distances. Between these two limits, the planes, after being repelled, attract
each other when the preceding expression is less than 21 [10214m]. We may
[10216]
[10216']
[10217]
[10218]
This may be much reduced by means of [6], Int., which gives
\/h-\-hcos.Q = ^/coPp = cos^Q ;
and then
l_ ^/ffl^ _l-cosj9_ [40], Int.
Substituting these and the similar reductions in the terms depending on 6', we get
1 , , , ^, ,> , 2cos.!i9' 2cos49
^'=2^-S21og.cang.i«'-21og.ta„g.i.5 +-^7^- ^^,
which is easily reduced to the form [10215].
We may here remark that we have seen, in [10195], that, when â– 5J = zj', there is a point
of inflexion A, fig. 152, page 910, midway between the two planes; and then these two
planes repel each other. If â– sj^to', as in [10196], and we begin to draw the planes nearer
to each other, we shall find, as in [10195'], that this point of inflexion will approach nearer and
nearer to the first plane, and will finally touch it, as in [10197]. If we continue still to draw
the planes nearer to each other, the fluid will continue to rise between the planes until it shall
attain to the height q, near the first plane, the repulsion of the planes continuing during all this
time. When the fluid has arrived at the height q, and near to the first plane, the repulsion
will cease, as in [10205] ; and by continuing to bring the planes still nearer to each other, the
repulsion will change into attraction, as in [10205]; and the distance of the planes where
this change takes place being represented by 2Z, its value is given by the formula [10215] ;
so that, if the planes are at a less distance from each other than the value of 21 [10215], they
will attract each other, and if they are at a greater distance than that value of 2l, they will
repel each other.
* (4357) When ^ = 0, we evidently have 2?=oo [10215]; and then, from [10214m],
it follows that, whatever be the distance of the planes, they will attract each other; and this
infinite distance is evidently the extreme limit at which the attraction can commence, as in
[10216']. On the other hand, when S = 6', and log.-^:;;J^=log.l = 0, the expression
u iijo uiuc. i.m.M, >.w^w , ....V. *"b-tang49 »
of 2? is equal to nothing; and this may be considered as the other limit, or point where the
planes begin to repel each other, as in [10214;n, 10217].
VOL. IV. 236
[10214c']
[10214^;]
[10214e ]
[10214/]
[10214g-]
[10214A]
[102141 ]
[10214ik]
[10214/ ]
[10214/n]
[10216a]
[1021G6]
[10216c ]
942
THEORY OF CAPILLARY ATTRACTION.
fM6c. C^I.
[10219]
Theorem
relative to
the attrac-
tive or re-
pulsive
forces of
planes.
[10220]
[10221]
[10222]
[10223]
[10224]
[10225]
[10226]
The action
and reaction
[10227]
of the planes
are equal.
[10228]
determine their attraction or repulsion, by means of the following theorem, which
may be easily deduced from the preceding theory [9552—9586].
" Whatever be the substances of which the planes are formed, the tendency of
each of them towards the other, is equal to the weight of a fluid prism, whose
height is the elevation above the level, of the extreme points of contact of the
interior fluid with the plane, minus that elevation without the plane ; whose
length is the half sum of these elevations, and whose width is that of the planes
in a horizontal direction. We must suppose the elevation to he negative, when it
changes into a depression below the level. If the product of these three
dimensions is negative, the tendency becomes repulsive."*
We shall here observe that the tendency is the same, and has the same sign
for both planes. For the two first factors being q — q^, and \{q-\-q}i for the
first plane; their product is ^(^^ — qf) [10220a]. The analogous product for
the second plane is \{q'^ — q'^) [10190»i, &c.]; thus, the width of the two
planes being supposed to be the same, the two fluid prisms whose weights are
equal to the tendencies of the one towards the other, are equal, provided
q'^ — qf is equal to q'^ — C[^', now this equality takes place in virtue of the
equation [10176], which, by substituting for sin.w, sin.^', their values 1 — ^f,
l__a9;2 [10203,10204], becomes
Thus, although the two planes act upon each other by the capillary force qf an
intermediate fluid, yet this reciprocal action is such that the action and the
reaction are equal to each other.
When the two planes are very near to each other, z will differ but very little
from q; so that, if we putf
z — q = z', or z = q-{-z',
[10220a]
[102206 ]
[10220c ]
* (4358) Considering q, q^ [10168, 10184'], as positive quantities, the height of the
prism [102J9] will be q — q^, its length ^{q-{-q) [10220]; and if we put its width equal
to unity [10220], its volume will be \{q^ — qj^), and its mass hgD.iq^ — qJ^). This
represents the ar/raca'ye force [10219]; and by changing its sign, we may consider it as a
repelling force represented by lgT)'{q^ — q^)^ being the same as that in [10187^].
Moreover the pressures on the two planes are equal, as is seen in [10191, Sic, 10222, Sic] ;
hence we easily perceive the correctness of the theorem in [10219 — 10221].
I (4359) Through the point U, fig. 165, page 927, draw the horizontal line Uct, cutting
[10228rt] the ordinate NC in c; then RN=:y [10163], NC=2 [10162], and RU=Nc = St is
equal to q [10202]; so that, if we put « = iVC=/2C/'-fCc = j + z', as in [10228], we
X.Suppl.2.] TWO PARALLEL PLANES DIPPED INTO A FLUID. 943
z' will be a very small quantity, whose square may be neglected. Then we [10229]
shall have
Z=sin.'5j — %^q7!\ , [10230]
consequently *
j2;=</2:' = — ^; [10231]
hence the equation [10172] will become
and by integration,
y= constant + ^ 2a<y '
To determine the constant quantity, we shall observe that, when y = 0,
we also have z' = [10228a], and thenf Z=sin.tjj; therefore we shall have
[10233]
[10234]
shall have Cc of the order z'; and its greatest value z'= Tt may be considered as of the
same order as the distance of the planes Ut=^'^l; and when 2/ is very small, this quantity [102286]
z' will be very small in comparison with q, as in [10229]. Now, substituting the value of
z = q-\-z' [10228], in [10171], we get [10230], neglecting the second and higher powers [10228c ]
of z'.
* (4360) The difTerential of [10228] gives dz=-dz'; substituting this in the differential
of [10230], and then dividing by — 2(i.q, we get [10231]. Substituting this value of dz '■"^
in [10172], we obtain [10232], whose integral gives [10233].
f (4361) We have generally, as in [10168/J, Z=iSm.-a/, -m^ being the acute angle [10235a]
formed by the ordinate Cc, fig. 165, page 927, and the arc of the curve at C. At the point
U, where y==0, we have to^ = zj [10166], and then the preceding value of Z becomes [10235a']
Z = Bm.vs, as in [10234] ; which is the same as is given by the equation [10230], putting
z' = 0. This last value of Z gives ^i — z^ = cos.-i^ ; and by substituting it in [10233], [102356]
we get at the point t7, = constant + ^^ , as in [10235]. Substituting this in [10233], [10235c]
i/i ygi COS.tU
we obtain, for the general value ofy, y = y~— — — — ; and at the point T, where [10235i]
Ts^=:vi' [10173], Z = sin.w' [10235a], y/i_Z2=cos.-5j', and y = 2l [10210], it becomes [10235e]
as in [10238], from which we easily deduce the value of q [10239]. If we put ^' = 0,
1 I Cos 2m
and TO = 2y, as in [10202i, Stc], the value of q [10239] will become q =
[10235/]
4aZ
Substituting -^a^ [9323p], cos.2y = l — 2y2 [44], Int., neglecting higher powers of v^^
it becomes q = , agreeing with the elevation q' computed in [10203^, 10204m], by
means of elliptical functions.
[10235g]
944 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
cos.w
[10235] constant = — - — ;
[1023G] moreover, 21 being the distance of the planes from each other [10210],
[10237] we shall have, when 2/ = 2/, z = q' [10175], and Z=sm.-a' [10235e] •,
consequently
_, COS.ts' COS.-!*
[10238] 21 = ^ ;
hence we get
n 09*^01 /7 1_ rEIevation 9 of a fluid between two very"!
[lU,«oJj Y A J ' ' L near planes whose distance is 2i. J
The height of the fluid between the planes is then in the inverse ratio of their
[10240] distance from each other. We may therefore, from this analysis, deduce the
following theorem.
"When the planes are very near to each other, the elevation of the fluid
[10241] between them is in the inverse ratio of their distance. This elevation is equal
to the half sum of the elevations which will occur if we suppose, in the first
place, that the first plane is of the same substance as the second, and, in the
next place, that the second plane is of the same substance as the first ; observing
to prefix the negative sign to the elevation, when it changes into a depression."*
This theorem is a corollary to that which we have before given upon the
elevation of a fluid between two prismatic surfaces of different substances,
[10242^ of which the one is included within the other.f
* (4362) It appears, from [9453,9454], that the elevation of a fluid between two parallel
planes, made of the same substance as the second plane, and whose distance from each other
[10241O] is 27, is very nearly equal to " •^=^' [9328]; and since &'=l'g-^' [9346,10213],
[102416 ] it becomes -~- . The depression of the fluid between two similar planes of the same
COS.tJ
[10241c] nature as that of the first plane, would in like manner be represented by — -, as is
evident from [9454, 10166], its sign being changed as in [10242]. The half sum of these
[10a41rf] two expressions is "" — -~j which is equal to the expression of q [10239], as in
[10241, Sic.]. This expression of q [10239] agrees with that given by M. Poisson in
page 182 of his Nouvelle Theorie, ^c, changing ^ into a^, 27 into 5, q into h, cos.'sr' into
sin.fA, and — cos.-n into sin./*', to conform to his notation.
t (4363) If we suppose the prisms treated of in [10102—10105] to be cylinders
[10242a] whose radii are infinite, we shall have -=1 [10095] ; and since the expression [10105'j
X. Suppl. 2.] TWO PARALLEL PLANES DIPPED INTO A FLUID.
945
We see by this theorem, and by that we have mentioned before
[10219 — 10221], that the repulsive force of the planes is much weaker than the
attractive force,* which commences when the planes are very near to each [10243]
other, and draws them towards each other with an accelerated motion. In this
case, the interior elevation of the fluid, or that between the planes, is very
great in comparison with the exterior elevation near the same planes. Therefore,
by neglecting the square of this last elevation in comparison with the square of
the first, we shall find that the fluid prism whose weight expresses the tendency [10244]
of one of the planes towards the other, in virtue of the first of the two preceding
theorems [10219 — 10221], will be equal to the product of the square of the
elevation of the interior fluid by half the width of the planes in a horizontal
direction.! This elevation being, by the second of these theorems [10245]
may be put under the form
it becomes h -.
9+?/
1 + 7
in which q, q, [10104], represent the elevations of the fluid in cylinders having the same
radius I, but composed successively of the two different substances. These quantities q, q^j
also represent the similar elevations between parallel planes of the like substances, whose
distance is I, as appears from [9998c]. Lastly, h represents, as in [10097'], the elevation
of the fluid between two of these planes of different substances, whose distance is I. These
values agree with [10241].
• (4364) It follows from [10220a], that the attractive force of two planes is
^gD.{q^ — ?/^)> and that it becomes repM?stve when the factor q^ — q^^ is negative; so
that its greatest repulsive force corresponds to 5'==0, and it then becomes, by neglecting its
sign, igD.q/^. Therefore the attractive force is to the greatest repulsive force as q^ — q/^
to q/^. Now q^, which represents the exterior elevation or depression of the fluid near the
planes [10184'], must always be very small ; but q [10239] may become very large, when
I is small ; therefore tlie repulsive force is much smaller than the attractive force, and the
ratio of these forces is nearly as qj^ to q^.
[102426 ]
[10242c ]
[10242rf]
[10243a]
[102435]
[10243c ]
[10243rf]
t (4.365) The attractive force ^gD.{q^ — q^) [10243a], when the planes are very [I0245a]
near to each other, and q^ is much smaller than q, becomes nearly equal to ^gD.q^
[10243c] ; and if we substitute the value of q [10239], it will become
, Tj /cos.ot' — cos-tsV
2 /l\a
[102456]
which is inversely proportional to the square of the distance 2^ of the two planes from each
other, as in [10246J.
VOL. IV. 237
[10249]
946 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
[10241, 10242], inversely proportional to their distance from each other, the
prism will be proportional to their horizontal width, divided by the square of
that distance ; the tendency of the two planes towards each other will therefore be
[10246] in the inverse ratio of the square of their distance [102456] ; consequently it will
follow the law of universal attraction, a law which seems to he followed by all
[10247] attractions and repulsions exerted at sensible distances, like electricity and
magnetism.
Wishing to determine by experiment the singular phenomenon of the
repulsion of planes, which changes into attraction by drawing the planes nearer
[10248] together, I requested M. Haiiy to make some experiments relative to this
curious result of the theory of capillary action. For this purpose he made some
experiments with planes of ivory, which, as is well known, may be moistened
by water, and with some talc laminae, having to the touch a sort of greasiness,
which prevents them from being moistened. These experiments have fully
confirmed the results of the theory, as may be seen in the following account
which he communicated.
" There was suspended, by a very delicate thread, a small square leaf of a
talc lamina, so that its lower end was dipped into the water. There was also
dipped into the same water, at the distance of a few centimetres, the lower
part of a parallelepiped or plate of ivory, so that one of its faces was parallel
to the leaf of talc ; and it was always kept in this parallel position, stopping the
plate of ivory occasionally, in order to be sure that the effect of the motion,
[10251] which it might impress upon the fluid, was insensible in the experiment ; then
the leaf of talc was repelled by the ivory.* Afterwards the ivory was moved
in a very slow manner towards the talc, until the distance between them was
very small, when suddenly the talc approached towards the ivory, and came in
[10252] contact with it. Upon separating the two bodies, it was found that the ivory
plate was moistened to a certain height above the level of the water ; and by
•• ^ ^ repeating the experiment before it was wiped, the attraction commenced sooner,
and sometimes it took place at the first moment, without being preceded by
any sensible repulsion. These experiments, repeated several times with care,
have always furnished the same results."
When the ivory plane is very moist, the water covers its surface, and forms
* (4366) In this experiment, the talc is what we have heretofore called the first plane
[10251a] [10158], to which the quantities q, q^, -a, &, correspond; and the ivory is the second plane,
[10250]
to which the quantities q', 5', to', 6', correspond.
X. Suppl. 2.] ADHESION OF A PLATE TO THE SURFACE OF A FLUID.
947
a new plane, which attracts the lamina of talc; and in this case, the corresponding
angle 6' of the formula [10215] is the greatest possible, being by the theory
equal to a right angle [9312/]. The value of 2/, given by the formula [10215],
which expresses the distance of the planes where the attraction begins, then
becomes greater* than when the ivory was not moistened, agreeing with the
experiment. Moreover, it may happen, by the effect of the friction of the fluid
against the talc lamina, when it sinks down, after it has been raised up between
the planes, that the angle ^ near the place of contact of the fluid with the talc
may become nothing or insensible, in the same manner as is observed relative to
the similar angle with mercury in a barometer, which decreases when the fluid
sinks ; then the preceding expression of 2/ becomes infinite,! and the attraction
is not preceded by any sensible repulsion.
[10254]
[10255]
[10256]
[10257]
ON THE ADHESION OP A PLATE TO THE SURFACE OF A FLUID.
When we lay a plate or disk upon the surface of a stagnant fluid, in a vessel
of great extent, we find that, to detach it, even in a vacuum, we experience a [io258]
resistance which increases with the size of the disk. The disk, as it rises, lifts
up with it a column of the fluid, which follows it till it gets to a certain limit.
* (4367) The expression of the distance 2Z between the planes, where the attraction
changes into repulsion, is given in [10215] ; and it may be put under the form
from which it is easy to show, that, while 6' increases, this value of 2/ will also increase;
for, by taking its differential, considering /, 6', only to vary, we get the first expression in
[10255<Z] ; and by making successive reductions, using [34', 31, 1], Int., which give
cos.^d' Aang.i6' =cos.ld' .sm.l&'=^sin.^6'y 1 — 2sin?^a'=cos.d',
we finally obtain the expression of '2dl [10255e],
2dl=
^2x (cos.2ie'.tang49
rffl' CI— 2sin2i5
-_sin.5«|
'^2a'(isin.i6'
â– sin.J^'^
rf9'
cos.S'
and as the factor
V/2x
C0S.9'
2sin.i9'
/2a'2sin.i9'*
sin.i6'
is always positive, when 6' increases from to a right angle, it
follows that 2Z will increase with d\ while it passes through those limits, agreeably to what
is stated in [10255].
f (4368) When ^ = 0, the expressron — log.tang.^4 becomes infinite; and the value of
2/ [10255fl] then becomes infinite, as in [10257].
[10255a]
[102556 ]
[10255c ]
[10255rf]
[10255e ]
[10255/]
[10257a]
948
THEORY OF CAPILLARY ATTRACTION.
[Mec. Cel.
when it falls back again into the vessel. At this limit, the column will be
sustained in equilibrium, if the force which raises the disk be exactly that which
corresponds to the state of equilibrium ; and it is evident that, for this to take
place, the force ought to be equal to the weight of the disk and that of the
[10259] column which is raised up. Thus the adhesion of the disk to the fluid is a
capillary phenomenon ; but to prove it incontestably, we shall compute this force
by analysis, and shall compare the results with experiments.
We shall consider a section of the surface of the column, by a vertical plane
[10260] passing through the centre of the disk, supposing it to be circular. This section
will be the generating curve, which, by its revolution about the vertical line or
axis passing through the centre of the disk, generates the external surface of the
column. Then we shall put
[10261] / = the radius of the disk ;
[10262] l'\-y = the distance from the vertical axis to any point of the generating curve ;
[10263] z = the height of the same point above the level of the fluid in the vessel.
The equilibrium of the columns will give, as in [9324], by observing
[10263'] that* 7 = 0,
[10264]
ddz
i+y'
dz
dz^
^dy'^
'i^z.
"Differential equation"
of the surface.
Tfi^.168.
* (4369) In the annexed
figures 168, 169, LECMH is
the level surface of the fluid in
[10264a] the vessel; FAR the diameter of
the disk, which is supposed to be
parallel to HL; AE the vertical
line or axis perpendicular to HL;
[102646] MmKkfF the curve whose revo-
lution about the axis AE generates
the external surface of the fluid ;
K, 7c, are two points of this curve
which are infinitely near to each
other; KD, kd, the corresponding
[10264c ] ordinates parallel to AF; and
the point m is infinitely near to
M, the point /' infinitely near to F; moreover the line Ff is on the continuatiort of
f'F. Then, in the present notation, we shall have
X. Suppl.2.] ADHESION OF A PLATE TO THE SURFACE OF A FLUID.
949
To integrate this equation, we shall put
[10264']
AF=:l, ED^z, DK=l-\-y, angle DKJc=z'a [10265].
At the point Mwe have the angle «==0; and at the point F, it becomes
AFf=^^ — AFf'='K—zi' [10279].
When the fluid is mercury, and the disk of glass, we shall have the angle AFf=^S°
[10353], and the surface of the fluid will have a gradual slope from F to M, as in
fig. 168. But if the fluid be water, and the disk of glass, well moistened with water, we
shall have the angle AFf' = 0^ [9359Z:], as in fig. 169; and the angle zi will gradually
increase from the point M, where « = 0, to the point /, where vs = ^7r, and the
surface of the fluid is vertical, as is evident from the equation [10266]; lastly, from / to JP,
the angle zs will continue to increase till it becomes 73 = 1^, at the point F; and then the
curve Ff will be horizontal, as appears from the same equation [10266]. Hence it is evident
that the surface which is formed by the revolution of this curve about its axis will be grooved
like a pulley. To find the equation of this surface, we shall suppose a canal KCBGH to
be drawn, with the vertical branches BCK, GH, and the horizontal branch BG^ the point
H being situated upon the level surface of the fluid in the vessel. Then putting, as in
[9310], R, R'y for the greatest and the least radii of curvature at the point K, the capillary
action at ^ will be equal to K — ^^•(r + ^/) [9315&] ; to which we must add the
pressure of the atmosphere P, and the weight of the column BK, namely gDxBK
[10187c], to obtain the pressure at the point B, in the canal BK, equal to
K-}-P — }Il.(-r-{-—j-\-gD.BK. Again, the capillary action at the point H, which
is represented by J5r[9259], being added to the pressure of the atmosphere P, and to the
weight gDxGH of the column GH, gives the pressure at G, at the bottom of the canal
HG, equal to K-{-P -\-gXGH. Putting the expressions [10264m, 0] equal to each
other, on account of the equilibrium of the canal B G, and neglecting K-\-P, which occurs
in both members of the equation, we get
iH.(^ + -^)^gD.{BK-GH)=gD.CK=gDz [102631
1 1
Substituting gB—H.^ [93722r'], and dividing by ^H", we get ;^+^.=2^^5
using the values of R, R', [9326, 9326'], it becomes
rf^2 ~^ u' du'\ ' du^J ,
and by
:2a2:;
observing that the quantity m [9320] represents the ordinate BK—l-\-y [10264c?], so that
du = dy. Substituting these values of m, du, in the preceding equation [102649-], it becomes
as in [10264], where du or dy is considered as constant [9327c]. The equation [102645-] is
equivalent to the equation (2), in page 228 of M. Poisson's Nouvelle Theorie, ^c., changing
u into t, and a into a'^ [9323p], to conform to his notation.
VOL. IV. 238
[1026id]
[10264e ]
[10264/]
[10264^]
[1026ih]
[10264i ]
[10264J]
[10264/ ]
[10264m]
[10264n]
[102640 ]
[10264p]
[10264p']
[10264? ]
[10264r ]
[10264* ]
950 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
[10265] « = the angle which an infinitely small arc of the generating section makes
with the horizontal line drawn from the lower end of this small arc
to the vertical line which passes through the centre of the disk. We
shall have *
dz
[10266] -= — tang.«;
and then the preceding equation will become f
dvi sin.OT -. , .
[10267] — - .COS.tn — — =2a2; (5)
Multiplying this by J?/. tang.*, or — dz [10266], we get, by integration,
[10268] cos.*+ / -y— — = constant — 0.2;.
We shall suppose that the integral commences ivith z; observing that, when
2 = 0, the arc of the surface will coincide with the level surface, which renders
[10270] w nothing; consequently cos.w = l, and we shall have, constant=l;
therefore
9 1 /^ ''dz. sin.-a .
O^Z = 1 COS.W — / — ; • (t) \ Equation of the surface.
J l + y w L
[10269]
[10271]
* (4370) Draw Jcl, fig. 168, 169, page 948, parallel and equal to dD; then it is evident
kl dz
[10266a] that — = tang.ytX/, or by [I0264td\, — 7-=tang.trf, whence we easily deduce [10266],
, dz
[102666] or dy=—:^^^^,
t (4371) The equation [10266] gives
[10268a] \/ ^ + J' = V/l+tang.2^ = ^;