An exceptional form of cell, but a beautiful manifestation of surface-tension (or so I take it to be), occurs in Trypanosomes, those tiny parasites of the blood that are associated with sleeping-sickness and many other grave or dire maladies. These tiny organisms consist of elongated solitary cells down one side of which runs a very delicate frill, or “undulating membrane,” the free edge of which is seen to be slightly thickened, and the whole of {267} which undergoes rhythmical and beautiful wavy movements. When certain Trypanosomes are artificially cultivated (for instance T. rotatorium, from the blood of the frog), phases of growth are witnessed in which the organism has no undulating membrane, but possesses a long cilium or “flagellum,” springing from near the front end, and exceeding the whole body in length[313]. Again, in T. lewisii, when it reproduces by “multiple fission,” the products of this division are likewise devoid of an undulating membrane, but are provided with a long free flagellum[314]. It is a plausible assumption to suppose that, as the flagellum waves about, it comes to lie near and parallel to the body of the cell, and that the frill or undulating membrane is formed by the clear, fluid protoplasm of the surface layer springing up in a film to run up and along the flagellum, just as a soap-film would be formed in similar circumstances.

Fig. 91. A, Trichomonas muris, Hartmann; B, Trichomastix serpentis, Dobell; C, Trichomonas angusta, Alexeieff. (After Kofoid.)

This mode of formation of the undulating membrane or frill appears to be confirmed by the appearances shewn in Fig. [91]. {268} Here we have three little organisms closely allied to the ordinary Trypanosomes, of which one, Trichomastix (B), possesses four flagella, and the other two, Trichomonas, apparently three only: the two latter possess the frill, which is lacking in the first[315]. But it is impossible to doubt that when the frill is present (as in A and C), its outer edge is constituted by the apparently missing flagellum (a), which has become attached to the body of the creature at the point c, near its posterior end; and all along its course, the superficial protoplasm has been drawn out into a film, between the flagellum (a) and the adjacent surface or edge of the body (b).

Fig. 92. Her­pe­to­mo­nas as­sum­ing the un­du­la­tory mem­brane of a Try­pa­no­some. (After D. L. Mac­kin­non.)

Moreover, this mode of formation has been ac­tual­ly wit­nessed and de­scribed, though in a some­what ex­cep­tional case. The little fla­gel­late monad Her­pe­to­mo­nas is nor­mal­ly des­ti­tute of an un­du­la­ting membrane, but possesses a single long terminal flagellum. According to Dr D. L. Mackinnon, the cyto­plasm in a certain stage of growth becomes somewhat “sticky,” a phrase which we may in all probability interpret to mean that its surface tension is being reduced. For this stickiness is shewn in two ways. In the first place, the long body, in the course of its various bending movements, is apt to adhere head to tail (so to speak), giving a rounded or sometimes annular form to the organism, such as has also been described in certain species or stages of Trypanosomes. But again, the long flagellum, if it get bent backwards upon the body, tends to adhere to its surface. “Where 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 trypanosome[316].” This in short is a precise description of the mode of development which, from theoretical con­si­de­ra­tions 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 “Herpetomonas stage.” 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 “equipotential surface”; and its char­ac­ter­is­tic 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 “screw-surface,” or helicoid, and, though we might think that nothing could well be more curved, yet its math­e­mat­i­cal properties are such that it constitutes a “ruled surface” whose “mean curvature” 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 axial rod and part a spiral wrapping round the axis, and then dipping the whole into a soapy solution.