ON THE SHAPES OF EGGS, AND OF CERTAIN OTHER HOLLOW STRUCTURES
The eggs of birds and all other hard-shelled eggs, such as those of
the tortoise and the crocodile, are simple solids of revolution; but
they differ greatly in form, according to the configuration of the
plane curve by the revolution of which the egg is, in a mathematical
sense, generated. Some few eggs, such as those of the owl, the penguin,
or the tortoise, are spherical or very nearly so; a few more, such
as the grebe’s, the cormorant’s or the pelican’s, are approximately
ellipsoidal, with symmetrical or nearly symmetrical ends, and somewhat
similar are the so-called “cylindrical” eggs of the megapodes and the
sand-grouse; the great majority, like the hen’s egg, are ovoid, a
little blunter at one end than the other; and some, by an exaggeration
of this lack of antero-posterior symmetry, are blunt at one end but
characteristically pointed at the other, as is the case with the eggs
of the guillemot and puffin, the sandpiper, plover and curlew. It is
an obvious but by no means negligible fact that the egg, while often
pointed, is never flattened or discoidal; it is a prolate, but never an
oblate, spheroid.
The careful study and collection of birds’ eggs would seem to have
begun with the Count de Marsigli[586], the same celebrated naturalist
who first studied the “flowers” of the coral, and who wrote the
_Histoire physique de la mer_; and the specific form, as well as the
colour and other attributes of the egg have been again and again
discussed, and not least by the many dilettanti naturalists of the
eighteenth century who soon followed in Marsigli’s footsteps[587]. {653}
We need do no more than mention Aristotle’s belief, doubtless old
in his time, that the more pointed egg produces the male chicken,
and the blunter egg the hen; though this theory survived into modern
times[588] and perhaps still lingers on. Several naturalists, such as
Günther (1772) and Bühle (1818), have taken the trouble to disprove it
by experiment. A more modern and more generally accepted explanation
has been that the form of the egg is in direct relation to that of the
bird which has to be hatched within—a view that would seem to have
been first set forth by Naumann and Bühle, in their great treatise
on eggs[589], and adopted by Des Murs[590] and many other well-known
writers.
In a treatise by de Lafresnaye[591], an elaborate comparison is made
between the skeleton and the egg of the various birds, to shew, for
instance, how those birds with a deep-keeled sternum laid rounded
eggs, which alone could accommodate the form of the young. According
to this view, that “Nature had foreseen[592]” the form adapted to and
necessary for the growing embryo, it was easy to correlate the owl
with its spherical egg, the diver with its elliptical one, and in like
manner the round egg of the tortoise and the elongated one of the
crocodile with the shape of the creatures which had afterwards to be
hatched therein. A few writers, such as Thienemann[593], looked at the
same facts the other way, and asserted that the form of the egg was
determined by that of the bird by which it was laid, and in whose body
it had been conformed.
In more recent times, other theories, based upon the principles of
Natural Selection, have been current and very generally accepted, to
account for these diversities of form. The pointed, conical egg of the
guillemot is generally supposed to be an adaptation, {654} advantageous
to the species in the circumstances under which the egg is laid; the
pointed egg is less apt than a spherical one to roll off the narrow
ledge of rock on which this bird is said to lay its solitary egg,
and the more pointed the egg, so much the fitter and likelier is it
to survive. The fact that the plover or the sandpiper, breeding in
very different situations, lay eggs that are also conical, elicits
another explanation, to the effect that here the conical form permits
the many large eggs to be packed closely under the mother bird[594].
Whatever truth there be in these apparent adaptations to existing
circumstances, it is only by a very hasty logic that we can accept
them as a _vera causa_, or adequate explanation of the facts; and it
is obvious that, in the bird’s egg, we have an admirable case for the
direct investigation of the mechanical or physical significance of its
form[595].
Of all the many naturalists of the eighteenth and nineteenth centuries
who wrote on the subject of eggs, one alone (so far as I am aware)
ascribed the form of the egg to direct mechanical causes. Günther[596],
in 1772, declared that the more or less rounded or pointed form of
the egg is a mechanical consequence of the pressure of the oviduct
at a time when the shell is yet unformed or unsolidified; and that
accordingly, to explain the round egg of the owl or the kingfisher, we
have only to admit that the oviduct of these birds is somewhat larger
than that of most others, or less subject to violent contractions.
This statement contains, in essence, the whole story of the mechanical
conformation of the egg.
Let us consider, very briefly, the conditions to which the egg is
subject in its passage down the oviduct[597].
(1) The “egg,” as it enters the oviduct, consists of the yolk only,
enclosed in its vitelline membrane. As it passes down the first portion
of the oviduct, the white is gradually superadded, {655} and becomes in
turn surrounded by the “shell-membrane.” About this latter the shell is
secreted, rapidly and at a late period; the egg having meanwhile passed
on into a wider portion of the oviducal tube, called (by loose analogy,
as Owen says) the “uterus.” Here the egg assumes its permanent form,
here it becomes rigid, and it is to this portion of the “oviduct” that
our argument principally refers.
(2) Both the yolk and the entire egg tend to fill completely their
respective membranes, and, whether this be due to growth or imbibition
on the part of the contents or to contraction on the part of the
surrounding membranes, the resulting tendency is for both yolk and egg
to be, in the first instance, spherical, unless otherwise distorted by
external pressure.
(3) The egg is subject to pressure within the oviduct, which is an
elastic, muscular tube, along the walls of which pass peristaltic waves
of contraction. These muscular contractions may be described as the
contraction of successive annuli of muscle, giving annular (or radial)
pressure to successive portions of the egg; they drive the egg forward
against the frictional resistance of the tube, while tending at the
same time to distort its form. While nothing is known, so far as I am
aware, of the muscular physiology of the oviduct, it is well known in
the case of the intestine that the presence of an obstruction leads to
the development of violent contractions in its rear, which waves of
contraction die away, and are scarcely if at all propagated in advance
of the obstruction.
(4) It is known by observation that a hen’s egg is always laid blunt
end foremost.
(5) It can be shown, at least as a very common rule, that those eggs
which are most unsymmetrical, or most tapered off posteriorly, are
also eggs of a large size relatively to the parent bird. The guillemot
is a notable case in point, and so also are the curlews, sandpipers,
phaleropes and terns. We may accordingly presume that the more pointed
eggs are those that are large relatively to the tube or oviduct through
which they have to pass, or, in other words, are those which are
subject to the greatest pressure while being forced along. So general
is this relation that we may go still further, and presume with great
plausibility {656} in the few exceptional cases (of which the apteryx
is the most conspicuous) where the egg is relatively large though not
markedly unsymmetrical, that in these cases the oviduct itself is in
all probability large (as Günther had suggested) in proportion to the
size of the bird. In the case of the common fowl we can trace a direct
relation between the size and shape of the egg, for the first eggs laid
by a young pullet are usually smaller, and at the same time are much
more nearly spherical than the later ones; and, moreover, some breeds
of fowls lay proportionately smaller eggs than others, and on the whole
the former eggs tend to be rounder than the latter[598].
――――――――――
We may now proceed to inquire more particularly how the form of the egg
is controlled by the pressures to which it is subjected.
The egg, just prior to the formation of the shell, is, as we have seen,
a fluid body, tending to a spherical shape and _enclosed within a
membrane_.
Our problem, then, is: Given a practically incompressible fluid,
contained in a deformable capsule, which is either (_a_) entirely
inextensible, or (_b_) slightly extensible, and which is placed in
a long elastic tube the walls of which are radially contractile, to
determine the shape under pressure.
If the capsule be spherical, inextensible, and completely filled with
the fluid, absolutely no deformation can take place. The few eggs that
are actually or approximately spherical, such as those of the tortoise
or the owl, may thus be alternatively explained as cases where little
or no deforming pressure has been applied prior to the solidification
of the shell, or else as cases where the capsule was so little capable
of extension and so completely filled as to preclude the possibility of
deformation.
If the capsule be not spherical, but be inextensible, then deformation
can take place under the external radial compression, {657} only
provided that the pressure tends to make the shape more nearly
spherical, and then only on the further supposition that the capsule is
also not entirely filled as the deformation proceeds. In other words,
an incompressible fluid contained in an inextensible envelope cannot be
deformed without puckering of the envelope taking place.
Let us next assume, as the conditions by which this result may be
avoided, (_a_) that the envelope is to some extent extensible, or (_b_)
that the whole structure grows under relatively fixed conditions. The
two suppositions are practically identical with one another in effect.
It is obvious that, on the presumption that the envelope is only
moderately extensible, the whole structure can only be distorted to a
moderate degree away from the spherical or spheroidal form.
At all points the shape is determined by the law of the distribution of
_radial pressure within the given region of the tube_, surface friction
helping to maintain the egg in position. If the egg be under pressure
from the oviduct, but without any marked component either in a forward
or backward direction, the egg will be compressed in the middle, and
will tend more or less to the form of a cylinder with spherical ends.
The eggs of the grebe, cormorant, or crocodile may be supposed to
receive their shape in such circumstances.
When the egg is subject to the peristaltic contraction of the oviduct
during its formation, then from the nature and direction of motion of
the peristaltic wave the pressure will be greatest somewhere behind the
middle of the egg; in other words, the tube is converted for the time
being into a more conical form, and the simple result follows that the
anterior end of the egg becomes the broader and the posterior end the
narrower.
With a given shape and size of body, equilibrium in the tube may be
maintained under greater radial pressure towards one end than towards
the other. For example, a cylinder having conical ends, of semi-angles
θ and θ′ respectively, remains in equilibrium, apart from friction, if
_p_ cos^2 θ = _p′_ cos^2 θ′, so that at the more tapered end where θ
is small _p_ is small. Therefore the whole structure might assume such
a configuration, or grow under such conditions, finally becoming rigid
by solidification of the envelope. {658} According to the preceding
paragraph, we must assume some initial distribution of pressure, some
squeeze applied to the posterior part of the egg, in order to give it
its tapering form. But, that form once acquired, the egg may remain
in equilibrium both as regards form and position within the tube,
even after that excess of pressure on the posterior part is relieved.
Moreover, the above equation shews that a normal pressure no greater
and (within certain limits) actually less acting upon the posterior
part than on the anterior part of the egg after the shell is formed
will be sufficient to communicate to it a forward motion. This is an
important consideration, for it shews that the ordinary form of an egg,
and even the conical form of an extreme case such as the guillemot’s,
is directly favourable to the movement of the egg within the oviduct,
blunt end foremost.
The mathematical statement of the whole case is as follows: In our egg,
consisting of an extensible membrane filled with an incompressible
fluid and under external pressure, the equation of the envelope is
_p_{n}_ + _T_(1/_r_ + 1/_r′_) = _P_, where _p_{n}_ is the normal
component of external pressure at a point where _r_ and _r′_ are the
radii of curvature, _T_ is the tension of the envelope, and _P_ the
internal fluid pressure. This is simply the equation of an elastic
surface where _T_ represents the coefficient of elasticity; in other
words, a flexible elastic shell has the same mathematical properties
as our fluid, membrane-covered egg. And this is the identical equation
which we have already had so frequent occasion to employ in our
discussion of the forms of cells; save only that in these latter we
had chiefly to study the tension _T_ (i.e. the surface-tension of the
semi-fluid cell) and had little or nothing to do with the factor of
external pressure (_p_{n}_), which in the case of the egg becomes of
chief importance.
The above equation is the _equation of equilibrium_, so that it must be
assumed either that the whole body is at rest or that its motion while
under pressure is not such as to affect the result. Tangential forces,
which have been neglected, could modify the form by alteration of _T_.
In our case we must, and may very reasonably, assume that any movement
of the egg down the oviduct during the period when its form is being
impressed upon it is very slow, being possibly balanced by the advance
of the {659} peristaltic wave which causes the movement, as well as by
friction.
The quantity _T_ is the tension of the enclosing capsule—the
surrounding membrane. If _T_ be constant or symmetrical about the axis
of the body, the body is symmetrical. But the abnormal eggs that a
hen sometimes lays, cylindrical, annulated, or quite irregular, are
due to local weakening of the membrane, in other words, to asymmetry
of _T_. Not only asymmetry of _T_, but also asymmetry of _p_{n}_,
will render the body subject to deformation, and this factor, the
unknown but regularly varying, largely radial, pressure applied by
successive annuli of the oviduct, is the essential cause of the
form, and variations of form, of the egg. In fact, in so far as the
postulates correspond near enough to actualities, the above equation
is the equation of _all eggs_ in the universe. At least this is so if
we generalise it in the form _p_{n}_ + _T_/_r_ + _T′_/_r′_ = _P_ in
recognition of a possible difference between the principal tensions.
In the case of the spherical egg it is obvious that _p_{n}_ is
everywhere equal. The simplest case is where _p_{n}_ = 0, in other
words, where the egg is so small as practically to escape deforming
pressure from the tube. But we may also conceive the tube to be so
thin-walled and extensible as to press with practically equal force
upon all parts of the contained sphere. If while our egg be in process
of conformation the envelope be free at any part from external pressure
(that is to say, if _p_{n}_ = 0), then it is obvious that that part
(if of circular section) will be a portion of a sphere. This is not
unlikely to be the case actually or approximately at one or both poles
of the egg, and is evidently the case over a considerable portion of
the anterior end of the plover’s egg.
In the case of the conical egg with spherical ends, as is more or
less the case in the plover’s and the guillemot’s, then at either end
of the egg _r_ and _r′_ are identical, and they are greater at the
blunt anterior end than at the other. If we may assume that _p_{n}_
vanishes at the poles of the egg, then it is plain that _T_ varies in
the neighbourhood of these poles, and, further, that the tension _T_ is
greatest at and near the small end of the egg. It is here, in short,
that the egg is most likely to be irregularly distorted or {660} even
to burst, and it is here that we most commonly find irregularities of
shape in abnormal eggs.
If one portion of the envelope were to become practically stiff before
_p_ ceases to vary, that would be tantamount to a sudden variation of
_T_, and would introduce asymmetry by the imposition of a boundary
condition in addition to the above equation.
Within the egg lies the yolk, and the yolk is invariably spherical or
very nearly so, whatever be the form of the entire egg. The reason
is simple, and lies in the fact that the yolk is itself enclosed in
another membrane, between which and the outer membrane lies a fluid
the presence of which makes _p_{n}_ for the inner membrane practically
constant. The smallness of friction is indicated by the well-known fact
that the “germinal spot” on the surface of the yolk is always found
uppermost, however we may place and wherever we may open the egg; that
is to say, the yolk easily rotates within the egg, bringing its lighter
pole uppermost. So, owing to this lack of friction in the outer fluid,
or white, whatever shear is produced within the egg will not be easily
transmitted to the yolk, and, moreover, owing to the same fluidity, the
yolk will easily recover its normal sphericity after the egg-shell is
formed and the unequal pressure relieved.
These, then, are the general principles involved in, and illustrated
by, the configuration of an egg; and they take us as far as we
can safely go without actual quantitative determinations, in each
particular case, of the forces concerned.
――――――――――
In certain cases among the invertebrates, we again find instances of
hard-shelled eggs which have obviously been moulded by the oviduct,
or so-called “ootype,” in which they have lain: and not merely in
such a way as to shew the effects of peristaltic pressure upon a
uniform elastic envelope, but so as to impress upon the egg the more
or less irregular form of the cavity, within which it had been for a
time contained and compressed. After this fashion Dr Looss[599] of
Cairo has {661} explained the curious form of the egg in _Bilharzia_
(_Schistosoma_) _haematobium_, a formidable parasitic worm to which
is due a disease wide-spread in Africa and Arabia, and an especial
scourge of the Mecca pilgrims. The egg in this worm is provided at one
end with a little spine, which now and then is found to be placed not
terminally but laterally or ventrally, and which when so placed has
been looked upon as the mark of a supposed new species, _S. Mansoni_.
As Looss has now shewn, the little spine must be explained as having
been moulded within a little funnel-shaped expansion of the uterus,
just where it communicates with the common duct leading from the ovary
and yolk-gland; by the accumulation of eggs in the ootype, the one
last formed is crowded into a sideways position, and then, where the
side-wall of the egg bulges in the funnel-shaped orifice of the duct, a
little lateral “spine” is formed. In another species, _S. japonicum_,
the egg is described as bulging into a so-called “calotte,” or
bubble-like convexity at the end opposite to the spine. This, I think,
may, with very little doubt, be ascribed to hardening of the egg-shell
having taken place just at the period when partial relief from pressure
was being experienced by the egg in the neighbourhood of the dilated
orifice of the oviduct.
This case of Bilharzia is not, from our present point of view, a very
important one, but nevertheless it is interesting. It ascribes to a
mechanical cause a curious peculiarity of form; it shews, by reference
to this mechanical principle, that two conditions which were very
different to the systematic naturalist’s eye, were really only two
simple mechanical modifications of the same thing; and it destroys the
chief evidence for the existence of a supposed new species of worm,
a continued belief in which, among worms of such great pathogenic
importance, might lead to gravely erroneous pathological deductions.
_On the Form of Sea-urchins_
As a corollary to the problem of the bird’s egg, we may consider for a
moment the forms assumed by the shells of the sea-urchins. These latter
are commonly divided into two classes, the Regular and the Irregular
Echinids. The regular sea-urchins, save in {662} slight details
which do not affect our problem, have a complete radial symmetry.
The axis of the animal’s body is vertical, with mouth below and the
intestinal outlet above; and around this axis the shell is built as a
symmetrical system. It follows that in horizontal section the shell is
everywhere circular, and we shall have only to consider its form as
seen in vertical section or projection. The irregular urchins (very
inaccurately so-called) have the anal extremity of the body removed
from its central, dorsal situation; and it follows that they have now
a single plane of symmetry, about which the organism, shell and all,
is bilaterally symmetrical. We need not concern ourselves in detail
with the shapes of their shells, which may be very simply interpreted,
by the help of radial co-ordinates, as deformations of the circular or
“regular” type.
The sea-urchin shell consists of a membrane, stiffened into rigidity
by calcareous deposits, which constitute a beautiful skeleton of
separate, neatly fitting “ossicles.” The rigidity of the shell is more
apparent than real, for the entire structure is, in a sluggish way,
plastic; inasmuch as each little ossicle is capable of growth, and the
entire shell grows by increments to each and all of these multitudinous
elements, whose individual growth involves a certain amount of freedom
to move relatively to one another; in a few cases the ossicles are
so little developed that the whole shell appears soft and flexible.
The viscera of the animal occupy but a small part of the space within
the shell, the cavity being mainly filled by a large quantity of
watery fluid, whose density must be very near to that of the external
sea-water.
Apart from the fact that the sea-urchin continues to grow, it is plain
that we have here the same general conditions as in the egg-shell, and
that the form of the sea-urchin is subject to a similar equilibrium
of forces. But there is this important difference, that an external
muscular pressure (such as the oviduct administers during the
consolidation of egg-shell), is now lacking. In its place we have the
steady continuous influence of gravity, and there is yet another force
which in all probability we require to take into consideration.
While the sea-urchin is alive, an immense number of delicate
“tube-feet,” with suckers at their tips, pass through minute pores
{663} in the shell, and, like so many long cables, moor the animal to
the ground. They constitute a symmetrical system of forces, with one
resultant downwards, in the direction of gravity, and another outwards
in a radial direction; and if we look upon the shell as originally
spherical, both will tend to depress the sphere into a flattened cake.
We need not consider the radial component, but may treat the case as
that of a spherical shell symmetrically depressed under the influence
of gravity. This is precisely the condition which we have to deal with
in a drop of liquid lying on a plate; the form of which is determined
by its own uniform surface-tension, plus gravity, acting against the
uniform internal hydrostatic pressure. Simple as this system is, the
full mathematical investigation of the form of a drop is not easy, and
we can scarcely hope that the systematic study of the Echinodermata
will ever be conducted by methods based on Laplace’s differential
equation[600]; but we have no difficulty in seeing that the various
forms represented in a series of sea-urchin shells are no other than
those which we may easily and perfectly imitate in drops.
In the case of the drop of water (or of any other particular liquid)
the specific surface-tension is always constant, and the pressure
varies inversely as the radius of curvature; therefore the smaller the
drop the more nearly is it able to conserve the spherical form, and the
larger the drop the more does it become flattened under gravity. We can
represent the phenomenon by using india-rubber balls filled with water,
of different sizes; the little ones will remain very nearly spherical,
but the larger will fall down “of their own weight,” into the form of
more and more flattened cakes; and we see the same thing when we let
drops of heavy oil (such as the orthotoluidene spoken of on p. 219),
fall through a tall column of water, the little ones remaining round,
and the big ones getting more and more flattened as they sink. In the
case of the sea-urchin, the same series of forms may be assumed to
occur, irrespective of size, through variations in _T_, the specific
tension, or “strength,” of the enveloping shell. Accordingly we may
study, entirely from this point of view, such a series as the following
(Fig. 328). In a very few cases, such as the fossil Palaeechinus,
we have an approximately spherical {664} shell, that is to say a
shell so strong that the influence of gravity becomes negligible as
a cause of deformation. The ordinary species of Echinus begin to
display a pronounced depression, and this reaches its maximum in such
soft-shelled flexible forms as Phormosoma. On the general question
I took the opportunity of consulting Mr C. R. Darling, who is an
acknowledged expert in drops, and he at once agreed with me that such
forms as are represented in Fig. 328 are no other than diagrammatic
illustrations
[Illustration: Fig. 328. Diagrammatic vertical outlines of various
Sea-urchins: A, Palaeechinus; B, _Echinus acutus_; C, Cidaris; D, D′
Coelopleurus; E, E′ Genicopatagus; F, _Phormosoma luculenter_; G, P.
_tenuis_; H, Asthenosoma; I, Urechinus.]
of various kinds of drops, “most of which can easily be reproduced
in outline by the aid of liquids of approximately equal density to
water, although some of them are fugitive.” He found a difficulty in
the case of the outline which represents Asthenosoma, but the reason
for the anomaly is obvious; the flexible shell has flattened down
until it has come in contact with the hard skeleton of the jaws, or
“Aristotle’s lantern,” within, and the curvature of the outline is
accordingly disturbed. The elevated, conical shells such as those of
Urechinus and Coelopleurus evidently call for some further explanation;
for there is here some cause at work {665} to elevate, rather than to
depress the shell. Mr Darling tells me that these forms “are nearly
identical in shape with globules I have frequently obtained, in which,
on standing, bubbles of gas rose to the summit and pressed the skin
upwards, without being able to escape.” The same condition may be at
work in the sea-urchin; but a similar tendency would also be manifested
by the presence in the upper part of the shell of any accumulation of
substance lighter than water, such as is actually present in the masses
of fatty, oily eggs.
_On the Form and Branching of Blood-vessels_
Passing to what may seem a very different subject, we may investigate
a number of interesting points in connection with the form and
structure of the blood-vessels, on the same principle and by help of
the same equations as those we have used, for instance, in studying the
egg-shell.
We know that the fluid pressure (_P_) within the vessel is balanced by
(1) the tension (_T_) of the wall, divided by the radius of curvature,
and (2) the external pressure (_p_{n}_), normal to the wall: according
to our formula
_P_ = _p_{n}_ + _T_(1/_r_ + 1/_r′_).
If we neglect the external pressure, that is to say any support which
may be given to the vessel by the surrounding tissues, and if we deal
only with a cylindrical vein or artery, this formula becomes simplified
to the form _P_ = _T_/_R_. That is to say, under constant pressure, the
tension varies as the radius. But the tension, per unit area of the
vessel, depends upon the thickness of the wall, that is to say on the
amount of membranous and especially of muscular tissue of which it is
composed.
Therefore, so long as the pressure is constant, the thickness of
the wall should vary as the radius, or as the diameter, of the
blood-vessel. But it is not the case that the pressure is constant,
for it gradually falls off, by loss through friction, as we pass from
the large arteries to the small; and accordingly we find that while,
for a time, the cross-sections of the larger and smaller vessels are
symmetrical figures, with the wall-thickness proportional to the
size of the tube, this proportion is gradually lost, and the walls
{666} of the small arteries, and still more of the capillaries,
become exceedingly thin, and more so than in strict proportion to the
narrowing of the tube.
――――――――――
In the case of the heart we have, within each of its cavities, a
pressure which, at any given moment, is constant over the whole
wall-area, but the thickness of the wall varies very considerably.
For instance, in the left ventricle, the apex is by much the thinnest
portion, as it is also that with the greatest curvature. We may assume,
therefore (or at least suspect), that the formula, _t_(1/_r_ + 1/_r′_)
= _C_, holds good; that is to say, that the thickness (_t_) of the
wall varies inversely as the mean curvature. This may be tested
experimentally, by dilating a heart with alcohol under a known
pressure, and then measuring the thickness of the walls in various
parts after the whole organ has become hardened. By this means it is
found that, for each of the cavities, the law holds good with great
accuracy[601]. Moreover, if we begin by dilating the right ventricle
and then dilate the left in like manner, until the whole heart is
equally and symmetrically dilated, we find (1) that we have had to use
a pressure in the left ventricle from six to seven times as great as in
the right ventricle, and (2) that the thickness of the walls is just in
the same proportion[602].
――――――――――
A great many other problems of a mechanical or hydrodynamical kind
arise in connection with the blood-vessels[603], and while these are
chiefly interesting to the physiologist they have also their interest
for the morphologist in so far as they bear upon structure and form.
As an example of such mechanical problems {667} we may take the
conditions which determine or help to determine the manner of branching
of an artery, or the angle at which its branches are given off; for,
as John Hunter said[604], “To keep up a circulation sufficient for
the part, and no more, Nature has varied the angle of the origin of
the arteries accordingly.” The general principle is that the form and
arrangement of the blood-vessels is such that the circulation proceeds
with a minimum of effort, and with a minimum of wall-surface, the
latter condition leading to a minimum of friction and being therefore
included in the first. What, then, should be the angle of branching,
such that there shall be the least possible loss of energy in the
course of the circulation? In order to solve this problem in any
particular case we should obviously require to know (1) how the loss
of energy depends upon the distance travelled, and (2) how the loss of
energy varies with the diameter of the vessel. The loss of energy is
evidently greater in a narrow tube than in a wide one, and greater,
obviously, in a long journey than a short. If the
[Illustration: Fig. 329.]
large artery, _AB_, give off a comparatively narrow branch leading
to _P_ (such as _CP_, or _DP_), the route _ACP_ is evidently shorter
than _ADP_, but on the other hand, by the latter path, the blood has
tarried longer in the wide vessel _AB_, and has had a shorter course
in the narrow branch. The relative advantage of the two paths will
depend on the loss of energy in the portion _CD_, as compared with that
in the alternative portion _CD′_, the latter being short and narrow,
the former long and wide. If we ask, then, which factor is the more
important, length or width, we may safely take it that the question is
one of degree: and that the factor of width will become much the more
important wherever the artery and its branch are markedly unequal in
size. In other words, it would seem that for small branches a large
angle of bifurcation, and for large branches a small one, is always the
better. Roux has laid down certain rules in regard to the branching of
arteries, which correspond with the general {668} conclusions which
we have just arrived at. The most important of these are as follows:
(1) If an artery bifurcate into two equal branches, these branches come
off at equal angles to the main stem. (2) If one of the two branches be
smaller than the other, then the main branch, or continuation of the
original artery, makes with the latter a smaller angle than does the
smaller or “lateral” branch. And (3) all branches which are so small
that they scarcely seem to weaken or diminish the main stem come off
from it at a large angle, from about 70° to 90°.
[Illustration: Fig. 330.]
We may follow Hess in a further investigation of this phenomenon. Let
_AB_ be an artery, from which a branch has to be given off so as to
reach _P_, and let _ACP_, _ADP_, etc., be alternative courses which
the branch may follow: _CD_, _DE_, etc., in the diagram, being equal
distances (= _l_) along _AB_. Let us call the angles _PCD_, _PCE_,
_x__{1}, _x__{2}, etc.: and the distances _CD′_, _DE′_, by which each
branch exceeds the next in length, we shall call _l__{1}, _l__{2}, etc.
Now it is evident that, of the courses shewn, _ACP_ is the shortest
which the blood can take, but it is also that by which its transit
through the narrow branch is the longest. We may reduce its transit
through the narrow branch more and more, till we come to _CGP_, or
rather to a point where the branch comes off at right angles to the
main stem; but in so doing we very considerably increase the whole
distance travelled. We may take it that there will be some intermediate
point which will strike the balance of advantage.
Now it is easy to shew that if, in Fig. 330, the route _ADP_ and _AEP_
(two contiguous routes) be equally favourable, then any other route on
either side of these, such as _ACP_ or _AFP_, must be less favourable
than either. Let _ADP_ and _AEP_, then, be equally favourable; that is
to say, let the loss of energy which the blood suffers in its passage
along these two routes be equal. {669} Then, if we make the distance
_DE_ very small, the angles _x__{2} and _x__{3} are nearly equal,
and may be so treated. And again, if _DE_ be very small, then _DE′E_
becomes a right angle, and _l__{2} (or _DE′_) = _l_ cos _x__{2}.
But if _L_ be the loss of energy per unit distance in the wide tube
_AB_, and _L′_ be the corresponding loss of energy in the narrow tube
_DP_, etc., then _lL_ = _l__{2} _L′_, because, as we have assumed, the
loss of energy on the route _DP_ is equal to that on the whole route
_DEP_. Therefore _lL_ = _lL′_ cos _x__{2}, and cos _x__{2} = _L_/_L′_.
That is to say, the most favourable angle of branching will be such
that the cosine of the angle is equal to the ratio of the loss of
energy which the blood undergoes, per unit of length, in the main
vessel, as compared with that which it undergoes in the branch.
While these statements are so far true, and while they undoubtedly
cover a great number of observed facts, yet it is plain that, as in
all such cases, we must regard them not as a complete explanation,
but as _factors_ in a complicated phenomenon: not forgetting that (as
the most learned of all students of the heart and arteries, Dr Thomas
Young, said in his Croonian lecture[605]) all such questions as these,
and all matters connected with the muscular and elastic powers of the
blood-vessels, “belong to the most refined departments of hydraulics.”
Some other explanation must be sought in order to account for a
phenomenon which particularly impressed John Hunter’s mind, namely the
gradually altering angle at which the successive intercostal arteries
are given off from the thoracic aorta: the special interest of this
case arising from the regularity and symmetry of the series, for “there
is not another set of arteries in the body whose origins are so much
the same, whose offices are so much the same, whose distances from
their origin to the place of use, and whose uses [? sizes][606] are so
much the same.”
{670}