THE SOLDIERLESS TERMITES OF AFRICA 15 



The Latent roots (eigenvalues) and vectors (eigenvectors) of the transformed 

 matrix a are then found, scaling each vector so that the sum of squares of its elements 

 equals its corresponding latent root. These vector elements provide the co-ordinates 

 of the set of points representing OTU's (species) in relation to the orthogonal (un- 

 correlated) principal axes of the entire set, whilst preserving unaltered their taxo- 

 nomic distances, defined by ^2(1 — Sjj). The effect of the transformation to a is 

 to remove the tendency for the first vector of the un-transformed similarity matrix 

 to have more or less constant elements which allow for the mean value of all the 

 elements of S. 



In the present study, as is usual, it was found that the relationships between 

 the points could be adequately summarized in a few dimensions by restricting 

 consideration to vectors corresponding to the largest roots of a. This involved 

 deciding where the variation represented by the smaller latent roots became in- 

 significant and could be taken as residual. The first root took up 19% of the trace 

 of at, the second 9%, and the third, 6%, together comprising only 34% of the total 

 sum of squares. The next three roots together brought this to 48 % of the trace. 

 To include further dimensions graphically would make presentation of the results 

 unwieldy and more difficult to interpret. Although collectively the residual 

 dimensions account for slightly more than half the total information content of the 

 transformed matrix, individually their contributions are small. They are therefore 

 ignored although it is realized that there must then be some distortion of the relation- 

 ships represented by the taxonomic distances which become approximations. 



The matrix of similarity coefficients calculated for the majority of species is shown 

 in Text-figs 7, 7a. Each species included is given a serial number indicating its 

 position in the systematic order of this paper. Some few species were discovered, 

 and one or two synonymies decided after the completion of the similarity analysis. 

 It did not seem worthwhile to re-run the analysis for so small a number of changes 

 and a few species are therefore omitted or appear twice. 



The three methods used to examine the similarity matrix gave results that agree 

 in general but differ in detail. Many published exercises in numerical taxonomy 

 have been undertaken to test either the conclusions of an existing conventional 

 classification, or the numerical methods themselves by comparison with it. In the 

 present investigation no conventional classification existed, and attempts to develop 

 one without the analysis of similarity were unsuccessful. Nevertheless where the 

 groupings suggested by the numerical results appeared wrong by conventional 

 standards, they have been adjusted to conform to the latter. The generic classi- 

 fication presented here therefore results from a blend of numerical and conventional 

 methods. 



Certain genera stand out clearly, the most distinctive being Ateuchotermes (serial 

 numbers 36-42 inclusive) which is separated in both types of cluster analysis (Text- 

 figs 8 & 9) by a wide gap from its nearest neighbours. The species of this genus 

 all unite above 88% phenon level apart from A. tranquillns (Silvestri) (42). At 

 the time of the analysis this species was represented by incomplete material and many 

 test comparisons could not be made. Although separated by the cluster analyses, 

 it was placed close to its congeners in several dimensions of the principal co-ordinates 



