Chapter V: The Forms of Cells (1)

Protoplasm, as we have already said, is a fluid or rather a semifluid substance, and we need not pause here to attempt to describe the particular properties of the semifluid, colloid, or jelly-like substances to which it is allied; we should find it no easy matter. Nor need we appeal to precise theoretical definitions of fluidity, lest we come into a debateable land. It is in the most general sense that protoplasm is “fluid.” As Graham said (of colloid matter in general), “its softness _partakes of fluidity_, and enables the colloid to become a vehicle for liquid diffusion, like water itself[264].” When we can deal with protoplasm in sufficient quantity we see it flow; particles move freely through it, air-bubbles and liquid droplets shew round or spherical within it; and we shall have much to say about other phenomena manifested by its own surface, which are those especially characteristic of liquids. It may encompass and contain solid bodies, and it may “secrete” within or around itself solid substances; and very often in the complex living organism these solid substances formed by the living protoplasm, like shell or nail or horn or feather, may remain when the protoplasm which formed them is dead and gone; but the protoplasm itself is fluid or semifluid, and accordingly permits of free (though not necessarily rapid) _diffusion_ and easy _convection_ of particles within itself. This simple fact is of elementary importance in connection with form, and with what appear at first sight to be common characteristics or peculiarities of the forms of living things.

The older naturalists, in discussing the differences between inorganic and organic bodies, laid stress upon the fact or statement that the former grow by “agglutination,” and the latter by {202} what they termed “intussusception.” The contrast is true, rather, of solid as compared with jelly-like bodies of all kinds, living or dead, the great majority of which as it so happens, but by no means all, are of organic origin.

A crystal “grows” by deposition of new molecules, one by one and layer by layer, superimposed or aggregated upon the solid substratum already formed. Each particle would seem to be influenced, practically speaking, only by the particles in its immediate neighbourhood, and to be in a state of freedom and independence from the influence, either direct or indirect, of its remoter neighbours. As Lord Kelvin and others have explained the formation and the resulting forms of crystals, so we believe that each added particle takes up its position in relation to its immediate neighbours already arranged, generally in the holes and corners that their arrangement leaves, and in closest contact with the greatest number[265]. And hence we may repeat or imitate this process of arrangement, with great or apparently even with precise accuracy (in the case of the simpler crystalline systems), by piling up spherical pills or grains of shot. In so doing, we must have regard to the fact that each particle must drop into the place where it can go most easily, or where no easier place offers. In more technical language, each particle is free to take up, and does take up, its position of least potential energy relative to those already deposited; in other words, for each particle motion is induced until the energy of the system is so distributed that no tendency or resultant force remains to move it more. The application of this principle has been shewn to lead to the production of _planes_[266] (in all cases where by the limitation of material, surfaces _must_ occur); and where we have planes, straight edges and solid angles must obviously also occur; and, if equilibrium is {203} to follow, must occur symmetrically. Our piling up of shot, or manufacture of mimic crystals, gives us visible demonstration that the result is actually to obtain, as in the natural crystal, plane surfaces and sharp angles, symmetrically disposed.

But the living cell grows in a totally different way, very much as a piece of glue swells up in water, by “imbibition,” or by interpenetration into and throughout its entire substance. The semifluid colloid mass takes up water, partly to combine chemically with its individual molecules[267], partly by physical diffusion into the interstices between these molecules, and partly, as it would seem, in other ways; so that the entire phenomenon is a very complex and even an obscure one. But, so far as we are concerned, the net result is a very simple one. For the equilibrium or tendency to equilibrium of fluid pressure in all parts of its interior while the process of imbibition is going on, the constant rearrangement of its fluid mass, the contrast in short with the crystalline method of growth where each particle comes to rest to move (relatively to the whole) no more, lead the mass of jelly to swell up, very much as a bladder into which we blow air, and so, by a _graded_ and harmonious distribution of forces, to assume everywhere a rounded and more or less bubble-like external form[268]. So, when the same school of older naturalists called attention to a new distinction or contrast of form between the organic and inorganic objects, in that the contours of the former tended to roundness and curvature, and those of the latter to be bounded by straight lines, planes and sharp angles, we see that this contrast was not a new and different one, but only another aspect of their former statement, and an immediate consequence of the difference between the processes of agglutination and intussusception.

This common and general contrast between the form of the crystal on the one hand, and of the colloid or of the organism on the other, must by no means be pressed too far. For Lehmann, {204} in his great work on so-called Fluid Crystals[269], to which we shall afterwards return, has shewn how, under certain circumstances, surface-tension phenomena may coexist with crystallisation, and produce a form of minimal potential which is a resultant of both: the fact being that the bonds maintaining the crystalline arrangement are now so much looser than in the solid condition that the tendency to least total surface-area is capable of being satisfied. Thus the phenomenon of “liquid crystallisation” does not destroy the distinction between crystalline and colloidal forms, but gives added unity and continuity to the whole series of phenomena[270]. Lehmann has also demonstrated phenomena within the crystal, known for instance as transcrystallisation, which shew us that we must not speak unguardedly of the growth of crystals as limited to deposition upon a surface, and Bütschli has already pointed out the possible great importance to the biologist of the various phenomena which Lehmann has described[271].

So far then, as growth goes on, unaffected by pressure or other external force, the fluidity of protoplasm, its mobility internal and external, and the manner in which particles move with comparative freedom from place to place within, all manifestly tend to the production of swelling, rounded surfaces, and to their great predominance over plane surfaces in the contour of the organism. These rounded contours will tend to be preserved, for a while, in the case of naked protoplasm by its viscosity, and in the presence of a cell-wall by its very lack of fluidity. In a general way, the presence of curved boundary surfaces will be especially obvious in the unicellular organisms, and still more generally in the _external_ forms of all organisms; and wherever mutual pressure between adjacent cells, or other adjacent parts, has not come into play to flatten the rounded surfaces into planes.

But the rounded contours that are assumed and exhibited by {205} a piece of hard glue, when we throw it into water and see it expand as it sucks the water up, are not nearly so regular or so beautiful as are those which appear when we blow a bubble, or form a drop, or pour water into a more or less elastic bag. For these curving contours depend upon the properties of the bag itself, of the film or membrane that contains the mobile gas, or that contains or bounds the mobile liquid mass. And hereby, in the case of the fluid or semifluid mass, we are introduced to the subject of _surface tension_: of which indeed we have spoken in the preceding chapter, but which we must now examine with greater care.

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Among the forces which determine the forms of cells, whether they be solitary or arranged in contact with one another, this force of surface-tension is certainly of great, and is probably of paramount importance. But while we shall try to separate out the phenomena which are directly due to it, we must not forget that, in each particular case, the actual conformation which we study may be, and usually is, the more or less complex resultant of surface tension acting together with gravity, mechanical pressure, osmosis, or other physical forces.

Surface tension is that force by which we explain the form of a drop or of a bubble, of the surfaces external and internal of a “froth” or collocation of bubbles, and of many other things of like nature and in like circumstances[272]. It is a property of liquids (in the sense at least with which our subject is concerned), and it is manifested at or very near the surface, where the liquid comes into contact with another liquid, a solid or a gas. We note here that the term _surface_ is to be interpreted in a wide sense; for wherever we have solid particles imbedded in a fluid, wherever we have a non-homogeneous fluid or semi-fluid such as a particle {206} of protoplasm, wherever we have the presence of “impurities,” as in a mass of molten metal, there we have always to bear in mind the existence of “surfaces” and of surface tensions, not only on the exterior of the mass but also throughout its interstices, wherever like meets unlike.

Surface tension is due to molecular force, to force that is to say arising from the action of one molecule upon another, and it is accordingly exerted throughout a small thickness of material, comparable to the range of the molecular forces. We imagine that within the interior of the liquid mass such molecular interactions negative one another: but that at and near the free surface, within a layer or film approximately equal to the range of the molecular force, there must be a lack of such equilibrium and consequently a manifestation of force.

The action of the molecular forces has been variously explained. But one simple explanation (or mode of statement) is that the molecules of the surface layer (whose thickness is definite and constant) are being constantly attracted into the interior by those which are more deeply situated, and that consequently, as molecules keep quitting the surface for the interior, the bulk of the latter increases while the surface diminishes; and the process continues till the surface itself has become a minimum, the _surface-shrinkage_ exhibiting itself as a _surface-tension_. This is a sufficient description of the phenomenon in cases where a portion of liquid is subject to no other than _its own molecular forces_, and (since the sphere has, of all solids, the smallest surface for a given volume) it accounts for the spherical form of the raindrop, of the grain of shot, or of the living cell in many simple organisms. It accounts also, as we shall presently see, for a great number of much more complicated forms, manifested under less simple conditions.

Let us here briefly note that surface tension is, in itself, a comparatively small force, and easily measurable: for instance that of water is equivalent to but a few grains per linear inch, or a few grammes per metre. But this small tension, when it exists in a _curved_ surface of very great curvature, gives rise to a very great pressure directed towards the centre of curvature. We can easily calculate this pressure, and so satisfy ourselves that, when the radius of curvature is of molecular dimensions, the {207} pressure is of the magnitude of thousands of atmospheres,—a conclusion which is supported by other physical considerations.

The contraction of a liquid surface and other phenomena of surface tension involve the doing of work, and the power to do work is what we call energy. It is obvious, in such a simple case as we have just considered, that the whole energy of the system is diffused throughout its molecules; but of this whole stock of energy it is only that part which comes into play at or very near to the surface which normally manifests itself in work, and hence we may speak (though the term is open to some objections) of a specific _surface energy_. The consideration of surface energy, and of the manner in which its amount is increased and multiplied by the multiplication of surfaces due to the subdivision of the organism into cells, is of the highest importance to the physiologist; and even the morphologist cannot wholly pass it by, if he desires to study the form of the cell in its relation to the phenomena of surface tension or “capillarity.” The case has been set forth with the utmost possible lucidity by Tait and by Clerk Maxwell, on whose teaching the following paragraphs are based: they having based their teaching upon that of Gauss,—who rested on Laplace.

Let _E_ be the whole potential energy of a mass _M_ of liquid; let _e__{0} be the energy per unit mass of the interior liquid (we may call it the _internal energy_); and let _e_ be the energy per unit mass for a layer of the skin, of surface _S_, of thickness _t_, and density ρ (_e_ being what we call the _surface energy_). It is obvious that the total energy consists of the internal _plus_ the surface energy, and that the former is distributed through the whole mass, minus its surface layers. That is to say, in mathematical language,

_E_ = (_M_ − _S_ ⋅ Σ _t_ ρ) _e__{0} + _S_ ⋅ Σ _t_ ρ _e_.

But this is equivalent to writing:

= _M_ _e__{0} + _S_ ⋅ Σ _t_ ρ(_e_ − _e__{0});

and this is as much as to say that the total energy of the system may be taken to consist of two portions, one uniform throughout the whole mass, and another, which is proportional on the one hand to the amount of surface, and on the other hand is proportional to the difference between _e_ and _e__{0}, that is to say to the difference between the unit values of the internal and the surface energy. {208}

It was Gauss who first shewed after this fashion how, from the mutual attractions between all the particles, we are led to an expression which is what we now call the _potential energy_ of the system; and we know, as a fundamental theorem of dynamics, that the potential energy of the system tends to a minimum, and in that minimum finds, as a matter of course, its stable equilibrium.

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We see in our last equation that the term _M_ _e__{0} is irreducible, save by a reduction of the mass itself. But the other term may be diminished (1) by a reduction in the area of surface, _S_, or (2) by a tendency towards equality of _e_ and _e__{0}, that is to say by a diminution of the specific surface energy, _e_.

These then are the two methods by which the energy of the system will manifest itself in work. The one, which is much the more important for our purposes, leads always to a diminution of surface, to the so-called “principle of minimal areas”; the other, which leads to the lowering (under certain circumstances) of surface tension, is the basis of the theory of Adsorption, to which we shall have some occasion to refer as the _modus operandi_ in the development of a cell-wall, and in a variety of other histological phenomena. In the technical phraseology of the day, the “capacity factor” is involved in the one case, and the “intensity factor” in the other.

Inasmuch as we are concerned with the form of the cell it is the former which becomes our main postulate: telling us that the energy equations of the surface of a cell, or of the free surfaces of cells partly in contact, or of the partition-surfaces of cells in contact with one another or with an adjacent solid, all indicate a minimum of potential energy in the system, by which the system is brought, _ipso facto_, into equilibrium. And we shall not fail to observe, with something more than mere historical interest and curiosity, how deeply and intrinsically there enter into this whole class of problems the “principle of least action” of Maupertuis, the “_lineae curvae maximi minimive proprietate gaudentes_” of Euler, by which principles these old natural philosophers explained correctly a multitude of phenomena, and drew the lines whereon the foundations of great part of modern physics are well and truly laid. {209}

In all cases where the principle of maxima and minima comes into play, as it conspicuously does in the systems of liquid films which are governed by the laws of surface-tension, the figures and conformations produced are characterised by obvious and remarkable _symmetry_. Such symmetry is in a high degree characteristic of organic forms, and is rarely absent in living things,—save in such cases as amoeba, where the equilibrium on which symmetry depends is likewise lacking. And if we ask what physical equilibrium has to do with formal symmetry and regularity, the reason is not far to seek; nor can it be put better than in the following words of Mach’s[273]. “In every symmetrical system every deformation that tends to destroy the symmetry is complemented by an equal and opposite deformation that tends to restore it. In each deformation positive and negative work is done. One condition, therefore, though not an absolutely sufficient one, that a maximum or minimum of work corresponds to the form of equilibrium, is thus supplied by symmetry. Regularity is successive symmetry. There is no reason, therefore, to be astonished that the forms of equilibrium are often symmetrical and regular.”

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As we proceed in our enquiry, and especially when we approach the subject of _tissues_, or agglomerations of cells, we shall have from time to time to call in the help of elementary mathematics. But already, with very little mathematical help, we find ourselves in a position to deal with some simple examples of organic forms.

When we melt a stick of sealing-wax in the flame, surface tension (which was ineffectively present in the solid but finds play in the now fluid mass), rounds off its sharp edges into curves, so striving towards a surface of minimal area; and in like manner, by melting the tip of a thin rod of glass, Leeuwenhoek made the little spherical beads which served him for a microscope[274]. When any drop of protoplasm, either over all its surface or at some free end, as at the extremity of the pseudopodium of an amoeba, is {210} seen likewise to “round itself off,” that is not an effect of “vital contractility,” but (as Hofmeister shewed so long ago as 1867) a simple consequence of surface tension; and almost immediately afterwards Engelmann[275] argued on the same lines, that the forces which cause the contraction of protoplasm in general may “be just the same as those which tend to make every non-spherical drop of fluid become spherical!” We are not concerned here with the many theories and speculations which would connect the phenomena of surface tension with contractility, muscular movement or other special _physiological_ functions, but we find ample room to trace the operation of the same cause in producing, under conditions of rest and equilibrium, certain definite and inevitable forms of surface.

It is however of great importance to observe that the living cell is one of those cases where the phenomena of surface tension are by no means limited to the _outer_ surface; for within the heterogeneous substance of the cell, between the protoplasm and its nuclear and other contents, and in the alveolar network of the cytoplasm itself (so far as that “alveolar structure” is actually present in life), we have a multitude of interior surfaces; and, especially among plants, we may have a large inner surface of “interfacial” contact, where the protoplasm contains cavities or “vacuoles” filled with a different and more fluid material, the “cell-sap.” Here we have a great field for the development of surface tension phenomena: and so long ago as 1865, Nägeli and Schwendener shewed that the streaming currents of plant cells might be very plausibly explained by this phenomenon. Even ten years earlier, Weber had remarked upon the resemblance between these protoplasmic streamings and the streamings to be observed in certain inanimate drops, for which no cause but surface tension could be assigned[276].

The case of amoeba, though it is an elementary case, is at the same time a complicated one. While it remains “amoeboid,” it is never at rest or in equilibrium; it is always moving, from one to another of its protean changes of configuration; its surface tension is constantly varying from point to point. Where the {211} surface tension is greater, that portion of the surface will contract into spherical or spheroidal forms; where it is less the surface will correspondingly extend. While generally speaking the surface energy has a minimal value, it is not necessarily constant. It may be diminished by a rise of temperature; it may be altered by contact with adjacent substances[277], by the transport of constituent materials from the interior to the surface, or again by actual chemical and fermentative change. Within the cell, the surface energies developed about its heterogeneous contents will constantly vary as these contents are affected by chemical metabolism. As the colloid materials are broken down and as the particles in suspension are diminished in size the “free surface energy” will be increased, but the osmotic energy will be diminished[278]. Thus arise the various fluctuations of surface tension and the various phenomena of amoeboid form and motion, which Bütschli and others have reproduced or imitated by means of the fine emulsions which constitute their “artificial amoebae.” A multitude of experiments shew how extraordinarily delicate is the adjustment of the surface tension forces, and how sensitive they are to the least change of temperature or chemical state. Thus, on a plate which we have warmed at one side, a drop of alcohol runs towards the warm area, a drop of oil away from it; and a drop of water on the glass plate exhibits lively movements when {212} we bring into its neighbourhood a heated wire, or a glass rod dipped in ether. When we find that a plasmodium of Aethalium, for instance, creeps towards a damp spot, or towards a warm spot, or towards substances that happen to be nutritious, and again creeps away from solutions of sugar or of salt, we seem to be dealing with phenomena every one of which can be paralleled by ordinary phenomena of surface tension[279]. Even the soap-bubble itself is imperfectly in equilibrium, for the reason that its film, like the protoplasm of amoeba or Aethalium, is an excessively heterogeneous substance. Its surface tensions vary from point to point, and chemical changes and changes of temperature increase and magnify the variation. The whole surface of the bubble is in constant movement as the concentrated portions of the soapy fluid make their way outwards from the deeper layers; it thins and it thickens, its colours change, currents are set up in it, and little bubbles glide over it; it continues in this state of constant movement, as its parts strive one with another in all their interactions towards equilibrium[280].

In the case of the naked protoplasmic cell, as the amoeboid phase is emphatically a phase of freedom and activity, of chemical and physiological change, so, on the other hand, is the spherical form indicative of a phase of rest or comparative inactivity. In the one phase we see unequal surface tensions manifested in the creeping movements of the amoeboid body, in the rounding off of the ends of the pseudopodia, in the flowing out of its substance over a particle of “food,” and in the current-motions in the interior of its mass; till finally, in the other phase, when internal homogeneity and equilibrium have been attained and the potential {213} energy of the system is for the time being at a minimum, the cell assumes a rounded or spherical form, passing into a state of “rest,” and (for a reason which we shall presently see) becoming at the same time “encysted.”

In a budding yeast-cell (Fig. 60), we see a more definite and restricted change of surface tension. When a “bud” appears, whether with or without actual growth by osmosis or otherwise of the mass, it does so because at a certain part of the cell-surface the surface tension has more or less suddenly diminished, and the area of that portion expands accordingly; but in turn the surface tension of the expanded area will make itself felt, and the bud will be rounded off into a more or less spherical form.

The yeast-cell with its bud is a simple example of a principle which we shall find to be very important. Our whole treatment of cell-form in relation to surface-tension depends on the fact (which Errera was the first to point out, or to give clear expression to) that the _incipient_ cell-wall retains with but little impairment the properties of a liquid film[281], and that the growing cell, in spite of the membrane by which it has already begun to be surrounded, behaves very much like a fluid drop. But even the ordinary yeast-cell shows, by its ovoid and non-spherical form, that it has acquired its shape under the influence of some force other than that uniform and symmetrical surface-tension which would be productive of a sphere; and this or any other asymmetrical form, once acquired, may be retained by virtue of the solidification and consequent rigidity of the membranous wall of the cell. Unless such rigidity ensue, it is plain that such a conformation as that of the cell with its attached bud could not be long retained, amidst the constantly varying conditions, as a figure of even partial equilibrium. But as a matter of fact, the cell in this case is not in equilibrium at all; it is in _process_ of budding, and is slowly altering its shape by rounding off the bud. It is plain that over its surface the surface-energies are unequally distributed, owing to some heterogeneity of the substance; and to this matter we shall afterwards return. In like manner the developing egg {214} through all its successive phases of form is never in complete equilibrium; but is merely responding to constantly changing conditions, by phases of partial, transitory, unstable and conditional equilibrium.

It is obvious that there are innumerable solitary plant-cells, and unicellular organisms in general, which, like the yeast-cell, do not correspond to any of the simple forms that may be generated under the influence of simple and homogeneous surface-tension; and in many cases these forms, which we should expect to be unstable and transitory, have become fixed and stable by reason of the comparatively sudden or rapid solidification of the envelope. This is the case, for instance, in many of the more complicated forms of diatoms or of desmids, where we are dealing, in a less striking but even more curious way than in the budding yeast-cell, not with one simple act of formation, but with a complicated result of successive stages of localised growth, interrupted by phases of partial consolidation. The original cell has acquired or assumed a certain form, and then, under altering conditions and new distributions of energy, has thickened here or weakened there, and has grown out or tended (as it were) to branch, at particular points. We can often, or indeed generally, trace in each particular stage of growth or at each particular temporary growing point, the laws of surface tension manifesting themselves in what is for the time being a fluid surface; nay more, even in the adult and completed structure, we have little difficulty in tracing and recognising (for instance in the outline of such a desmid as Euastrum) the rounded lobes that have successively grown or flowed out from the original rounded and flattened cell. What we see in a many chambered foraminifer, such as Globigerina or Rotalia, is just the same thing, save that it is carried out in greater completeness and perfection. The little organism as a whole is not a figure of equilibrium or of minimal area; but each new bud or separate chamber is such a figure, conditioned by the forces of surface tension, and superposed upon the complex aggregate of similar bubbles after these latter have become consolidated one by one into a rigid system.

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Let us now make some enquiry regarding the various forms {215} which, under the influence of surface tension, a surface can possibly assume. In doing so, we are obviously limited to conditions under which other forces are relatively unimportant, that is to say where the “surface energy” is a considerable fraction of the whole energy of the system; and this in general will be the case when we are dealing with portions of liquid so small that their dimensions come within what we have called the molecular range, or, more generally, in which the “specific surface” is large[282]: in other words it will be small or minute organisms, or the small cellular elements of larger organisms, whose forms will be governed by surface-tension; while the general forms of the larger organisms will be due to other and non-molecular forces. For instance, a large surface of water sets itself level because here gravity is predominant; but the surface of water in a narrow tube is manifestly curved, for the reason that we are here dealing with particles which are mutually within the range of each other’s molecular forces. The same is the case with the cell-surfaces and cell-partitions which we are presently to study, and the effect of gravity will be especially counteracted and concealed when, as in the case of protoplasm in a watery fluid, the object is immersed in a liquid of nearly its own specific gravity.

We have already learned, as a fundamental law of surface-tension phenomena, that a liquid film _in equilibrium_ assumes a form which gives it a minimal area under the conditions to which it is subject. And these conditions include (1) the form of the boundary, if such exist, and (2) the pressure, if any, to which the film is subject; which pressure is closely related to the volume, of air or of liquid, which the film (if it be a closed one) may have to contain. In the simplest of cases, when we take up a soap-film on a plane wire ring, the film is exposed to equal atmospheric pressure on both sides, and it obviously has its minimal area in the form of a plane. So long as our wire ring lies in one plane (however irregular in outline), the film stretched across it will still be in a plane; but if we bend the ring so that it lies no longer in a plane, then our film will become curved into a surface which may be extremely complicated, but is still the smallest possible {216} surface which can be drawn continuously across the uneven boundary.

The question of pressure involves not only external pressures acting on the film, but also that which the film itself is capable of exerting. For we have seen that the film is always contracting to its smallest limits; and when the film is curved, this obviously leads to a pressure directed inwards,—perpendicular, that is to say, to the surface of the film. In the case of the soap-bubble, the uniform contraction of whose surface has led to its spherical form, this pressure is balanced by the pressure of the air within; and if an outlet be given for this air, then the bubble contracts with perceptible force until it stretches across the mouth of the tube, for instance the mouth of the pipe through which we have blown the bubble. A precisely similar pressure, directed inwards, is exercised by the surface layer of a drop of water or a globule of mercury, or by the surface pellicle on a portion or “drop” of protoplasm. Only we must always remember that in the soap-bubble, or the bubble which a glass-blower blows, there is a twofold pressure as compared with that which the surface-film exercises on the drop of liquid of which it is a part; for the bubble consists (unless it be so thin as to consist of a mere layer of molecules[283]) of a liquid layer, with a free surface within and another without, and each of these two surfaces exercises its own independent and coequal tension, and corresponding pressure[284].

If we stretch a tape upon a flat table, whatever be the tension of the tape it obviously exercises no pressure upon the table below. But if we stretch it over a _curved_ surface, a cylinder for instance, it does exercise a downward pressure; and the more curved the surface the greater is this pressure, that is to say the greater is this share of the entire force of tension which is resolved in the downward direction. In mathematical language, the pressure (_p_) varies directly as the tension (_T_), and inversely as the radius of curvature (_R_): that is to say, _p_ = _T_/_R_, per unit of surface. {217}

If instead of a cylinder, which is curved only in one direction, we take a case where there are curvatures in two dimensions (as for instance a sphere), then the effects of these must be simply added to one another, and the resulting pressure _p_ is equal to _T_/_R_ + _T_/_R′_ or _p_ = _T_(1/_R_ + 1/_R′_)[285].

And if in addition to the pressure _p_, which is due to surface tension, we have to take into account other pressures, _p′_, _p″_, etc., which are due to gravity or other forces, then we may say that the _total pressure_, _P_ = _p′_ + _p″_ + _T_(1/_R_ + 1/_R′_). While in some cases, for instance in speaking of the shape of a bird’s egg, we shall have to take account of these extraneous pressures, in the present part of our subject we shall for the most part be able to neglect them.

Our equation is an equation of equilibrium. The resistance to compression,—the pressure outwards,—of our fluid mass, is a constant quantity (_P_); the pressure inwards, _T_(1/_R_ + 1/_R′_), is also constant; and if (unlike the case of the mobile amoeba) the surface be homogeneous, so that _T_ is everywhere equal, it follows that throughout the whole surface 1/_R_ + 1/_R′_ = _C_ (a constant).

Now equilibrium is attained after the surface contraction has done its utmost, that is to say when it has reduced the surface to the smallest possible area; and so we arrive, from the physical side, at the conclusion that a surface such that 1/_R_ + 1/_R′_ = _C_, in other words a surface which has the same _mean curvature_ at all points, is equivalent to a surface of minimal area: and to the same conclusion we may also arrive through purely analytical mathematics. It is obvious that the plane and the sphere are two examples of such surfaces, for in both cases the radius of curvature is everywhere constant, being equal to infinity in the case of the plane, and to some definite magnitude in the case of the sphere.

From the fact that we may extend a soap-film across a ring of wire however fantastically the latter may be bent, we realise that there is no limit to the number of surfaces of minimal area which may be constructed or may be imagined; and while some of these are very complicated indeed, some, for instance a spiral helicoid screw, are relatively very simple. But if we limit ourselves to {218} _surfaces of revolution_ (that is to say, to surfaces symmetrical about an axis), we find, as Plateau was the first to shew, that those which meet the case are very few in number. They are six in all, namely the plane, the sphere, the cylinder, the catenoid, the unduloid, and a curious surface which Plateau called the nodoid.

These several surfaces are all closely related, and the passage from one to another is generally easy. Their mathematical interrelation is expressed by the fact (first shewn by Delaunay[286], in 1841) that the plane curves by whose rotation they are generated are themselves generated as “roulettes” of the conic sections.

Let us imagine a straight line upon which a circle, an ellipse or other conic section rolls; the focus of the conic section will describe a line in some relation to the fixed axis, and this line (or roulette), rotating around the axis, will describe in space one or other of the six surfaces of revolution with which we are dealing.

If we imagine an ellipse so to roll over a line, either of its foci will describe a sinuous or wavy line (Fig. 61B) at a distance alternately maximal and minimal from the axis; and this wavy line, by rotation about the axis, becomes the meridional line of the surface which we call the _unduloid_. The more unequal the two axes are of our ellipse, the more pronounced will be the sinuosity of the described roulette. If the two axes be equal, then our ellipse becomes a circle, and the path described by its rolling centre is a straight line parallel to the axis (A); and obviously the solid of revolution generated therefrom will be a _cylinder_. If one axis of our ellipse vanish, while the other remain of finite length, then the ellipse is reduced to a straight line, and its roulette will appear as a succession of semicircles touching one another upon the axis (C); the solid of revolution will be a series of equal _spheres_. If as before one axis of the ellipse vanish, but the other be infinitely long, then the curve described by the rotation {219} of this latter will be a circle of infinite radius, i.e. a straight line infinitely distant from the axis; and the surface of rotation is now a _plane_. If we imagine one focus of our ellipse to remain at a given distance from the axis, but the other to become infinitely remote, that is tantamount to saying that the ellipse becomes transformed into a parabola; and by the rolling of this curve along the axis there is described a catenary (D), whose solid of revolution is the _catenoid_.

Lastly, but this is a little more difficult to imagine, we have the case of the hyperbola.

We cannot well imagine the hyperbola rolling upon a fixed straight line so that its focus shall describe a continuous curve. But let us suppose that the fixed line is, to begin with, asymptotic to one branch of the hyperbola, and that the rolling proceed until the line is now asymptotic to the other branch, that is to say touching it at an infinite distance; there will then be mathematical continuity if we recommence rolling with this second branch, and so in turn with the other, when each has run its course. We shall see, on reflection, that the line traced by one and the same focus will be an “elastic curve” describing a succession of kinks or knots (E), and the solid of revolution described by this meridional line about the axis is the so-called _nodoid_.

The physical transition of one of these surfaces into another can be experimentally illustrated by means of soap-bubbles, or better still, after the method of Plateau, by means of a large globule of oil, supported when necessary by wire rings, within a fluid of specific gravity equal to its own.

To prepare a mixture of alcohol and water of a density precisely equal to that of the oil-globule is a troublesome matter, and a method devised by Mr C. R. Darling is a great improvement on Plateau’s[287]. Mr Darling uses the oily liquid orthotoluidene, which does not mix with water, has a beautiful and conspicuous red colour, and has precisely the same density as water when both are kept at a temperature of 24° C. We have therefore only to run the liquid into water at this temperature in order to produce beautifully spherical drops of any required size: and by adding {220} a little salt to the lower layers of water, the drop may be made to float or rest upon the denser liquid.

We have already seen that the soap-bubble, spherical to begin with, is transformed into a plane when we relieve its internal pressure and let the film shrink back upon the orifice of the pipe. If we blow a small bubble and then catch it up on a second pipe, so that it stretches between, we may gradually draw the two pipes apart, with the result that the spheroidal surface will be gradually flattened in a longitudinal direction, and the bubble will be transformed into a cylinder. But if we draw the pipes yet farther apart, the cylinder will narrow in the middle into a sort of hourglass form, the increasing curvature of its transverse section being balanced by a gradually increasing _negative_ curvature in the longitudinal section. The cylinder has, in turn, been converted into an unduloid. When we hold a portion of a soft glass tube in the flame, and “draw it out,” we are in the same identical fashion converting a cylinder into an unduloid (Fig. 62A); when on the other hand we stop the end and blow, we again convert the cylinder into an unduloid (B), but into one which is now positively, while the former was negatively curved. The two figures are essentially the same, save that the two halves of the one are reversed in the other.

That spheres, cylinders and unduloids are of the commonest occurrence among the forms of small unicellular organism, or of individual cells in the simpler aggregates, and that in the processes of growth, reproduction and development transitions are frequent from one of these forms to another, is obvious to the naturalist, and we shall deal presently with a few illustrations of these phenomena.

But before we go further in this enquiry, it will be necessary to consider, to some small extent at least, the _curvatures_ of the six different surfaces, that is to say, to determine what modification {221} is required, in each case, of the general equation which applies to them all. We shall find that with this question is closely connected the question of the _pressures_ exercised by, or impinging on the film, and also the very important question of the limitations which, from the nature of the case, exist to prevent the extension of certain of the figures beyond certain bounds. The whole subject is mathematical, and we shall only deal with it in the most elementary way.

We have seen that, in our general formula, the expression 1/_R_ + 1/_R′_ = _C_, a constant; and that this is, in all cases, the condition of our surface being one of minimal area. In other words, it is always true for one and all of the six surfaces which we have to consider. But the constant _C_ may have any value, positive, negative, or nil.

In the case of the plane, where _R_ and _R′_ are both infinite, it is obvious that 1/_R_ + 1/_R′_ = 0. The expression therefore vanishes, and our dynamical equation of equilibrium becomes _P_ = _p_. In short, we can only have a plane film, or we shall only find a plane surface in our cell, when on either side thereof we have equal pressures or no pressure at all. A simple case is the plane partition between two equal and similar cells, as in a filament of spirogyra.

In the case of the sphere, the radii are all equal, _R_ = _R′_; they are also positive, and _T_ (1/_R_ + 1/_R′_), or 2 _T_/_R_, is a positive quantity, involving a positive pressure _P_, on the other side of the equation.

In the cylinder, one radius of curvature has the finite and positive value _R_; but the other is infinite. Our formula becomes _T_/_R_, to which corresponds a positive pressure _P_, supplied by the surface-tension as in the case of the sphere, but evidently of just half the magnitude developed in the latter case for a given value of the radius _R_.

The catenoid has the remarkable property that its curvature in one direction is precisely equal and opposite to its curvature in the other, this property holding good for all points of the surface. That is to say, _R_ = −_R′_; and the expression becomes

(1/_R_ + 1/_R′_) = (1/_R_ − 1/_R_) = 0;

in other words, the surface, as in the case of the plane, has _no {222} curvature_, and exercises no pressure. There are no other surfaces, save these two, which share this remarkable property; and it follows, as a simple corollary, that we may expect at times to have the catenoid and the plane coexisting, as parts of one and the same boundary system; just as, in a cylindrical drop or cell, the cylinder is capped by portions of spheres, such that the cylindrical and spherical portions of the wall exert equal positive pressures.

In the unduloid, unlike the four surfaces which we have just been considering, it is obvious that the curvatures change from one point to another. At the middle of one of the swollen portions, or “beads,” the two curvatures are both positive; the expression (1/_R_ + 1/_R′_) is therefore positive, and it is also finite. The film, accordingly, exercises a positive tension inwards, which must be compensated by a finite and positive outward pressure _P_. At the middle of one of the narrow necks, between two adjacent beads, there is obviously, in the transverse direction, a much stronger curvature than in the former case, and the curvature which balances it is now a negative one. But the sum of the two must remain positive, as well as constant; and we therefore see that the convex or positive curvature must always be greater than the concave or negative curvature at the same point. This is plainly the case in our figure of the unduloid.

The nodoid is, like the unduloid, a continuous curve which keeps altering its curvature as it alters its distance from the axis; but in this case the resultant pressure inwards is negative instead of positive. But this curve is a complicated one, and a full discussion of it would carry us beyond our scope.

In one of Plateau’s experiments, a bubble of oil (protected from gravity by the specific gravity of the surrounding fluid being identical with its own) is balanced between two annuli. It may then be brought to assume the form of Fig. 63, that is to say the form of a cylinder with spherical ends; and there is then everywhere, owing to the convexity of the surface film, a pressure inwards upon the fluid contents of the bubble. If the surrounding liquid be ever so little heavier or lighter than that which constitutes the drop, then the conditions of equilibrium will be accordingly {223} modified, and the cylindrical drop will assume the form of an unduloid (Fig. 64 A, B), with its dilated portion below or above,

as the case may be; and our cylinder may also, of course, be converted into an unduloid either by elongating it further, or by abstracting a portion of its oil, until at length rupture ensues and the cylinder breaks up into two new spherical drops. In all cases alike, the unduloid, like the original cylinder, will be capped by spherical ends, which are the sign, and the consequence, of the positive pressure produced by the curved walls of the unduloid. But if our initial cylinder, instead of being tall, be a flat or dumpy one (with certain definite relations of height to breadth), then new phenomena may be exhibited. For now, if a little oil be cautiously withdrawn from the mass by help of a small syringe, the cylinder may be made to flatten down so that its upper and lower surfaces become plane; which is of itself an indication that the pressure inwards is now _nil_. But at the very moment when the upper and lower surfaces become plane, it will be found that the sides curve inwards, in the fashion shewn in Fig. 65B. This figure is a catenoid, which, as

we have already seen, is, like the plane itself, a surface exercising no pressure, and which therefore may coexist with the plane as part of one and the same system. We may continue to withdraw more oil from our bubble, drop by drop, and now the upper and lower surfaces dimple down into concave portions of spheres, as the result of the _negative_ internal pressure; and thereupon the peripheral catenoid surface alters its form (perhaps, on this small scale, imperceptibly), and becomes a portion of a nodoid (Fig. 65A). {224} It represents, in fact, that portion of the nodoid, which in Fig. 66 lies between such points as O, P. While it is easy to

draw the outline, or meridional section, of the nodoid (as in Fig. 66), it is obvious that the solid of revolution to be derived from it, can never be realised in its entirety: for one part of the solid figure would cut, or entangle with, another. All that we can ever do, accordingly, is to realise isolated portions of the nodoid.

If, in a sequel to the preceding experiment of Plateau’s, we use solid discs instead of annuli, so as to enable us to exert direct mechanical pressure upon our globule of oil, we again begin by adjusting the pressure of these discs so that the oil assumes the form of a cylinder: our discs, that is to say, are adjusted to exercise a mechanical pressure equal to what in the former case was supplied by the surface-tension of the spherical caps or ends of the bubble. If we now increase the pressure slightly, the peripheral walls will become convexly curved, exercising a precisely corresponding pressure. Under these circumstances the form assumed by the sides of our figure will be that of a portion of an unduloid. If we increase the pressure between the discs, the peripheral surface of oil will bulge out more and more, and will presently constitute a portion of a sphere. But we may continue the process yet further, and within certain limits we shall find that the system remains perfectly stable. What is this new curved surface which has arisen out of the sphere, as the latter was produced from the unduloid? It is no other than a portion of a nodoid, that part which in Fig. 66 lies between such limits as M and N. But this surface, which is concave in both directions towards the surface of the oil within, is exerting a pressure upon the latter, just as did the sphere out of which a moment ago it was transformed; and we had just stated, in considering the previous experiment, that the pressure inwards exerted by the nodoid was a negative one. The explanation of this seeming discrepancy lies in the simple fact that, if we follow the outline {225} of our nodoid curve in Fig. 66 from O, P, the surface concerned in the former case, to M, N, that concerned in the present, we shall see that in the two experiments the surface of the liquid is not homologous, but lies on the positive side of the curve in the one case and on the negative side in the other.

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Of all the surfaces which we have been describing, the sphere is the only one which can enclose space; the others can only help to do so, in combination with one another or with the sphere itself. Thus we have seen that, in normal equilibrium, the cylindrical vesicle is closed at either end by a portion of a sphere, and so on. Moreover the sphere is not only the only one of our figures which can enclose a finite space; it is also, of all possible figures, that which encloses the greatest volume with the least area of surface; it is strictly and absolutely the surface of minimal area, and it is therefore the form which will be naturally assumed by a unicellular organism (just as by a raindrop), when it is practically homogeneous and when, like Orbulina floating in the ocean, its surroundings are likewise practically homogeneous and symmetrical. It is only relatively speaking that all the rest are surfaces _minimae areae_; they are so, that is to say, under the given conditions, which involve various forms of pressure or restraint. Such restraints are imposed, for instance, by the pipes or annuli with the help of which we draw out our cylindrical or unduloid oil-globule or soap-bubble; and in the case of the organic cell, similar restraints are constantly supplied by solidification, partial or complete, local or general, of the cell-wall.