1. The aim of the paper has been to show how far differences in proportions may safely be used in systematic grouping. The use of percentage sizes is unsatisfactory since they generally change with absolute size, and Huxley suggested the allometry formula as a basis for analysis. But no study has been made of the best statistical methods for (a) discovering the nature of the growth-relationship between two organs from small samples, (6) estimating the most probable values of Huxley's growth-coefficients and comparing these in different samples.
2. Difficulties in (a) and (b) are discussed. While it is usually impossible to determine the nature of the growth-relationship from small samples, the systematist can make use of the allometry formula if the points appear to show a linear trend when plotted on a double logarithmic grid. Reasons are given for supposing the normal regression methods applied to the logarithms of the measurements to be the best way of estimating the values of the growth-coefficients.
3. To illustrate the methods, measurements were made of lengths of maxilla;, nasals, and cranium in samples of the skulls of the three genera of Xenarthrous Anteaters, Myrmecophaga, Tamandua, and Cyclopes. Their adult facial indices and total skull-lengths are:–
Myrmecophaga: facial index=l-6, skull-length 36 to 38 cm.
Tamandua: facial index=0–80, skull-length 13 to 14 cm.
Cyclopes: facial index =0–5, skull-length up to 5 cm.
The systematic problem is “Do genetic differences directly affecting relative face-length intervene between the three genera, or are the laws of relative growth in face-length the same throughout ?”
4. Convenient changes in notation are suggested and the regression formulae described. Calculated values of a and their standard errors are given Table I. p. 59. t-tests applied to these suggest that Myrmecophaga has a higher relative growth-rate in face length than the other two genera, which do not differ very much. The initial growth index and its standard error are, however, found to be quite useless for purposes of comparison.
5. A better method of analysis is proposed. This consists in comparing the fit of a single line, separate lines, and parallel lines to the samples by an analysis of Residual Variance. Comparison of parallel lines through the sample means and separate lines will test differences in slope, and comparison of a single line with parallel lines will test differences in other factors, here termed “positional.”
6. These tests show no differences between the two samples of Tamandua measured by different observers. Applied to the three genera they show clearly that allometric differences intervene between them, both slope and positional differences being significantly large. Further tests show no significant differences in slope between Tamandua and Cyclopes, while positional differences between them appear only for the allometry of the maxillaæ. If this means that the snout has the same average relative growth-rate in both genera, then calculation shows that Tamandua has on the average maxillae 1–36 times and nasals 1–09 times as long as Cyclopes with the same cranium length. Difficulties are discussed in interpreting the test for positional differences when slope differences are significant.
7. Size-increase in the skull of Myrmecophaga compared with Tamandua appears to have been concentrated in length. The former has relatively a much slimmer rostrum and more sharply-pointed posterior projections to the nasals and anterior projections to the frontals.
8. Most of the main differences between the skulls of Cyclopes and the two larger genera are probably the result of the marked size-differences. The reduction in facio-cranial flexure and sharpness of the dorsal edges to the orbits, and the loss of the eoronoid process, which occur in the larger genera, may be due to this factor. The systematic value of these differences is therefore doubtful, and the evidence for classifying Cyclopes in a separate subfamily after Winge, or in a separate family after Pocock, is unsatisfactory. A return is proposed to the original classification of the group as three separate genera forming the family Myrmecophagidse.
9. These anteaters present the unusual case of a positive correlation between the relative growth-rate of an allometric organ and final size.