moreover, the differential of [10266] gives, by observing that dy is constant [10264r],
ddz dvs
substituting these in [10264], we get [10267]. Multiplying the first term of [10267] by
dy.tang.w, and the other terms of that equation by the equivalent expression — dz [10266],
we get
[10268c] — <?«.sin.« + %4^ = - 2az£Zz,
i-\-y
whose integral gives [10268]. When z = 0, the angle to corresponding to the point iW,
fig. 168, 169, page 948, will be «=0, and the integral f ' ' then commences
[10268e] [10269]; so that the equation [10268] becomes, at that point, l = constant. Substituting
this in [10268], we easily obtain the value of az^ [10271].
[10268rf]
X. Suppl. 2.] ADHESION OF A PLATE TO THE SURFACE OF A FLUID.
951
When the disk is very large, / will be great in comparison with* -j=\ thus we [10272]
shall have a first approximate value of z, by neglecting the integral
— / -YT — ' ^^ ^^® equation [10271], which gives
rApproximate valueof I. I [10^73]
V/2 . ,
V/a
dz.s'm.zi
Then we may substitute this value of z, in the integral — /
by this means becomes f
l+y V o-i/o l-\-y
this integral is easily reduced to the following form,
— / —Tx = — .,;, , , .?! — cos/^w] — 2 « Xf /
J l-\-y 3(/4-y) ^ ^ ^ ^ V ^J ^
, which [10273']
[10274]
[10275]
[10272a]
* (4372) While zs varies from 0^ to 48^ [10264/], if the fluid be mercury ; or from
Trf = 0° to w=200° [10264A], if the fl.uid be water; the ordinate z will vary from z=0,
to z = EA, its greatest value; and if Z be very large, we may consider the integral
/dz.sin.zi . r i , AE AE . . . ., ,
—T- , as not exceeding a quantity of the order — -, or -—=, which must evidently [102726]
i-\-y I AF
be much less than unity, from the nature of the capillary action ; so that this term must, in
general, be much less than either of the terms 1,cos.ot, which occur in [10271] ; and if we retain
only these two terms, we shall have nearly a2;2 = l — cos.zs = '2,s\xi.^^zs [1], Int. Dividing
this by a, and extracting the square root, we get z [10273], which is of the order
1
y- , mentioned in [10272], and is much less than I, as is observed in [10272], and is also
[10272c ]
[10272c']
evident from the experiments which are mentioned in [10317,10333, &;c.]. If we substitute
a-4=a [9323^] in [10273], it will become
2; = l/2.a.sin.jCT. ' [Approximate value of z.] [10272i]
t (4373) The differential of [10273] gives dz^^^.^disi.cos.^vs', substituting this in [10274a]
^/a
[10273'], and then putting sin.'ztf=2sin.jCT.cos4zs [31], Int., we get [10274]. The [102746]
integral of [10274] can easily be reduced to the form [10275] ; their differentials being equal
to each other, as is easily perceived by observing that the differential of the last term of [10274c ]
[10275] is destroyed by the part of the differential of the first term depending on the
differential of l-\-y, as is evident by inspection ; lastly, the differential of the first term [10274rf]
depending on cos.^^w, produces the differential of [10274].
952
THEORY OF CAPILLARY ATTRACTION.
[Mec. Cel.
[10276]
[10277]
[10278]
[10278']
The element of this last integral is never infinite ; for, although ^ becomes
U'SJ
infinite, when « is nothing, because we have*
dy dz. cos.-a 1 cos.-a
d^a dTS.sin.zi 2^/2a*sin.^'5j'
yet, as it is multiplied, in the preceding integral [10275], by — d^.(l — cos.^^w), the
coefficient of d^s in this product is never infinite. If we neglect the term divided
by (l-{-yy, in comparison with that which is divided by l-^-y^ we shall have
''dz.s'm.zi 2^2.(1 — cos.^'a)
-/:
The integral must be taken from 73 = to vs = nf — w', supposing thatf
[10276a]
[10276o']
[102766]
[10276c ]
[10276c']
[10276i]
[10276e ]
[10276/]
* (4374) We have, in [10266],
1
^y=-^'-vi^ ''^•i5:5=-(^-J''«-o<»-i«)-iSS [10274a];
now, substituting s\n.zs=2s\n.\-ui .cos.^iHi and dividing by d-a, we get
dy 1 cos.zs
d-a 2^2a*8in.iCT'
as in [10276], The value of dy, deduced from this expression, being substituted in the
part of the second member of [10275], under the sign/, gives
/dy.[\ — cos.s^ra) 1 /•rfra.cos.'5i.{l — cos.3^'5J)
[l+yf "^"~V^*J [l^y)^.Bm.hzi *
Now the factor (1 — cos.^Jts) is equal to
(1 — cos.Jzs) . (1 -|- cos.Jot-}-cos.2J73) = 2sin.2J-cy.(l+cos.|'ztf+cos.2J'5j),
and
sin.i^'5y=2sin.^«.cos.:j7rf [1,31], Int.;
hence, by substitution in [102766], it becomes
2^2a;/(Z
d-m
; . cos.w . ( 1 -}- cos .^•sj -f- cos.^^ct) . tang-ij-cr ;
and as zi never exceeds 200° [10272a], the coefficient of dvi must always be finite.
IVIoreover, as the elements of this integral are divided by the great quantity il-\-yY, it is
evident that the whole integral is very small; and by neglecting it, we obtain [10278].
t (4375) Substituting [10278] in [10271], then changing z into z' [10279'] and
[10280o] zi into * — vi' [10278'], so as to correspond to the upper point F of the curve,
fig. 168, 169, page 948, where we have the angle AFf'^=-ts', it becomes as in
[102806] [10280]; observing that, at this point, the ordinate l-\-y becomes simply equal to /,
because the fluid is supposed to terminate at the circular border of the lower surface of
[102806'] the disk. Now, dividing [10280] by a, substituting l4-cos.'5*'=2cos.2i'5j' [6], Int.,
X.Suppl.2.] ADHESION OF A PLATE TO THE SURFACE OF A FLUID.
953
•a' = the angle which is formed, in the plane of the generating section of the
fluid, by two lines drawn through the upper point of contact of the fluid
with the plane, in directions respectively parallel to the tangents of the
surface of the fluid, and of the surface of the attracting solid, at an
insensible distance from that point of contact; both these tangents
being drawn in a downward direction, or towards the interior part ot
the fluid;
z'=z the extreme value of z, or the whole height of the column EA, which
is raised by the disk.
[10279]
[10279']
[10280d]
[10280rf']
and then extracting the square root, we get z', as in [10281], neglecting the third and
higher powers of a~^
We must not use the value zi='r( — zs' [10280a], for the upper limit of the surface, if [10280c]
the fluid rises ever so little on the outer rim FOP<^ of the disk, fig. 170, 171. This will
be evident hy considering the simple case^
where a glass disk, whose figure is that of a '-
right cylinder with a circular base, is dipped
into a vessel of water, to a considerable
depihj and then gradually raised up in a
vertical direction; taking care that the opera-
tion is performed with sufficient steadiness to
enable the fluid on the surface to adjust itself
to its successive states of equilibrium. In
order to form a more distinct idea of the figure
of the surface of the fluid, in the difierent
situations of the cylinder, we shall suppose,
as in [10012m — a], that the rim or intersec-
tion of the base of the cylinder with its
curved surface, is rounded off in the form of
a quadrant of a circle FOP, whose radius
is extremely small. The cylinder being well
moistened, the fluid will ascend on the vertical sides PQ, TU, of the cylinder, fig. 170, and
will adhere, so as to form with them, at the upper part of the fluid, the angle zs'. In this
case, the general value of the angle DKlc^-a will become, at the side of the cylinder,
zs = ^'r; and it will retain the same value, while the cylinder is gradually raised up in a
vertical direction, until the upper part of the fluid has descended along the side QP to the
point P. When the fluid has attained to the point P, the angle zi, corresponding to its
summit, will increase by passing over the curved surface POF, as the cylinder is slowly
elevated; so that, when this summit is at the point F, fig. 171, where the angle AFf'==-a':=:0
[10279, lO^SOd, 9359k], we shall have â– a=iAFf=.-r. Hence it appears that, while the
summit of the fluid descends through the quadrantal arc POF, the angle zi increases firom
VOL. IV. 239
[10280c ]
[10280/]
[10280g-]
[10280g^]
[10280A]
[10280i ]
954
THEORY OF CAPILLARY ATTRACTION.
[M^c. Cel.
Then the equation [10271] will give
[10280] 0.Z
' II COG.- 2/2.n-sin.VL
[10280A]
[102804']
[10280? ]
[10280Z']
[10280m]
[10280n]
[10280o ]
[10280;)]
[10280pq
[10280g ]
[10280g']
[10280r ]
[10280/]
[10280* ]
[10280f ]
[10280u]
[10280t) ]
TO = ^* to zi = TT, the whole increment being ^*, or a quadrantal arc. During the whole
of this descent along the arc POF, the angle -us', which is fornned by the upper part of the
section of the fluid and the section of the cylinder at the same point, remains invariable, and,
in the present hypothesis, is w' = [10280i]; so that, if we put i for the acute angle which
a vertical line, drawn through any point O of the quadrantal arc POF, forms with the arc
at O, we may put generally â– ^s = ^*-|-^, for the value of trf, corresponding to the upper
limit of the surface of the fluid, when it is at the point O. The values of the angle cr,
corresponding to the points P, O, F, which we have found to be ^tt, J-*-(-i, t [I0280Ar, Z'],
respectively, have been computed upon the supposition that the angle zs' = [10280^] ; but
if this angle has a finite value, the preceding expressions of to [10280wi] must be decreased
by to', and then they will become respectively ^t — to', 5-*-|-* — w', ir — th' ; so that we
shall have for the extreme limit of â– ttf [10280/'], corresponding to any point O of the arc
POF, the following expression;
•c:r = ^*-|-i — to';
in which i represents the angle corresponding to the circular arc PO, its least value being
i = 0, its greatest value *=|7r [10280»i]. This extreme limit of the value to is to be
used instead of 7s:^=7r — to', given by the author in [10278'] ; and the effect of making this
alteration is equivalent to that of changing his value of to', in [10280p'], into to'-j-|« — i;
since, by this means, the expression TO = 5r — to' [10280^'] changes into
^=a* (to'-}-!* *)=|T-{-i to',
as in [10280o]. Making the same change in the value of to;' in the formulas [10280, 10281],
they become respectively
az^ = l-^-sin.(^-TO')-V^•^l-^^° f(^^^+^— ^^)^
s' = \ /^.cos.(Jto' + |* — Ji)
0^ 3/a.cos.(^TO'-[-43- — lii)
These expressions may be simplified by putting
i = TO' — ^'T!'-\-'2v, or v = ^ir-{-^i — ^to',
from which we get i — to' = — (|ir — 2v), ^to'-j-:|* — ii = ^'!r — v. Substituting these in
[10280r], and then multiplying by (xr^ = a^ [9323j3], we get, by observing that
l-f-sin.(i — to') becomes 1 — sin.(^'ff — 2u) = l — cos.2u = 2sin.% [1], Int.,
z ' 2= 2a2 . sin.2 V — ?V:^1^\ ( I _ cos.3y) .
This expression is the same as the equation (6) in page 232 of M. Poisson's Nouvelle
Theorie, ^c, neglecting terms of the order a^. If we neglect terms of the order a^, in
[10280m], and extract the square root, we shall get very nearly
X. Suppl. 2.] ADHESION OF A PLATE TO THE SURFACE OF A FLUID.
955
whence we obtain very nearly
2 , (1 — sin.^^d')
-.cos.A^ — -hrj 7-r
a 3/oL.cos.Jra
jHeight of the elevated column. | [10281]
z'=/2.a.sin.'U = ^2.a.sin.(Jir + |j: — |z3') [10280s].
From this last expression of z', it is evident that, while * increases, from i = to its greatest
value « = 2^ [10280p], the value of 2;' will increase, and that z' will attain its maximum
when i = ^'T(, and then «'= ^2.a.sin.(J'jr — J'5j')=\/2.a.cos.J'3i'; which is the case
treated of by La Place in [10280, &z,c]. This maximum value corresponds to the state
where the summit of the fluid is situated on the lowest point J?" of the arc POF, as in fig. 171,
page 953. If we put â– 5j' = 0j as in the case of water with a glass disk [10280iJ, this
maximum value of z' will become z'=^2-a' In the same case of w'=0, if we put
t = 0, we shall have, from [10280^], v=^^if, whence sin.v=^^, and then the expression
of z' [10280zi;] becomes z' = a; this corresponds to the case where the summit of the
fluid ascends to the point P, and agrees with the calculation in [9425^]. Hence it appears,
that, when zi'=0, the value of z' increases with the angle i, from the upper point P,
where i = 0, and z' = a [10281a], to the lower point F, where i = ^*, and z' = ^.a
[IO28O2;], being its greatest value. This maximum value of z' must correspond very nearly
to the situation of the disk when the greatest quantity of fluid is elevated, the radius of the
disk being supposed much greater than z' or a; and it being evident that this volume would
be decreased, if the outer limit of the fluid were removed from F towards A, on the lower
surface of the disk.
In the case we have just considered, where water is elevated by a glass disk or cylinder,
the external surface of the fluid MKP, fig. 170, 171, page 953, is always concave, and the
lower part of the surface, near Mor L, has the level line of the fluid in the vase MEL for
Its asymptote. As the summit of the fluid descends along the arc POF, in consequence of the
gradual elevation of the disk, the external surface of the fluid becomes grooved in fig. 171,
at the point V, where the ordinate YV, or .the distance of the surface from the axis AE, is a
minimum, and the tangent to the curve is vertical, or is = ^'K. The figure of the surface
FVM may be very easily investigated by means of the equations [10271, 10266]; and if
we neglect the last term of [10271], on account of its smallness, we shall have, as in
[10272f^J,
2: = ^2.a.sin.J'S3'.
At the point F, where «^* — ot'=w; [10280c, 9980/ft'j, it becomes E A^=\/^ . a.sm.^w,
and at the point F, where -a^^ir [10281^], or ^2'Sin.^-a==l, we get EY=a [10281i].
The difference of these two expressions is the depression of the point V below the lower
surface of the disk, namely,
v4 r=a.|\/2.sin.iOT — 1^.
We have, from [34, 31, 1], Int., =-: — =^r^—. r") substituting this, and the
•• ' •" tang.OT sm.'rt 2sin,^w.cos.it3 °
differential of [10281iJ, dz= -y^.d-a.cos.^zi, in [102666], we get
[10280«;]
[10280a; ]
[102801/ ]
[IO28O3/']
[102802 ]
[10281a]
[102816]
[10281c ]
[10281rf]
[]0281e]
[10281/]
[1028]g-]
[10281/1 ]
[10281i ]
[10281ft ]
[10281/ ]
[10281m]
[1028 In]
956 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
To obtain the volume of the whole column raised up, we must first multiply
[10282] this value of 2' by the lower surface of the disk, or by */^, and then add to
, dz /«•. 1 \ / 1 — 2sin.2^isr \ a lidtir , a , . ,
[IO28I0] dy = — = — [ -r- ' dzF. COS. izs- ].[-——, — == 7^'~^~r 4-—7='dzT.SlQ.izr,
â– â– â– â– ^ tang.ar \^2 ) \2sm.!|o'.cos.iar/ ^2 sin.iw ' /2
whose integral is
[10281;? ] y=— r^ •/ ^iii + 1^ '-^^"^ • sin.Jar=— -^ . hyp.log.tang.Jzr-y2 • « • cos. Jar-fconstant j
observing that, from [59, 54, 31], Int., we have
, ,, y 1 N rf. tariff.^ or i(/tir.{C0S4ra-)~^ ^rfo" ic?o>'
[10281g] f^.(hyp.log.tang.i^)= ^^^^^^^ = ^^^^^^^ == 2sin4^.cos.i.. = sna^-
The constant quantity in [10281^] must be so taken that, at the point JP, fig. 171, page 953,
where w = t^ [10281A:], we may have y=0; hence we get
[10281r] constant = -^.hyp.log.tang.|tf>-|-^.a.cos.^t^;
substituting this in [10281p], we obtain the general value of y, in the following form ;
a , , tang.4t« , ._ , , , .
[10281s ] y^'U^' ^^'P-'°S-toi^ + v/2 . a . (cos.iw; — cos.^^r) .
At the point F, where ^â– sr = ^'n', we have sin.^z3- = cos.jOT- = i/^; consequently
[102810 tang.i>=' = f:p^^ [41], Int., becomes tang.^^ = -^_= ^_^ Substituting these
in [10281s], we get the value of y corresponding to the point P, namely,
[10281U] y=-^.hyp.log.|(l-f ^2).tang.^w;^+a.f/2.cos.^w; — Ij.
In the case of water with a glass cylinder, we have ot'= [10280i], and z^ = * [10281Z;] ;
[10281j;] hence sin.^w = l, cos.^tf? = 0, tang.|t(; = l; substituting these in [10281»i, m], we get
very nearly
[1028111^] ^Y=a.(/2 — l) = aX0,4142.
[10281X] y = ^.hyp.log.(l + ^/^) — a = ~aX0,3768.
If we suppose, as in [9372o], that a = 3'"S88972, we shall get AY=l"'\6ll;
y = — 1"" ,466 ; which correspond to the case where the radius of the disk or cylinder is
excessively large in comparison with a. The expressions [1028b', m, s, u, w, x] agree with
those of M. Poisson, in page 230 of his Nouvelle Theorie, Sfc, by making some slight
reductions.
The supposition that the lower edge of the surface of the cylinder is terminated by the
small circular arc POF, may be applied when the external surface of the fluid is convex, as
in the case of mercury with a wet glass cylinder. In this case, we shall suppose that the
'â– *-' mercury forms the acute angle w with the vertical side PQ of the cylinder, when the upper
surface of the fluid rests against the point P ; and if we slowly raise up the cylinder in a
X. Suppl. 2.] ADHESION OF A PLATE TO THE SURFACE OF A FLUID.
957
this product the volume which surrounds the fluid cylinder whose base is the
lower surface of the disk. This last volume is equal to the integral*
— 2'^f(l-{-y).zdy^ taken from ct = to TO = 'jr — w'; thus we shall have, for
the expression of the whole volume V of the elevated column,
r=w^\/ f.cos.i^'-
Sa.cos.i^ra'
'^^J(l^y)>zdy,
[Volume of the"]
elevated fluid.J
We may rigorously determine this last integral in the following manner.
The equation [10267], being multiplied by — (^ + 2/) '^2/5 gives, by
integration, t
— 2a ./(/ + y) . zdy = (/ -f y) . sin.w + constant.
To determine the constant quantity, we shall observe that the integral must be
taken between the limits w = and w = «' — w' [10283]; and we shall now
show that (/ + 7/).sin.ra is nothing when i3 = 0. For l-\-y becomes infinite
when OT is nothing; therefore, by reducing its expression into a series ascending
according to the powers of to, the first term of this series will be of the form
A.Ts~^. Moreover, z being nothing when ^ = 0, if we also reduce its value
into a series ascending according to the powers of zs, the first term of it will be
of the form ^'.ra'', r and r' being positive. Then the equation
[10283]
[10284]
[10285]
[10286]
[10287]
[10288]
vertical direction, so as to bring the point O, which corresponds to the angle i, against the
upper surface of the fluid, its inclination with the vertical will be i-\- w ; and when this is
equal to J-ir, the fluid will become horizontal, and upon a level with the surface of the fluid [10282c ]
in the vessel. Continuing still to elevate the cylinder, so that the upper part of the fluid may
fall upon a point between O and JP, the angle i-\-w will exceed ^it, and the surface of the [l0282rf]
fluid near that point will become concave.
* (4376) Continue the line kl, fig. 168 or 169, page 948, to meet ME in c; then the
area CKkc=CKxKl== — zdy; multiplying this by the circumference described by the [10283a]
point K, in its revolution about the axis AE, it becomes — 2'jr.{l-{-y).zdy, which [102836]
represents the element of the volume of the fluid in question. Its integral is as in [10283] ; [10283c]
its limits being from the point M, where «==0, to the point F, where •« = * — -a' [10280a]. [10283rf]
Adding this to the quantity ifPsf [10282], we get the whole volume V o( the fluid, namely, [10283e]
V=^Pz' — 2'jr.f{l-\-y).zdy; and by substituting the value of «' [10281], it becomes as in [10283/]
[10284]. We may remark that, for convenience, we have inserted the symbol V in [10284], [10283g]
but it is not in the original work.
t (4377) This product is
integral gives [10285].
VOL. IV.
d-a. cos.-a. {I -\- y) -\- dy. s'm. -a =— 2a.. {I -\-y). zdy, whose [10285a]
240
958
THEORY OF CAPILLARY ATTRACTION.
[Mec. Cel.
[10289]
[10290]
[10291]
[10292]
[10293]
dz
dy
= — tang.w [10266]
will give, by noticing only these first terms, and observing that tang.w becomes
nearly equal to « when it is very small,*
r'.A'.
r.A.-a-
-a:
whence we deduce, by the comparison of the exponents of -a,
1 — r = r';
(Z+2/).sin.'« will therefore become f A.x/, by substituting A.-ss"^ for l-{-y,
and -a for sin.«; so that (l+y^.sm.-a is nothing when « = 0; consequently
[10291a]
[102916]
[10291c ]
* (4378) We have, in [10287, 10288], /+y = ^.-a-'4-&c.; z = A'.'sf'-\-hc.i
— tang.'5j== — -a — he. [45], Int. Substituting these in [10289], it becomes
r'. ^'. «'•'-' rft3-f&z;c. „
! = — -Ef — &c. ;
— r.A. vs~^~^d-a -f- &.C.
and, as the first term of this equation must be the same in both members of the equation, we
shall have
r'.A'
'd-a
•vi\
r'.A'
— r.A.T;i~''~^dvs
whence we easily deduce [10291]. This may also be put under the form '— ^.'bj'''+''=zj;
[10291<i] and, to make the exponents of -a equal to each other, we must put r'+r = l, as in
[10292].
f (4379) The first term of Z-j-y [10291a] is A.vs-% and the first term of sin.-rtf is zt
[10293a] [43], Int. ; therefore the first term of their product (Z-j-y).sin.'5j is A.T!i^~''=A.'Bf'
[10292]; and as r' is positive [10288], this will vanish when t3=:0; consequently the
second member of the expression [10285] must give the constant quantity equal to nothing,
[102936] and we shall have — 2<3..f[l-\-y).zdy=(l-{-y).s\v\.vs. At the second limit, where «=* — -a'
[10286], and /+y = Z [102806], corresponding to the point F, fig. 168, 169, page 948,
[10293c] the integral becomes — 2o.f{l-\-y).zdy = l.s\n.'si'. IVIultiplying this by - , we get [10294],
[10293c'] and by putting sin.w'=2sin.^Bj'.cos.^'5i' [31], Int., it becomes
[10293<f] - 2*./(Z + T/).;?(Zy = -£.Z.sln.^H-'.cos.^z5'.
Instead of restricting the calculation to the particular value of «! = «' — w', which is used by
[i0293e] \^^ Place in [10278', &,c.], we may use the more general expression ■5J = ^ii'-}~* — "^
[I0280o], which requires that we should change w' into -5^' + ^* — * [10280^], in
La Place's formulas; and by this means the expression [10294] becomes
X. Suppl.2.] ADHESION OF A PLATE TO THE SURFACE OF A FLUID.
959
the constant term of the preceding integral is nothing. We shall therefore
have, by observing that, at the other limit of the integral, w becomes * — w', and
y = [102806],
_2^.y; - '(Z + 3/).zJy = ^./.sin.^'; [10294]
consequently the whole volume of the elevated column will be *
Substituting this in the expression of the volume F" [10283/], it becomes
F==*Z^«' + ^.Z.COS.(t— isr').
Multiplying this by the gravity g, and the density D, we get the following expression of the
mass M of the fluid, which is elevated by the disk ;
M=ffgD.l^z'-]-l.gD.Lcos.{i—z>r').
This may be reduced to the same form as the equation (1) in page 226 of M. Polsson's
Nouvelle Theorie, ^c, by changing M, I, z', D, ■&', a, into A, r, Jc, p, v — w, a~^,
respectively, to conform to his notation. In the case treated of by La Place, where i = ^t
[10280y'], the value of V [10293^] becomes
V=^Pz'-Jrl.Ls\n.vr',
which, by substituting the values of z', sin.*' [10281, 10293c'], becomes as in [10295a]; or,
by reduction, as in [10295], corresponding to the case where the summit of the fluid is
situated at F, fig. 171, page 953, on the lowest point of the quadrantal arc POF. If i=0,
and cr'=0, the expression [10293^'] becomes V^Trf^z' -\ - A\ and in this case, we have
z' = a. [10281a], also a=a~2 [10280^]; therefore this value of F becomes
The first term of the second member of this expression vPa. is equal to the volume of the
fluid immediately under the disk, which is in the form of a cylinder whose altitude Is z'=a,
and the radius of Its base equal to Z; the second term ttIo? Is equal to the volume of the
elevated fluid which Is situated without this cylinder, being the same as that we have already
computed in [lOlOTi] ; and these two expressions ought evidently to be equal to each other,
because the action of the fluid cylinder [10293^] upon the external mass [10293r], is exactly
the same as that of a glass tube of the same dimensions well moistened [9220] .
[10293/]
[10293g]
[10293A]
[10293i ]
[10293ft]
[10293i ]
[10293m]
[10293n]
[10293o]
[10293p]
[10293?]
[10293r ]
[10293* ]
* (4380) Substituting [10293c?] in [10284], it becomes
jjt^ 'jtI?. s/ 2. cos.W .
\\ — sin.^^:a' — Gsin.^-a'.cos.s^-s'j ;
^/g^ 3a . cos.i'zs'
and, by putting cos.^J«'=l — sln.^^a', in the last term, it becomes as in [10295].
[10295a]