Chapter V: The Forms of Cells (2)

Before we pass to biological illustrations of our surface-tension figures, we have still another preliminary matter to deal with. We have seen from our description of two of Plateau’s classical experiments, that at some particular point one type of surface gives place to another; and again, we know that, when we draw out our soap-bubble into and then beyond a cylinder, there comes a certain definite point at which our bubble breaks in two, and leaves us with two bubbles of which each is a sphere, or a portion of a sphere. In short there are certain definite limits to the _dimensions_ of our figures, within which limits equilibrium is stable but at which it becomes unstable, and above which it {226} breaks down. Moreover in our composite surfaces, when the cylinder for instance is capped by two spherical cups or lenticular discs, there is a well-defined ratio which regulates their respective curvatures, and therefore their respective dimensions. These two matters we may deal with together.

Let us imagine a liquid drop which by appropriate conditions has been made to assume the form of a cylinder; we have already seen that its ends will be terminated by portions of spheres. Since one and the same liquid film covers the sides and ends of the drop (or since one and the same delicate membrane encloses the sides and ends of the cell), we assume the surface-tension (_T_) to be everywhere identical; and it follows, since the internal fluid-pressure is also everywhere identical, that the expression (1/_R_ + 1/_R′_) for the cylinder is equal to the corresponding expression, which we may call (1/_r_ + 1/_r′_), in the case of the terminal spheres. But in the cylinder 1/_R′_ = 0, and in the sphere 1/_r_ = 1/_r′_. Therefore our relation of equality becomes 1/_R_ = 2/_r_, or _r_ = 2 _R_; that is to say, the sphere in question has just twice the radius of the cylinder of which it forms a cap.

And if _Ob_, the radius of the sphere, be equal to twice the radius (_Oa_) of the cylinder, it follows that the angle _aOb_ is an angle of 60°, and _bOc_ is also an angle of 60°; that is to say, the arc _bc_ is equal to (1/3) π. In other words, the spherical disc which (under the given conditions) caps our cylinder, is not a portion taken at haphazard, but is neither more nor less than that portion of a sphere which is subtended by a cone of 60°. Moreover, it is plain that the height of the spherical cap, _de_,

= _Ob_ − _ab_ = _R_ (2 − √3) = 0·27 _R_,

where _R_ is the radius of our cylinder, or one-half the radius of our spherical cap: in other words the normal height of the spherical cap over the end of the cylindrical cell is just a very little more than one-eighth of the diameter of the cylinder, or of the radius of the {227} sphere. And these are the proportions which we recognise, under normal circumstances, in such a case as the cylindrical cell of Spirogyra where its free end is capped by a portion of a sphere.

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Among the many important theoretical discoveries which we owe to Plateau, one to which we have just referred is of peculiar importance: namely that, with the exception of the sphere and the plane, the surfaces with which we have been dealing are only in complete equilibrium within certain dimensional limits, or in other words, have a certain definite limit of stability; only the plane and the sphere, or any portions of a sphere, are perfectly stable, because they are perfectly symmetrical, figures. For experimental demonstration, the case of the cylinder is the simplest. If we produce a liquid film having the form of a cylinder, either by

drawing out a bubble or by supporting between two rings a globule of oil, the experiment proceeds easily until the length of the cylinder becomes just about three times as great as its diameter. But somewhere about this limit the cylinder alters its form; it begins to narrow at the waist, so passing into an unduloid, and the deformation progresses quickly until at last our cylinder breaks in two, and its two halves assume a spherical form. It is found, by theoretical considerations, that the precise limit of stability is at the point when the length of the cylinder is exactly equal to its circumference, that is to say, when _L_ = 2π_R_, or when the ratio of length to diameter is represented by π.

In the case of the catenoid, Plateau’s experimental procedure was as follows. To support his globule of oil (in, as usual, a mixture of alcohol and water of its own specific gravity), he used {228} a pair of metal rings, which happened to have a diameter of 71 millimetres; and, in a series of experiments, he set these rings apart at distances of 55, 49, 47, 45, and 43 mm. successively. In each case he began by bringing his oil-globule into a cylindrical form, by sucking superfluous oil out of the drop until this result was attained; and always, for the reason with which we are now acquainted, the cylindrical sides were associated with spherical ends to the cylinder. On continuing to withdraw oil in the hope of converting these spherical ends into planes, he found, naturally, that the sides of the cylinder drew in to form a concave surface; but it was by no means easy to get the extremities actually plane: and unless they were so, thus indicating that the surface-pressure of the drop was nil, the curvature of the sides could not be that of a catenoid. For in the first experiment, when the rings were 55 mm. apart, as soon as the convexity of the ends was to a certain extent diminished, it spontaneously increased again; and the transverse constriction of the globule correspondingly deepened, until at a certain point equilibrium set in anew. Indeed, the more oil he removed, the more convex became the ends, until at last the increasing transverse constriction led to the breaking of the oil-globule into two. In the third experiment, when the rings were 47 mm. apart, it was easy to obtain end-surfaces that were actually plane, and they remained so even though more oil was withdrawn, the transverse constriction deepening accordingly. Only after a considerable amount of oil had been sucked up did the plane terminal surface become gradually convex, and presently the narrow waist, narrowing more and more, broke across in the usual way. Finally in the fifth experiment, where the rings were still nearer together, it was again possible to bring the ends of the oil-globule to a plane surface, as in the third and fourth experiments, and to keep this surface plane in spite of some continued withdrawal of oil. But very soon the ends became gradually concave, and the concavity deepened as more and more oil was withdrawn, until at a certain limit, the whole oil-globule broke up in general disruption.

We learn from this that the limiting size of the catenoid was reached when the distance of the supporting rings was to their diameter as 47 to 71, or, as nearly as possible, as two to three; {229} and as a matter of fact it can be shewn that 2/3 is the true theoretical value. Above this limit of 2/3, the inevitable convexity of the end-surfaces shows that a positive pressure inwards is being exerted by the surface film, and this teaches us that the sides of the figure actually constitute not a catenoid but an unduloid, whose spontaneous changes tend to a form of greater stability. Below the 2/3 limit the catenoid surface is essentially unstable, and the form into which it passes under certain conditions of disturbance such as that of the excessive withdrawal of oil, is that of a nodoid (Fig. 65A).

The unduloid has certain peculiar properties as regards its limitations of stability. But as to these we need mention two facts only: (1) that when the unduloid, which we produce with our soap-bubble or our oil-globule, consists of the figure containing a complete constriction, it has somewhat wide limits of stability; but (2) if it contain the swollen portion, then equilibrium is limited to the condition that the figure consists simply of one complete unduloid, that is to say that its ends are constituted by the narrowest portions, and its middle by the widest portion of the entire curve. The theoretical proof of this latter fact is difficult, but if we take the proof for granted, the fact will serve to throw light on what we have learned regarding the stability of the cylinder. For, when we remember that the meridional section of our unduloid is generated by the rolling of an ellipse upon a straight line in its own plane, we shall easily see that the length of the entire unduloid is equal to the circumference of the generating ellipse. As the unduloid becomes less and less sinuous in outline, it gradually approaches, and in time reaches, the form of a cylinder; and correspondingly, the ellipse which generated it has its foci more and more approximated until it passes into a circle. The cylinder of a length equal to the circumference of its generating circle is therefore precisely homologous to an unduloid whose length is equal to the circumference of its generating ellipse; and this is just what we recognise as constituting one complete segment of the unduloid.

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While the figures of equilibrium which are at the same time surfaces of revolution are only six in number, there is an infinite {230} number of figures of equilibrium, that is to say of surfaces of constant mean curvature, which are not surfaces of revolution; and it can be shewn mathematically that any given contour can be occupied by a finite portion of some one such surface, in stable equilibrium. The experimental verification of this theorem lies in the simple fact (already noted) that however we may bend a wire into a closed curve, plane or not plane, we may always, under appropriate precautions, fill the entire area with an unbroken film.

Of the regular figures of equilibrium, that is to say surfaces of constant mean curvature, apart from the surfaces of revolution which we have discussed, the helicoid spiral is the most interesting to the biologist. This is a helicoid generated by a straight line perpendicular to an axis, about which it turns at a uniform rate while at the same time it slides, also uniformly, along this same axis. At any point in this surface, the curvatures are equal and of opposite sign, and the sum of the curvatures is accordingly nil. Among what are called “ruled surfaces” (which we may describe as surfaces capable of being defined by a system of stretched strings), the plane and the helicoid are the only two whose mean curvature is null, while the cylinder is the only one whose curvature is finite and constant. As this simplest of helicoids corresponds, in three dimensions, to what in two dimensions is merely a plane (the latter being generated by the rotation of a straight line about an axis without the superadded gliding motion which generates the helicoid), so there are other and much more complicated helicoids which correspond to the sphere, the unduloid and the rest of our figures of revolution, the generating planes of these latter being supposed to wind spirally about an axis. In the case of the cylinder it is obvious that the resulting figure is indistinguishable from the cylinder itself. In the case of the unduloid we obtain a grooved spiral, such as we may meet with in nature (for instance in Spirochætes, _Bodo gracilis_, etc.), and which accordingly it is of interest to us to be able to recognise as a surface of minimal area or constant curvature.

The foregoing considerations deal with a small part only of the theory of surface tension, or of capillarity: with that part, namely, which relates to the forms of surface which are {231} capable of subsisting in equilibrium under the action of that force, either of itself or subject to certain simple constraints. And as yet we have limited ourselves to the case of a single surface, or of a single drop or bubble, leaving to another occasion a discussion of the forms assumed when such drops or vesicles meet and combine together. In short, what we have said may help us to understand the form of a _cell_,—considered, as with certain limitations we may legitimately consider it, as a liquid drop or liquid vesicle; the conformation of a _tissue_ or cell-aggregate must be dealt with in the light of another series of theoretical considerations. In both cases, we can do no more than touch upon the fringe of a large and difficult subject. There are many forms capable of realisation under surface tension, and many of them doubtless to be recognised among organisms, which we cannot touch upon in this elementary account. The subject is a very general one; it is, in its essence, more mathematical than physical; it is part of the mathematics of surfaces, and only comes into relation with surface tension, because this physical phenomenon illustrates and exemplifies, in a concrete way, most of the simple and symmetrical conditions with which the general mathematical theory is capable of dealing. And before we pass to illustrate by biological examples the physical phenomena which we have described, we must be careful to remember that the physical conditions which we have hitherto presupposed will never be wholly realised in the organic cell. Its substance will never be a perfect fluid, and hence equilibrium will be more or less slowly reached; its surface will seldom be perfectly homogeneous, and therefore equilibrium will (in the fluid condition) seldom be perfectly attained; it will very often, or generally, be the seat of other forces, symmetrical or unsymmetrical; and all these causes will more or less perturb the effects of surface tension acting by itself. But we shall find that, on the whole, these effects of surface tension though modified are not obliterated nor even masked; and accordingly the phenomena to which I have devoted the foregoing pages will be found manifestly recurring and repeating themselves among the phenomena of the organic cell.

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In a spider’s web we find exemplified several of the principles {232} of surface tension which we have now explained. The thread is formed out of the fluid secretion of a gland, and issues from the body as a semi-fluid cylinder, that is to say in the form of a surface of equilibrium, the force of expulsion giving it its elongation and that of surface tension giving it its circular section. It is prevented, by almost immediate solidification on exposure to the air, from breaking up into separate drops or spherules, as it would otherwise tend to do as soon as the length of the cylinder had passed its limit of stability. But it is otherwise with the sticky secretion which, coming from another gland, is simultaneously poured over the issuing thread when it is to form the spiral portion of the web. This latter secretion is more fluid than the first, and retains its fluidity for a very much longer time, finally drying up after several hours. By capillarity it “wets” the thread, spreading itself over it in an even film, which film is now itself a cylinder. But this liquid cylinder has its limit of stability when its length equals its own circumference, and therefore just at the points so defined it tends to disrupt into separate segments: or rather, in the actual case, at points somewhat more distant, owing to the imperfect fluidity of the viscous film, and still more to the frictional drag upon it of the inner solid cylinder, or thread, with which it is in contact. The cylinder disrupts in the usual manner, passing first into the wavy outline of an unduloid, whose swollen portions swell more and more till the contracted parts break asunder, and we arrive at a series of spherical drops or beads, of equal size, strung at equal intervals along the thread. If we try to spread varnish over a thin stretched wire, we produce automatically the same identical result[288]; unless our varnish be such as to dry almost instantaneously, it gathers into beads, and do what we can, we fail to spread it smooth. It follows that, according to the viscidity and drying power of the varnish, the process may stop or seem to stop at any point short of the formation of the perfect spherules; it is quite possible, therefore, that as our final stage we may only obtain half-formed beads, or the wavy outline of an unduloid. The formation of the beads may be facilitated or hastened by jerking the stretched thread, as the spider actually does: the {233} effect of the jerk being to disturb and destroy the unstable equilibrium of the viscid cylinder[289]. Another very curious phenomenon here presents itself.

In Plateau’s experimental separation of a cylinder of oil into two spherical portions, it was noticed that, when contact was nearly broken, that is to say when the narrow neck of the unduloid had become very thin, the two spherical bullae, instead of absorbing the fluid out of the narrow neck into themselves as they had done with the preceding portion, drew out this small remaining part of the liquid into a thin thread as they completed their spherical form and consequently receded from one another: the reason being that, after the thread or “neck” has reached a certain tenuity, the internal friction of the fluid prevents or retards its rapid exit from the little thread to the adjacent spherule. It is for the same reason that we are able to draw a glass rod or tube, which we have heated in the middle, into a long and uniform cylinder or thread, by quickly separating the two ends. But in the case of the glass rod, the long thin intermediate cylinder quickly cools and solidifies, while in the ordinary separation of a liquid cylinder the corresponding intermediate cylinder remains liquid; and therefore, like any other liquid cylinder, it is liable to break up, provided that its dimensions exceed the normal limit of stability. And its length is generally such that it breaks at two points, thus leaving two terminal portions continuous with the spheres and becoming confluent with these, and one median portion which resolves itself into a comparatively tiny spherical drop, midway between the original and larger two. Occasionally, the same process of formation of a connecting thread repeats itself a second time, between the small intermediate spherule and the large spheres; and in this case we obviously obtain two additional spherules, still smaller in size, and lying one on either side of our first little one. This whole phenomenon, of equal and regularly interspaced beads, often with little beads regularly interspaced between the larger ones, and possibly also even a third series of still smaller beads regularly intercalated, may be easily observed in a spider’s web, such as that of _Epeira_, very often with beautiful regularity,—which {234} naturally, however, is sometimes interrupted and disturbed owing to a slight want of homogeneity in the secreted fluid; and the same phenomenon is repeated on a grosser scale when the web is bespangled with dew, and every thread bestrung with pearls innumerable. To the older naturalists, these regularly arranged and beautifully formed globules on the spider’s web were a cause of great wonder and admiration. Blackwall, counting some twenty globules in a tenth of an inch, calculated that a large garden-spider’s web comprised about 120,000 globules; the net was spun and finished in about forty minutes, and Blackwall was evidently filled with astonishment at the skill and quickness with which the spider manufactured these little beads. And no wonder, for according to the above estimate they had to be made at the rate of about 50 per second[290].

The little delicate beads which stud the long thin pseudopodia of a foraminifer, such as _Gromia_, or which in like manner appear upon the cylindrical film of protoplasm which covers the long radiating spicules of _Globigerina_, represent an identical phenomenon. Indeed there are many cases, in which we may study in a protoplasmic filament the whole process of formation of such beads. If we squeeze out on to a slide the viscid contents of a mistletoe berry, the long sticky threads into which the substance runs shew the whole phenomenon particularly well. Another way to demonstrate it was noticed many years ago by Hofmeister and afterwards explained by Berthold. The hairs of certain water-plants, such as Hydrocharis or Trianea, constitute very long cylindrical cells, the protoplasm being supported, and maintained in equilibrium by its contact with the cell-wall. But if we immerse the filament in some dense fluid, a little sugar-solution for instance, or dilute glycerine, the cell-sap tends to diffuse outwards, the protoplasm parts company with its surrounding and supporting wall, {235} and lies free as a protoplasmic cylinder in the interior of the cell. Thereupon it immediately shews signs of instability, and commences to disrupt. It tends to gather into spheres, which however, as in our illustration, may be prevented by their narrow quarters from assuming the complete spherical form; and in between these spheres, we have more or less regularly alternate ones, of smaller size[291]. Similar, but less regular, beads or droplets may be caused to appear, under stimulation by an alternating current, in the protoplasmic threads within the living cells of the hairs of Tradescantia. The explanation usually given is, that the viscosity of the protoplasm is reduced, or its fluidity increased; but an increase of the surface tension would seem a more likely reason[292].

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We may take note here of a remarkable series of phenomena, which, though they seem at first sight to be of a very different order, are closely related to the phenomena which attend and which bring about the breaking-up of a liquid cylinder or thread.

In some of Mr Worthington’s most beautiful experiments on {236} splashes, it was found that the fall of a round pebble into water from a considerable height, caused the rise of a filmy sheet of water in the form of a cup or cylinder; and the edge of this cylindrical film tended to be cut up into alternate lobes and notches, and the prominent lobes or “jets” tended, in more extreme cases, to break off or to break up into spherical beads (Fig. 70)[293]. A precisely similar appearance is seen, on a great scale, in the thin edge of a breaking wave: when the smooth cylindrical edge, at a given moment, shoots out an array of tiny jets which break up into the droplets which constitute “spray” (Fig. 71, _a_, _b_). We are at once reminded of the beautifully symmetrical notching on the calycles of many hydroids, which little cups before they became stiff and rigid had begun their existence as liquid or semi-liquid films.

The phenomenon is two-fold. In the first place, the edge of our tubular or crater-like film forms a liquid ring or annulus, which is closely comparable with the liquid thread or cylinder which we have just been considering, if only we conceive the thread to be bent round into the ring. And accordingly, just as the thread spontaneously segments, first into an unduloid, and then into separate spherical drops, so likewise will the edge of our annulus tend to do. This phase of notching, or beading, of the edge of the film is beautifully seen in many of Worthington’s experiments[294]. In the second place, the very fact of the rising of the crater means that liquid is flowing up from below towards the rim; and the segmentation of the rim means that channels of easier flow are {237} created, along which the liquid is led, or is driven, into the protuberances: and these are thus exaggerated into the jets or arms which are sometimes so conspicuous at the edge of the crater. In short, any film or film-like cup, fluid or semi-fluid in its consistency, will, like the straight liquid cylinder, be unstable: and its instability will manifest itself (among other ways) in a tendency to segmentation or notching of the edge; and just such a peripheral notching is a conspicuous feature of many minute organic cup-like structures. In the case of the hydroid calycle (Fig. 72), we are led to the conclusion that the two common and conspicuous features of notching or indentation of the cup, and of constriction or annulation of the long cylindrical stem, are phenomena of the same order and are due to surface-tension in both cases alike.

Another phenomenon displayed in the same experiments is the formation of a rope-like or cord-like thickening of the edge of the annulus. This is due to the more or less sudden checking at the rim of the flow of liquid rising from below: and a similar peripheral thickening is frequently seen, not only in some of our hydroid cups, but in many Vorticellas (cf. Fig. 75), and other organic cup-like conformations. A perusal of Mr Worthington’s book will soon suggest that these are not the only manifestations of surface-tension in connection with splashes which present curious resemblances and analogies to phenomena of organic form.

The phenomena of an ordinary liquid splash are so swiftly {238} transitory that their study is only rendered possible by “instantaneous” photography: but this excessive rapidity is not an essential part of the phenomenon. For instance, we can repeat and demonstrate many of the simpler phenomena, in a permanent or quasi-permanent form, by splashing water on to a surface of dry sand, or by firing a bullet into a soft metal target. There is nothing, then, to prevent a slow and lasting manifestation, in a viscous medium such as a protoplasmic organism, of phenomena which appear and disappear with prodigious rapidity in a more mobile liquid. Nor is there anything peculiar in the “splash” itself; it is simply a convenient method of setting up certain motions or currents, and producing certain surface-forms, in a liquid medium,—or even in such an extremely imperfect fluid as is represented (in another series of experiments) by a bed of sand. Accordingly, we have a large range of possible conditions under which the organism might conceivably display configurations analogous to, or identical with, those which Mr Worthington has shewn us how to exhibit by one particular experimental method.

To one who has watched the potter at his wheel, it is plain that the potter’s thumb, like the glass-blower’s blast of air, depends for its efficacy upon the physical properties of the medium on which it operates, which for the time being is essentially a fluid. The cup and the saucer, like the tube and the bulb, display (in their simple and primitive forms) beautiful surfaces of equilibrium as manifested under certain limiting conditions. They are neither more nor less than glorified “splashes,” formed slowly, under conditions of restraint which enhance or reveal their mathematical symmetry. We have seen, and we shall see again before we are done, that the art of the glass-blower is full of lessons for the naturalist as also for the physicist: illustrating as it does the development of a host of mathematical configurations and organic conformations which depend essentially on the establishment of a constant and uniform pressure within a _closed_ elastic shell or fluid envelope. In like manner the potter’s art illustrates the somewhat obscurer and more complex problems (scarcely less frequent in biology) of a figure of equilibrium which is an _open_ surface, or solid, of revolution. It is clear, at the same time, that the two series of problems are closely akin; for the {239} glass-blower can make most things that the potter makes, by cutting off _portions_ of his hollow ware. And besides, when this fails, and the glass-blower, ceasing to blow, begins to use his rod to trim the sides or turn the edges of wineglass or of beaker, he is merely borrowing a trick from the craft of the potter.

It would be venturesome indeed to extend our comparison with these liquid surface-tension phenomena from the cup or calycle of the hydrozoon to the little hydroid polype within: and yet I feel convinced that there is something to be learned by such a comparison, though not without much detailed consideration and mathematical study of the surfaces concerned. The cylindrical body of the tiny polype, the jet-like row of tentacles, the beaded annulations which these tentacles exhibit, the web-like film which sometimes (when they stand a little way apart) conjoins their bases, the thin annular film of tissue which surrounds the little organism’s mouth, and the manner in which this annular “peristome” contracts[295], like a shrinking soap-bubble, to close the aperture, are every one of them features to which we may find a singular and striking parallel in the surface-tension phenomena which Mr Worthington has illustrated and demonstrated in the case of the splash.

Here however, we may freely confess that we are for the present on the uncertain ground of suggestion and conjecture; and so must we remain, in regard to many other simple and symmetrical organic forms, until their form and dynamical stability shall have been investigated by the mathematician: in other words, until the mathematicians shall have become persuaded that there is an immense unworked field wherein they may labour, in the detailed study of organic form.

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According to Plateau, the viscidity of the liquid, while it helps to retard the breaking up of the cylinder and so increases the length of the segments beyond that which theory demands, has nevertheless less influence in this direction than we might have expected. On the other hand, any external support or adhesion, such as contact with a solid body, will be equivalent to a reduction of surface-tension and so will very greatly increase the {240} stability of our cylinder. It is for this reason that the mercury in our thermometer tubes does not as a rule separate into drops, though it occasionally does so, much to our inconvenience. And again it is for this reason that the protoplasm in a long and growing tubular or cylindrical cell does not necessarily divide into separate cells and internodes, until the length of these far exceeds the theoretic limits. Of course however and whenever it does so, we must, without ever excluding the agency of surface tension, remember that there may be other forces affecting the latter, and accelerating or retarding that manifestation of surface tension by which the cell is actually rounded off and divided.

In most liquids, Plateau asserts that, on the average, the influence of viscosity is such as to cause the cylinder to segment when its length is about four times, or at most from four to six times that of its diameter: instead of a fraction over three times as, in a perfect fluid, theory would demand. If we take it at four times, it may then be shewn that the resulting spheres would have a diameter of about 1·8 times, and their distance apart would be equal to about 2·2 times the diameter of the original cylinder. The calculation is not difficult which would shew how these numbers are altered in the case of a cylinder formed around a solid core, as in the case of the spider’s web. Plateau has also made the interesting observation that the _time_ taken in the process of division of the cylinder is directly proportional to the diameter of the cylinder, while varying considerably with the nature of the liquid. This question, of the time occupied in the division of a cell or filament, in relation to the dimensions of the latter, has not so far as I know been enquired into by biologists.

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From the simple fact that the sphere is of all surfaces that whose surface-area for a given volume is an absolute minimum, we have already seen it to be plain that it is the one and only figure of equilibrium which will be assumed under surface-tension by a drop or vesicle, when no other disturbing factors are present. One of the most important of these disturbing factors will be introduced, in the form of complicated tensions and pressures, when one drop is in contact with another drop and when a system of intermediate films or partition walls is developed between them. {241} This subject we shall discuss later, in connection with cell-aggregates or tissues, and we shall find that further theoretical considerations are needed as a preliminary to any such enquiry. Meanwhile let us consider a few cases of the forms of cells, either solitary, or in such simple aggregates that their individual form is little disturbed thereby.

Let us clearly understand that the cases we are about to consider are those cases where the perfect symmetry of the sphere is replaced by another symmetry, less complete, such as that of an ellipsoidal or cylindrical cell. The cases of asymmetrical deformation or displacement, such as is illustrated in the production of a bud or the development of a lateral branch, are much simpler. For here we need only assume a slight and localised variation of surface-tension, such as may be brought about in various ways through the heterogeneous chemistry of the cell; to this point we shall return in our chapter on Adsorption. But the diffused and graded asymmetry of the system, which brings about for instance the ellipsoidal shape of a yeast-cell, is another matter.

If the sphere be the one surface of complete symmetry and therefore of independent equilibrium, it follows that in every cell which is otherwise conformed there must be some definite force to cause its departure from sphericity; and if this cause be the very simple and obvious one of the resistance offered by a solidified envelope, such as an egg-shell or firm cell-wall, we must still seek for the deforming force which was in action to bring about the given shape, prior to the assumption of rigidity. Such a cause may be either external to, or may lie within, the cell itself. On the one hand it may be due to external pressure or to some form of mechanical restraint: as it is in all our experiments in which we submit our bubble to the partial restraint of discs or rings or more complicated cages of wire; and on the other hand it may be due to intrinsic causes, which must come under the head either of differences of internal pressure, or of lack of homogeneity or isotropy in the surface itself[296]. {242}

Our full formula of equilibrium, or equation to an elastic surface, is _P_ = _p_{e}_ + (_T_/_R_ + _T′_/_R′_), where _P_ is the internal pressure, _p_{e}_ any extraneous pressure normal to the surface, _R_, _R′_ the radii of curvature at a point, and _T_, _T′_, the corresponding tensions, normal to one another, of the envelope.

Now in any given form which we are seeking to account for, _R_, _R′_ are known quantities; but all the other factors of the equation are unknown and subject to enquiry. And somehow or other, by this formula, we must account for the form of any solitary cell whatsoever (provided always that it be not formed by successive stages of solidification), the cylindrical cell of Spirogyra, the ellipsoidal yeast-cell, or (as we shall see in another chapter) the shape of the egg of any bird. In using this formula hitherto, we have taken it in a simplified form, that is to say we have made several limiting assumptions. We have assumed that _P_ was simply the uniform hydrostatic pressure, equal in all directions, of a body of liquid; we have assumed that the tension _T_ was simply due to surface-tension in a homogeneous liquid film, and was therefore equal in all directions, so that _T_ = _T′_; and we have only dealt with surfaces, or parts of a surface, where extraneous pressure, _p_{n}_, was non-existent. Now in the case of a bird’s egg, the external pressure _p_{n}_, that is to say the pressure exercised by the walls of the oviduct, will be found to be a very important factor; but in the case of the yeast-cell or the Spirogyra, wholly immersed in water, no such external pressure comes into play. We are accordingly left, in such cases as these last, with two hypotheses, namely that the departure from a spherical form is due to inequalities in the internal pressure _P_, or else to inequalities in the tension _T_, that is to say to a difference between _T_ and _T′_. In other words, it is theoretically possible that the oval form of a yeast-cell is due to a greater internal pressure, a greater “tendency to grow,” in the direction of the longer axis of the ellipse, or alternatively, that with equal and symmetrical tendencies to growth there is associated a difference of external resistance in {243} respect of the tension of the cell-wall. Now the former hypothesis is not impossible; the protoplasm is far from being a perfect fluid; it is the seat of various internal forces, sometimes manifestly polar; and accordingly it is quite possible that the internal forces, osmotic and other, which lead to an increase of the content of the cell and are manifested in pressure outwardly directed upon its wall may be unsymmetrical, and such as to lead to a deformation of what would otherwise be a simple sphere. But while this hypothesis is not impossible, it is not very easy of acceptance. The protoplasm, though not a perfect fluid, has yet on the whole the properties of a fluid; within the small compass of the cell there is little room for the development of unsymmetrical pressures; and, in such a case as Spirogyra, where a large part of the cavity is filled by a fluid and watery cell-sap, the conditions are still more obviously those under which a uniform hydrostatic pressure is to be expected. But in variations of _T_, that is to say of the specific surface-tension per unit area, we have an ample field for all the various deformations with which we shall have to deal. Our condition now is, that (_T_/_R_ + _T′_/_R′_) = a constant; but it no longer follows, though it may still often be the case, that this will represent a surface of absolute minimal area. As soon as _T_ and _T′_ become unequal, it is obvious that we are no longer dealing with a perfectly liquid surface film; but its departure from a perfect fluidity may be of all degrees, from that of a slight non-isotropic viscosity to the state of a firm elastic membrane[297]. And it matters little whether this viscosity or semi-rigidity be manifested in the self-same layer which is still a part of the protoplasm of the cell, or in a layer which is completely differentiated into a distinct and separate membrane. As soon as, by secretion or “adsorption,” the molecular constitution of the surface layer is altered, it is clearly conceivable that the alteration, or the secondary chemical changes which follow it, may be such as to produce an anisotropy, and to render the molecular forces less capable in one direction than another of exerting that contractile force by which they are striving to reduce to an absolute minimum the {244} surface area of the cell. A slight inequality in two opposite directions will produce the ellipsoid cell, and a very great inequality will give rise to the cylindrical cell[298].

I take it therefore, that the cylindrical cell of Spirogyra, or any other cylindrical cell which grows in freedom from any manifest external restraint, has assumed that particular form simply by reason of the molecular constitution of its developing surface-membrane; and that this molecular constitution was anisotropous, in such a way as to render extension easier in one direction than another.

Such a lack of homogeneity or of isotropy, in the cell-wall is often rendered visible, especially in plant-cells, in various ways, in the form of concentric lamellae, annular and spiral striations, and the like.

But this phenomenon, while it brings about a certain departure from complete symmetry, is still compatible with, and coexistent with, many of the phenomena which we have seen to be associated with surface-tension. The symmetry of tensions still leaves the cell a solid of revolution, and its surface is still a surface of equilibrium. The fluid pressure within the cylinder still causes the film or membrane which caps its ends to be of a spherical form. And in the young cell, where the surface pellicle is absent or but little differentiated, as for instance in the oögonium of Achlya, or in the young zygospore of Spirogyra, we always see the tendency of the entire structure towards a spherical form reasserting itself: unless, as in the latter case, it be overcome by direct compression within the cylindrical mother-cell. Moreover, in those cases where the adult filament consists of cylindrical cells, we see that the young, germinating spore, at first spherical, very soon assumes with growth an elliptical or ovoid form: the direct result of an incipient anisotropy of its envelope, which when more developed will convert the ovoid into a cylinder. We may also notice that a truly cylindrical cell is comparatively rare; for in most cases, what we call a cylindrical cell shews a distinct bulging of its sides; it is not truly a cylinder, but a portion of a spheroid or ellipsoid. {245}

Unicellular organisms in general, including the protozoa, the unicellular cryptogams, the various bacteria, and the free, isolated cells, spores, ova, etc. of higher organisms, are referable for the most part to a very small number of typical forms; but besides a certain number of others which may be so referable, though obscurely, there are obviously many others in which either no symmetry is to be recognized, or in which the form is clearly not one of equilibrium. Among these latter we have Amoeba itself, and all manner of amoeboid organisms, and also many curiously shaped cells, such as the Trypanosomes and various other aberrant Infusoria. We shall return to the consideration of these; but in the meanwhile it will suffice to say that, as their surfaces are not equilibrium-surfaces, so neither are the living cells themselves in any stable equilibrium. On the contrary, they are in continual flux and movement, each portion of the surface constantly changing its form, and passing from one phase to another of an equilibrium which is never stable for more than a moment. The former class, which rest in stable equilibrium, must fall (as we have seen) into two classes,—those whose equilibrium arises from liquid surface-tension alone, and those in whose conformation some other pressure or restraint has been superimposed upon ordinary surface-tension.

To the fact that these little organisms belong to an order of magnitude in which form is mainly, if not wholly, conditioned and controlled by molecular forces, is due the limited range of forms which they actually exhibit. These forms vary according to varying physical conditions. Sometimes they do so in so regular and orderly a way that we instinctively explain them merely as “phases of a life-history,” and leave physical properties and physical causation alone: but many of their variations of form we treat as exceptional, abnormal, decadent or morbid, and are apt to pass these over in neglect, while we give our attention to what we suppose to be the typical or “characteristic” form or attitude. In the case of the smallest organisms, the bacteria, micrococci, and so forth, the range of form is especially limited, owing to their minuteness, the powerful pressure which their highly curved surfaces exert, and the comparatively homogeneous nature of their substance. But within their narrow range of possible diversity {246} these minute organisms are protean in their changes of form. A certain species will not only change its shape from stage to stage of its little “cycle” of life; but it will be remarkably different in outward form according to the circumstances under which we find it, or the histological treatment to which we submit it. Hence the pathological student, commencing the study of bacteriology, is early warned to pay little heed to differences of _form_, for purposes of recognition or specific identification. Whatever grounds we may have for attributing to these organisms a permanent or stable specific identity (after the fashion of the higher plants and animals), we can seldom safely do so on the ground of definite and always recognisable _form_: we may often be inclined, in short, to ascribe to them a physiological (sometimes a “pathogenic”), rather than a morphological specificity.

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Among the Infusoria, we have a small number of forms whose symmetry is distinctly spherical, for instance among the small flagellate monads; but even these are seldom actually spherical except when we see them in a non-flagellate and more or less encysted or “resting” stage. In this condition, it need hardly be remarked that the spherical form is common and general among a great variety of unicellular organisms. When our little monad developes a flagellum, that is in itself an indication of “polarity” or symmetrical non-homogeneity of the cell; and accordingly, we {247} usually see signs of an unequal tension of the membrane in the neighbourhood of the base of the flagellum. Here the tension is usually less than elsewhere, and the radius of curvature is accordingly less: in other words that end of the cell is drawn out to a tapering point (Fig. 73). But sometimes it is the other way, as in Noctiluca, where the large flagellum springs from a depression in the otherwise uniformly rounded cell. In this case the explanation seems to lie in the many strands of radiating protoplasm which converge upon this point, and may be supposed to keep it relatively fixed by their viscosity, while the rest of the cell-surface is free to expand (Fig. 74).

A very large number of Infusoria represent unduloids, or portions of unduloids, and this type of surface appears and reappears in a great variety of forms. The cups of the various species of Vorticella (Fig. 75) are nothing in the world but a beautiful series of unduloids, or partial unduloids, in every gradation from a form that is all but cylindrical to one that is all but a perfect sphere. These unduloids are not completely symmetrical, but they are such unduloids as develop themselves when we suspend an oil-globule between two unequal rings, or blow a soap-bubble between two unequal pipes; for, just as in these cases, the surface of our Vorticella bell finds its terminal supports, on the one hand in its attachment to its narrow stalk, and on the other in the thickened ring from which spring its circumoral cilia. And here let me say, that a point or zone from which cilia arise would seem always to have a peculiar relation to the surrounding tensions. It usually forms a sharp salient, a prominent point or ridge, as in our little monads of Fig. 73; shewing that, in its formation, the surface tension had here locally diminished. But if such a ridge or fillet consolidate in the least degree, it becomes a source of strength, and a _point d’appui_ for the adjacent film. We shall deal with this point again in the next chapter. {248}

Precisely the same series of unduloid forms may be traced in even greater variety among various other families or genera of the Infusoria. Sometimes, as in Vorticella itself, the unduloid is seen merely in the contour of the soft semifluid body of the living animal. At other times, as in Salpingoeca, Tintinnus, and many other genera, we have a distinct membranous cup, separate from the animal, but originally secreted by, and moulded upon, its semifluid living surface. Here we have an excellent illustration of the contrast between the different ways in which such a structure may be regarded and interpreted. The teleological explanation is that it is developed for the sake of protection, as a domicile and shelter for the little organism within. The mechanical explanation of the physicist (seeking only after the “efficient,” and not the “final” cause), is that it is {249} present, and has its actual conformation, by reason of certain chemico-physical conditions: that it was inevitable, under the given conditions, that certain constituent substances actually present in the protoplasm should be aggregated by molecular forces in its surface layer; that under this adsorptive process, the conditions continuing favourable, the particles should accumulate and concentrate till they formed (with the help of the surrounding medium) a pellicle or membrane, thicker or thinner as the case might be; that this surface pellicle or membrane was inevitably bound, by molecular forces, to become a surface of the least possible area which the circumstances permitted; that in the present case, the symmetry and “freedom” of the system permitted, and _ipso facto_ caused, this surface to be a surface of revolution; and that of the few surfaces of revolution which, as being also surfaces _minimae areae_, were available, the unduloid was manifestly the one permitted, and _ipso facto_ caused, by the dimensions of the organisms and other circumstances of the case. And just as the thickness or thinness of the pellicle was obviously a subordinate matter, a mere matter of degree, so we also see that the actual outline of this or that particular unduloid is also a very subordinate matter, such as physico-chemical variants of a minute kind would suffice to bring about; for between the various unduloids which the various species of Vorticella represent, there is no more real difference than that difference of ratio or degree which exists between two circles of different diameter, or two lines of unequal length. {250}

In very many cases (of which Fig. 80 is an example), we have an unduloid form exhibited, not by a surrounding pellicle or shell, but by the soft, protoplasmic body of a ciliated organism. In such cases the form is mobile, and continually changes from one to another unduloid contour, according to the movements of the animal. We have here, apparently, to deal with an unstable equilibrium, and also sometimes with the more complicated problem of “stream-lines,” as in the difficult problems suggested by the form of a fish. But this whole class of cases, and of problems, we can merely take note of in passing, for their treatment is too hard for us.

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