density, is of an insensible thickness, and if we suppose the
point O to be situated beneath the lower surface of this
stratum, and distant from it h\j the insensible quantify X, which expresses the limit of the
distance to which the corpuscular action is sensible [9258c?J, the whole length of the column
OCy = >^', will yet be of an insensible length. We may consider the stratum of the fluid,
contained between the two hmiting spherical surfaces N'O'M', NOM, to increase in
density, from the point O', where it is nearly equal to nothing [lOSOS', Sic], to the lowest point
O, where it is very nearly equal to the internal density of the fluid, which we shall represent
by unity; then, taking any point Z of the column OO, and putting OZ=z, we may
suppose (he density D of the spherical stratum, of the thicTcness dz, included between the
surfaces which correspond to z and z-^-dz, to be a function of z; and the mass of the fluid
contained in the part dz of the colunm OO ^ will be represented by the product of the two
quantities D and dz, which is Ddz, and this also will be a function of z. Moreover the
action of the whole spherical mass of the fluid N'O'M', upon a particle at z, may be
represented by a function of z, which we shall denote by T{z). Multiplying this by the
mass Ddz of the column [9260/n], we get Ddz . T{z), for the action of the whole sphere
N'O'M' upon the part dz of the column (JO; and its integral f^'Ddz .T[z), or, as it may
be xvv\iien,f^Ddz.T[z) [9240a], represents the whole action of the sphere N'O'M' upon
the fluid of a variable density in the column O'O. When 6==co, or e = [9260c], the
spherical surfaces N'O'M', NOM, change into the horizontal planes K'OT, KOI, and
the pressure at the point O, in the case of the uniform density 1 [9260<?J, becomes equal
to K, which is independent of — or e [9260c] ; and the slightest consideration will make
it manifest, that, when the density varies, in these horizontal strata, from the upper surface at
O towards the lower surface at O, the pressure at this last point, which we shall represent
by K, will also be independent of e. Now in order to compare together the pressures at the
base O of the column OO, in the case of a variable density D, with that which corresponds
VOL. IV. 177
[9260g-]
[9260g^
[9260A]
[92607i']
[9260t ]
[9260&]
[9260^']
[9260Z ]
[9260m]
[9260n]
[9260o]
[926q;>]
[9260?]
[9260r]
[9260*]
706 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
[9262^ denoted by the same letter, in the theory of astronomical refraction explained
[9260f]
[9260m]
[9260p]
[9260w;]
[9260x]
[9260t/]
[9260z]
[9261a]
[92616]
[9261c]
[9261rf]
[9261c]
[9261/]
[9261g]
[9261fe]
[9261i]
[When the fluid in the column O'O is of the variable densitj"!
D, and is terminated by the plane surface K'<y I . J
[When the fluid in the column O'O is of the uniform deaiit;*!
1 , and is terminated by the plane surface K'Qfl', J
to a constant density Ij the surfaces in both cases being supposed to be the horizontal
planes K' O I\ KOI, we shall suppose, in the first place, that the density D [9260//], and the
attraction r (z) [9!260n] correspond to the fluid of a variable density between these two
parallel planes, and that r (^r) changes into r^(«)> when the fluid is of the uniform density 1,
Then we shall have, from [9260p, r, 5],
J^Ddz.T{z) = K,
r,dz.TXz)=K.
In the case of uniform density, the integral function f^dz.T^^z) is expressed by the symbol
K [9260u], if 6=00, or if the terminating surfaces be the planes K'OI\ KOI; but the
same integral function f^dz.vX^) is expressed by K-\-Ke [9260cZ], when 6 is of a finite
magnitude, or, in other words, when the limiting planes K'OI\ KOI, change into the
limiting spherical surfaces N'OM', NOM, respectively. Hence it appears that, when the
density of the fluid is constant, the effect of changing the infinite number of horizontal strata,
parallel to K'OT, and of the thickness dz, into the same number of infinitely near
concentrical spherical strata, of the thickness dz, and described about the centre C, is merely
to add a term of the order e [9253A:], to the function /^tZz-r^s;); or, in other words, the
function r (2;) may be supposed to be increased by a term of the order e. A similar change
must evidently obtain in the expression with a variable density [9260m], when the
horizontal strata are changed into a spherical form. For, as in the preceding case, T'^{z) must
be increased by a term of the order e ; and the density D, which depends on the variations of
the compressing force [9n3m], will also vary in consequence of the change of pressure, in
taking the spherical form, as has been already shown, when the density is uniform [9260rf],
where K changes into K-{- Ke, in consequence of this change of form ; so that, by neglecting
terms of the order c^, which, according to observation, are not of sufficient importance to be
retained, we may consider the function D X r(z) [9260m] as being augmented by a term
of the order e, and this will increase the value of K, by a term of the same order as c, which
we shall represent by Ke. Therefore the expressions of the pressure on the point O of the
column 00, will be finally reduced to the following forms, in the case of the spherical
surfaces N'OM', NOM;
fadz . r (z) ^ K. (1 -{- e) ; [in the case of uniform density 1.]
f^Ddz . r (z) = K . ( 1 -f- e) . [In the case of a variable density D.J
Hence it appears that the only effect of the variableness of the density of the fluid is an
alteration in the value of the symbols -BTand l=Ke [9260c], which express the effect of
the corpuscular and the capillary attraction, changing them respectively into K and - = Ke.
This produces, however, no alteration in the calculations of the effects of the capillary attraction,
because K, H, can be determined by observation alone, and not by analytical deductions, the
law of attraction being unknown. Similar remarks may be made in case the surface is
concave, and the effect will be as in [9258s — w], merely to change the signs of c, e.
X. Suppl. 1. <^ 2.] ATTRACTION OF A SPHERE. 707
in the tenth book * [8168]. [9262"]
2. It is easy to deduce from what has been said, the action of a sphere [of
uniform density] upon an infinitely narrow column of fluid, situated within the
sphere, and perpe7idicular to its surface. Suppose two equal spheres MON,
POQ, fig. 115, to be in contact at the point O; that ^ — ^ ^9263]
10 K is a plane touching both spheres in that point, and
that 0«S is the fluid column. The particle q of the
lower meniscus lOQPK will act upon the column OS,
to elevate it. For if we draw the isosceles triangle ^
Oqr, it is evident that the attractions of the particle q ( \ [9264]
upon the part Or of the column mutually destroy each
other; but the action of q upon rS tends to elevate
the fluid, in the same manner as a point ^ similarly situated in the upper
meniscus lOMNK, The two meniscuses act, therefore, with the same force to [9265]
elevate the column of fluid ; and we have seen, in the preceding article, that the
IT
action of the upper meniscus, in producing this elevation, is — [9260] ; there- [^^^]
fore this quantity also expresses the action of the lower meniscus.
Now the action of an indefinite mass above OS, and terminated by the plane
I OK, is the same upon the column O S sts that of a mass situated below, and [9267]
terminated by the same plane; for any point r of this column is equally attracted
by both masses, but in opposite directions, since it is in equilibrium in virtue of
these attractions [9258/, &c.]. And as K denotes, by the preceding article [9259], [9268]
[92671
Finally we may remark, that, in almost all the formulas of the author, where he uses the
symbols K, H, as appertaining to the calculation of a fluid of uniform density, we can [9261A:]
use the same formulas, supposing the fluid to vary in density near its surface, considering his
values of Kj H, to be the same as those we have named K, H, in the present article. For [9261i ]
example, the general equation of the surface of the fluid [9318], corresponds to the case of
a variable density, by using for H the value resulting from actual observation, which must [9261m]
necessarily be the same as H of the present article ; and we shall use the author's formulas [9261n]
in this way throughout these notes.
* (4124) The quantity K, in the interior of the fluid mass, where the density is equal to
unity, is supposed by La Place to be much greater than — [9253A:, g], and he seems to [9262a]
consider this value of jRTas a representation of the corpuscular intensity in the interior of the [92626]
fluid mass; while H is proportional to that part of it upon which the capillary intensity [9262c]
depends [9360, &.c.]. At an insensible distance from the surface of the fluid, the quantity [9262rf]
K may be extremely small, or negative, as has been observed in [9174aJ.
[9270]
708 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
the action of the superior mass upon the column OS, it will also denote the
[9269] action of the inferior mass upon that column, drawing doivnwards; now this
action is composed of two parts, namely, that of the sphere POQ, and the
action of the meniscus lOQFK; therefore, by putting S for the action of the
sphere, and observing that the meniscus attracts the column upwards [9258w],
[927J] and that its action upon it is — , we shall have,
rQ9701 H rTTiis is conect even when we~|
L^^'^J O ___ J^ m I notice the change in the den- I
- *^ 1 ■**■J I »ity of the fluid near its I
^ l_8uiface. J
or, by transposition,
rnoiyoi C« C I -"^ I This is correct even when we |
[y/«7u] O ^=^ jCL ~f~ — • I notice the change of density I
f) I in the fluid near its surface. I
Hence it follows, that the action of a body terminated by a sensible portion of a
convex spherical surface, upon a fluid column placed within it, and perpendicular
[9274] to the middle of that surface, is represented by K-\- -r-, drawing downwards
from O to S.
If the surface of the body, instead of being convex, is concave (as in fig. 112,
page 695), then the action of the mass ME FN upon the canal OZ will be,
[9274'] CIS we have just seen, equal to K — — [9258w] ; therefore the action of a body
^sphTricai terminated by a sensible portion of a spherical surface [9274', 9274], loill be
segment.
TT
[9275] K — — == the internal action of a concave spherical segment ;
6
H
These expressions are
correct even when we
notice the change of
density near the surface
of the fluid, as is seen
[9276] K-\- — =: the internal action of a convex spherical segment, in [oaeie, &c.], or in
3. We may noiv determine generally the action of a body terminated by any
[9277] curve surface, upon a column situated within it, and contained in an infinitely
narrow canal drawn perpendicular to any point of that surface. If we suppose
an ellipsoid to be drawn through this point, so as to touch the surface, its
action will be very nearly the same as that of the solid, since, this action being
supposed to extend only to insensible distances, the meniscus which represents
'• ■' the difference between the solid and the ellipsoid, will have no sensible action
upon the column, at the points where these two bodies differ sensibly from each
other. We have seen in [9260], that the action of the meniscus which is
[9279] formed between the sphere and its tangent plane, is — , and that, [in the case
of a fluid of uniform density,] it is of the order - [9257], relative to the
action jfiTof this solid, z being equal to, or less than, the radius of the sphere of
X. Suppl. 1. <§. 3.] ATTRACTION OF AN ELLIPSOID. ^09
[9280]
sensible activity of the body. It is evident, for the same reason, that the
action of the meniscus which is the difference between the ellipsoid and the body,
H z
will be relative to the action — of the order * -, therefore it may be neglected
JT
in comparison with — ; we shall therefore determine the action of this ellipsoid
upon the column. One of the axes of this ellipsoid is in the direction of the
column, and this axis we shall put equal to 2a. If we suppose two planes to be
drawn through this axis and the other two axes of the ellipsoid, their sections
will be two ellipses, each of which will have 2a for an axis, and we shall
represent the other two axes by 2a' and 2a". The radius of curvature of the [9282]
first ellipsis, at the point of contact of the body with the ellipsoid, will be f
«^ ' . . a"^
— , and that of the second at the same point will be — ; and by putting these
radii equal to 6, 6', respectively, we shall have 6 = — , b' = — . Through the
same point of contact and the axis 2a, if we draw a plane, which is inclined
by the angle d to the plane passing through the two axes 2a and 2a', the section [9284]
* (4125) If we refer to the general expression of the value of z (9769), corresponding
to the meniscus, we shall have, as in [9770], z = A . x^ -\- "K . xy -\- B . y^^ for the part of z [9280a]
comprised in the ellipsoidal or parabolical part of the meniscus ; and the remaining terms of
z, namely, C.a?-{-D.x^y-{- Sic. [9769], for the part of z corresponding to the difference [92806]
between the meniscus and ellipsoid ; and this last part may be considered as of the order
X X
— relative to the first part, or of the order - [9240a] ; therefore this part must be so very [9280c]
small, that it may be neglected in comparison with the other part.
t (4126) In the ellipsis whose axes are 2a, 2a', if we take a very small absciss sr, counted
from the vertex, corresponding to the coordinate y, we shall have, when z,y, are infinitely
1/3 a"^
small, by the nature of the ellipsis [379c, 378s], |- = p = — ; and if we suppose r to be the [9283a]
radius of curvature corresponding to this absciss and ordinate, we shall evidently have, from
the properties of the circle, 'Hrz — z^ = y^, or, on account of the smallness of z, 5- =r ; hence [92836]
we haver= — , as in [9283]. In like manner, we get — for the radius of curvature at the [9283c]
vertex, in the ellipsis whose axes are 2a, 2a". These axes are represented in [9283] by
h,h', respectively; so that we shall have 6 = — , 6' = — , which are evidently the greatest [9283(f]
and the least radii of curvature of all the ellipses formed by the sections of the ellipsoid by
a plane passing through the axis a.
VOL. IV. 178
710 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
[9285] of the ellipsoid, by this last drawn plane, will be an ellipsis, having 2a for one
of the axes ;* and the other semi-axis, which we shall denote by A, will be
given by the following formula ;
a'^a"^ 11 1
The radius of curvature of this ellipsis at the point of contact, being
A'^ I 1
[9287] represented by B, we have B := — , or — = ct . — [9284e] ; and, by
substituting [9286, 9283], we successively obtain,
[9288] i = a . ) i- sin.254-4 -cos.^n = ]-, . sm.^6 + - . cos.^^.
The action of an infinitely small portion of the ellipsoid formed by the plane
[9289] which passes through the axes 2a and 2A, and by another plane which is
inclined to this by the angle di, and passes through the axis 2a, will be very
nearly the same as that of the like portion of a sphere whose radius is B ; and
H
[9290] as the action of this sphere is, by what has been said [9273], equal to K-\-—,
1 r TT J
[9291] that of the infinitely small portion in question will be f ^ » ds .< K-{- — > ;
^iir \, Jo 1
therefore the whole action of the ellipsoid upon the canal will he
* (4127) Changing o, p, y, into a, a', a", to conform to the piesent notation, we get
x2 w2 22
[9284o] the equation of an ellipsoid [142S6J, under the fonn — -|-^+-^=l; »> y, ^j being
the rectangular coordinates of the surface of the ellipsoid, whose semi-axes, parallel to those
coordinates, are «, a', a", respectively ; the centre of the ellipsoid being the origin of the
coordinates. If we now suppose a plane to be drawn through the axis 2a, so as to form the
angle ^, with the axis of y, the section of the ellipsoid will be an ellipsis whose axes are
2a, 2.4 [9285] ; and the coordinates of the extremities of the semi-axis A will evidently bie
[9284&] a? = 0, y==,A.cos.&, z = A .sm.L Substituting these in the equation of the ellipsoid
[9284c] [9284a], we get — ^^— -\- - ^^ = 1 ; which bemg multiplied by ^^,3i„2^^«^^.co8.2fl
gives [9286]. The radius of curvature, corresponding to this ellipsis, at the point of contact,
[9284fr| is found, as in [9283], by changing, in the expression of 6, a' into A ; by this means it
w32 1 1
becomes, as in [9287], 5 = — , or ^==a. — . Substituting the expression [9286], we
[9284c]
get the first value of - [9288]. The second of those expressions is deduced from the
first, by using the values of 5, V [9283].
â– a
[9292a] f (4128) The whole action of a sphere of the radius B being JT-f- - [9290], that of
X. Suppl. 1. §3.] ATTRACTION OF AN ELLIPSOID. 711
the integral being taken from 6 = to ^ = 2*, which gives for that action the
following expression ; [9293]
C\ 1'\ r* Action of an ellipsoid, which is~|
K^_4_ 4- FJ J I K I correct even when the density I rnfttt*i
u\.-j- 2 J-t .\ T-f- ,,>, of the fluid is variable near its I ly^iiii]
C ^ J l_8urface. J
If the surface be concave, we must suppose b and b' negative. If it be in part [9294']
concave, and in part convex, like the circumference of a pulley, ice must suppose
the radius of curvature corresponding to the convex part to be positive, and that 'â– ^^^^
corresponding to the concave part to be negative.
Putting B, B', for the radii of curvature of the sections of the surface of the r^'
body by two planes inclined to each other by a right angle, we shall have, by what
has been said [9288],
•4 = ^.sin.2^ + ^•cos.^^• t^^^
Jf Q Q
hence we find, by changing 6 into | + ^ which changes B into B, 1^^298]
|j = ^,.cos.24 4-^.sin.2^; [9299]
consequently
s+i=\+F t^J
Therefore the preceding action [9294] may also be put under the following
form;
TT IT ["Action of any body upon an internal column,"!
V" \_ I , • I the rectangular radii of curvature being B,B'. I [93011
•**■1 Q Tj I ' Q TV 7 I This is correct even when we notice the change |
'^■O i6£i Lof density near the surface of the fluid. J
General
theorem.
[9302]
that is, the action of a body, of any convex form, upon a fluid contained in
an infinitely narrow canal, perpendicular to any point whatever of its surface, is
equal to the half sum of the actions of tivo spheres whose radii are respectively
equal to the following ones, namely, the radius of curvature of any section of the
a section of the sphere corresponding to the angle d6, will be found by multiplying it by
— , as in [9291]. Substituting the variable value of - [9288], and prefixing the sign of [92926]
tiv Jo â–
integration, we obtain the whole action of the ellipsoid as in [9292]. This is reduced to the
form [9294], by observing that we have, as in [1544a], J^'^dd . cos.^a = «- ; /^"dd . sin.^^ = if j [9292cl
also/o2''rf4 = 2T. ^
* (4129) Adding together the expressions [9297, 9299], and putting sin *4 -f cos.*4 = 1, [9300a]
we get [9300] ; substituting this in [9294], we get [9301].
712 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
surface by a plane drawn perpendicular to the surface through that pointy and
the radius of curvature of the section formed by a plane perpendicular to the
[9303] preceding plane*
4. We shall nmv determine the form of the surface of water included within
[9304] a tube of any form. We may, in this investigation, as is well known, use either
. the principle of a curvilinear canal terminated at two points of this surface, or
the principle of the perpendicularity of the force at the surface. In the present
question, the first of these principles has a great advantage over the second,
[9305] because it requires only the determination of the two actions JTand — . ( 7 + t; )
[9294], and in fact only the last of these two forces, since the first ^disappears
from the equation of the surface [9318], as we shall soon see. Although the
force which produces this second action, is, at the surface, incomparably greater
[9306] than gravity ; yet, as it acts only upon an insensible interval, its action upon a
fluid column of a sensible length, may be compared with the force of gravity
upon the column. But if we wish to make use of the principle of the
perpendicularity of the resultant of all the forces, at the surface, we must consider
[9307] not only the action which produces the forces K and ~^'\T'^t\ which are
perpendicular to the surface, but also the force of gravity, and the force which
arises from the attraction of the meniscus corresponding to the difference
[9308] . . .
between the ellipsoid and the body ; for, although the action of this part upon
a fluid column is insensible, because it acts sensibly only at an insensible
[9308'] distance, yet it is of the same order as gravity. On account of the difficulties
in estimating all these forces and their directions, it is much more convenient to
use the principle of the equilibrium of the canals. f
* (4130) The actions of the two spheres whose radii are B, "B, are respectively, as in
[9301a] [9273], ^-f -, JT-f- -, whose half sum is K -\- - -{ - -, \ being the same as the
JS B ^tt tin
action of the ellipsoid of curvature [9301], as in [9302 — 9303]. If the surface be concave,
we must put 6, b', negative, as in [9294'J ; consequently B, B, [9297, 9299], will be
[93016] negative, and then the action [9301] will become K — ^^']~B + '^\'
f (4131) La Place has used this second method in his supplement to this theory
[9309o] [9812 — 9845], and it serves to prove, a posteriori, the identity of the two results.
I
X.Supp!.!. §4.] EQUATION OF THE SURFACE OF A FLUID IN A TUBE. 713
We shall suppose O, fig. 116, to be the lowest wigMG, [9308"]
point of the surface AOB oi the water contained in
the tube, and we shall use the following symbols ;
z, the vertical coordinate OM; \^ ^" — f)^ - ^^ [9309]
a:, y, the horizontal rectangular coordinates of any
point N of the surface ; 'â– â– '
jR the greatest, and R the least, radius of curvature of the surface, at the [9310]
point Nf then R, R, will be the roots of the following quadratic
equation ;
* (4132) We have a?, y, sr, for the rectangular coordinates of the point iVTof the surface
for which the radius of curvature is to be computed ; and we shall suppose that, for ^ •^JUaj
another extremely near point of the surface, these coordinates become respectively
x-\-}ij y-\-lc, z-\-l; then, from Taylor's theorem [610], we shall obtain the value of
z-\-l [9310e], neglecting terms of (He third order, which are not wanted in the present [93106]
computation. We shall now suppose that the same symbols, accented, namely,
x'ji/fZ', p', q', r', s', t', /', correspond to the osculatory sphere at the point iV, without [9310c]
altering the increments A, k; hence we shall have the expression of z'-f-?' [9310/"], which
is similar to that of 5; -j- ^' Lastly, by putting a, ^, 7, for the coordinates of the centre of mojo/ii
the osculatory sphere, parallel respectively to the axes of x, y, z, also R for the radius of
the sphere, we shall have the equation of this osculatory spherical surface at iV, as in [9310^],
being conformable to [19e] ;
z-\-l=:z-j-(ph-{-qk)-[-i (rA^ -f ^skk + tk"^) ; [9310e ]
sf-\-r = sf-\- (p'A + c{k) + ^ (r'A^ -|- 2s'hk + I'kr^) ; [9310/]
i22 = (a,' _ a)2 -I- (y' — p)8 + (;?' — 7)2. [9310g]
Taking the differentials of the equation [9310o-], considering jR, cl, (3, 7, as constant, we
get, by successive reductions, the following values of p', q', /, s', t', relative to the spherical
surface ;
^'-m
(x'— a) , /rfA (y-f3)
z'-y ' ^ W.
q = U; =—
[9310A]
, /ddz'\ 1 {x' — a)^ -(l + p'2)
, /ddz'\ (x'~a).q' (/— 3)Y_ pV .
, /ddz^ 1 { y'-^f — n+q"^)
'=V^V=~^W""F^=^3="^'=7"- t9310Z]
Now if we compare the expressions [9310e,/], we shall find that, when A = 0, A; = 0, we
must put z' = z, to make the spherical surface pass through the point N, whose coordinates
are x, y, z. Moreover if we put p' = p, q' = q, we shall find that the terms pA + q^j
VOL. IV. 179
714 THEORY OF CAPILLARY ATTRACTION. [M^c. Cel.
[9311] i22.(rt— s^)— i?.v/I+?+?-{(l+q')r— 2pqs+(14-p^).t}4-(l+p'+q^^