8/ being the distance of the two planes [9452]. Now by [9526], the inclination of the two
796 THEORY OF CAPILLARY ATTRACTION. [M6c. Cel.
In the preceding experiment, the planes were distant from each other -j\ of an
English inch, at 20 inches distance from their line of intersection [9691, 9689],
or at their axis situated at the extremity of these planes. Therefore their distance
[9699'] from each other was only -^^^ of an inch, at the distance of 10 inches; and we
[9700] shall suppose 6 [9696] to be equal to ten English inches. As a half millimetre
of distance between two vertical and parallel planes, corresponds, by the
[9701] preceding article, to an elevation of the oil of orange of 6'°'-,7389,* we shall
have
[^02] (0,5x6,7389) square millimetres =(-gLx A) square inches.
[96986] planes is 2o; therefore the distance of these planes from each other, at a point whose distance
from their common section is b, will be nearly 2Z=26.tang.ra, on account of the smallness
H' sin.rt'
of the angle ra. Substituting this in the preceding value of h, it becomes h =
[9698c]
g.2b. tang.'O
[9698rf] IMultiplying this by — , we get — =— — ^ — '- ; hence, by [9549], g=sin.F', or,
as it may be written, sin.F^=-.— [9698]. In this demonstration, we might have used
[9698c] sin.« instead of tang.-s, by measuring the half interval of the planes, from the centre of the
drop to either of the planes which are used in the experiment ; then the formula [9549/J
would agree with [9698].
[9702a]
* (4236) The elevation of any fluid in a tube whose diameter is I"'", or radius 0"^,5, is
the same as between two parallel planes whose distance is 0â„¢-,5 [9410]. Now by [9670'],
[97026] the elevation of oil of oranges in a tube of the diameter 1â„¢, is 6â„¢,7389; therefore, by the
[9702c] principle used in [9665, &;c.], we must have O'^jS X 6""-,7389, for the constant quantity
[9702rf] representing the product of the distance of two parallel planes, by the elevation h of the oil
of oranges between the two planes. To reduce the constant quantity [9702c] from square
millimetres to square inches of English measure, we must divide it by the square of
[9702e] 25""-,3918, and then the constant quantity becomes as in the first member of [9704], which
is to be put equal to the product of the distance of the planes, ^'j of an English inch
[9699'], by the corresponding elevation of the fluid h', hence we get the equation [9704],
from which we deduce the value of h [9705]. Substituting this value of h, and that of
[9705^] 5 = 10- [9700], in [9698], we get [9706].
The quantity 6â„¢-,7389 [9702c], deduced from the experiments of Haiiy, is much too
small, from his not having well moistened the tubes. In an experiment made by Gay-Lussac,
the temperature was 15°,5 of the centigrade thermometer, the diameter of the tube
[9702t] 1^.^296 = 2/ [9343], and the elevation of the fluid 9= 10â„¢, 4 [9353]. From these we
[9702A] get 5,= 9 + iZ=10â„¢-,616 [9372a:], and ^ = 2a2 = 2/^, = 1 3'^-,758 [9372y], instead of
6'"',7389 found by Haiiy [9670'] ; and if we use this result in [9706], we shall have
[9702A]
X. Suppl. 1. -^ 14.] EXPERIMENTS WITH INCLINED PLANES. 797
h being estimated in English inches. We have seen, in the preceding article,
that the English inch contains 25'"S3918 [9678]; therefore we shall have ^^^^^
0,5x6,7389 1
(25 3918'\2 "^ 32 ' [Expressed in square inches.] [9704]
which gives
, 16X6,7389
^=(25,3918)^- ' [munearinohes.] [9705]
Hence the formula [9698] becomes
. j^ 16x6,7389 100 1,6723
sm.F = -. =:—
10 X (25,3918)2 a2 ^2
[9706]
a being estimated in English inches.
The angle formed by the two glass planes, in the experiment, having for its
sme jgTT^j this angle is 10â„¢- 44'-. The lower plane having been placed [9707]
horizontally at the commencement of the experiment, it is evident that, to
obtain the inclination V of the intermediate plane, we must decrease by 5'"' 22'-
all the inclinations of Hauksbee's table. We must then subtract all the
numbers of that table from 20'"-, to obtain the successive values of a. This
being premised, we shall have the following table :
[9708]
. „ 16X13,758 100 3,4142
This formula makes the sines of F^more than double of those which are deduced from the third
column of the table [9709] from the formula [9706], the ratio of their sines being as [9702m]
1,6723 to 3,4142, or as 1 to 2,041 ; and as the computed angles in the table are generally
too large, it must follow that the results of Hauksbee's experiments must differ very much [9702n]
from those derived from Gay-Lussac's experiment in [9702?]. These differences arise chiefly
from the difBculty of making accurate observations of this kind, and in part from the terras [97020]
neglected in the expression of sin. F [9549, &c.] ; and we may add, that we have supposed, in
the theory, that the drop is nearly circular, whereas by observation it is found to be oblong when
a is small. We may finally observe that, instead of [9702Z], Poisson uses, in page 260 of
his work, sin.F= ' „ , as the result of Gay-Lussac's experiment ; the difference between [9702p]
these formulas arises chiefly from his deducing the value of the numerator 3,4117 from the rg^Qgoi
formula [9372/J, instead of using that in [9372y]; but the difference is of no importance in
comparison with the much greater errors of the observation.
VOL. IV. 200
798
THEORY OF CAPILLARY ATTRACTION.
[M6c. Cel.
[9709]
[9710]
[9711]
[9712]
[9713]
Distances a, in inches, from the
middle of the drop to the in-
tersection of the planes.
18"
16
14
12
10
8
6
5
4
3
2
Observed values of F, in
sexagesimals.
g-"- 38'
. . . 19 38
. . . 29 38 ,
...39 38,
...54 38
1^- 39 38,
2 39 38 ,
3 54 38,
54 38,
54 38
.5
.9
21 54 38
Values of V, calculated by the
formula [9706].
17â„¢- 44'
....22 27,
....29 20.
....39 55.
....57 29,
. 1^- 29 50 ,
.2 39 45,
.3 50 8
. 5 59 58
10 42 31
24 42 49,
Difference between the calcula-
ted and the observed angles,
expressed in aliquot parts of
the observed values.
1
• 1,2*
1
• T*
_1_
•99*
_ 1_
1 To
1
1
•T"o*
1
1 3 6 {
_1_
•52'
1
• TT*
. -J-.
'12'
The calculated values of V agree with the observed values, as well as could
be expected in a formula which is only approximative, and in observations in
which the fractions of a quarter of a degree were found by mere estimation.
Towards the limits of the least and greatest distances of the drop from the line
of intersection of the planes, the difference is the greatest, and it is evident, from
^ 10, that this ought to be the case ; because, in the greatest distances, the drop
has not sufficient width in comparison with its thickness; and in the least
distance, its width bears too great a ratio to its distance from the line of
intersection.
It is this experiment of Hauksbee which Newton refers to in his Optics,
question 31. *' If two plane polished plates of glass, 3 or 4 inches broad, and 20
or 25 long, be laid, one of them parallel to the horizon, the other upon the first,
so as at one of their ends to touch one another, and contain an angle of about
10 or 15 minutes; and the same be first moistened on their inward sides with
a clean cloth dipped into oil of oranges or spirit of turpentine ; and a drop or
two of the oil or spirit be let fall upon the lower glass at the other end ; so soon
as the upper glass is laid down upon the lower, so as to touch it at one end, as
above, and to touch the drop at the other end, making with the lower glass an
angle of about 10 or 15 minutes; the drop will begin to move towards the
concourse of the glasses, and will continue to move with an accelerated motion
till it arrives at that concourse of the glasses. For the two glasses attract the
drop, and make it run that way towards which the attractions incline. And if,
when the drop is in motion, you lift up that end of the glasses where they meet,
X. Suppl. 1. «^ 15.] EXPERIMENTS WITH BENT TUBES. 799
and towards which the drop moves, the drop will ascend between the glasses,
and therefore is attracted. And as you lift up the glasses more and more, the
drop will ascend slower and slower ; and at length rest, being then carried
downward by its weight, as much as upwards by the attraction. And by this
means you may know the force by which the drop is attracted at all distances
from the concourse of the glasses."
"Now by some experiments of this kind (made by the late Mr. Hauksbee),
it has been found that the attraction is almost reciprocally in a duplicate
proportion of the distance of the middle of the drop from the concourse [97141
of the glasses, viz. reciprocally in a simple proportion, by reason of the
spreading of a drop, and its touching each glass in a larger surface; and
again reciprocally in a single proportion, by reason of the attractions growing
stronger within the same quantity of attracting surface. The attraction,
therefore, within the same quantity of attracting surface, is reciprocally as the
distance between the glasses. And therefore, where the distance is exceeding
small, the attraction must be exceeding great."
The explanation which Newton gives of the capillary phenomena, in this [97151
extract and in that we have before given, is very proper to show the advantages
of the mathematical and precise theory explained in the first section.
15. We have seen that the water rises in a capillary tube by the effect of the r97iQi
concavity of its interior surface. The effect of the convexity of the surfaces
becomes sensible in the following experiments :
If we dip a capillary tube into water to a small depth, and close the lower
part of the tube with the finger, then draw it from the water, we shall find that,
by taking away the finger, the fluid will descend in the tube, and form a drop of t^'^^'^l
water at its lower extremity. But when it has ceased to descend, the height
of the column will still remain greater than the elevation of the water in the
tube above the level, when it was dipped into that fluid. This excess arises
from the action of the drop of water upon the column ; for it is evident that, in
this experiment, the concavity of the interior surface of the column, and the
convexity of the external surface, or that of the drop itself,
contribute to raise the water in the tube. g.
ABC, fig. 135, is a curved capillary tube, whose branches
are of unequal lengths. By dipping it vertically into the water, so
that its shortest branch AB may be wholly immersed, the water ^cp=i
will rise in the branch B C above the level to a height which we \S - iX^
shall represent by FG. Then drawing the tube from the water, -^
there will be formed at the extremity J, a drop ANO; and when the fluid is [9719]
5-
jsr
[9718]
[9721]
800 THEORY OF CAPILLARY ATTRACTION. [Mec. C61.
stationary in the tube, we shall find that, by drawing through the summit TV of
[9719'] the drop, the horizontal line iV/', the height I'C of the water, in the longest
branch, will exceed FG. If with the finger we wipe away successively the
drops which are formed in A^ this height will gradually decrease ; and when we
have rendered the surface of the water at this point plane and horizontal, the
elevation of the water in the branch BC, above the horizontal line A I, will be
[9720] equal to FG. Lastly, if we successively apply drops of water at the extremity
A, the surface of the water at this extremity will again become convex, and
the fluid will rise more and more in the branch BC, so that the preceding
phenomena will be produced again in an inverted order. The excess of the
height of the column in the branch B C, above the height FG, appears in these
experiments to correspond to the convexity of the surface ANO; we must, to
ascertain the exact correspondence, measure the width and the chord of that
[9721'] surface. But the great difficulty in taking these measures has prevented its
being done.
The effect of a greater or less convexity in the surface, is also sensible in the
following experiment: ABC, fig. 136, is a capillary siphon
which contains a column of mercury ABC. By inclining
the tube on the side A, the mercury moves to the height ^' in
[9722] the branch AB, and to the point C in the branch BC. By
raising up the tube slowly, the mercury of the branch AB will
return towards A, whilst that of the branch BC will return
towards C. Then we find that the surface of the mercury in
the branch AB is less convex than that of the mercury in the branch BC; and
[9723] if^ through the summit of the first of these surfaces, we suppose a horizontal
plane to be drawn, the summit of the second surface will be below this plane.
This difference in the convexity of the two surfaces, arises from the friction of
the mercury against the sides of the tube ; the parts of the surface in the branch
AB, which return towards A, and which touch the tube, are retarded a little
by the friction, whilst the parts in the middle of this surface, do not experience
[9724] the same obstacle ; whence it follows that the surface must be less convex ; on
the other hand, the friction must produce a contrary effect upon the surface of
the mercury in the branch BC. Now as soon as the first of these surfaces is
less convex than the second, it will follow that the mercury will suffer,
by its action upon its own particles, a less pressure in the branch BA
than in the branch BC, and therefore its height in the first of these two
branches must exceed a little its height in the second, which is conformable to
experiment; a similar effect is observed in a barometer, when it is rising or
falling.
[9725]
[9727]
[9728]
[9729]
[9730]
X. Suppl. 1. § 15.] EXPERIMENTS WITH BENT TUBES. 801
Capillary siphons also furnish some phenomena which are a consequence of
the theory. They may be reduced to this general phenomenon deduced from
experiment: If we dip into a vessel of water any siphon ABC, fig. 137,
whose two branches are of equal or unequal widths, and then [9726]
draw it forth, the water will not run out from the longest branch ^-^^-^
BC, if the difference of the two branches of the siphon be less kf y
than the height FG, to which the fluid would rise in a tube of the -^- a-
same width as the branch AB. To prove that this result is a ^
consequence of the theory, we shall suppose that the fluid, whilst ^iff'^7.
running from the branch C, has assumed the position oiaBC, the
point a being very near the end A. Let q be the height of B
above the surface aio; the pressure which the fluid suffers at i,
the middle point of the surface aio, will be equal, Jlrst, to the
pressure of the atmosphere, which we shall denote by P; second, to the action
of the fluid upon its own particles, which is equal to K — g.FG,* g being the
force of gravity; third, to the pressure of the column q, taken with the
sign — , or to — gq. Thus an infinitely narrow canal, passing from i through
the axis of the siphon, will be pressed upwards by the force
P-\-K—g.FG-^gq, [9731]
q' being the height of the point B above the point C, the fluid at the point C
will likewise be pressed upwards, by the force P-{-K — gq', if the surface of
the fluid be plane in C, or by a greater force if that surface be convex [9276] ;
and the one or the other of these two cases must take place, when the fluid [^733]
runs from C, or has a tendency to run from it. In this hypothesis, this second
force must be less than the preceding; their difference
g.(q'^q^FG) . [9734]
must therefore be a positive quantity; consequently the excess q' — q of the
longer over the shorter branch, must he equal to or exceed FG, which is found
to he the case hy experiment.
H H
* (4237) This action is, by [9258], equal to ^—-, but by [9354], J=g'q, 5 being, [9728a]
in this formula, the quantity denoted by FG, so that the capillary action at i is K — g'FG
pressing upwards; from this we must subtract the gravity of the column iB=g.q [9727], and
add the pressure of the atmosphere P, and it gives the whole action upwards as in [9731].
In the supplement to this theory [10005—10023, he], several additional cases are
mentioned of the effects of the capillary action depending on the drop which forms at the [9728c]
extremity of a tube as in [9717, &tc ], and in other similar phenomena. >,
VOL. IV. 201
[9732]
[9735]
[97285]
[9739]
[9740]
802 THEORY OF CAPILLARY ATTRACTION. [IVIec. Cel.
In general, if we compare with tliis theory the diflferent phenomena which
[9736] have been carefully noticed by philosophers, we shall find that they appear like
corollaries deduced from the theory.
16. It now remains to give the experiments which have been made to determine
the concavity or convexity of the surfaces of the fluids in capillary tubes.
Philosophers having heretofore considered the curvature of the surfaces only as
[9737] a secondary effect, and not as the principal cause of capillary attraction, they
have taken but little pains to determine the curvature. Messrs. Haiiy and
Tremery have endeavored, at my request, to determine that of the surface
[9738] of water. They introduced into a tube AB, fig. 138, whose j^. ^^g
interior diameter is two millimetres, a column of water MmnN;
and, after having closed the tube at both ends, they held it j»r
vertically, and then carefully measured the two lengths Mm
and li; I and i being the nearest points of the two surfaces
MIN, min. The difference Mm — li, is equal to the sum of
the two lines IP, ip ; and they found this sum equal to ^ . MN.
According to the analysis in [9350, &c.], this sum must be equal ""35"
to MN, if the angle which we have denoted by 6', in [9346], be a right angle,
or if the surface of the water be a tangent to the sides of the tube.* But we
must observe, that, if we suppose them to be tangents, we cannot accurately
observe the points of contact. That which has been taken for the point M, is a
point where the surface of the water begins sensibly to quit the sides of the tube;
[974(ri and it is easy to prove that, to make IP + ip = |^| . MN, it is only necessary to
[^41] take, instead of M, m, the points which aref 0"'-,0226 distant from the tube,
which is not an improbable error. The preceding experiment seems, therefore,
[9742] to indicate that the angle ^' is a right angle, for water in a glass vessel. A
similar experiment made with oil of oranges produces the same result. Thus
* (4238) If we neglect the small quantity a in [9350], we shall have z=Ltang.ld'', and
[9739a] when ^'=90"-, it becomes z = l, or IP = MP, fig. 138. In like manner we have
ip=pn=zPNi hence IP -\-ip==,MP -\-PN=:MN, as in [9739].
[9741a] t (4239) Having IP = ip, MN==2MP, the equation [9740'] gives /P = |^.MP.
Now the surface MIN being supposed spherical, with the radius l"^-, and arc MI=6',
[97416] we have IP={l — cos.^')=^2s\n.^^', MP = sinJ'=2sin.i^'.cos.|^' [31, 1], Int.
Substituting these in the last expression of IP [9741a], and dividing by 2sIn4d'.cos.J^', we
[9741c] g^' tang.i(J'=t| = tang.40''-30"-; hence ^4'=40''-30'"-, or 6'—8V-; consequently
MP = sin.81'' =0,9877 = 1—0.0123; which is less than the radius by 0™,0123, instead
of 0â„¢S0226 [9741].
u.—
[9744]
X.Suppl. l.<^16.] CAPILLARY ACTION IN A BAROMETER. 803
ive have reason to believe that the surfaces of tvater, oil, and generally of the [9743]
fluids which moisten glass, are very nearly hemispherical in capillary tubes. llriTL
Determining, in the same manner, the convex surface of mercury in a very lie'con-^'
cave and
narrow glass tube, we have found that it is very nearly a hemisphere. If we gp^'fj^a,
compare this result with that which we have given in [9673], upon the depression
of mercury below its level, in very narrow glass tubes, we may correct the effect
of capillary action in the heights of the barometer. This effect is nothing in
barometers with branches of equal diameters ; but in a barometer formed by
a tube dipped into a large cistern, the capillary effect becomes so much the more
sensible as the diameter of the tube is decreased. The barometrical height,
counted from the summit of the column, is always less than that which depends
upon the pressure of the atmosphere ; thus we see how inaccurate the method
of those observers is, who measure the height of the barometer from the level [9746]
to the points where the upper surface of the column touches the tube. To
reduce the heights of the barometer to those which depend upon the pressure of
the atmosphere, and thus to render different barometers comparable with each
other, we must correct these heights for the capillary effect ; and we can do [9747]
this by the approximate integration of the differential equation [9324].
Integrating this equation, we get*
dz
H H du ^S /* ,
7-=- . %,rzudu; ' [9748]
the ordinate z being counted downwards, from the summit of the surface of [9749]
Ti-
the column. The quantity -r is the capillary effect,\ or what we must add to [9750]
* (4240) This integral is given in [9329] ; and by substituting a [9328], and supposing
the integrals to commence with m=0, we get
v/
iL^ ^.fzudu = t. r. â– [9748.)
multiplying this by — , we obtain — [9748].
f (4241) The effect of the capillary action is to alter the level by the quantity q [9353],
H 1
id this is equal to ~7 > or — [9354], which is to be added totl
to obtain its height corrected for the capillary action, as in [9750],
XT J
and this is equal to -;> or — [9354], which is to be added to the height of the barometer, [9750o]
804 THEORY OF CAPILLARY ATTRACTION. [Mec. Cel.
the height of the barometer^ to obtain the height depending upon the pressure of
the atmosphere. Now we have, bj what has been said,*
[9751] ?^^'=^.7».i.333.
Let / be the semi-diameter of the tube, estimated in millimetres. At the point
where u = l, we have
dz
\P"^^ ^= = sin.«' [9389];
\/' + %
the value of ^ [9748, 9750] will therefore be f
[9753] — __ 1 ^ / ^ZUdu, rCapillary effect in a barometer.!
gb 21 P^^ •- -'
To obtain this integral, we must find z in terms of u. We may determine
[97541 ^^ ^y ^^P^riment, observing that 2'^.fozudu is the volume included between the
surface of the mercury at the upper extremity of the column,! the surface of
H.s'm.d'
[9751o1 * (^^^^) ^^ ^® multiply the first expression of q [9360] by g, we get — - — =gq,
where I represents the radius of the tube [9343], and q the corresponding elevation or
[97516] depression [9353]. Now by [9673], we find, for mercury, that when Z= 0â„¢-,5, q = 7'^-,333 ;
substituting these in [9751a], we get [9751].
[9753a] t (4243) Substituting sin.d' [9752], in [9748], we get - =— .sin.d' — ^Jzudu,
H H.Bm.d' 2 /» , ,
Dividing this by g, and putting m = Z [9752], it becomes -7 = ', ji-J^udu; but
[97536]
[9753c]
r««^,-. , . H.8m.6' 1"5X7'°',333 ,, , • • • • u j- 1
fi-om [9751], we obtam = ; and by substitutmg it m the preceding value
gl 21
of the capillary action upon a barometer — [97536, 9750], it becomes as in [9753].
X (4244) Let MIISI, fig. 139, be the surface of the mercury, A.IB i mgjL39.
[9754a] meeting the sides of the tube in M,N\ Jits vertex; IBbA, the /jC ^^^\
horizontal tangent drawn from the point /. Upon this tangent let ^ -^ ^
fall, fi-om the points C, c, M, of the curve /iV/, the perpendiculars CJ?, c6, MA. Then,
[97546] if ^e put IB=Uf Bh = du, BC = z, we shall have the space BbcC=zdu; the
[9754c] volume formed by the revolution of this space about the axis IP will be equal to the quantity
zdUy multiplied by the arc 2*m, which it describes; so that the volume will be ^ir.zudu,
[9754rfl whose integral 2'^.flzudu evidently represents the whole volume described by the space
lAMCI, in revolving about the axis IP. This agrees with [9754].
The subject of the capillary action in a barometrical tube is again resumed by the
[9755]
X.Suppl. 1. <^ 16.] CAPILLARY ACTION IN A BAROMETER. 805
the tube, and a horizontal plane drawn through the top of the column. This volume
may be accurately measured by the weight of the mercury necessary to fill it.
We may therefore form, either by analysis or by experiment, a table of the
correction for the capillary effect in a barometer, relative to the different
diameters of the tube [as in 10443z]. In this calculation, it is supposed that the
tubes are of the same nature ; but there may be a slight difference in them ; [9756]