[Illustration: FIG. 11. Diagrams of the geometrical relations which are generally exhibited by extra pairs of appendages in Arthropoda. The sections are supposed to be those of the apex of a tibia in a beetle. _A_, anterior, _P_, posterior, _D_, dorsal, _V_, ventral. _M_^{1}, _M_^{2} are the imaginary planes of reflexion. The shaded figure is in each case a limb formed like that of the other side of the body, and the outer unshaded figures are shaped like the normal for the side on which the appendages are. On the several radii are shown the extra pairs in their several possible relations to the normal from which they arise. The normal is drawn in thick lines in the center.] Next the geometrical relations of the halves of the supernumerary pair are determined by the position in which they stand in regard to the original appendage. These relations are best explained by the diagram (Fig. 11), from which it will be seen that the two supernumerary appendages stand as images of each other; and, of them, that which is adjacent to the normal appendage forms an image of it. Thus if the supernumerary pair arise from a point on the dorsal surface of the normal appendage, the two _ventral_ surfaces of the extra pair will face each other. If they arise on the anterior surface of the normal appendage, their morphologically posterior surfaces will be adjacent, and so on. These facts give us a view of the relations of the two halves of a dividing bud very different from that which is to be derived from the exclusive study of normal structures. Ordinary morphological conceptions no longer apply. The distribution of the parts shows that the bud or rudiment which becomes the supernumerary pair may break or open out in various ways according to its relations to the normal limb. Its planes of division are decided by its geometrical relations to the normal body. Especially curious are some of the cases in which the extra pair are imperfectly formed. The appearance produced is then that of two limbs in various stages of coalescence, though in reality of course they are stages of imperfect separation. The plane of "coalescence" may fall anywhere, and the two appendages may thus be compounded with each other much as an object partially immersed in mercury "compounds" with its optical image reflected from the surface.