THE FORMS OF CELLS
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.
――――――――――
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.
――――――――――
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.”
――――――――――
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.”
[Illustration: Fig. 60.]
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.
――――――――――
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.
[Illustration: Fig. 61.]
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.
[Illustration: Fig. 62.]
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.
[Illustration: Fig. 63.]
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,
[Illustration: Fig. 64.]
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
[Illustration: Fig. 65.]
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
[Illustration: Fig. 66.]
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.
――――――――――
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.
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.
[Illustration: Fig. 67.]
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.
――――――――――
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
[Illustration: Fig. 68.]
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.
――――――――――
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.
――――――――――
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].
[Illustration: Fig. 69. Hair of _Trianea_, in glycerine. (After
Berthold.)]
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].
[Illustration: Fig. 70. Phases of a Splash. (From Worthington.)]
――――――――――
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.
[Illustration: Fig. 71. A breaking wave. (From Worthington.)]
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.