its surface. ãWhere the flagellum was pretty long and active, its efforts to continue movement under these abnormal conditions resulted in the gradual lifting up from the cytoplasm of the body of a sort of pseudo-undulating membrane (Fig. 92). The movements of this structure were so exactly those of a true undulating membrane that it was {269} difficult to believe one was not dealing with a small, blunt trypanosome53.ã This in short is a precise description of the mode of development which, from theoretical conôÙsiôÙdeôÙraôÙtions alone, we should conceive to be the natural if not the only possible way in which the undulating membrane could come into existence.
There is a genus closely allied to Trypanosoma, viz. Trypanoplasma, which possesses one free flagellum, together with an undulating membrane; and it resembles the neighbouring genus Bodo, save that the latter has two flagella and no undulating membrane. In like manner, Trypanosoma so closely resembles Herpetomonas that, when individuals ascribed to the former genus exhibit a free flagellum only, they are said to be in the ãHerpetomonas stage.ã In short all through the order, we have pairs of genera, which are presumed to be separate and distinct, viz. Trypanosoma-Herpetomonas, Trypanoplasma-Bodo, Trichomastix-Trichomonas, in which one differs from the other mainly if not solely in the fact that a free flagellum in the one is replaced by an undulating membrane in the other. We can scarcely doubt that the two structures are essentially one and the same.
The undulating membrane of a Trypanosome, then, according to our interpretation of it, is a liquid film and must obey the law of constant mean curvature. It is under curious limitations of freedom: for by one border it is attached to the comparatively motionless body, while its free border is constituted by a flagellum which retains its activity and is being constantly thrown, like the lash of a whip, into wavy curves. It follows that the membrane, for every alteration of its longitudinal curvature, must at the same instant become curved in a direction perpendicular thereto; it bends, not as a tape bends, but with the accompaniment of beautiful but tiny waves of double curvature, all tending towards the establishment of an ãequipotential surfaceã; and its charôÙacôÙterôÙisôÙtic undulations are not originated by an active mobility of the membrane but are due to the molecular tensions which produce the very same result in a soap-film under similar circumstances.
In certain Spirochaetes, S. anodontae (Fig. 90) and S. balbiani {270} (which we find in oysters), a very similar undulating membrane exists, but it is coiled in a regular spiral round the body of the cell. It forms a ãscrew-surface,ã or helicoid, and, though we might think that nothing could well be more curved, yet its mathôÙeôÙmatôÙiôÙcal properties are such that it constitutes a ãruled surfaceã whose ãmean curvatureã is everywhere nil; and this property (as we have seen) it shares with the plane, and with the plane alone. Precisely such a surface, and of exquisite beauty, may be produced by bending a wire upon itself so that part forms an