical divisions of Fig. 2, the straight line which gives the best linear approxima- 
tion to their distribution is 
y-yo = —I- - *o), 
where ?/„ is the mean radial length in the group, a-y the standard deviation of radial 
lengths ; Xf, is the mean angular distance of the group from the standard columellar 
radius, the standard deviation of angular distances, and r^y is Galton's function, 
measuring the correlation between radial length and angular distance from the 
standard radius*. The standard deviation of an " array " of radii within the group, 
whose angular distance is constant, is <ry Vl — r^,,; and this will be taken as the 
measure of the variability of spiral radii at any point. 
The values of r„j, ay, and Cy'^l-r^xy ft>r the six lowest groups of radii 
measured in the young shells, and for the corresponding groups of radii in adults, 
are given in Table IV. I wish here to express my gratitude to Miss Alice 
Lee, D.Sc, who has generously spent much time and labour in checking the whole 
* Cf. Yule; Roy. Soc. Proc. Vol. 60, pp. 477—489. 
