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Pierre Simon Laplace.

# Mécanique céleste (Volume 4)

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[10017'] column^ the drop will burst, and spread over the lower base of the tube, where it

[10016^]
[10016^]

[lOOlGr ]

[10016s I
[10016f ]

[10016u]

[10016V ]

[10016m>]
[10016m/]

[10016x]

[1001 6r']
[10016y]

which we shall suppose to be represented by AOB, in figure 158, considering the points
A, B, as the lowest parts of the tube, and supposing the inner lines of the tube R'A, U'B,
to be continued so as to meet the line WTV in the points ^and V, In this case, the fluid
being situated above the surface AOB, the corpuscular action at any point N of the surface

[93156], will change from K — ^H.i- + - j to K-\-^H.(— + — j , as we have seen

in [9301a, 6] ; and the corpuscular action at O will be K-\-^HA - \-r)\. Then, if an

infinitely small canal NMO, similar to nsoj be drawn, with a vertical branch MO, and a

horizontal branch NM, the capillary action at N will be K-\-^H.( — -\-~j, and this

is continued along the horizontal branch of the canal NM to M; and in descending in the
branch MO, the pressure is increased at O by the weight of the column MO, which is
gXMO=g.{z — h) [10016J']; so that the action at O will be

and this must be equal to the expression of that action in [10016g'] ; hence we have

so that, if we substitute g=^Ha, [10016Z], and then divide by ^H, we shall get the
following equation of the surface AOB;

We may elimmate r + r; from this equation, by observing that the corpuscular action at o,
at the summit of the vertical canal Oo, is K — iH.f— + T7); and by adding to this the
weight of the column Oo, which is represented by gXOo=*>g.s [I00l6d], we get the
pressure at O, equal to K — ^H.( - { - rA-{-g.s. Now this must be equal to the

corpuscular action at O, which is K-{-^H.( - \'r^ [10016^]^ hence we have

Rejecting K from both members of this equation, and then dividing by ^H, or ig^~^
[10016Z], we get

and by substituting [I0016»i], we obtain

^+b..= 2a.(c.

■A)>

X. Suppl. 2.]

FLUID SUSPENDED IN A TUBE.

879

wUl form a new drop^ which will become more and more convex, until it
forms a hemisphere whose diameter is equal to the exterior diameter of the
tube. Then, if the column be in equilibrium, its length will be equal to tlie

substituting lliis in [10016m], it becomes

R+R' = — 2^•(^•

■c)',

and by using the values of -, — [9326, 9326'], we finally obtain the following differential
equation of the lower convex surface of the fluid AOB;

— -1-1 ^ /i J_'^^^^ /" / dz^\l

du^~^u'Tu'\ '^d^y=^~^'^'^-^~^^'\}'^^^) ' l^"'' """"'^ '"''■"''•]

which differs from the equation of the upper surface aob [IOOI60], only in the sign of a

so that the integrals of the equations of both these surfaces may be obtained by the methods

pointed out by the author in [9328—9379], and it is unnecessary to enter into any further

explanation of them. We may remark that similar results are obtained when we suppose the

upper surface to be convex, or the lower surface concave.

The volume aohVW, included between the upper surface aoh and the base WTF, is

evidently represented by '2'jf .fzudu ; and tlie similar volume AOBVW, included between

the lower surface AOB and the same base, is 2'jr.fzudu; the difference of these two

integrals is equal to the volume V of the fluid aob BOA; hence we have

V= 25r .fzudu — 2'ff .fzudu.

The integrals of these expressions may be obtained by approximation, In a series arranged

according to the powers of a, by methods similar to those which are used in [9328 — 9379],

restricting the calculation to terms of the first order in a, in the value of V. We have not

thought it expedient to insert these general calculations, which are of no practical use in

comparing with experiments ; we shall therefore restrict our remarks to the notice of two of

the most simple and remarkable cases. First, when the elevation of water in a tube Po, is

exactly equal to that arising from the capillary action of the same tube, when well wet and

dipped by its lower end AB into a vessel of water. In this case, it is evident that the figure

of the lower surface AOB will coincide very nearly with that of the plane APB. If some

of the water is taken from the tube, so as to decrease the altitude Po, the point O will

ascend above P, and the lower external surface will become concave. On the contrary, if

more water is inserted in the tube, so as to increase the altitude Po, the lower surface will

become convex; and this convexity will increase by the addition of more fluid, until the surface

AOB becomes nearly hemispherical, like the upper surface aob. If we suppose the

upper and lower surfaces to be hemispheres whose radius is Z, the elevation Po, produced by

[10017"]

77

the capillary action at the upper surface, will be Q^=~j [9360], and the similar action of

It

the lower surface ABO will likewise be 0- = — ; hence their combined action will produce

gl

the double elevation 2q, spoken of in [10016, Sic.].

[10016z ]

[10017a]
[100176]

[10017c ]

[10017rf]
[10017e ]

[10017/]
[10017ff]

[10017AJ

[10017i]
[100174]

880

THEORY OF CAPILLARY ATTRACTION.

[Mec. Ccl.

sum of the elevations of the fluid in two other glass tubes dipped into the vessel

L10018] hy their lower ends, their diameters being respectively equal to the internal and

external diameters of the first tube. Lastly, if Hie column be of a greater

[10017/

[10017m]

[10017n]

[10017o;
[lOOlTp;

[10017^

[10017r
[10017s

[10017f

[10017tt]

[1001 7r
[10017w]

[10017a:
[100171/

[100172

In like manner the volume V of a drop of the fluid, formed at tlie bottom of the tube
-4 J?, is represented by V = tc .Jv?dz^ taken between the limits z=.TO and z=TP.
This may likewise be reduced to a series of terms in a, as in [1001 7e, &.c.]. If the figure
AOB be a hemisphere, we shall have V'=^%'k.P, which is the value of the actual volume,
if we neglect terms of the order a; but if we retain terms of the order a, the volume of the
drop will become ^^^^^^.(l-f-^aZ^), as we may easily prove by means of the formula
[10017/], using values of dz, u, he, similar to those in [9339 — 9350], and differing only
by writing — a for a; but we have not thought it worth the trouble to insert this calculation,
as it is of no importance, on account of the difficulty of ascertaining the mass of a drop with
any great degree of accuracy [lOOlTr]. If the density of the drop be D, its gravity^, its
mass m, supposing it to be a hemisphere, we shall have

m = ^'!r.gD.P.
According to this theory, the mass of the drop m must be proportional to the density of the
fluid ; but this does not agree with the observations of Gay-Lussac. For, by some
experiments made with the same tube, and in similar circumstances, he found that 100 drops
of water, whose density was 1,0000, weighed 8^'*™",9875; and 100 drops of alcohol, whose
density was 0,8453, weighed only 3'"'°",0375 ; so that the drop of alcohol was about one
third of that of water, while their densities differed only one fifth part. The radius of the
tube used in these experiments was t==3'"'\09; hence V = 62 cubic millimetres nearly
[10017m], or F^' = 0,062 cubic centimetres. Multiplying this successively by the densities
1,0000, 0,8453, we get for the mass of the drop of water m=;0""""-,06, nearly ; and for that
of alcohol, m = 0''*'"-505 nearly. These differ very much from the experiments [10017r, s] ;
and the differences will not be avoided by introducing the lemi of the order a [10017n];
for, though it will decrease the error of the mass of the drop of water, it will increase that
of the drop of alcohol ; hence we see that the uncertainty of the observations makes it
unnecessary to notice the terms of the order a. We may finally remark, that the figure of
the drop is calculated when in a state of equilibrium, but the drop separates from the tube
only when the equilibrium is destroyed ; and it is probable that from this source arises the
great difference between the calculated and observed mass in a drop. i. , a

In the preceding calculations, it is supposed that the water is poured into the top of the tube ;
but we may suppose the tube to be inserted in the bottom of a large vessel of water. Then
the fluid will not run out from the bottom of the tube, if the distance fi-om the level of the
fluid in the vessel to the bottom of the tube be less than the quantity q [I0017i],
corresponding to the elevation in a capillary tube; but a drop will begin to form at the bottom
of the tube when that distance exceeds q; finally, when the distance exceeds 2q [10017Z:],
the water may flow out fi-ora the tube, neglecting in these estimates terms of the order a, as
in [10017OT, &;c.]. . ' „• y^, nirrfofp

X. Suppl. 2.]

SEVERAL FLUIDS IN A TUBE.

881

length, part of the fluid ivill fall from the tube. All these remits of the theory
are confirmed hy experiment.

We shall noio consider an indefinite vase filled with any number of fluids, placed
horizontally, the one above the other. ''If we dip vertically into the vase the lower
end of a right prismatic tube, the excess of the weight of the fluids contained in
the tube, above the weight of the fluids ivhich it would have contained independent
of the capillary action, is the same as the weight of the fluid ivhich ivould rise
above the level, in case there ivas in the vessel only that fluid in which the lowest
part of the tube is dipped.^''* In fact, the action of the prism, and of this fluid,
upon the same fluid contained in the tube, is evidently the same as in this last
case. The other fluids contained in the prism being sensibly elevated above its
lower base, the action of the prism upon each of them can neither elevate nor
depress them. As to the reciprocal action of these fluids upon each other,

* (4307) For the purpose of illustration, we may take the case in which there are only
two fluids included in the vertical cylindrical tube RR'U'U, fig. 158, page 877, where aol is
the upper surface; AOB, tlie common surface of the two fluids; WtTV, a horizontal plane,
drawn within the tube, and at a sensible distance above its lower extremities R, U, and at a
finite distance below the surface AOB. To estimate the upward pressure upon any point
t of this plane, arising from the action of the fluid in the vase, and in the lower part of the
tube RWVU, we shall suppose a slender filament or canal nNtzrv to be drawn, having
two vertical branches rv, zn, connected by the horizontal branch rzy the termination v
being at the level surface of the fluid in the vase. Then the pressure at the bottom of the
canal v?-, upon a unit of surface, will be represented, as in [9555], by P-{-K-\^gXvr.
This same pressure is continued in the horizontal direction of the canal rz; and in ascending
in the vertical branch zt, it is decreased by taking away the pressure of the fluid in that
branch, corresponding to the ascent; so that, at the point t, we have for the expression of the
vpward pressure, the function F -\-K-{-g.(vr — zt). Now this quantity remains the same
whatever changes may be mad€ in the number or nature of the fluids in the tube above the
surface AOB; always supposing the fluid in the lower part of the tube and in the vase to
remain unaltered, as well as its point of level v. For, in these changes, no alteration is made
in the process for finding the expression of the pressure [10020e], which depends wholly on
the elevation of the fluid in the vase, and on that of the fluid in the tube below the plane
WJ^'^ so that its action upon the fluid situated immediately above fVV, and within its sphere
of action, must remain the same; and the action of the tube upon the same part of the fluid
must also remain unaltered. Hence it follows that, whatever be the number of fluids in the
tube, the mass in the branch nt must remain unaltered; and as this is true for every canal
passing through the surface WF^, the whole mass above the plane fVV must remain
invariable, whatever changes may take place in the number of the fluids; therefore this mass
must remain the same as if there were only one fluid in the tube and in the vase, as in [10024J.

VOL. IV. 221

[10019]

[10020]

Action of
several
fluids in a
tube in the
same as if
only the
lower fluid
were con-
tained in it.

[10021]

[10022]

[10020ffl]
[10020&]

[10020c ]
[10020^/]

[10020e ]
[10020/]

[10020g-]
[10020A]

[10020i ]

882

THEORY OF CAPILLARY ATTRACTION.

(Mec. Cel.

[10023] they will evidently destroy each other, if they form together a solid mass,
which may be supposed without affecting the equilibrium.

Hence it follows that, if we dip into a jluid the lower end of a prismatic tube,
[10024] and then pour into the tube another fluid which remains above the first* the weight

* (4308) If the volume of the upper fluid be given, as well as the densities of the two

[10024a] fluids, we may thence deduce the increment of the altitude of the central point of the upper

surface, in consequence of the introduction of the upper fluid into the tube. For this purpose

we shall suppose the surfaces of the two fluids to be spherical, as in [10027, 10027ff, jp', &c.],

and we shall use the following symbols, referring to fig. 158, page 877 ;

[100246] w, «', ^, are the same angles as those which are defined in [10028 — 10030] ;

[10024c] 7= the internal radius of the tube, as in [10035'j; its circumference being 2^/;

[10024c'] ^ = rO=the elevation of the lowest point O of the lower spherical surface AOB

above the horizontal plane TV TV;
[lOQQid] h-\-£'=To = the elevation of the lowest point o of the upper spherical surface aob

above the horizontal plane ^TF; hence Oo = To — TO=s';
[10024e] Z) = the density of the lower fluid, or the fluid in the vase;

[10024/] D'=the density of the upper fluid; and for brevity we shall put

[10024^] f(.)=?2!^?±^^^Ill; f.(,)^ cos.^^+Nn.3.-| .

[10024A.] -Kps^ihe volume of the upper fluid included between the spherical surfaces

aob, AOB, this quantity being supposed to be given.
Now it is evident that this volume of fluid, contained between the spherical surfaces
aob, AOB, is composed of the three following parts, namely,

[10024i ] The cylinder IkKL + the annulus Ikhoa — the annulus LKBOA ;

ri0024A:l and we have the cylinder ?fcJCZ# = ^Z^£', the annulas lkboa = vP.F{z3) [93360,10024^^];

and in like manner, the annulus LKBOA = 'xP.F{f). Substituting these values in the

expression [10024i], and putting the resulting quantity equal to irZ^s [10024A], we get
[10024Z] 'KT^s = 'Klh'^'sl^.F{vi) — ^P.F{&).

Dividing by the common factor vP, we obtain
[10024m] s = £'+Z.F(7;i) — Z.F(^).

Again, since the volume WAOBV is equal to the sum of the volumes of the cylinder
[10024n] WLKV, and of the annulus LKBOA, we shall have, by substituting the values of these

quantities, in symbols,
[100240] The volume WA0BF=''rP.h-{-vP.F(6).

Multiplying the volume of the upper fluid irPs [10024A] by its density D', and the
volume of the lower fluid [10024o] by its density D, then taking the sum of the two products,
we obtain the expression of the whole mass of the fluid in the tube, above the plane WTV,
namely,
[•100242,] ^^^^^ WaobV=D'.-KT^s^D.icP.\h-\-l.F{d)\.

l-:^ ./I .Mil

X. Supp].2.]

HEIGHT OF TWO FLUIDS IN A TUBE.

883

of the two fluids cantained in the tube will be the same as thai of the fluid which it
contained before [10020e]. It is evident, by the first method, that the surface of
the upper fluid will be the same as in the case where the lower end of the
tube is dipped into that fluid. At the points of contact of the two fluids, they will

[10025]

The value of this mass must remain the same as when there is only one fluid in the tube

[10020i]; and if we suppose, in this case, that the elevation of the point o above the plane

WV is H, and the angle formed by the fluid at the surface and near the side of the tube is

«', as in [10029], the volume WaobV will become -KT^H-{-ml^.F{'a'), as is evident from [10024?]

[10024o] ; multiplying this by the density D, we get another expression of the same mass,

namely,

The mass of the fluid WaohV=D.'Kl^.\H-{-LF(vi')]. [10024r ]

Putting this equal to the expression [10024jo], and then dividing by D.'!d% we get by
successive reductions, and using [10024/w],

H:i-LF{-:,')~.s+h-{-LF{6)

[D-D' )

:s'+Z.jP(^)

[D-D')

D

.e + Aj

whence we obtain

(A+s')-iJ=^-^-^.s-z.iF(^)-FMI;

and by putting, for brevity, x equal to the increment of the elevation of the point o, in
consequence of pouring into the tube the column of the upper fluid, whose volume is *Pfe,
and density D\ over the mass of the lower fluid, we shall have x={h-\-s'^ — H, and
the expression [10024mJ will give

D

'.s—L\F{zs) — F(zs')\.

If w = '5j', this value of x will become

x=

(D-D')

D

.s:

[10024a ]
[10024f ]

[10024u]

[10024V ]
[10024u>]

[10024a:]

and as D'^D', the expression of a? will be positive; so that, in this case, the elevation of the

point of the upper surface, will be increased by pouring into the tube the upper

fluid, whose density is D'. If we put zs=0, zi'=60\ we shall have, from [10024^], [10024y]

F('us) =.0,Sd3f F{6) =0,131; substituting these in [10024ti;], we shall get very

nearly,

{D—D'

D

.e-

•i?.

[100242 ]

The expressions of x [10024w, z] agree with those which are given by M. Poisson, in
pages 140, 141, of his Nouvelle Theorie, ^c, changing D into p, D' into p', / into a.

884

THEORY OF CAPILLARY ATTRACTION.

[MecCel.

[1002G] hate a common surface; hut this surface will he different from that which the

two fluids would ham separately^ and it is interesting to determine tlie nature of it.

For this purpose, we shall suppose the interior surface of the prism to be a

right cylinder, of a very small diameter, and placed vertically. In this case, it is

evident, by what has been said in the preceding theory, that the common surface

[10027] of the two fluids, and those which they would separately have in this tuhe, will he
spherical surfaces of different radii* We shall put

[10025o] and putting cos.'n'= — — , cos.-s

H'

El

to conform to his notation; and if we suppose

[100255 ]
[10025c ]

[10025</]

[10025e ]
[10025/]

[10025^]

[10027a ]
[100276 ]

[10027c ]
[10027(1]
[10027e ]

[10027c']

D'=^jjD, we shall have, from [10024z], a; = J(j8 — ^I, which may become negativei{
£<:^2Z; and M. Poisson remarks, that this result may serve to explain the phenomenon
observed by Dr. Young, who dipped a capillary tube into a vessel containing water, and
suffered the fluid to ascend in it; he afterwards poured into the top of the tube a small drop of
oil, and found that the upper surface of the oil finally settled at a less elevation above the level
of the water in the vessel, than that which the water in the tube had before the addition of
the oil. M. Poisson supposes, in this experiment, that the tube had not been previously
moistened by the vpater in its upper part; and upon pouring in the oil, it became moistened
by the oil, and by this means the upper angle, or that formed by the lower side of the tube
and the tangent to the surface of the oil, became -^ = 0. On the contrary, as the tube had
not been previously moistened by the water, he supposes that the surface of this fluid was not
a tangent to the side of the tube, so that -a' might have a finite value; and in this case, x could
become negative, as in the experiment of Dr. Young. M. Poisson finally observes, that this
experiment was brought by Dr. Young as an objection to La Place's theory of the capillary
action, supposing it to be incompatible with this theory; instead of which it serves to furnish
an interesting verification of its accuracy.

* (4309) When there is only one fluid in a cylindrical tube RR'U'U^ fig. 158,
page 877, the surface will be nearly spherical, as is proved in [9336o-, Sic], from the general
equation [9342], or rather from [9334] ; so that the surface corresponding to the angles ts, to',
in the hypothesis [10027, &;c.], must be nearly spherical. The same may be proved relative
to the common surface of the two fluids AOB, fig. 158, by pursuing the same method as in
[9309 — 9342], and using the figure in like manner as we have used fig. 116, page 713, or
fig. 112, page 695; that is, by drawing an infinitely slender and uniform canal nI^tT03Io,
with the vertical branches tNn, TO Mo, and the horizontal branches tT, NM; the area of
a section of this canal perpendicular to its length, being taken for unity. Then it is evident
that the fluid in the canal nNMo must be in equilibrium ; so that the vertical pressure of the
fluid in the branch niV, at the point iV, must be the same as at the point M, in the branch
oM, being equal to that in the horizontal branch NM; and in like manner, from the
equilibrium of the fluid in the canal nl^i TO Mo, the vertical pressure at t must be the
same as at T. Hence it follows that the part of the pressure at the point t, arising from the
capillary action at iV, and the weight of the column Nt, must be equal to the part of the

X, Suppl.2.] COMMON SURFACE OF TWO FLUIDS IN A TUBE. 885

•a = the angle which the surface of the upper fluid forms with the lower [10028]

surface of the tube, supposing no other fluid to be used ;

to' = the similar angle when the lower fluid only is used; [10029]

6 == the angle which the common surface of the two fluids forms with the [10030]

lower surface of the tube.

pressure at T arising from the capillary action at O, and the weight of the column MT; and
from this principle we shall deduce the equation of the surface AOB in the following
manner:

Putting, as in [9310], R, R', for the greatest and the least radii of curvature at the point [10027/"]
N of the common surface ANOB of the two fluids; and, as in [9316], b, h', for the
greatest and the least radii of curvature at the point O ; also

tN=TM = z, tn = z% MN=Tt = Uy TO = h,hc.', [I0027g]

we shall have, as in [10032,9275], K'—iH'.(~^^\ for the downward action of [10027/i]

\R R J

the fluid below the surface AOB, upon a portion of the same fluid in the canal at iV;

and the upper fluid will attract the external lower fluid at N with the upward force

K,—iH,.(^^^) [10033,9254]. The difference of these two forces gives the whole [10027f]
downward force at iV" equal to K^^ — ^^'V » ^" »') » ^ being the measure of the capillary [10027t]
intensity; and at the point O, the preceding expression will become K^^ — ^^'\b~^b')' ^^^^^H

Then, the density of the lower fluid being represented by D, and that of the upper by D',
we shall have DgXtN=Dgz for the mass of the fluid in the part tN of the canal
in. Adding to this the capillary action at iV [10027A:], it becomes

K-iG-{i+^)+J^g^' [10027m]

Again, the mass of the column MO is represented by D'gXMO=D'g.{z—h), and [10027n]
that of the column TO by Dgh; their sum being added to the capillary action at O
[10027/], gives

K^-^.G.(^- + l)j + D'g.{z-h) + Dgh, [10027O]

for the action in the column MT, which, as in [10027e'J, is to be put equal to the action in
the column tN [10027m]. Hence we have

K^_^G.(^^ + '-}l + Dgz=K,^iG.Q + l^ + D'g.{z-h) + Dgh; [10027o']

and if we put, for abridgment, c = iG.(j^ + l^ — (D-D').gh, we shall get [10027o"]

^GJ —-{-—,)=■ {D D') .gZ -\- C. [Equation of the common surface AOB.'j [10027p ]

\R R'/

From this equation we may deduce others, like those in [9318, 9334, 9442, Sic.]; whence
VOL. IV. 222

886

THEORY OF CAPILLARY ATTRACTION.

[Mec. C6I.

We may here observe, that these angles are not those which the surfaces form at
[10030] their points of contact with the tube ; but they are formed by the planes which are
tangents to those surfaces at the limit of the sphere of sensible activity of the tube,
as we have said several times. We shall suppose that

[I0027p']

[10027?]
[10027?']

[10027r]

[10027* ]
[10027* ]

[10027u]

[10027u']
[10027D ]

[10027t>']
[10027u>]

[10027X ]
[10027yl
[10027Z ]

we may conclude, as in [9336^], that the common surface of the two fluids, or that which
corresponds to the angle 6 [10030], will also be very nearly spherical.

The whole mass of fluid contained in the vertical branch nNt is evidently represented by
D'gXnN-{-DgxNt, or, in symbols, D'g.{z' — z)-{-Dgz={D—D').gz-}-D'gz';
and as the ordinates z, z', are measured from the level surface of the fluid in the vessel,

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