drawn through F, perpendicular to the plane of the figure, and continued infinitely above and
below it; this value is
Z^^lI.sin.«;.{tang.Ku'-ct) - tang.Kw' -«.')} . [^''^Tpof aV^^n!"^]
X. Suppl. 2.] ACTION OF A WEDGE ON A TRIANGLE. 853
capillary forces are only modifications of these attractive forces, depending [9976]
Changing w into w', we get the corresponding value of Z relative to the angle ti>',
namely,
Z= Jif . sm.w'. \ tang4 {w' — a) — tang.^ {w' — a') \ . TAction of^e. ^^^^^,_^'-^'\
Subtracting the expression [9976c] from [9976cZ], we get, for the action of the wedge a' — a
upon the part of the plane included by the angle w' — w, the following expression,
Z = AH 5 + ^^"•^'- ^ + tang4(«;' — a) — tang.i(w' — a') ) pAction of a wedge a'-an
^ '^-j-sin.u;.! — tang.Kw; — a)-ftang4(w;— a')5' L "?<>" -^ pi^^^^ -'-«'• J
this action ieing upwards in a vertical direction. The angles w' — w, a! — a, are here
considered as positive quantities. The positive values of the angles w, w', relative to the
attracted plane, are counted from the line FQ^ towards the right, and the positive values of
the angles a, a', relative to the attracting wedge, are counted from the line FE towards
the left. If we wish to count the angles a, a', from the line FE towards the right, we
must change the signs, putting a = — b', a' = — a', considering a', b', as positive quantities.
If we also, for the sake of symmetry, put w = a, w'= h, and then substitute ^-ff= q [9972e],
we shall find that the expression [9976e] will become
; -f sin.b . 1 + tang.^(b + b') — tang-Kb + a')
' -\- sin.a . \ — tang. J (a -J" b') -f- tang. J(a -{- a')
Z=q.
[
Action of a wedge b' — a''
upon a plane b — a.
^
representing the action of a wedge A'FB', fig. 151, included by the xfigJ^L
angle b' — a' upon a triangular part of the plane AFB included by ^
the angle b — a ; the angles a, b, a', b', being considered as positive
when falling to the right of the line EFQ^, otherwise negative; the
origin of the angles a', b', being the upward vertical line FE, and the
origin of the angles a, b, being the line .FQ on the continuation of EF,
or in the direction of gravity. If we use, for abridgment, the values of
A, B, A', B' [9976A:], and make a slight change in the arrangement of
the terms of the expression [9976^], we shall find that it will become of
the form [9976/], from which we may easily derive the equivalent
expression [9976»i], as we shall soon show.
' Z=^^sin.a.(tang.^-tang.B') + sln.b.(tang.5-tang.^')h ^^^,, ,, , ^,,„, ,,
, . . , ^ ^i> . . I / / -Tk . ■m\-i L upon a plane b — a.
Z= ^.f sin.a'. (tang.^ — tang.^') -f sm.b'.(tang.J5 — tang.5')^ ;
The identity of the two expressions [9976Z, m], is easily proved ; for if we change 6 into
— a', in [38], Int., we can easily deduce cos.a' — cos.a = tang.^(a'+a) . (sin.a — sin.a'),
which, by using the notation [9976A:], becomes as in [9976o] ; and in like manner we get
[9976p, y, r]. The sum of these four equations gives [9976.?]; observing that the first
member of this sum vanishes. Multiplying this last expression by — q, and adding the
product to [9976Z], we get [9976ml. '
VOL. IV. 214
[9976J]
[9976e ]
[9976c']
[9976/]
] [9976^]
T
[9976ft]
[9976Z ]
[9976m]
[9976n]
854 THEORY OF CAPILLARY ATTRACTION. [M6c. C6I.
[9976'] upon the curvature of the jiuid surface in the first method, and upon
[9976o]
[9976/7]
[9976?]
[9976r]
[9976*]
[9976* ]
[9976u]
[9976v]
[9976w]
[9976x]
[9976x']
[9976y]
-tang.J5')
â– tang.^').
[9976z]
cos .a' — cos.a = tang.-4 . (sin. a — sin. a') ;
cos.b' — cos.b = tang.S . (sin.b — sin.b') ;
cos.b — cos.a'=tang.^'. (sin.a' — sin.b) ;
cos.a — cos.b'=tang.J5'. (sin.b' — sin.a) ;
= — sin.a'. (tang.^ — tang.^') — sin.b'. (tang.B —
-\- sin.a . (tang.-4 — tang.B ) -j- sin.b . (tang. J? —
We may also deduce the equation [9976/n] from [9976?], by observing that the action of a
solid wedge, included between the angular space b — a, upon a triangular plane included Id
the angular space b' — a', is equal and contrary to the reaction of the plane upon the wedge ;
we may therefore change the symbols a, b, into a', b', reciprocally, without altering the
value of Z. This change does not alter the values of A, B [9976A:], but changes A' into
B'f and B' into A'; and by making the same changes in [9976/], we get [9976w]. These
two values of Z are equivalent to those given by M. Poisson, in page 89 of his Nouvelle
Theorie, ^c, where they are marked (5) and (4); a', b', he, being changed into a, b, &c.
We may in this connection remark that whenever any one of the angles
A, B, A', B', becomes equal to a right angle, which can happen when either of the lines
FA, FB, fig. 151, is in the same direction as one of the lines A'F, or B'F, so that the plane
and solid can be in contact with each other upon an infinitely long line, the expression of Z
may become infinite by means of the tangent of this angle ; but when this tangent is multiplied
by the sine of an angle a, b, a', or b', which then becomes equal to nothing, the value of Z
may be finite, though it may appear under the illusory form of Z:
kind occurs when we suppose the lines FA, FA', FQ, to coalesce, as in fig. 152, and the
line FB to be on the continuation of BF'. In this case, a*, b',
will be negative, and we shall have
a' = — .180% a=^0, b':
hence we get from [99762:]
A = — 90\ B=0, A' = — 90^-^ih,
Here tang.-4 becomes negative and infinite, but being multiplied by
sin.a, or sin.a', in the expressions of Z [9976/, m], it becomes
illusory. To ascertain its true value, we shall, in the first place,
suppose b = J*, making the line B'FB coincide with the horizontal
line MFM'. In this case, the vertical action of the body
MFQ^ upon the plane M'FQ vanishes, as we may prove in
the following manner: Suppose the vertical Wnes M'N'P', MNP, to be drawn parallel to
FQ, the line M'N'P' being in the attracted plane, and the line MNP in the attracting
body ; this last line being situated either in the plane of the figure, or at any distance above
or below it. Then, through any points N, P, of the line MNP, draw the hnes NN', PP',
;-. An example of this
:-b;
jB'=_^b.
parallel to the horizontal plane, passing through the line MM' ; and if we consider the action
X. Suppl.2.]
ACTION OF A WEDGE ON A TRIANGLE.
855
the position of the attracting planes in the second method; whereas the [9976"]
of two equal particles of the attracting body situated at the points N, P, upon the two equal
particles of the attracted plane at the points N', P', we shall evidently perceive that the upward or
vertical action of the particle N upon P' is balanced by the downward action of the particle
P upon N' ; and as the same result holds good for all the particles, the vertical action of
MF(^ upon M'FQ must vanish. Now
The triangle QFJ5 = plane M'F Q — triangle M'FB.
And as the vertical action of the body MFQ^ upon the plane M'FQ^ vanishes, it is evident
that we shall have
The action of MFQ^ upon the triangle QFB = — the action of MFQ^ upon the triangle M'FB.
Adding to this the action of the part B'FM upon the triangle Q^FB, we get for the whole
vertical action of the body B'F^ upon the triangle Q^FB, the following expression, namely,
The action of B'Fq upon QjP£=the action of B'FM upon QJf 5 — action of MFQ upon M'FB.
We shall now compute the values of these two portions. . In the first place, ki finding the
action of B'FM upon Q,FB, we have
a = Q, a'=;— 90', b'=^ — b;
and then, from [99T6/(:], we get
u4 = — 45% B = 0, ^' = — (45"'— ^b), J5'== — ib.
Substituting these in [9976/J, we get
The action of B'FM upon QFB = g'.sin.b.tang.(45*'— ^b).
Id computing the action of MFQ^ upon M'FB, we have
a'=— 180% b'=— 90''-, b = 90%
and a must be changed into b ; hence, from [9976A:], we have
^ = — 90" -fib, S=0, ^'=-45% B'^ — (45'' — ih).
Substituting these in [9976»i], we get
The action of MFQ upon iW'FJ5= — ^.tang.(45'*- — ^b).
Substituting [9977/, h], in [9977<;j, and using [6, 34', 31], Int., we get successively
The action o^B'Fq upon qFB=^q. tang.(45''- — ^b) . ^ I + sin.b \
=^9.tang.(45''— ib).|2cos.2(45''— ib)|
=^2^ .sin.(45<'- — ^b) .cos.(45''— ib) =^ 5.sin.(90*'-^b)
==5^.cos.b.
We may apply this formula to the calculation of the action of the successive
concentrical lamina of a homogeneous fluid, elevated in a cylindrical capillary
tube, treated of by the author in [9902, &-c.}. For this purpose we shall
suppose BADM) fig. 153, to be the tube, whose vertical axis is OZ;
AZD the level of the suface of the fluid in the vase in which the tube is
dipped; ZA=;^ZD the inner radius of the tube; ZQ^ = ZC the inner
Mff.IS3,
JM
[9977a]
[99776]
[9977c]
[9977rf]
[9977c]
[9977e1
[9977/]
[9977/"]
[9977g]
[9977A]
[9977t ]
[9977*]
^eu Si CJQ
856 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
[9976'"] affinities seem to me to be the attractive forces themselves, acting with
[9977A;']
[9977Z]
[9977m]
[9977n]
[9977o]
[9977p]
[9977?]
[9977r]
[9977«]
[9977* ]
[9977u]
[9977t>]
[9977t)']
[9977«']
radius of the concentrical stratum, whose section by the plane of the figure is represented by
the parts ABFQ^ and DMEC. If we draw a tangent G'FG to the point F, and neglect,
as in [9926], the curvature of the sides of the tube, we may consider the fluid contained in
the part ABFQ^ as being limited by two planes, drawn through QF, FB, perpendicular
to the plane of the figure, and continued infinitely above and below it ; and if we put the
angle QFG=&, we shall have, from [9977i],
The vertical action of the external mass Q,FB upon the internal plane GFQ^C=sq.cos.b;
so that, if we suppose the thickness of this attracted plane to be dc, the action upon it will be
q.cos.b.dc. Integrating this relative to the whole circumference c of the stratum whose
radius is ZQ^, we get
The whole action of the external stratum upon the internal mass of the fluid FQ^CE 0=qc.cos.b.
At the side of the tube, or rather at an insensible distance from it, where b = zs [9892], this
action becomes qc. cos.-a; and by resubstituting q = ^H [9972e], it is reduced to the
following expression;
Vertical action of the annulus=^flc.cos.w,
being the same as the weight of the internal mass of the homogeneous fluid FQ^CE,
as in [9898].
It is easy to extend this demonstration to the case of nature, where the fluid, near its
surface, is covered with an excessively thin pellicle of variable density, whose upper surface
B'F'O'M', fig. 154, is so rare that its density may be considered
as nearly equal to nothing. Through all the points of this surface,
we shall suppose perpendiculars O'oO, F'fF, &ic. to be erected.
Then, taking, on any one of these perpendiculars, as, for example, on
the vertical axis O'oOZ, a point o, at an insensible distance below
the surface, but sufficiently far from it to have the density equal to
that of the internal mass of the fluid, which is represented by
unity ; we shall draw through o the surface bfom, cutting all the
perpendiculars O'oO, FfF, Sic, at right angles in o,f, &,c. In
Hke manner, we shall draw the surface BFOM, cutting the same perpendiculars at right
angles in O, F, he; this surface being at an insensible distance below bfom, but sufficiently
far from it to render the corpuscular actions of the fluid, which are actually situated in the
surfaces BFOM, bfom, wholly insensible upon each other. Taking any point F of the
surface BFOEM, and drawing through it the vertical line FQ parallel to the axis OZ, we
shall suppose the line i^Q to be at svch a distance from the side AB of the tube, that the
fluid near the tube, whose density is variable, can have no corpuscular action upon the fluid
which is situated beyond FQ^, or in the plane QFEC. Then the figure AB'F'FQ^, being
supposed to revolve about the axis O'Z, will form a figure, somewhat like an annulus, whose
action on the fluid contained within it will be exactly equal to tl)e weight of the internal mass
jPiy.lSA,
Mf
M
^ a
c JJ
X. Suppl.2.] ACTION OF A FLUID IN A TUBE. 857
all their energy, [9976""}
of the fluid Q^FF'O'E'EC, which is elevated above the level AD of the fluid in the vase .^^
where the tube is dipped. To prove this, we shall, for the sake of distinctness, denote the
different portions of the fluid, and of the plane, in the following manner :
iS^=the portion of the fluid formed by the revolution of the figure ABFQ^ about the [9977y]
axis O'Z;
S'=zthe portion of the fluid formed by the revolution of the figure BB'F'F about the [9977z]
axis O'Z',
P=the portion Q^FOEC of the attracted plane, whose thickness is supposed to be [9978a]
extremely small, and represented by unity ;
P' = the portion FF'O'E'EOF of the same attracted plane; [99786]
6 = the angle Q^FG, formed at the point F by the vertical line FQ, and the tangent [9978c]
FG to the section of the surface FOE.
In calculating the action of the annulus S-\-S' upon the attracted plane P-j-P', it is
convenient to take separately into consideration the four distinct portions. First, the action [9978d]
of S upon P; second, the action of S upon P'; third, the action of »S^' upon P; fourth,
the action of S' upon P'. In computing these attractions, we shall neglect, as in [9926], the [9978c]
curvature of the side of the tube, and consider it as being developed in a plane passing
through AB', perpendicular to the plane of the figure. Moreover we may consider the [9978/]
mass S, in its action upon P, as being of the uniform density unity, because the part of the
plane P which is nearest to the side of the tube AB, or nearest to the surface bfoc, Is
beyond the sphere of any sensible action upon Pj therefore we shall have the action of S
upon P by the same process as in [9977i], which gives
The vertical action of S upon P =3'. C0S.5. [9978A]
In calculating the action of 6" on P', we may, as in [9978/], consider the masses S and P' TaaJQ.'^
as being of the uniform density unity, because the part of (S near the side of the tube is
beyond the sphere of any sensible corpuscular action upon P'; and the same holds good
relative to the upper part of P', which is beyond the sphere of any corpuscular action of S. [9978i]
We may therefore calculate the action of S upon P' by either of the formulas [9976?, m].
Then, according to the notation in [9976i, Zc] , we have
a=&, b = 90''+J, a'=— 130% b'=:— &; ^^^^^^
A=^—W'-{-^b, P = 45% ^'= — (45''—^), P' = 0.
Substituting these in the value of Z [9976»i], it becomes — q.sm.b-, so that we shall
have
The vertical action of »S upon P' = — g'.sin.S. [9978m]
In like manner, in calculating the action of S' upon P, we may consider the masses S' and
P as being of the uniform density unity, because the part S' near the side of the tube, and [9978n]
VOL. IV. 215
858
THEORY OF CAPILLARY ATTRACTION.
[Mec. Cel.
[9977] We shall now resume the equation [9930], observing that, if, tvilh
[99780]
[9978o']
[9d78p]
[9978gl
[9978r]
[9978* ]
[9978* 1
[9978«]
[9978u]
[9978u>]
[9978x]
near its upper surface, is beyond the sphere of any sensible action upon F. Tlien, according
to the notation in [9976 J, Ar], we have
a=0, h = b, a' = — 6, h'=90'- — b;
A= — ib, B = 45% A' = 0, JB' = 45''— i6.
Substituting these in [9976/], we get
2r= g- . sin.6 . tang.jB = g- . sin.6 ;
so that we shall have
The vertical action of S' upon P=g'.sin.6.
Lastly, if we represent the direct action of S' upon P' by q^ , this direction being inclined to
the vertical by the angle b, we shall have, by resolving it in a vertical direction, the following
expression :
The vertical action of S' upon P'= q^.cos.b.
Adding together the four parts of this action [9978^, m, p, r], and neglecting the terms which
destroy each other, we finally get, for the vertical action of S-j-S' upon P-\-P'f the
following expression :
The vertical action of the external mass B'F'FQ^A upon the internal plane QFF'CyE'E C=(q-\-q,) -coaA
In computing that part of the mass of the fluid which is elevated by the capillary action,
and surrounded by the mass )S-}-5'; and whose section is represented by P-{-P', or, in the
figure 154, page 856, by Q^FF'O'E'EC; we may, on account of the insensible thickness of P'j
neglect wholly the consideration of this pellicle or mass of fluid of variable density near the
surface; and then, taking FOE for the limiting surface where the density becomes equal to
unity, we may consider the internal elevated mass to be limited by the space whose section is
FQ^CEO, being the same as in fig. 153, page 855 [9977o, &c.]. Now if we put, in like
manner as in [9977p],
?==|ff, q + q, = m>
and then compare together the expressions [9977m, 9918t], we shall find that the effect of
noticing the variation of density near the surface of the fluid, in computing the vertical action
of the mass S-\-S' upon the internal mass whose section isP-|-P', is merely to change
q into q-\-q^, or H into H [9978y], as in [9261e,/J ; and by making this change in [9977g],
we get
The vertical action of the annulus = ^Hc.cos.w,
being the same as the weight of the internal mass of the elevated fluid FQ^CEO [9898],
changing H into H, as above.
By the same process which we have used in computing the action of the external annulus
B'F'FqA, fig. 154, p. 856, upon the internal plane qPFO'E'EC [9918t], we may find the
action of the internal mass of fluid whose section is QFF'O'E'EC upon the external plane,
or upon the section of the annulus B'F'FQA. The only change will be to insert in the
X. Suppl.2.] ACTION OF A FLUID IN A TUBE. 859
a cylindrical tube whose internal radius is /, we put q for the mean [9977']
^KK" Q,
formula [9978^] the angle Q^FG' = w, instead of its supplement QFG^=5 = 180''- — w
hence we get
The vertical action of the internal mass Q^FFO'E'EC upon the external plane B'F'FQ^A={q-{-q,).coa.w
In these expressions of the vertical action, we have
supposed the line AQ^CD [9977a:] to be on the
horizontal level of the surface of the fluid in the vase
in which the tube is dipped; but the same formulas
hold good when we suppose the masses 5* and P to be
limited by any horizontal plane A'Q^'C, fig. 155,
parallel to -4Q, but much nearer to the surface of the
fluid, taking care, however, to have it sufficiently distant,
so that the corpuscular action of the surface BFO
shall not extend to the plane A'(^'. In this case, the
formula [99782;] will give
The vertical action of tlie internal mass C'Q'FFO'E'E upon the external plane B'FF^A' ={q-\-q^) . coa.w.
We may here remark that the whole mass of the fluid B'ADM' which is situated
above the level AD of the mass of fluid in the vase in which the tube is dipped, is in
equilibrium by the mutual action and reaction of the fluid and the tube ; and we shall now
proceed to show how we may make use of this circumstance to investigate the relation
between the angle w and the forces which act on the fluid in the plane B'A'Q'FF', which,
for brevity, we shall represent by N; supposing the thickness of this plane to be infinitely
small) and represented by unity. The forces acting on the plane N, supposing them all to be
resolved in a vertical or upward direction^ are the following :
First. The upward action of the part of the tube near B'A\ which is within the sphere
of the corpuscular action on the plane N, This force we shall represent by T.
Second. The upward action of the mass S-\-S' contained in the space which is formed by
the revolution of the plane N, or B'A'^'FF', about the axis O'Z. This force
we shall represent by S.
Third. The upward action' of the mass S^, which is included in the space formed by the
revolution of the plane AA'^^' about the axis O'Z. This force we shall
represent by -cj^.
Fourth. The upward action of the mass P^, formed by the revolution of the plane
C'Q'Q Z about the axis O'Z. This force we shall denote by P^.
Fifth. The upward action of the internal mass P-f-P', formed by the revolution of the
plane C'Q^'FF'O' about the axis O'Z. The value of this action has been
computed in [99796], and found to be equal to {q -{-q^). cos. lo, or ^H.cos.m;.
The sum of these five forces must be put equal to nothing, upon the principle adopted ia
[9978y]
[9978z I
[9979a]
[99796]
[9979c]
[9979rf]
[9979«]
[9979/]
[9979g-]
[9979A]
[9979i )
[9979ft]
[9979/)]
860 THEORY OF CAPILLARY ATTRACTION. [M6c. Cel.
[9977"] height to which the fluid rises above the level, the volume of the
[9979c], namely, that there is an equilibrium in the actions of the fluid and tube upon any
portion S-{-S' of the mass of the fluid; hence we have
[9979^] = T-j-S-f c<^ + P, + (5' + ^J.cos.M?;
and it now remains to compute the values of T, S, P, observing that zs^ cannot be determined,
because the law of the corpuscular action on the part of the fluid of variable density near
the side of the tube is unknown.
[9979m] If we suppose the plane N to be divided into filaments parallel to -4' Q', or perpendicular
to the side of the tube, the vertical action of the part of the tube which is situated above
any filament, will evidently be balanced by the contrary action of the part of the tube situated
below the same filament; therefore the sum of all these actions will give
[9979n] T=0.
In computing the force S, we shall suppose the plane N to be divided into vertical filaments
Jcl, JcT, &c., and which, at the points /, /', &c., are bent, in the plane of the figure, into
[9979o] the directions IH, TH', Uc, perpendicular to the curve BIFF, at the points /, /', &c.,
respectively. If this vertical plane, with its filaments, revolve about the axis O'Z, it will
divide the whole space S-\-S' [9979^] into similar and equal filaments. To distinguish the
plane and filaments in their new positions, we shall suppose that, after the plane iV has
revolved about the axis O'Z, by any given angle a, it is denoted by {N) in its new position;
and in like manner the filament klH changes into (klH), h'TH' into {Jc'I'H'), &c.
Then it is evident that, as the filaments JcIH, {1c TH), are at the same distance from the side
of the tube, they must have the same density, and be similar and equal to each other; and
[9979^] tj^jg jjolds good even for the filaments which are extremely near to the tube, supposing
always that their bases are infinitely small and equal to each other ; similar remarks may be
made relative to any other corresponding filaments, as Jc'I'H', {Tc'I'H'), &,c. As the action
of the plane (iV) upon the plane iV must be insensible when their distance is of any sensible
•■^l magnitude, the angle a may be considered as excessively small ; so that, without any sensible
error, we may consider the planes iV, (-/V), as parallel, and the side of the lube corresponding
to A'B' as a plane, as is supposed by the author in [9926, &tc.]. Then, on account of the
symmetry of the situation of the four filaments TcIH (JcIH), Jc'I'H {Jc'I'H), and having,
moreover, the mass JcIH=maiSS {JcIH), the mass Jc' TH=\.\\e mass {Jc'I'H), we shall
have, by considering only their vertical actions upon each other,
[9979r ] Action of {JcIH) upon fc '/'!?'= action of JcIH upon {Jc'FH),
[9979* ] = — action of {JcTH) upon JcIH;
observing that the equation [9979r] can be demonstrated by a method similar to that which is
used in [9976y, z, he], and that the expression [9979;?] can be derived from the second
[9979< ] member of [9979r], by considering that the action of JcIH upon (JcTH) is equal and
contrary to the action of {Jc'I'H') upon JcIH. Transposing the expression [99795] to its
first member, we get
[9979U] ^^jiQ^ of {JcIH) upon Ar'J'H' -faction of {Jc'I'H) upon JcIH=0.
X, Sappl. 2.] ANGLE OF CONTACT « NEAR THE SIDE OF A TUBE. 861
elevated mass will be iiFq [as in 9985a], and the circumference c of [9978]
COS.t^ = — ; — = 77 .
9 + ?/ ' H
[9979»]
[99791/]
From this equation it follows, that the vertical action of the two filaments {klH), (JcTH'),
of the plane (iV), upon the two corresponding filaments klH, k'l'H', of the plane N,
mutually destroy each other; and, as this must hold good for the filaments into which the
plane N or (iV) can be divided, the whole sum of these actions will give