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Pierre Simon Laplace.

# Mécanique céleste (Volume 4)

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'a
instead of the value in [9534]. The effect of this modification will be to change tang.w
into sin.w, in all the formulas where it occurs from [9534] to [9549], as well as in
the corresponding notes.

[9534g]

[9535a] * (4194) The expression of h [9534] gives -, as in the first formula in [9535], or, as

it may be written.

[95356]

1 I y + -.tang.^fl'.tang.w

h a.tang.tsi , x

a

Developing this, according to the powers of x, and neglecting a?^, &ic., it becomes of the
same form as the second expression in [9535].

X. Suppl.l. <^ 10.] FIGURE OF A DROP BETWEEN TWO INCLINED PLANES. 769

We shall suppose a canal to be so situated in the intermediate plane, that one
of its extremities is at the point of the section whose coordinates are a;, ?/, the [9536]
other at the point of the section through which the axis of y passes; the
equilibrium of the fluid in this canal will give the equation *

K-^,-{-gx.sm.V=^K-^^, ' [9537]

placing one accent below the letters relative to this last point. But we have, by [9537']
what precedes! [9523],

1 sin.d' Q 1 sin.^' Q

b" h b" b"~ h, 6;'

h' being, in this case, the radius of curvature of the curve formed by the
intersection of the intermediate plane with the surface of the drop. Moreover,!
1 1

[9538]
[9539]

A a.tang.-sf o^.tang
1 1 ,

A^ a.tang.-za'

-.n+^.tang.i^'.tang.w j , [9540]

[9541]

* (4195) We shall suppose, in fig. 125, page 766, that BaLt is an extremely narrow [9537o]
canal, of uniform diameter, passing along the intermediate plane, and terminated at the points
B,L; and that the radii of curvature at B are b' for the arc GBL, and b for the arc
passing through B, perpendicular to the plane of the figure; also b',, 6^, for the radii of the [95376]
similar arcs at the point L. Then the capillary action at B upon the canal BaL, is as in

[9485/J, equal to ^-|-f.Q, — ^); and at L, this action is K^^.Q—^Y Now [9537c]

the point B is elevated above L by the quantity C^Xsin.F [9527], or oj.sin.F. [9537rfJ
Multiplying this by g, we get the pressure of the column of fluid, to be added to the action
at B, to obtain that at L, when in equilibrium, as in [9485e, he.]. Hence we get

^+f.(f-J)+^^-sin.F=^+f.(^, - ^). Substituting ^— ^=^ [9500], and [9537e]

the similar expression - — -=-, it becomes as in [9537]. [9537/]

t (4196) The first equation [9538] corresponding to the point E, fig. 125, page 766,
is the same as [9523], which is found by a calculation similar to that in [9486—9523], [9538o]
using, instead of/, h [9495, 9494], the values CE, EE' [9526i— A:]. When the point E falls

in Kj the quantity h changes into A, [9537'J, and the value of — [9538] changes into [95386]
that of ^, [9538].

X (4197) The equation [9540] is the same as [9535], and at the point L, fig. 125,
page 766, where a? = 0, and h = h, [9537'], it becomes as in [9541].
VOL. IV. 193

770 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.

f9542]

[9546]

therefore we shall have*

Hx.s'mJ'

£ . (^1 + ^. tang.i^'. tang.:.) + iQ^. (^1,— i) +^a:. sm.F== 0.

2fl^.tang.

As the section differs but little from a circle, b' is very nearly equal to the half
width of the drop a; b" is, by what has been said [9522, 9514c], very nearly

[9543] equal to - — ;, and h is half the thickness of the drop; b' is therefore very

sinJ ^ "^

large in comparison with 6"; consequently — is very small in comparison with
[9544] _; therefore the difference - — - may be neglected in comparison with

Tj, — 77;. This may be done with more propriety, since, b\ being a mean

[9545] ' • .11

between the extreme values of 6', the greatest value of the difference t, — r,

is only about half of the difference of the extreme values of -, . Moreover,
the figure of the drop being very nearly circular, as is found by experiment, the
difference 77 — 7-^ is nearly insensible. We may also, in the preceding

[9547] equation, neglect the fraction -.tang.i^'.tang.^, in comparison with unity;

a

because, 2a.tang.^ being the thickness of the drop whose width is 2a, this
[9548] fraction does not exceed the ratio of the thickness of the drop to its width,
which ratio, by hypothesis, is very small. This being premised, the preceding
equation will give f

* (4198) Transposing all the terms of [9537] to the first member, we get
[9542a] lU. ^,—\i\ +^a?.sm.l^=0;

and by substituting the values [9538], it becomes
[95426] ^H.sm.4'.^l_l^-|-^QH.^~i^+^^.sin.F=0.

Now if we subtract [9540] from [9541], we get

and by substituting it in [9542&], we obtain [9542].

t (4199) Neglecting the second and third terms of [9542], on account of their smallness
l- ** "■• [9546, Sic], it becomes — ^ ^'^'°' 1- sfj? . sin. F= : dividing this by s:x. and

X. Suppl. 1.^11.] ATTRACTION BETWEEN TWO VERTICAL PLANES. 771

sm.F= - . [9549]

2a2 .g . tang.OT

[9550]

[9551]

Thus the angle Fis very nearly in the inverse ratio of the square of «, as in a
drop suspended in equilibrium in a cone [94756]. Comparing this expression
of sin.F with that of the preceding article, w^e see that, the angle formed by
the two planes [9526] being supposed equal to the angle formed by the axis [S^SO']
of the cone and its side [9462], the sine of the angle V relative to the
intermediate plane, is equal to the sine of the angle relative to the axis of the
cone.* We ought not, however, to forget, in comparing the results of this
analj^sis with experiments, that the expressions of sin.F are only approximate
values.

1 1 . The preceding analysis furnishes an explanation and the measure of a
singular phenomenon discovered by experiment ; namely, that, ivhether the fluid he
elevated or depressed between two vertical and parallel planes, dipped into a fluid [9552]
at their lower extremities, the planes ivill tend towards each other ; so that, if two
small glass vases, in the form of a parallelepiped, floating upon water or mercury,
happen to approach near to each other, they will immediately unite together.
To prove that this must take place, we shall consider the two planes MB

[9552']

transposing the first term, we get [9549]. Poisson, in computing this problem in his Nouvelle
Theorie, &/'C., uses the same fundamental equation as in [9318], carrying on the approximation [95496]
so as to include some terms which are neglected by La Place, as being within the limits of
the errors of the observations. These terms are, however, omitted by M. Poisson, in his final raxAQc-]
expression of sin.F, which is nearly the same as that given in [9549], as may be seen by

changing, in his formula (10), page 258 of that work, the symbols a^, c, i, 6, -a, into

If

— , a,zs, V, ^'-|-90'*, respectively, in order to conform to the present notation ; putting also, as

^ [9549c]

in [95345-], si"-* ^o'" tang.tj, in [9534, &z;c.] ; this last change being within the limits of the

degree of accuracy aimed at by La Place, throughout his whole calculation ; so that, instead

of [9549], we may put very nearly s'm.V=r-Y- — r—\' r " [9549/"]

[9549rf]

Sa^.g'.sin.'ztf

* (4200) If we suppose that the angle formed by the axis of the cone and its sides
is 2t3, instead of « [9462], we shall find that the expression [9475a] will become

[9550o]

2 H. sin.,

nearly, because, on account of the smallness of vi, we have sin.2« = 2sin.OT [95506]

a 2a2.g'.sin.'5J

nearly. This being equal to the expression [9549/], proves the correctness of the remarks

in [9551].

772

THEORY OF CAPILLARY ATTRACTION.

[Mec. Cel.

[9553]

[9554]

[9554']
[9555]

[9556]

[9557]

[9558]

[9558']
[9559]

2r
JL

^ -f- ^y

T'i^.lZT. JB'

^^

H

G-

Qi

and iV/?, fig. 127, and shall suppose, in the

first place, that the fluid is elevated between

them. The infinitely small external part of

the plane NR, situated at i2, below the level

VP^ will be pressed by a force which may be

thus estimated: Imagine a canal VSR, of

which the branch VS is vertical, and the branch

SR horizontal ; then the force acting upon the

fluid in the canal VS is equal to g. VS, ^

augmented by the force acting in V, either by

the action of the fluid upon the canal, or by the pressure of the atmosphere.

The first of these two forces is represented by K [9259], and we shall suppose

that the pressure of the atmosphere is P ; then the whole force of the column

VSWiWhQ P + K+g.VS. The action, with which the fluid in the canal

RS IS urged, is composed of two parts. First, the action of the fluid upon the

canal, which is equal to K [9259]. Second, the action of the plane upon the

same canal ; but this action is destroyed by the attraction of the fluid upon the

plane, and there cannot arise from this source any tendency in the plane to

move. For, by considering only these reciprocal attractions, the fluid and the

plane will be at rest, the action being equal and contrary to the reaction ; these

attractions can produce only an adhesion of the plane to the fluid, which need

not be noticed. Hence it follows that the fluid presses the point R with a

force equal to P-\-K-\-g.VS—'K, or simply P-\-g.VS.

We shall now determine the corresponding internal pressure. For this
purpose, we shall suppose the canal OQR to have a vertical branch OQ,
terminated at the surface O, and a horizontal branch QR. The force with
which the fluid is urged in the branch O Q, is equal to g.OQ, increased by
the pressure of the atmosphere P, also by the force with which the fluid acts

upon the column O Q ; and this is, by what has been said,* equal to K — ^ ,

b being the radius of curvature at O. Therefore the force with which the

M H

[9558a] * (4201) This force is represented in [93156] by K — — — — ; and as h'=cr,

[95585] for reasons similar to that of putting h infinite in [9414], it becomes K — — , as in [9452fl].
[9558c]

In the original work, it is erroneously printed K — -r ; to correct this mistake we have
changed h into 26, in all the formulas [9559—9578], h being the radius of curvature at O.

[9562]

[9563]

[9564]

X.Suppl.l.'^.ll.] ATTRACTION BETWEEN TWO VERTICAL PLANES. 773

T7

fluid OQ is urged, is P-\-K — -7-\-g.0Q. Now, by what has been said,* [9559/]

S=g,OP. [9560]

Therefore the force of the canal OQ is P + K+g.PQ. The force of the [9561]
canal QR is equal to K [9259] ; the point R will therefore be pressed on
the inner surface by the difference of these forces, which is P-\-g.PQ, or
P-\-g. VS. Thus the plane is pressed with equal forces [9557, 9562] on the
outer and inner surfaces, and it will therefore be in equilibrium in virtue of
these pressures.

The exterior fluid rises as high as Z, forming a curve VZ' Z\ and between
the planes, it rises as high as A", forming the curve ON'N. The parts of the
plane extremely near to Z and N, and similarly placed at a distance from Z and
N equal to or less than the sensible sphere of activity of the plane, are equally
pressed within and without ;t because the surfaces of the fluid, comprised
within this sphere, and near the points Z, N, are very nearly of the same form.
Besides, the extremely small difference, which may exist between the internal
and external pressures of the fluid, being limited to an insensible extent, we
may neglect it, and notice only the pressure exerted by the fluid at the points
where the action of the plane upon the surface ceases to be sensible. Therefore
let Z' be one of the points of this surface, and suppose the horizontal canal Z'q

IT

to be formed. The force in Z' will bet P + K — ^, R being the radius [9566]

* (4202) The expression [94526] gives —=g.q'; and in like manner, in the

JJ ^* [9559o]

present case, we get, as in [9560], —=g.OPj by changing 6' into b, and q' into OP,

to conform to the present notation. Substituting this in [9559'], it becomes

P + K-^g.{Oq-OP) = P + K-\-g.Pq, [95596]

as in [9561].

t (4203) If the angle 6' [9346] is the same on both sides of the plane, as the author
supposes in this article, the part of the pressure here spoken of will be the same on both [9565a]
sides of that plane ; but if there is a difference in the value of this angle, on account of the
plane being more or less wet on the one side than on the other, or being made of a different [95656]
substance on opposite sides, the pressures may be unequal, as we shall see when treating of this
subject in the second supplement [9983m?, &.c.].

J (4204) The capillary action, which in [9558'] is called K—-, here becomes

[9565]

VOL. IV.

194

774 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.

[9567] of curvature in Z'. If we put TZ' = x, the equilibrium of the fluid in the
canal Z'LFV will give, by using [9416],*

[9568] g=^^.

The point V being placed at a distance from the plane, such that the radius of
[9568'] curvature of the surface in V may be considered as infinite, the external

pressure in q will be
[9569] P-^K—gX.

The corresponding internal pressure will be f

[95701 P + K—^+g.{OP — x),

or P + K — gx ; therefore the internal and external pressures are equal through
the whole extent of ZG.

We shall now consider the pressure above the point Z, The external pressure

[9572] is reduced to P [9554']. The internal pressure upon a point R' is determined

by considering a canal OQ'R', Q'R' being horizontal. The pressure of the

[9566a] K — r-, because the radius of curvature b changes into R at the point Z'. Adding to
this the atmospherical pressure P [9558], and we get the whole external action at g', equal to
[95666] p^K-^^, as in [9566],

[9568a] * (4205) Substituting — [9326] in [9416], we get -— 2a2; = -; but at the point V,

we have 6'=oo, or |,=0 [9558a, 9568'], also ^=^ [9328]; hence -^ — 2.|.5r=0,
[95685] jf H H

or —=gz] and by changing z [9309] into x [9567], to conform to the present notation,

it becomes as in [9568]. Substituting this in [9566], it becomes as in [9569].

[9570o] t (4206) This is found in precisely the same manner as in [9559'], supposing the line
Z'q to be continued till it meets O Q in Q". For then the action at q, on the canal ^''g^,
is found by adding together the atmospherical pressure P [9554'] ; the gravity of the column

[95706] OQ", which is equal to ^.OQ"; and the capillary action at O, which is equal to

JT— I [9558']. The sum is P + X— ^ + ^.OQ"; and as

[9570c] Oq;'=OP — Pq"=OP'-TZ'=OP — x [9567],

it becomes as in [9570]; and by using —-=g.OP [9560], it becomes P-\^K — gx, as
in [9571].

[9575]

X.Suppl.l.<^ll.] ATTRACTION BETWEEN TWO VERTICAL PLANES. 775

column Oq' is* P + ^— ^-f ^.OQ', or P J^ K^g .OP+g.OQ', or, [9573]

lastly, P + K—g.P Q'. The contrary pressure of the canal R'Q' is K [9259], [9574]
so that the point R' is pressed at its inner surface by the force P — g.PQ';
therefore the plane is pressed inwards , at that point, by the force \ g.PQ'.

XT

In the part NN'O, the pressure at N' is J P-\-K — — ,, h' being the [9576]
radius of curvature in N'; and by supposing the horizontal canal N'p' to be
formed, the interior pressure in p' will be P — -jj. Now putting x' for the [9577]
height of the point N' above IK, we shall have, from [9416],^

* (4207) This is the same as [95706], changing OQ" into OQ', and then substituting
jy [9573a]

—=g.OP [9560]; for by this means it becomes P -\'K—g,OP -{■g.0(^', as in

[9573].

t (4208) The difference between the internal pressure P — g-PQ,^ [9575], and the
external pressure P [9572], is g.PQ^', as in [9575], in the direction -R'Q'.

[9575a]

t (4209) This is similar to the force at Z' [9566], changing the radius JR, relative to mg^g^i

ing to the point N'. The force of the canal N'p' ,

the point Z', into b' [9576], corresponding to the point N'. The force of the canal N'p
at p' is K [9259], and by subtracting it from the preceding value [9576], we get the

pressure in p', in the direction N'p' equal to P — ^,) as in [9577].

[95766]

§ (4210) The equation [9416] is, in [9568a], put under the form - — 2az = - ; [9577a]

R, b', being the radii of curvature corresponding, in the present case, to the points N% O,
respectively [9326, 9413'] ; and to conform to the notation here used, we must change them
into b',b [9576, 9559]; also z [9309] must be changed into x' [9577], and then the

preceding equation will become - — 2(u;'=7-. Substituting a-==g [95686], multiplying [95776]

Tf H

by ^H, and transposing the second term, we get 2b''^Qb'^^^'' ^^ '° [9578].

H H

Substituting the value of - [9560], it becomes ^,=g-{OP -\-x')=g. Gp', as in the

jj

second form of [9578]. Substituting this in the function P —^, [9577], denoting the

internal pressure at p', it becomes P — g. Op'; taking the difference between this and the [9577rf]
external pressure at jo', namely P [9572], we obtain the whole inward pressure g-Gp' [9577c]
[9579], in the direction p'Ni

776 THEORY OF CAPILLARY ATTRACTION. [Mec. C^I.

therefore the plane at the point p' will be pressed inwards, in the direction p'N\
by the force g. Gp'.

Hence we easily perceive that the force which presses the plane NR inwards.
[9580] is equal to the pressure of a column of fluid ivhose height is ^.(GiV-f GZ),
and ivhose base is the part of the plane included between Z and * N.

r9580al * i^'^^^) From what has been proved in [9575, 9579], it follows that, at any part of the
line ZN, as at R\ the pressure will be as g.GR' ; and by putting GR'=w, it becomes
g.w; therefore the whole pressure on the part dw of lo, will be givdw, whose integral,

[95806] supposing it to commence at the point Z, will be ^^.(t^^ — GZ^) ; and the whole pressure
upon ZN is found by putting w= GN; hence it becomes

[9580c] ^g.{GN^—GZ^)^^g.{GN+GZ).{GN—GZ)=ig.{GN-i-GZ).NZ,

which is evidently equal to the pressure of a column whose base is NZ, and height
[9580rf] ^i^Qjsf_^QZ)^ asm [9580].

In all these calculations, it is supposed, as in [9434, Sic.], that the point iV, which is taken
[9580e ] as the extreme point of elevation of the fluid near the plane, is not in fact the actual point of

contact with the plane, or with the upper surface of the fluid, but is distant from it by an
[9580/*] insensible quantity of the order X, corresponding to the radius of the sphere of activity of

the corpuscular attraction, or to the distance between the upper surface nom of the fluid, and
[9580g] the assumed surface NOM; the fluid varying in density, from the upper surface nom,

where the density is very small or nothing, to the lower surface NOM, where the density
[9580ft] is the same as that of the internal fluid mass, which is represented by unity. We may also

remark, relative to the actual measures of the elevation of the point N above the horizontal
[9580t] level of the fluid in the vase, which we shall represent by GN='k, that it is a matter of

perfect indifl^erence whether we consider that point as being situated in the surface nom, or
[9580A:] in the surface NOM, the interval between these surfaces being wholly insensible to our

senses, on account of its smallness, so that the one may be used for the other without any
[9580Z ] appreciable error. Similar remarks may be made relative to the surface of the fluid on the

outside of the plane at Z, supposing its elevation above the level of the fluid in the vase to
[9580m] be represented hy GZ=1c,.

The values of k, Tc, [9580i, m\, being substituted in the difference of the pressures on the

opposite sides of the plane [9580c], it becomes
[9580n] hg-{^^ — ^^/^) =the difference of the pressures computed in [9580].

We shall see, in the second supplement to this book [9983w, &c.], that this expression of the
[9580o] pressure requires some modification when the planes are of a different nature, or, in other

words, when the angle -a [9346'] corresponding to the inner side of the plane at N, differs

from the similar angle w^ relative to the outer side of the plane at Z ; the correction of this
[9580p] pressure being ^g,o?.{s\n,-a — sin.w^); so that we shall have, as in [9983u>],
[9580y] hg'Q^^- ^.') +^5-«'.(sin.«~sin.^,) = {Ti^^corr^^^^^^^^

This expression is the same as that given by IVI. Poisson, in page 172 of his Nouvelle
Theorie, ^c, and it is reduced to a much more simple form in [9984/*].

X. Suppl. 1.^11.] ATTRACTION BETWEEN TWO VERTICAL PLANES.

777

A similar result holds good for the plane MB ; thus we have the force with
which the two planes tend towards each other, and we see that this force increases
in the inverse ratio of their distance from each other*

In a vacuum, the two planes will also tend towards each other ; the adhesion
of the plane to the fluid produces then the same effect as the pressure of the
atmosphere.

We may prove in the same manner, that when the fluid is depressed between
two planes, the pressure which each plane suffers in an inward direction, is equal
to the pressure of a column of the fluid whose height is the half sum of the
depressions below the level, at the points of contact of the internal and external
surfaces of the fluid with the plane, and whose base is the part of the plane
included between the two horizontal lines drawn through those points.^

[9581]

Pressure
on two
planes dip-

[9582]

ped in a
fluid.

[9583]

[9584]

[9585]
[9586]

* (4212) When the planes are very near to each other, we shall have GZ^ very small
in comparison with GN^; and then the whole pressure [9580c], ^g.(GN^ — CrZ^), is
nearly equal to ^g.GN^; and as GN is nearly in an inverse ratio to GX [9454], the
whole pressure will be very nearly in the inverse ratio of the square of GX. Moreover, if
we divide the whole pressure ^g.GN^ [95826], by the height GN, we shall get ^g.GN,
for the mean pressure on any given point of the column ZN, which will therefore be
inversely as the distance of the planes G X.

t (4213) The annexed figure 128 is similar to fig. 127,
but is adapted to a convex surface NOM, depressed below
the level VX of the fluid in the vessel; in this case, the
demonstration is nearly the same as in [9552 — 9578], merely
changing the signs of the terms. Thus upon the principles
mentioned in [9276, 9294'], we must change the sign of b,
in [9559'], and by this means it becomes

Also, in [9560], we must change the signs of 6, OP; and we

H

get, as in that formula, —=g.OP. Substituting this in the preceding expression, it

becomes P + K-{-g.{OP-{- OQ)=zP + K-{-g.P q, asm [9561]; and the rest of the
calculation [9561,9562] isnotahered; so thatwe find, as in [9562], that the opposite actions at
JR, and of course upon any point of the plane NR below N, mutually balance each other. The
same takes place on the part GZ, the pressures of the atmosphere on opposite sides mutually
destroying each other. It now remains to examine the action upon the part N Z. To determine
this, we shall draw through any point -R', a narrow horizontal canal jR'jP, which is bent upwards
in jPF^ to meet the level surface of the water in V. Then the pressure at V is P-\- K; and at
JR' this will produce a force similar to that computed in [9557], namely, P-\-K-\-g. VF — K,
VOL. IV. 195

XT T

Mff.128.
G- F

X-

!Z"\

1
z

^' a

I

IK

T'

Jsr

-7^
R

[9582a]
[95826]

[9582c]

[9586a]
[95866]

[9586c]

[9586rf]
[9586c ]

[9586/]

[9586g-]
[9586^]

778 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.

12. It now remains, in order to complete this theory of capillary attraction, to
inquire into what produces the concavity or convexity of the fluid contained in the
tube or between two planes. The principal cause is the reciprocal attraction of
the tube and fluid, compared with the action of the fluid upon its own particles.
We shall here suppose that these attractions follow the same law, tvith respect to

[9587'] the distance, both for the particles of the tube and for those of the fluid, and that
they differ only by their intensities at the same distance ; moreover we shall put

[9588] p = the intensity of the action of the particles of the tube upon those of

the fluid; ' k

[9588'J p'= the intensity of the particles of the fluid upon each other.

This being premised, we shall suppose, in fig. 131, that
the lower part of the vertical tube ABCD is dipped into
the fluid contained in the vessel, and that MN is the line
of the level of the fluid in the vessel. We shall also
suppose that the whole surface of the fluid in the tube is
plane, and at the same level. The point O of this surface

[9589] included within the sphere of the sensible activity of the

1  ...  86
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