The growth in width (
W
) of the segments of the abdomen relative to carapace size (
S
), and the graded distribution of growth along the abdomen, are analyzed by the method of fitting, to the observed values of
W
, polynomial regressions of progressively higher power in
S
. The simplest (linear) relation reveals the main features and each closer approximation furnishes further detail. The second object of the method, to select the lowest power of polynomial which adequately represents the data, gives the quadratic, though it is found that its adequacy varies in the different segments, which demand, for uniform adequacy, a non-affine set of polynomials. Adequacy is determined from the residual variance. The set of quadratics for the seven segments of the abdomen are combined, by a modification of Medawar’s transformation method, to give a single key relation which, within the scope of the data, defines abdomen width completely, spatially and temporally. This step involves the definition of the parameters of the quadratics as continuous functions of abdomen width at selected body size. It is suggested that the key relation to the transformation might, by analogy, be termed the ‘form-cinematogram’ for abdomen width. The equation: ‘form = shape+size’ is useful in the present context and is advocated for general recognition. The ‘shape-cinematogram’ may be derived from the form-cinematogram . Alternative attempts to derive a satisfactory form-cinematogram from the data are outlined. The form change is surprisingly simplified by the excision of the initial width measurements from all subsequent width measurements. The overall change in shape of the abdomen is visualized by the co-ordinate transformation method applied reciprocally between initial and final proportions.
A generalization of the quadratic cone of $$\mathop {\mathrm{PG}}(3,q^n)$$ and its relation with the affine set of the Lüneburg spread