Royal Society (Great Britain).

The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) online

. (page 34 of 85)
Online LibraryRoyal Society (Great Britain)The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) → online text (page 34 of 85)
Font size
QR-code for this ebook


wards those parts whither the force tends. Therefore the weight of the in-
cumbent column, generates a like quantity of motion, in a given time, in the
effluent water, as it could generate in the same time in the column itself, fall-
ing freely through a vacuum.

Now because, by the hypothesis, the particles of water find no resistance
for want of lubricity, and all those particles, which are just going out in the
very hole, are urged by an equal pressure of the superincumbent water, it is
plain that the velocity of all these is equal.

Let V be that common velocity; a the height, in falling from which in
vacuo that velocity would be acquired ; A the height of the water above the
hole; V the velocity acquired by falling in vacuo from the height a; t the time
of falling from the same height; f the area of the hole; and let, the water flow
out of the hole in the time t. Now, because in the time t, with the velocity v,
the space 2a will be run over, the space — will be run over in the same time
with the velocity v. Therefore this will be the length of the column of water,
which flows out of the hole in the time t; and the magnitude of this column,
or the measure of the water flowing out in the time t, will be , and the

motion of the same will be _1£L. But the motion which can be generated in
the column of water over the hole, in the same time, x, if carried by its own
weight through a vacuum, is thus. Its velocity will be v, and as its magnitude
is AF, its motion will be afv. But that motion, from what has been said above,
is equal to the motion of the column of water flowing out in the time t, or
afv = . Hence v = vy' ■ .

V

Also, the measure above assigned, of the water running out in the time x,
or = AFv/2. a. E, I.

V

Carol. 1. — Since a : a :: v^ : v% therefore a = ^ = ^A. Therefore the
height a, which the effluent water can reach, by turning the motion upwards,
is half the height of the water in the vessel above the hole, which is the very
height determined by Sir I. Newton, Princip. ed. 3, lib. 2, pr. 36.

Carol. 2, — If we ascribe to the effluent water, that velocity which is acquired
by falling from the whole height of the water above the hole, that is, if we

suppose t; := V, then the above determined motion of the water , is :=

2afv, or double that motion which can be generated by the column over the

oo 2



284 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739.

hole, and therefore not to be generated but by double this column, as in the
Princip. ed. 1 and 3, lib. 3, pr. 36.

Scholium. — ^This measure here determined kf^I, or 2af X O.707, as it
falls far short of that which is generally determined by mathematicians, viz.
2af, so it far exceeds that measure which is shown by Poleni's experiments,
or 2af X 0.57 1 ; and no wonder, for what is supposed in this problem, that
the particles of water find no resistance in running down, the hypothesis is far
from the true state of things.

Prob. II. — To determine the Motion, Measure, and J^elocity of water, running
out into a Vacuum, through a Circular Hole in the middle part of the bottom of a
cylindrical vessel, where the particles of water find Some Resistance for want of
a Lubricity, but so small that the decrease of the motion of the effiuent water occa-
sioned, cannot be accounted any thing.

Let abcd, fig. 4, pi. 7, be an immense cylindrical vessel: ef a circular hole
made in the middle part of the bottom; and, the water being perfectly at rest
and unmoved in the vessel, let the stopper be removed from the hole, that a
passage may be opened for the water through it.

Then because the water was at rest, and now begins to run out through the
hole, and the water placed above follows that which runs out, and the natural
motion of the water is not disturbed by pouring any over it, and the hole is in
the very middle of the bottom, that portion of water which is in motion, and
descends towards the hole, will necessarily assume some regular figure ahefkb,
of which the lower base is the hole itself, and the upper base, the upper sur-
face of the water ab, and all the horizontal sections are circular. This is called
a cataract; but we do not yet examine what is the figure of the cataract: it is
sufficient for our present design, to observe that it is regular, and that the same
quantity of water passes in a given time through each of its horizontal sections.

Now because all that water which tends downwards, is contained in the cata-
ract, it follows that the rest of the water ahec, bkpd, which is without the
cataract, has no motion at all, and is perfectly at rest. Therefore in any hori-
zontal section of the cataract hck, whose centre is c, the points h, k, shall
represent the bounds between the water descending towards the hole, and the
surrounding quiescent water.

Also, as the point k is the bound of motion and rest, and the particles of
water, while they are in motion, find a resistance for want of lubricity, the
particle of water a. within the cataract, fig. 5, next to the point k, must be
carried downwards only with the least velocity. Otherwise it would necessarily
carry with it the next particle a, placed without the cataract, contrary to the
hypothesis. But the particle (3, which is contiguous within to the particle «,



YOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 285

will only descend with the least relative velocity, with regard to the particle a;
because otherwise it would carry the particle a. away with it, by accelerating it,
and this particle a, being now in a quicker motion, would carry away with it
the particle a. In like manner the particle y being placed more within, and
contiguous to the particle (3, will descend with the least relative velocity with
regard to the particle |3 ; and the other particles S, i, &c. being placed one more
within than another, will descend with the least relative velocity, with regard
to each of the particles lying next to each of them without. And by this
means the absolute velocity of the particles must necessarily increase gradually
from the bound towards the centre c, that the velocity of the water may be
greatest in the very centre, and least at each bound k and h.

But it is necessary that the resistance which each quicker particle finds from
the friction of the adjacent slower one, placed without, should be perpetu-
ally equal through the whole section of the cataract. Otherwise that particle
which finds the greater resistance, will accelerate the adjacent slower particle,
till the resistance is by this means diminished, and becomes equal to that resist-
ance which is found by the other particles. But if the resistance be every
where equal through the whole section of the cataract, the relative velocity of
the particles will be also equal every where, when one of them necessarily fol-
lows another.

Therefore the absolute velocity of every particle, which is the sum of all the
relative velocities, from the circumference of the section to that very particle,
taken all together, is in the ratio of the distance of the same particle from the
circumference of the cataract.

Now let r be the radius of the hole, m to 1 in the proportion of the circum-
ference to the diameter, mr^ the area of the whole, v the velocity with which
the water descends in the centre of the hole, a the height by falling from which
in vacuo the velocity v is acquired, a the height of the water above the hole, v
the velocity acquired by falling in vacuo.from the height a, t the time of fall-
ing from the same, z the distance of any particle from the centre of the hole,
and let the water run out in the time t.

Now the measure of the water, which goes out of the hole in the time t,
will be found after this manner: z will be the radius of any circle within the
hole, 2mz its circumference, 2mzz the nascent annulus adjacent to that circum-
ference, and X V the velocity of the water in that annulus.



Since v : — u :: 2a : 2ai' X the length of the stream flowing through

the nascent annulus in the time x; the measure of that water will be 2mzz



■286 PHILOSOPHICAL TRANSACTIONS. [aNNW I739.



r—x , ^^ rzx—z^z .,^a . „f .I- 1 ..■ 2OTAD.



X 2av X = 4mAv X ; the fluent of which, viz. X3?V— 2z',

vr \r 3vr

when z = r, gives — - — for the measure of the water passing through all the
hole in the time t.

But the motion of the same water will be also found thus. The measure of
the water running through the nascent annulus, in the time t, being

-^^ X rzz — z'z, and its velocity being v X — — , its motion will be — — X



4mAt)*



rzz — z'^i X n X = o- X r'^zz — 2rz^z + z'z, the fluent of which,

T V/*

when z = r, gives — — for the motion of the water, running out in the time
T, through all the hole.

But this motion is equal to that which the column over the hole can acquire,
in the same time t, by falling by its own weight through a vacuum, that is to
the motion afv, or av X mr^; therefore — - — = mavr^. Hence v = v \/ 3.

Also the abovementioned measure of the water issuing at the hole in the
time T, viz.—- - = „ — X vv^3 = —75-. a, e. i,

' 3v 3v v3

Carol. 1. — Since v^ : v^ :: a: a, therefore a = ^ = ^ X 3v^ = 3a. Tliere-
fore the height to which the water can rise, with that velocity with which it
runs out in the centre of the hole, is triple the height of the fluid above
the hole.

Carol. 2. — ^The figure of the cataract will be determined in the following
manner :

Let HK, fig. t), be any section of the cataract, having the centre c; and let
its radius ck = y, the height of the water above that section, or ci = x, t the
time of falling in vacuo from the height x, and as before let lf = r, and

LI = A.

Now the water passes through this section hk in the same quantity as it runs
out of the hole ef. But if the vessel be shortened, so that its height be
reduced from il to ic, and so that the section now becomes the very hole in the
bottom of the vessel, the water will pass through this section in a given time,
in the very same quantity as it passed through the same before the vessel was
shortened.

Now the vessel being shortened, the measure of the water issuing by the
hole HK, in the time t, by the preceding solution, is — ^ , and the measure of

the fluid issuing in the time t, is —'^ x j = -j^ X — ^: for t : r ::/A:y^ar.



i



VOL. XL!.] PHILOSOPHICAL TRAi^SACTIONS. •2€7

But, from what has been said above., the measure of the water issuing by the
hole HK, in the given time t, when the vessel is shortened, is equal to the
measure of the fluid passing in the same time through the section hk, when
the vessel is entire, or equal to the measure of it issuing by the hole ep in the
same time. Iheretore — -|- X — ;— = . ■, or y^^x= r v/a, or xy* = Ar ,
which is the same equation of the hyperbolical curve, by the rotation of which
he formerly showed that the figure of the cataract was generated.

Scho/.'l. — The measure of the water now found, 2mAr^\/^, or ImAr^ X
0.57735, rather exceeds the measure Imxr^ X 0.571, obtained from Poleni's
experiments. But this difference, in some measure, arises from not considering
the decrease in the motion of the water from resistance, in this problem.

Schoi. 1. — The measure of the effluent water, as determined by this solution,
is accurate, if the height of the vessel be considered as infinitely greater than
the diameter of the hole. But as this height has a finite ratio to the diameter
of the hole, the measure will be something less, so that, when the height is 5
times greater than the diameter, it will diff^er from the truth only the 320(X)th
part, and when it is double, only about the 5 1 20th part, which differences are
smaller than can be discovered by any experiment. And this small difference
proceeds from hence, that the abovementioned relative velocity, and therefore
the absolute velocity of the particles of water, which have been considered as
in a direction perpendicular to the horizon, are really in a direction somewhat
oblique, when every particle comes near the axis of the cataract in descending.

But if a true and accurate solution be desired, when the altitude of the
water has any ratio whatever to the diameter of the hole, it may be done as
follows.

From the property of the cataract curve, in corol. 2 of this problem, viz.
xy* = Ar*. the subtangent of this curve at the place of the hole will be 4a, and
at the place of any section the subtangent will be 4x, that is, 4 times the
height of the water above that section. But such a cataract curve is described
not only by the exterior water, which flows beyond the hole, but also by that
part of the water which flows through any annulus of the hole, that is, every
particle of water describes such a curve.

Now let 2 be the distance of any particle, in the hole, from 'its centre, and
let this particle descend through the smallest space in a tangent to the cataract
curve. Hence its velocity in the direction of the tangent, or the velocity
— ^^ V, explained in this problem, will be to the velocity in the perpendicular
direction, as V^i6a* + z^ is to 4a; therefore the velocity in the perpendicular
direction will be -7==== x — - " " ' „.



288 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739.

And hence, after the manner of the above solution, the measure of the
water passing through the nascent annulus will be — ^ X —p==L=z. The

fluent of which being taken, either by Cotes's forms or by infinite series, when
properly 'corrected, will give the whole quantity run out by the hole in the time
t; which in a series is the quantity ?^ X 1 - ■— - , + Jq^, — &c.
Hence, by supposing a to be infinitely greater than r, or the height than the
hole, the measure comes out barely 2A.mr^\/-y, the same as was deternjined be-
fore. Hence also.

When A = lOr, the measure is ^Amr^v^-r X (1 — ttttto) nearly.

And when a = 4r, it is 2AOTr^'/4- X (1 — -nVo) nearly.

So that, instead of the true measure, we may always take 2AmT^\/-^, without
any sensible error, even in so small an altitude, and much more in an altitude
many times greater, as it usually is in experiments; which makes the computa-
tion very easy,

Prob. III. — Supposing again the same thing as before, and neglecting the acce-
leration of the water without the hole; required to determine the Diameter oj" the
F^ein of water at the small distance without the hole, where the vein is most con-
tracted, and the Velocity of the water in the Vein so contracted.

In the solution of the former problem it was observed, that the particles of
water passing through the hole, do not all issue with the same velocity, but
every one with a greater velocity as it is nearer the centre ; and that the relative
velocity of the inner particles, with respect to the particles that touch each of
them on the outside, is constantly equal through all the hole ; and this relative
velocity proceeds from the resistance given to the particles, by the surrounding
water, as they descend towards the hole.

But after the water has passed the hole, and its outer surface is no longer
resisted by the surrounding fluid, nor by the ambient air, because moving in a
vacuum by the hypothesis, that relative velocity, or inequality of absolute velo-
city, can no longer obtain. For now the swifter particles must necessarily acce-
lerate the slower contiguous ones, and must also themselves be retarded by the
slower, till all the particles have acquired one common velocity, which will
happen at a small distance without the hole.

But while all the particles are acquiring this common velocity, the diameter
of the vein must necessarily be contracting. This happens in the same manner,
as when a rapid river is joined with a slower, as the Rhone with the Saone: in
the common channel, the velocity of the water from both rivers is equal, and
the water passes through a section of this channel in like quantity as before,
through the sections of both rivers ; though a section of the Rhone below the



VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 289

junction is much less than the sum of the two sections of both rivers above
the same.

Therefore let the radius of the contracted vein of water, where all the par-
ticles in its section have acquired an equal velocity, be f, and let that common
velocity be called u: then as v : y :: 2a : — the length of the vein, and there-
fore — " X m^^ Js the measure of the water passing through the section in the

time t; the motion of which in that time is therefore — .

But the measure of the water passing through the section of the vein, must
be equal to that passing through the hole in the same time, that is,

— V— = -::^'°'^?"-^'

Also the motion of the water through the hole, as it is not altered by the
action of the particles on each other, must be equal to the motion of the water

through the section of the vein, that is, Avmr'^ = — , or 2f^u^ = r^v'^.

Hence, dividing this equation by that immediately above, it gives

Hence j" = ~ ='^x ■^, = ^r\ and j = r/^. q. e. i.

Corol. — Since v^ = 4v*, and the altitudes are in the duplicate ratio of the
velocities generated by falling through them, therefore this is the velocity of
the water in the contracted vein, by which it can jet upwards in vacuo to ^ of
the height of the fluid above the hole.

Scholium. — This extraordinary contraction of the vein of water was first dis-
covered about 30 years before, by Sir I. Newton, when he was considering the
motion of effluent water more attentively, on account of some difficulties pro-
posed by Mr. Cotes, who was then taking care of the 2d edition of the Prin-
cipia; and Poleni afterwards confirmed it by many experiments. From that
time this phenomenon has greatly exercised the wits of philosophers, without
however detecting the true cause of it.

The radius of the vein, ry^^, orO.Sldor, determined by this problem, is a
little less than 0.84r, as delivered by Sir Isaac; and a little greater than 0.78r,
according to Poleni's measure, being indeed nearly a mean between them both.

Prob. IV. — Having given the measure of effluent water, through a circular
hole in the bottom of a cylindrical vessel; to determine the motion of the same,
and the velocity in the centre of the hole.

Let the given measure of the water, issuing in the time t, be 1mr\q; to
which the measure assigned by the analysis in prob. 2 will be equal, viz.

vol. VIII. P p



ago PHILOSOPHICAL TRAKSACTIONS. [aNNO 1739-

Imr^A.q = — - — , or t; = 3vq. But the motion of the same water, assigned

by the analysis in the same problem, is — — ; and by substituting here, instead
of tJ^, its value just now found, that motion becomes Sq'^mr^AV, a. e. i.

Carol. — If from the motion which can be generated in the time t, by the
column of water over the hole, viz. mr^Av, be subtracted the motion of the
water running out in the same time, viz. Zq'^mr^Av, there remains mr'^AV X 1 — 3q'^
for the motion lost by the resistance in the time t.

Prob. V. — fVlth the same data and suppositions as before, and neglecting the
acceleration of the water without the hole; it is proposed to determine the dia-
meter of the vein of water at a small distance without the hole, where the vein is
most contracted, and also the velocity of the water in the vein so contracted.

By prob. 3, the measure of the water passing through a section of the vein,

in the time T, is — - — ^; which is also equal to the given measure Imr^Aq;
hence f'^u = r'^vq.

Again, by prob. 3, the motion of the water passing through a section of the
vein, in the time t, is — - — , to which is equal the motion determined by the
former problem, viz. Sq^mrlAV; which gives 2fV = S^Vv^. But
M = -^ — -^-7- = -l^v; and j'' = — i = r^vo X rr- = t'' ; therefore

= r\/^. Q. E. I.

Corol. 1 . — The same ratio remains between the radius of the hole and that
of the contracted vein, whether the motion of the effluent water be any how
diminished by resistance, as in this problem, or not diminished as in prob. 3,
being both ways the same, j = r\/-^.

Corol. 1. — When the motion of the effluent water is diminished by resistance,
the velocity is at the same time diminished in the contracted vein. For in prob.
3 it was u = -^v\/3, but now it becomes u = -l^v; so that u is diminished from
0.866v to 0.856v, taking q = 0.571, according to Poleni's experiments.

Prob. VI. — Supposing the water issuing through a circular hole in the middle of
the bottom of a cylindrical vessel, when the particles of water, as they floiv down-
wards within the vessel, suffer so great a resistance from a want of lubricity, that
the motion of the fluid is much diminished by it, and also the measure of the ef-
fluent water being given; it is proposed to determine the motion of the same, and
the velocity with which it passes through the middle of the hole.

Let the given measure of the water, issuing in the time t, be Imr'^Aq, as in
prob. 4 ; and by that prob. we have for the motion of the same Sq'^mr^Av, also
the velocity with which it passes through the centre of the hole, or ti = 39V.
a. E. I.



VOL XLI.j PHILOSOPHICAL TRANSACTIONS. 1Q\

Corol. — When q is given, v is as v, that is, as v'a.

Prob. VII. — Supposing the water issuing into the air, and neglecting the acce-
leration of the water without the hole, proceeding from gravity; when any two of
the following are given, it is pioposed to determine the third, viz. the measure of
the effluent water, its velocity in the axis of the contracted vein, and the diameter
of the same vein.

When the water issues through the hole into a vacuum, it is shown, in the
solution of prob. 3, that the velocity of the particles of water is the same in the
whole section of the contracted vein ; but now, when the vein passes through
the air, the velocity is no longer equal in all parts of the section: for the outer
parts of the vein put the surrounding air into motion, and are retarded by it,
so that they cannot acquire the same velocity as the rest. But the outer parts,
when they are retarded by the air, retard the inner contiguous parts, and these
the next ; and so by this means every outer particle is carried slower than the
contiguous inner one, so that the velocity is greatest in the axis of the vein,
and least at the circumference. For which reason it is. Dr. Jurin thinks, that
the middle parts of the water in fountains rise much higher in the open air than
they would rise in vacuo.

Also, those parts of the air that are contiguous to the vein of water, when
they are put into motion by this fluid, they put the adjacent ones into motion,
that lie near them on the outside, and these the next outer ones, and these
again the next, and so on successively to some distance without the circumfe-
rence of the vein.

But the velocity of the particles of water must necessarily so decrease, from
the axis of the vein to its circumference, that the relative velocity of every par-
ticle, wherever placed, must be every where the same, with respect to the par-
ticle lying on the outside. For if any particle had a greater relative velocity
than the rest, it would find a greater resistance from the attrition of the adja-
cent outer particles, and thus would be reduced to an equal relative velocity with
the rest. In like manner, every particle of the surrounding air, which is put
into motion, will have all the same relative velocity with respect to the adjacent
particles of the air outwards.

But the relative velocity of the particles of water among themselves, is very
different from the relative velocity of the particles of air; as may be conceived
in this manner. Any particle of water in the outer part of the vein, is solicited
by the next particle inwards, to accelerate its motion; and is also retarded by
the next particle of air; and when that outer particle has acquired the due velo-
city, these two contrary forces must needs be equal, one of which retards the
particle, and the other accelerates it. But that cannot be done, unless the pro-

p p 2



2Q2 PHILOSOPHICAL TRANSACTIONS. [aNNO 173Q.

duct of the relative velocity, and of the density of the accelerating particle of
water, be equal to the product of the relative velocity, and of the density of
the retarding particle of air. But the density of air is to the density of water
as 1 to 900 nearly. Therefore the relative velocity between the outer particle
of water and the next of air, is to the relative velocity of the two next particles
of water, as 90O to I nearly,

Also, that inmost particle of^ air is solicited by the next contiguous particle



Online LibraryRoyal Society (Great Britain)The Philosophical transactions of the Royal society of London, from their commencement in 1665, in the year 1800 (Volume 8) → online text (page 34 of 85)