Pierre Simon Laplace.

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moreover the action of the tubes upon the mercury must be very small, to render
the surface of the fluid in very narrow tubes nearly hemispherical [9650]. This
difference cannot, therefore, have any sensible influence upon the heights of the
barometer.

author in [10454, Stc.]. It is also treated of in the notes [10442a — 10444a, 10456a— fc] ; [9754c]
and a table [10443z] is given, showing the depressions of the mercury for tubes of various [9754/]
diameters from SI""'* to 2™'. This table is computed upon the supposition that the angle of
contact of the mercury with the upper side of the tube is 48° = 43^' IS"'; and the numbers [9754g-]
would vary if this angle were increased or decreased by its friction against the sides of the
tube in its ascent or descent, or from any other cause. It was observed many years since by
M. Casbois, professor of medicine at Metz, that, by boiling the mercury in a barometrical [9754^]
tube, the convexity of its surface will be gradually diminished, and that, by continuing the
boiling a sufficient length of time, the surface will become plane, and finally concave ; and he
suggested that this process might be used in obtaining a barometer with a plane surface. This
experiment was afterwards confirmed by La Place and Lavoisier, who succeeded in constructing [9754^ ]
a barometer with a plane surface ; and they adopted the opinion of M. Casbois, that this change
in the convexity of the surface was produced by the expulsion of the moisture from the [9754A]
mercury by the continued process of boiling. But M. Dulong has lately given a much more [97541 ]
satisfactory explanation of this phenomenon, by observing that, in the operation of boiling, the
mercury in contact with the air becomes oxydized, and that this part, by adhering to the sides
of the tube, or by mixing with the other parts of the fluid, produces the change in the capillary
action which had been discovered by M. Casbois. The correctness of this explanation has
been verified in several ways, namely, by viewing with a microscope the sides of the tube [9754n]
where the particles of the oxyde were visible and of a reddish hue ; by agitating the barometer
in an acid which decomposes the oxyde, since it was found that the surface of the mercury
then resumed its convex form ; finally by boiling the mercury in an atmosphere of hydrogen
gas, which does not oxydize the mercury; for it was then found that no change whatever was
produced in the surface of the mercury, however long the boiling was continued.
VOL. IV. 202

[9754m]

[9754o]

806 THEORY OF CAPILLARY ATTRACTION. [Mec. C61.

SUPPLEMENT

TO THE

THEORY OF CAPILLARY ATTRACTION.

The objects of this supplement are, to complete the theory which I have given
of the capillary phenomena ; to extend its application ; to confirm its results by a
comparison w^ith experiment; and to present, in a new point of view, the effects
of the capillary action, so as to render more evident the identity of^ the
attractive forces, upon which this action depends, with those which produce the
affinities of bodies.

ON THE FUNDAMENTAL EaUATION OP THE THEORY OF CAPILLARF ACTION.

The equation of partial differentials, in this theory of capillary attraction,
[9318, or 9324], is deduced from the principle of the equilibrium of canals.

[9757] This principle consists in the supposition, that a homogeneous fluid mass, when
acted upon by attractive forces, will be in equilibrium, if the equilibrium takes place
in any canal whatever, whose extremities are situated at the surface of the fluid.
We may prove it easily in this manner : Suppose, in the interior of the fluid, a
re-entering canal, of uniform width, but infinitely small. If, from an attracting
point,* taken as a centre, with any radius, we describe a spherical surface cutting
this canal ; and upon the same centre, with a radius which differs but infinitely

[9758] little from the first, we describe a second surface ; each of these surfaces will
cut the canal at least in two points, and will intercept at least two infinitely

* (4245) This attraction may arise from any particle of the fluid, or from any other body ;
[9757a] thus the earth's centre, or rather the point to which bodies upon the earth's surface lend, by
means of the force of gravity, may be considered as one of the points here spoken of.

X. Suppl.2.]

EQUATION OF THE CAPILLARY ACTION.

807

small portions of this canal. It is evident that the two columns of the fluid
included in these portions, will be acted upon by equal attractive forces, and as
thej have the same height in the direction of these forces, they will be in
equilibrium with each other.* Hence we see that the whole canal will be in
equilibrium by the action of the attracting body; therefore it follows that the
equilibrium will hold good, whatever be the number of these points. Suppose

[9758']

[9759]

* (4246) To illustrate what is here said, let FLKI, fig. 140, be the tube or vessel
containing the fluid of uniform density [9757], whose surface is
FGBHI. The proposed canal is represented by GAHB,
whose interior part is GAH, and the part which is bent upon
the surface of the fluid is GBH. Then, if C be the attracting
point; MIS OP, mnop, ihe spherical surfaces drawn about C
as a centre, cutting off the parts of the canal in the places
represented by MNnm, OopP; the action of the point C
upon the matter contained in these two parts of the canal will
be equal, and will balance each other. For if we suppose the
uniform area of the base of the canal, measured perpendicularly to its sides, to be A, and that
the line CM forms the angle 6 with that base, the area corresponding to the spherical surface

A

MN, or mn, will evidently be -r— ; and this, being multiplied by the difference of the radii

Cm, CM, represented by dr, will be the volume contained in the part corresponding to
MNn m, which will therefore be — — . Now the force at C, which we shall represent by
F, draws in the direction MC,. and when resolved in the direction Mm, it will evidently be
F.sinJ; multiplying this by the volume -— , we obtain the whole action upon the particles

MNnm, in the direction of the canal Mm, equal to - — xF.sinJ = AFdr; and as the

quantities A, F, dr, are the same at the point P as at M, the action upon the particles
OopP will in like manner be represented by AFdr, the variable angle 6 having vanished
from both these expressions ; hence the action at these two points must be equal, and opposite
to each other, and they must therefore balance each other. The same may be proved in
other parts of the canal ; so that the action of the force C upon the canal produces an
equilibrium. In like manner we may prove, if the attracting point be at c', and the spherical
surfaces cut the canal at the four points near Q, R, S, T, that the forces acting on the canal
near Q, R, will balance each other, and in like manner those near S, T; whence we may
conclude that the canal will be in equilibrium by means of the force c'. Lastly, if we suppose
the force to be that of gravity, acting at c, we may draw about c, as a centre, the arcs
MP', m'p', and then proceed as we have done with the force at C, and thus prove that the
action of gravity alone produces also an equilibrium in the canal.

[9759a]

[97596]

[9759c]

[9759rf]
[9759c]

[9759/]

[9759g-]
[9759A]

[9759i ]
[9759A]
[9759/ ]

[9760"J

[9761]

[9762]

808 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.

now that a portion of this canal is situated in the surface of the fluid, and that
it is there bent down in the direction of that surface ; the equilibrium in the
canal will still continue. Therefore, if we suppose that the equilibrium exists
separately in the interior part of the canal, it will also take place separately in

[9760] the portion which is situated at the surface. This last equilibrium can be
maintained in only two ways; either because, at each point of the canal, the
force by which the fluid is urged, is perpendicular to its sides ; or because that,

[9760'] while the fluid is pressing in one direction at one end of the canal, this pressure
is destroyed by a contrary one at the other end. But in this last case, there will
not be an equilibrium in the part of the canal which is situated upon the surface,
if the two ends of this canal are terminated in that part of the surface of the
fluid which presses in the same direction. Therefore, upon the principle that
there is generally an equilibrium in an internal canal whose extremities are at
the surface of the fluid, if we suppose a re-entering canal to be formed, of which
a portion is situated upon the surfa^ce of the fluid, the resultant of the forces
which act upon the fluid in this portion, must be perpendicular to the sides of
the canal. Now this cannot take place, for every direction of the canal, unless
this resultant is perpendicular to the surface of the fluid; for, by reducing it to
two forces, the one perpendicular, and the other parallel to the surface, this last
force will not be destroyed by the sides of a canal which are situated in the
direction of that force. The equilibrium in any internal canal is, therefore,

[9763] necessarily connected with the condition of the perpendicularity of the force to
the surface ; and this condition, if it be satisfied, assures the equilibrium of the
whole fluid mass as we have seen in [138''']. The equations deduced either
from the equilibrium of the canals, or from the perpendicularity of the force to

[9764] the surface, must therefore be identical, or, at least, the one must be the
differential of the other ; and it is evident that the second is the differential of
the first. For the equation given by the equilibrium of the canals [9318, 9324]
contains only differentials of the second order; instead of which the tangential
force, at a capillary surface [9308, &c.], arises from two causes, namely, the

[9765] action of gravity, resolved in a direction parallel to that surface, and the
attraction of the mass corresponding to the difference between the whole mass
of the fluid and that of the osculatory ellipsoid ; and it is evident that this last
^ action depends upon differentials of the third order [9308] ; therefore the
equation resulting from the condition that the tangential force is nothing, or, in
other words, that the force is perpendicular to the surface, must contain

[9766] differentials of the third order [9810] ; consequently it must be the differential of
the equation given by the equilibrium of the canals [9810/t]. But it is interesting

[9767]

[9768]

X. Suppl.2.] EQUATION OF THE CAPILLARY ACTION. 809

to prove this a posteriori. It is what we shall now do, and the result will be a
confirmation of the fundamental equation of this theory [9318], and will also
be a simple method of obtaining that equation.

We shall take for the origin of the rectangular coordinates x, y, 2, any point
of the surface of the fluid, which we shall denote by O ; and we shall define the
axes of X, y, z, in the following manner :

For the axis of z, we shall take the line drawn through O, perpendicular

to the surface of the fluid ;
For the axes of x, y, we shall take two rectangular lines, drawn through the rg^^n

point O, perpendicular to the axis of z ;

Then the value of z, considered as a function of x, y, will represent the
equation of the surface of the fluid, and this value can be developed in a series [9768"]
ascending according to the powers and products of x, y, and of the following
form :*

+ C:t^-\'D3fy + Exf + Ff P^ofThefl^l."""*""''] [9769]

+ &c.

The three first terms of this expression of 2, namely, Ax^-j-'>^xy-\-Bif,
correspond to the ellipsoid which touches the surface, or, to speak more strictly, it

[9770]

* (4247) If we put z^ for the terms in the second member of [9769], it is evident that rgygg^-i
the general value of z, developed as in [610, 611], in a series ascending according to the
powers and products o( x, y, will be fully expressed by z=a-\-bx-\-cy-\-z^. But the [97696]
supposition that the ordinates a?, y, z, commence together at the point O, will give, at that
point, a? = 0, y = 0, z = 0; consequently z^ — [9769a, 9769], and then [97696]
becomes = a; hence the general value of z becomes z=:hx-{-cy-\-z^. Now the [9769c]
differentials of z, z^, give

(£) = ^+(S)' (l)=^ + (|')' ^g,,g,3

(|) = 2^a;-fXy+&c.; (^)=Xy + 25y + &c.;

and at the point O, where x=0, y = 0, these partial differentials of z^ vanish; therefore
we have, at that point, (^^^ = b, f^^\=zc; but as the tangent of the surface at the point [9769e ]
O, is taken for the plane of x, y, we must evidently have, at that point, f^J = 0, ^- j = 0; [9769/]

hence we get = &, = c; substituting these in the value of z [9769c], we finally obtain
z=z^, as in [9769].

VOL. IV. 203

810 THEORY OF CAPILLARY ATTRACTION. [M^c.C^I.

corresponds to the osculatory paraboloid. Now the attraction of this paraboloid
upon the point O, is evidently in the direction of the axis of z, since the solid

[9771] is symmetrical on the opposite sides of this axis ; therefore the tangential force
at the point 0, arising from the action of the whole mass, must depend wholly
upon the attraction of the solid, whose surface is defined by the following
equation :*

z = Cs^ + Dx^y + Exf + Ff

^^^^^ -|.&C.

This solid is the same as the diflference between the whole mass arid the
osculatory paraboloid. To determine the tangential force, depending upon this
solid, upon the point O, we shall put / for the distance of one of the elements

[9773] of the solid from that point ; also 6 for the angle which this right line makes
with the axis of x. The attractions upon the point O being sensible only in a
very small space, we may here consider the three right lines x, y,.f, as being all
in the plane which is a tangent to the surface in the point O, and we may neglect

[9774] the powers and products of x and «/, superior to the third order. Thus we
shall have, for an element or differential of this solid,!

* (4248) If the body attracting the point O is symmetrical about the axis z, that is,
if the points are so situated that the value of z remains unaltered when x, y, are changed into

[9772a] — X, — y, respectively, it will evidently produce no tangential force, because the particles

[97726] similarly situated on opposite sides of the axis z, act with equal forces in opposite tangential
directions, and thus mutually balance each other. Now if we change x into — x, and y
into — y, in [9769], the powers and products of x, y, of the even dimensions 2, 4, 6, &xj.,
will remain unaltered, and will therefore be symmetrical, and may be neglected in computing
the tangential force; and it will be only necessary to retain the uneven dimensions 3,5,7, he.

[9772rf] [9772]. Indeed, we may neglect the 5th, 7th, he., dimensions, since, by a calculation similar
to that in [9783, &Lc.], they will produce terms depending on ff^df.cp(f), ff^df.cp{f), &£C.,
which must be incomparably smaller than the terms depending on ff^df.(p(f) [9780, &c.],
depending on the third dimension of x, y, on account of the extreme smallness of the limit

[9772/*] of/ at which the attraction is sensible; so that we may put z= Cx^ -{- Dx^y -\- Exy^ -^ Fy^,
as in [9774, Sic], in computing the tangential attraction.

t (4249) Let AOD, fig. 141, be the tangent plane;
OA, OD, the axes of x,y, respectively;

[9775a] AOB = 6, BOC^d&, OB=f, Bb = df, BC=^fd6,

[9775o'] and the area BCcb=fdf.dd, representing the base of the

attracting particle, whose height is z [9772]. The product of
[97756] this height by the base, gives the whole mass as in [9775];

multiplying this by <?(/), we get its attraction, in the direction

[9772c]

[9772e ]

X. Suppl.2.] EQUATION OF THE CAPILLARY ACTION. 811

fdf.d&,\Ca?-\-Dx'y^Exy^-{-Ff], [9775]

If we denote the law of attraction by (p(/), the attraction of this element upon
the point O, resolved in a direction parallel to the axis of x, will be i^T^Q]

fdf'<p(f).d6,cosJ.{Cx' + D:i^y-}-Exf + Ffl; [9777]

and parallel to the axis of y, it will be

fdf.(p(f).dd,sm.6.lC3^-{-Dx^y + Exf + Ffl, [9778]

Moreover we shall have *

a;=/.cos.^, y=/.sin.^; [9779]

the tangential force of the point O, depending upon the attraction of the fluid
mass, parallel to the axis of x, will be

fffW''P(f)-dd'{^-co^'^^ + D'Cos.^6.sm.6 + E.cos,^6.sin.^d+F.cos.d.sm.^6l, [978O]

and parallel to the axis of y, it will be

yX/"'^-'?(/)-^^-!C'.cos.3a.sin.d4-Z).cos.2^.sin.25 + E.cosJ.sin.3^ + i^.sin.^5|. [9781]

The integrations of [9780,9781], relative to 5, must be taken from ^ = to
d = 2*, * being the semi-circumference whose radius is unity; hence these
expressions become respectively f

^* . { 3 C -{- £| 'ff^df. 9 (/) , [TangenUal force in the direction *.] [9783]

;|-* . \ 3F -{- jD I 'jy^df. <p (f) . [Tangential force in the direction y.] [9784]

O B, equal to fdf. (p(/) .d6.\Cx^-{- Dxy^ -}- Exy^ + Fy% To resolve this in the directions [9775c ]
OA, OD, we must multiply it respectively by cos.^, and sin.^; and then it becomes as in
[9777, 9778]. We may here remark, that the principle adopted in [9774] is equivalent to [9775rfJ
the supposition that all the particles of fluid, situated on the ordinate z, are projected on the
tangent plane ccy, in a single point, corresponding to the coordinates x, y.

* (4250) In the triangle OBA, fig. 141, page 810, we have OA—OB.cos.AOB,
AB=OB.sm.AOB; and by substituting the symbols [9775a, &;c.], we get the values of [9779a]
x,y [9779]. Substituting these in [9777], we obtain [9780], and by the same means [9778J
changes into [9781].

t (4251) From [8], Int., we have

fd6. cos *6.=zfd6.{i-{-icos.2d-{-^cosA6]= ^6 -\-is\n.2d-{-^\smAd. [9783a]

This vanishes when ^ = 0; and when d = 25r, it becomes f^''d6.cos.*6==^.'2T=^-if; [97836]
and by changing 6 into 6 — 90% we get f^'dd. sin.* 6 = ^1:. Again, by using [31, 1], Int.
we have
fdd , cos.2 6 . sin,2 6 =fdd . {cos.6 . sin,fl)2 z=fdd . Q sin.2^)2 z=fd6 . (i — i cos.4^) = ^5 — ^ . sin.4^, [9783c ]

812 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.

The integral relative to / may be taken from /= to /= co, so that it is
[9785] independent of the dimensions of the attracting mass [9240a — /]. This is what
characterizes this kind of attractions, which, being sensible only at insensible
distances, allows us to notice or neglect, at pleasure, the attractions of bodies
situated beyond their sphere of sensible activity. We shall put, as in [9233'],

[9786] ^(f)=C—fidf.cp(f),

the integral fdf.<?(f) being taken from f==0, and c being its value when
[9787] f is infinite [9232']. n(/) will be a positive quantity, decreasing with extreme
rapidity [9233"] ; and we shall have, by taking the integrals from * f= ;

[W88] frdf.f(f) = -f'n(f) + y_pd/.n(f).

— f^ . n (/) is nothing when f= co ; for although f^ then becomes infinite,
[9789] ^jjg extreme rapidity with which n(y*) is supposed to decrease, renders

f^.n(f) nothing [9240A;]. The functions (f)(f) and ^(f) may be very ivell
'■ ^ compared with exponentials like c~^- [9240A, &c.] ; c being the number

whose hyperbolic logarithm is unity, and i being a very great positive and
[9791] integral number. For c~'^ is finite when f= 0, and becomes nothing when

y*is infinite; moreover it decreases with extreme rapidity, and in such a manner
[9792] that the product /". c"*'' always vanishes when / is infinite [9240A:], whatever
[9793] be the value of the exponent n. We shall now put, as in [9241'],

[9794] fif4f.Tl(f) = c' - ^(f),

c' being the value of that integral when f is infinite [9240']. "^(f) will also
[9795] ^g ^ positive quantity decreasing with extreme rapidity [9241"] ; and we shall
have t

[9783rf]
[9783e]

which vanishes when 5 = 0; and when 5 = 2*, it becomes J^''dd.cos.^d.s\n.^& = ^.2if=lT.
IVIoreover, we have fdd.cos.^6.smJ= — fcos.^d.d.{cos.6)=^ — icos^6, which vanishes
when 6=0, and also when 5 = 2*; so that we have f^''dd.cos.^6.sm.6=:0; and if we
change 6 into 6 — 90''-, we get f^''d6.sm.^&.cosJ—0. Substituting these integrals in
[9780], it becomes as in [9783], and [9781] becomes as in [9784].

[9788a] * (4252) The differential of [9786] is d.Tl{f) = — df.(p{f); substituting this in
[9788J»] the first member of [9788], we get ffW'^{f)=—ff^-d'^{f)- Integrating by parts, it
becomes as, in the second member of [9788], as is easily proved by differentiation ; and by
[9788c] neglecting the term — /^n(/) [9789], it becomes /oTV/. (p(/) = 4 ./o7^^/. n(/).

t (4253) The differential of [9794] is fdf.Tl{f)=z — d.*(f)'y substituting this in
[9796a] ^jjg |j^g^ ^gj.|^ qJ* [9788], and then integrating by parts, we get successively

X. Suppl.2.] EQUATION OF THE CAPILLARY ACTION.

813

When /is infinite, /^.y(/) becomes nothing; therefore we shall have, by
taking the integral from /= to /= od, [9797]

V:pdf,Tl{J) = ^SJdf.^{f). [9798]

Lastly, if we put, as in [9253a'],

§=/oy^/.^(/), [9799]

we shall have,

f:rdf'^(f)=-^foydM{f)=^'^^, [98oo]

Thus the two preceding tangential forces [9783, 9784], parallel to the axes of
X and y, will become

CuCy -\~ £jj , II J [Tangential force parallel to the axis of K.] [9801]

, (•JT -\- U) , jti « [Tangential force parallel to the axis of y.] [9801'!

Now, by observing that the axis of z is perpendicular to the surface of the fluid
[9768], we shall have at the point O, as in [9769/],

the expression of z being developed in a series, according to the powers and
products of a:, y, by means of the theorem [610, 611], becomes*

y.

._fddz\ ^^ fddz\ ,fddz\ ;

'^\dx')' Q'^KdxHy)' 2 '^\dxdyy' 2 '^\dy^)'Q'

+ &C.;

[9802]
[9803]

[9804]

^frdf.n(f)=.-4fp.d,^f) = -^4p.^f)+8ffdf.^(J'), [97966]

as in [9796]. The part — 4f^.^(f), vanishes at the limits of the integral, as in [9796c]
[9240i, k, he], and by neglecting it in the equation [9796&], we get [9798] ; substituting it

in [9788c], we obtain, by using ~ [9199], f^f^,df.(p{f)=8foydf ,-*{/) = ^ , as in [9796rf]

[9800]. Lastly, by substituting this integral in [9783, 9784], we obtain the tangential forces
[9801,9801'].

* (4254) The general development of z [610, 611], contains the terms in [9804] ; the
three first terms of the form a-{-bx-\-cy being neglected, as in [9769a — f]. Comparing [9804a]
the expression of z [9804] with the assumed form [9769], we obtain the values of
C, D, E, F [9805, 9806]; hence the tangential forces [9801, 9801'], become as in [98045]
[9807, 9807'], respectively.

VOL. IV. 204

814 THEORY OF CAPILLARY ATTRACTION. [Mec. C^l.

which gives

[9806] ^=i-(£ji^' -^^-^-CJ)-

Consequently the preceding tangential forces [9801, 9801'], will become, as in
[98046J,

[98071 hH- ^ I I "4" I ) r » fTangential force parallel to the axil x.l

^ ( \<ix^J \dxdy^J )
C / dH \ f dPz \ }

[9807*] h-H» \ ( ——^ ) ~\~ { 3 ) C ' [Tangential force parallel to the axis y.l

We shall put g for the force of gravity, and — du for the element of its
[9807"] direction. Then the condition that the whole force acting at the surface must
be perpendicular to it, or, in other words, that the resultant of the tangential
forces is nothing, is reduced, as we have seen in [138, 138'], to the following
formula, namely, that the sum of the products of each force, by the element of its
direction, is 7iothing. Multiplying, therefore, by dx the force parallel to the axis
of X ; by dy the force parallel to the axis of y ; and the gravity g by — dw^
then taking the sum of these products, and putting it equal to nothing, we shall
obtain the following equation :

[9809] \H. ^(^^.dx + (^^^.dy + {£l^^.dx+(^^.dy\-gdu = 0.

[9809^ From the formula [9314], we have at the point O, where* p = r-^j = 0,

[9809"] q = ('^^=0 [9312,9803],

[9810a] * (4255) Substituiing p = 0, q = [9809', 9809"], in [9314], we get \-\-~ = t^t\

R R

[98106] and by substituting the values of r, t, [9313], we obtain ( 7^) + ( 7^) = d +■«;• The

differential of this equation is the same as in [9310], observing that z being a function of

[9810c] x,y, we have, by the usual rules of differentiation, ^* (ti) ='(77)-^'^+ (j^ j-^y?

[9810^] ^•(^2) = (^o)-^^+(j3)-^y- Substituting [9810] in [9809], we get [9811];

2

multiplying this by — , we get [9811']. Now the quantities K, h, b', which occur in the

2

fundamental equation [9315], are constant; so that if we multiply this equation by — •— ,

H

[9810e] it may be put under the form ( d + ^/) ^.^ = constant ; or hy changing gz [9309]

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