there must be an infinite number of laws of attraction, which, by substitution
in these formulas, will give that result. The knowledge of these laws is
however the most delicate and important part of this theory : it is indispensably
necessary to connect together the different phenomena of capillary attraction, [9177]
and Clairaut himself would have seen the necessity of it, if he had attempted,
for example, to extend his investigations from tubes to the capillary spaces
where it becomes very nearly equal to that of the fluid in the interior of its mass ; observing
that, though the distance X' is insensible, it may be supposed much greater than the distance [9173f]
to which the corpuscular action of any one particle of the fluid upon any other particle of its [9173u]
mass extends. Similar remarks may be made relative to the change of density in the fluid
near the sides of the tube. We shall see in [9260/, &,c. 9841c, Stc], that this change of density [9173r]
produces a corresponding change in the value of the capillary intensity H [9262c] ; but as
this quantity can be found only from actual experiments, and not from the analytical expression [9173m>]
[9253', &ic.], it leaves the results of La Place's theory unimpaired in all the formulas
depending on this quantity ; and the calculation of the effects of the capillary attraction is in [9173x]
almost every case the same as if the change of density had been noticed ; the slight differences
which occur will be pointed out, wherever they happen to be found, with the methods of [9173?/]
correction as in [95805'] ; and we may in this connection observe, that the first or greatest force
spoken of by the author in [9184] is that which corresponds to the interior of the fluid, as
computed by him upon the supposition that the corpuscular action (p (/) is always positive
and the density uniform; it is also evident that this result might be very much modified near [9l73r]
the surface of the fluid, by taking into view the repulsive force of the heat, and the change
of density near that surface. For, in a stratum of variable density, at an insensible distance
from the surface, we may conceive of such an arrangement of the particles of the fluid, as
will render this action very different from that which the author supposes ; making it there
either large or small, positive or negative, according to the nature of the functions which are [9174oJ
assumed, to express the corpuscular action [9l73ej and the density in that part of the fluid.
688 CAPILLARY ATTRACTION. [Mec.Cel.
included between parallel planes, in order to deduce from analysis, the ratio
[9178] q£ equality, indicated by experiment, between the ascent of the fluid in a
cylindrical tube, and its ascension between two parallel planes, whose distance
from each other is equal to the semi-diameter of the tube ; which no person
has yet attempted to explain. A long while ago, I endeavored in vain to
determine the laws of attraction which would represent these phenomena ;
but some late researches have rendered it evident that the whole may be
[9179J represented by the same laws, which satisfy the phenomena of refraction ; that
is, by laws in which the attraction is sensible only at insensible distances ; and
from this principle we can deduce a complete theory of capillary attraction.
[9180] Clairaut supposes that the action of a capillary tube may be sensible upon
the infinitely thin column, which passes through the axis of the tube. Upon
this point I differ wholly from him, and think, with Hawksbee and other
philosophers, that the capillary attraction is, like the force producing refraction,
[9181] and all the chemical affinities, sensible only at insensible distances. Hawksbee
observed that in glass tubes, whether the glass is very thick, or very thin, the
water rises to the same height, if the interior diameters are the same. Hence
it follows that the cylindrical strata of glass, which are at a sensible distance
from the interior surface, do not aid in raising the water, though in each
one of these strata, taken separately, the fluid ought to rise above the level.
It is not the interposition of the strata, which they include between them,
[9182] which prevents their action upon the water ; for it is natural to suppose that
the capillary attraction, like the force of gravity, is transmitted through other
bodies; this attraction must therefore disappear solely by reason of the distance
of the fluid from these strata ; whence it follows that the attraction of the
glass upon the water is sensible only at insensible distances.
Making use of this principle, 1 have determined the action of a fluid mass,
terminated by a portion of a spherical concave or convex surface, upon a
[9183] column situated within it, contained in an infinitely narrow canal, and directed
towards the centre of that surface. By this action, I mean the pressure which
the fluid, contained in the canal, would exert, by means of the attraction of
the whole mass, upon a plane base situated within the canal, perpendicular to
its sides, and at any sensible distance from the surface : this base being taken
for unity, I shall show that this action is less or greater than if the surface
were plane ; less if the surface be concave [9275] ; greater if the surface he convex
[9184] [9276]. Its analytical expression is composed of two terms: the ^r5^, which
is much greater than the second [9262a,c], denotes the action of a mass
terminated by a plane surface ; and / think that upon this term depends the
[9187]
X. Suppl. 1.] INTRODUCTION. 689
suspension of the mercury, in a barometrical tube, at a height two or three times
greater than that which is produced by the pressure of the atmosphere ; also the [9185]
refractive poiver of diaphanous bodies, cohesion, and in general the chemical
affinities. The second term denotes the part of the action, depending on the
spherical form of the surface ; or in other words, the action of the meniscus,
included betioeen that surface and the plane which touches it. This action is to [9186]
be added to, or subtracted from, the preceding one, according as the surface
is convex or concave. It is inversely proportional to the radius of the spherical
surface ; for it is evident that the less this radius is, the greater will be the
meniscus, near the point of contact. This second term produces the capillary [9188]
action, which differs therefore from the chemical affinities corresponding to the
first term. -^
From these results, relative to bodies terminated by sensible segments of a [9189]
spherical surface, I have deduced this general theorem [93021. "7w all the General
^ ' f5 L J theorem.
laws which render the attraction insensible at sensible distances, the action of a
body terminated by a curve surface, upon an infinitely narrow interior canal, 'â– â– '
which is perpendicular to that surface, at any point whatever, is equal to the half
sum of the actions upon the same canal, of two spheres which have the same
radii as the greatest and the least radii of curvature of the surface at that [9191]
point.^^ By means of this theorem, and of the laws of the equilibrium of fluids,
we can determine the figure which a fluid mass must have, when it is included
within a vessel of a given figure, and acted upon by gravity. It depends upon
an equation of partial differentials of the second order [9318], whose integral
cannot be obtained by any known methods. If the figure be of revolution, [9192]
this equation is reduced to common differentials [9324], and may be integrated
by a very approximate method, when the surface is very small. By this
means I shall prove that, in tubes of a very small diameter, the surface of the
fluid will approximate the more towards the form of a spherical segment, as
the diameter of the tube shall be decreased [9342, &c.]. If these segments be [9193]
similar in different tubes of the same matter, the radii of their surfaces will
be in the direct ratio of the diameters of the tubes. Now this similarity of
the spherical segments will appear evident by considering that the distance, at
which the action of the tube ceases to be sensible, is imperceptible ; so that if,
by means oj a very powerful microscope, we should be able to make it appear
equal to a millimetre, it is probable that the same magnifying poiver would give ^ "^^
to the diameter of the tube an apparent length of several metres. The surface
of the tube may therefore be considered as very nearly a plane surface, for an [9195']
VOL. IV. 173
690 CAPILLARY ATTRACTION. [Mec.Cel.
â– extent which is equal to that of the sphere of its sensible activity ; the fluid loill
therefore be elevated or depressed near that surface, in almost the same manner
as if it were a plane. Beyond this point, the fluid ivill be subjected ordy to the
[919GJ force of gravity and its oivn action on its particles; its surface will be very
nearly that of a spherical segment, of which the extreme tangent planes, being
those of the fluid surface at the limits of the sensible sphere of activity of the
tube, will he very nearly, in the different tubes, equally inclined to their sides ;
[9197] whence it follows that all these segments will be similar.
The comparison of these results gives the true cause of the elevation,
or depression, of fluids, in capillary tubes, in the inverse ratio of their
diameters. If, in the axis of a glass tube, we imagine an infinitely narrovjr
canal to be placed, which, after being continued downwards, in a vertical
[9198] direction, a little below the bottom of the tube, is then turned in a horizontal
direction below the tube, and afterwards in a vertical direction upwards, until
it meets the horizontal surface of the water in the vessel, where the lower
extremity of the tube is immersed ; the action of the water in the tube, upon
this canal, will be less on account of the concavity of its surface, than the
action of the water in the vessel upon the same canal ; therefore the fluid must
[9199] rise in the tube to compensate for this difference ; and as it is, by what has
been said, in the inverse ratio of the diameter of the tube, the elevation of
the fluid above its level must follow the same law.
If the interior surface of the fluid be convex, which is the case with
mercury in a glass tube, the action of the fluid upon the canal will be greater
[9200] than that of the fluid in the vessel ; the fluid must therefore sink in the tube,
in proportion to this difference, and the depression will therefore be in the
inverse ratio of the diameter of the tube.
Therefore the attraction of a capillary tube has no other influence upon the
elevation or depression of the fluid which it contains, than that of determining
[9201] the inclination of the first tangent planes of the interior fluid surface, situated
very near to the sides of the tube ; and it is upon this inclination that the
concavity or convexity of the surface depends, as loell as the magnitude of its
radius. The friction of the fluid against these sides, may increase or diminish,
a little, the curvature of its surface, as we see daily in the barometer ; and then
[9202] the capillary effects increase or diminish in the same ratio. These effects
increase in a very sensible manner, by the combined forces arising from the
concavity of one surface and the convexity of the other surface. We shall
see hereafter [97l7,&c.], that we may thus raise the water in capillary tubes to a
X. Suppl. 1.] INTRODUCTION. 691
greater height above its level, than when they are dipped into a vessel filled
with that fluid.
The differential equation of the surface of a fluid included within capillary
tubes of revolution [9383], leads to this general result, namely; that if, in a
cylindrical tube, we introduce a cylinder, having the same axis as the tube, [9203]
and with such a diameter that the interval or space between its surface
and the interior surface of the tube may be very small, the fluid will ascend,
in the space between these tubes, to the same height as in a tube whose radius
is equal to the interval between them [9410]. If we suppose the radii of the
tube and cylinder to be infinite, it will correspond to the case of a fluid,
included between two vertical and parallel planes which are very near to each
other. The preceding result is verified at that limit, by experiments [9204]
[9658, &c.] formerly made in presence of the Royal Society of London,
and under the inspection of Newton, who has quoted them in his Optics, an
admirable work, where that profound genius has put forth many original ideas,
elevated far above the science of his time, and which have been confirmed by
modern chemistry. M. Hauy has consented, at my request, to make some
experiments near the other limit, with tubes and cylinders of very small [9205]
diameters, and he has found the preceding result to be equally correct at this
limit, as at the first.
The phenomena observed in a drop of fluid, in motion, or suspended in
equilibrium, either in a conical capillary tube, or between two planes a little
inclined to each other, are very proper for verifying this theory. A small [9206]
column of water in a conical tube, open at both ends, and supported
horizontally, flows towards the vertex of the tube : and we easily perceive that
this ought to take place. For the surface of the fluid column is concave at
both its extremities ; but the radius of the surface which is nearest the vertex,
is less than the radius of the other surface nearest to the base ; therefore the
action of the fluid upon its own particles, is the least on the side nearest the
vertex, consequently the column must tend towards that part. If the fluid be [9207]
mercury, then its surface will be convex, and its radius will, in like manner,
be less near the vertex than near the base ; but, on account of its convexity,
the action of the fluid upon its particles will be greatest towards the summit,
and the column must therefore tend towards the base of the tube.
We may balance this action, by the weight of the column itself, and keep
it suspended in equilibrium, by inclining the axis of the tube to the horizon.
A very simple calculation shows, that, if the length of the column be small, [9208]
692 CAPILLARY ATTRACTION. [Mec. Cel.
the sine of the inclination of the axis is then, very nearly, in the inverse ratio
of the square of the distance from the middle of the column to the vertex of
the cone [9475]. A similar result takes place, if we put a drop of fluid
between two planes, which are inclined to each other by a very small angle,
[9209] and are in contact at their horizontal borders [9550]. These results are
entirely conformable to experiment, as we may see in Newton's Optics
(question 31). This great mathematician endeavored to explain them ; but
his explanation, when compared with that which is here given, shows the
advantages of an accurate mathematical theory.
By calculation we also find, that the sine of the inclination of the axis of
[9210] the cone to the horizon, is then very nearly equal to a fraction, having for its
denominator the distance from the middle of the drop to the vertex of the
cone, and for its numerator the height to which the fluid will rise in a
cylindrical tube whose diameter is equal to that of the cone in the middle of the
column [9474]. If the two planes, enclosing a drop of the same fluid, are inclined
to each other by an angle, which is equal to the angle formed by the axis of
[9211] the cone and one of its sides, the plane which bisects the angle formed by the
two preceding planes, must have the same inclination to the horizon, as the
axis of the cone, for the drop to remain in equilibrium. Hawksbee made
very carefully an experiment of this kind [9709], which we shall hereafter
compare with the preceding theorem ; the very little difference which is
found, between this experiment and the theorem, is an incontestible proof
of its accuracy.
The theory furnishes an explanation, and the measure, of a singular
phenomenon noticed in experiments ; namely, that, whether the fluid be
elevated or depressed, between two vertical and parallel planes, dipped into the
fluid by their lower extremities, these planes have a tendency to approach towards
each other. It is shown by analysis [9580, Stc], that, if the fluid be elevated
[9213] between them, each plane suffers a pressure, tending inwardly, and equal to
that of a column of the same fluid, having for its height the half sum of the
elevations of the fluid above the level, at the points of contact of the interior
and exterior surfaces of the fluid with the plane, and for its base the part of
the plane which is included between the two horizontal lines drawn through
these points. If the fluid is depressed between these planes, each of them, in
like manner, will suffer a pressure tending inwards, and equal to a column of the
[9214] same fluid, having for its height the half sum of the depressions below the
level of the points of contact of the interior and exterior surfaces of the
[9212]
X. Suppl. 1.] INTRODUCTION. 693
fluid with the plane, and for its base the part of the plane included between
the two horizontal lines drawn through those points* [9586].
The concavity or convexity of the surface of a fluid, included within capillary
spaces, has heretofore been considered as nothing more than a secondary effect
of the capillary action, and not as the principal cause of such phenomena ; so [9215]
that but little attention has been paid to the curvature of these surfaces ; but as
the preceding theory makes the phenomena depend chiefly on the curvature, it
becomes interesting to determine it. Several experiments made very carefully^ by
M. Hauy, indicate that, in glass capillary tubes of a very small diameter, the [92161
concave surface of water and oil, and the convex surface of mercury, differ but
very little from that of a hemisphere.
Clairaut made this singular remark ; namely, that, if the law of the attraction
of the matter of the tube upon the fluid, differs only by its intensity from the
law of the attraction of the fluid upon its own particles, the fluid will rise above [9217]
the level, so long as the intensity of the first of these attractions shall exceed the
half of the intensity of the second. If the intensity of the first of these attractions [9217']
be exactly equal to half\ of the intensity of the second, it will be easy to show that
the surface of the fluid in the tube will be horizontal, and that it will not
rise above the level. If these two intensities be equal, the surface of the
fluid in the tube will be concave, and of a hemispherical form; then the [9218]
fluid will be elevated in the tube. If the intensity of the attraction of the
* (4113) We shall see in [9580^, &£c.] that the values of the pressures here given require
some modification, when the angles formed by the vertical planes and the tangents of the
surfaces of the fluid near to them are different, on different sides of the same plane ; as
sometimes happens on account of the planes being more or less moistened on the one side
than on the other.
[9214a]
f (41 14) This ratio of the intensities of the attraction of the tube and fluid, when the [9218a]
surface is horizontal, is computed upon the supposition that the density of the fluid is uniform,
and that it does not vary near the surface of this fluid or near the sides of the tube. Now [92186]
this is not the case, as the author himself has remarked in [10502'] ; therefore this demonstration
is defective, as will more fully be seen in [9596/] ; but it is probable, however, that this ratio [9218c]
is nearly correct. We shall also see in [9626n, &.c. 9655«, he], that the remarks made in
[9218], for a concave hemisphere, and in [9219], for a convex hemisphere, are very nearly
correct. Lastly, what is stated in [9219'], for concave or convex segments, may be considered L^'^^""]
as very nearly correct, except that in [9219'] we must not suppose that the ratio of the
intensities at the hmit of the surfaces, is strictly equal to one half, as the author asserts, [9218e]
though it is probably very nearly equal to it [9596A:, &:c.]
VOL. IV. 174
694 CAPILLARY ATTRACTION. [Mec. Cel.
tube be nothim, or insensible, the surface of the fluid in the tube will be convex
[92191 . .
and hemispherical ; and then the fluid will be depressed. Between these two
limits, the surface of the fluid will be that of a spherical segment ; and it will be
concave or convex, according as the intensity of the attraction of the matter of
[9219'] the tube upon the fluid shall be greater or less than the half of the attraction
of the fluid upon its own particles [9587—9655, 921 8e].
If the intensity of the attraction of the tube upon the fluid exceeds that of
the attraction of the fluid upon its own particles, it appears to me probable that
[9220] tken the fluids by attaching itself to the tube, forms an interior tube, ivhich produces
alone the elevation of the fluid, whose surface is concave and hemispherical. We
have reason to suppose, that this is the case with water, and some kinds of oil,
in a glass tube.
The case of a fluid which rises up between vertical planes, forming with each
other a very small angle, or that where a fluid flows out from a capillary
siphon, aflbrds several phenomena which are merely corollaries of this theory.
In general, if we take the trouble to compare the numerous experiments of
[9221] observers upon capillary action, we shall find that the results, obtained in these
experiments, when they have been made with the proper precautions, may be
deduced from the theory ; not by vague, and always uncertam considerations,
but by a series of geometrical reasoning, which seems to leave no doubt about
the truth of the theory. I hope that this application of analysis to one of the
most curious objects of physics, may interest mathematicians, and excite them
to increase more and more these applications, which unite the advantage of
confirming physical theories and improving analysis itself, by requiring new
processes of calculation.
X. Suppl. 1. <§> 1.] THEORY OF CAPILLARY ATTRACTION.
695
SECTION I.
THEORY OF CAPILLARY ATTRACTION.
1. We shall suppose a vase ABC D (fig. 112,) to be filled with water, as
high as AB, and that a glass capillary tube, NMEF, open at both ends, has its
lower end immersed in the water, which will
rise up into the tube to O, and the surface
will form a concave figure M ON; O being
the lowest point of this surface. We shall
also suppose that an infinitely narrow canal
O ZRV, composed of a single filament of
water, passes through the point O and the
axis of the tube ; then it is evident, from jd
the principles of capillary attraction, which we have just explained, that the
action of the water below the horizontal line I OK, will be the same upon
the column Z, as the action of the water in the vase upon the column VR.
But the meniscus MIOKN will act upon the column O Z upwards, and will
therefore tend to raise the fluid. Hence it follows, that, in the state of equilibrium,
the water in the canal O ZRV must be elevated higher in the tube than in the
vase, so that it may compensate, by its weight, for the action of the meniscus.
The law of this ascent, in tubes of different diameters, depends upon the
attraction of the meniscus ; and in this case, as in the theory of the figure of the
planets, there is a mutual relation between the figure and the attraction of the
body, which renders the calculation difficult. In the investigation of this subject,
we shall consider the action of a body of any figure, upon a column of fluid,
contained in an infinitely narrow canal, drawn perpendicularly to its surface, and
whose base we shall take for unity.
We shall suppose, in the first place, that the body is a sphere, and we shall
ascertain its action upon a fluid, contained in a canal, situated without the sphere,
and perpendicular to its surface. For this purpose we shall resume the analysis
[9222]
[9223]
[9223')
[9224|
[9227]
696 - CAPILLARV ATTRACTION. [Mec. Cel.
[9225] given in [470^', &c.] ; putting r for the distance of the attracted point, from the
centre of a spherical stratum, whose radius is m, and thickness du ; also d for the
[9226] angle which the radius u forms with the right line r, and rs for the angle which the
plane passing through the lines u^ r, makes with a fixed plane passing through the
right line r ; then the element of the spherical stratum will be u^du ,dzs.di. sin.d