6 PROCEEDINGS OF THE NATIONAL MUSEUM vol. 124 



time no general solution to the problem was known. The transforma- 

 tion established by Gower (1966) now provides a simple and elegant 

 solution. We write di 3 for the average NFD value between individuals 

 i and j; we form a matrix (au) such that aii=ajj = and au= — K(du) 2 . 

 Let the row-means of this matrix be the vector (ai.), the columns 

 means (a.j), and the grand mean a..; we then form the matrix (b u ), 

 where bij = aij— ai.— a.j+a..l. The eigenvalues and eigenvectors of 

 this matrix are extracted and standardized so that the length of 

 each vector is equal to the value of its corresponding root. Gower 

 demonstrates that these vectors define a Euclidean space in such a 

 way that the distance between two individuals is equal to its original 

 dij value, and in such a way that the space has been reduced as 

 efficiently as is possible with a linear transformation. (The space is 

 not everywhere real, but this is of no importance in practice.) In 

 our case, three axes were found to suffice for the general configuration, 

 but any substantial deviations in the next three axes were noted. 



It is possible to simplify the configuration further by moving 

 overtly into the techniques of factor analysis. Given that three axes 

 are all that is required, the requirement is to reduce the values of 

 the principal diagonal of the "h,/' matrix so that the least possible 

 information remains in the matrix after the extraction of three 

 positive roots. The method is explained in standard books on factor 

 analysis (e.g., Cattell, 1952; Thomson, 1951); it is iterative and 

 somewhat time-consuming in computation. Automatic programs 

 exist on the Control Data Corporation 3600 computer at Canberra 

 for the basic ordination (program GOWER) and the factor-analysis 

 version (NEWGOWER). These two programs accept the upper 

 triangle of the original (du) matrix as a string of coefficients and 

 carry out all subsequent adjustments and calculations automatically. 



Material Examined 



All known American species of Portunus, Callinectes , 3 and Arenaeus 

 were examined. Also examined were: (1) certain Indo-West Pacific 

 species of Portunus, comprising P. pelagicus, P. sanguinolentus, P. 

 pubescens, P. convexus, and P. cf. trituberculatus; (2) the known non- 

 American species of Callinectes; and (3) Scylla serrata for comparison 

 with the distinctness of other genera. 



The species examined are listed in table 2. Extensive series of western 

 American forms were examined, as recorded in Garth and Stephenson 

 (1966), but fewer specimens of Atlantic species were seen, and there 

 were no critical examinations of difficult groups. Atlantic species were 



8 Since this paper has been completed, Williams (1966) has described a new 

 species of Callinectes, C. similis, which is commented upon later (p. 18) 



