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

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of water to accelerate the motion, and retarded by the next particle of air out-
wards. And as here two contrary forces are equal to each other, the product
of the relative velocity and density of the accelerating particle of water, will be
equal to the product of the relative velocity and density of the retarding particle
of air. Therefore the relative velocity between those two particles of air, will
be to the relative velocity between the inmost particle of air and the next of
water, as gOO to 1 nearly ; and it will be to the relative velocity between the two
next particles of water, as QOO X QOO to 1 nearly. And this great relative
velocity will be always the same through the whole thickness of the ring of
air, which is drawn into motion by the effluent water.

Now let r, m, v, a, v, a, t denote the same things as in prob. 2. Also let
V be the velocity of the water in the axis of the contracted vein, p the radius
of the same vein, and r the radius of an imaginary vein, by which the velocity
u, by decreasing gradually, in like manner as it decreases in the true vein, is
at length reduced to nothing. Also let the measure of the water passing
through the hole in the time t, be Iqmr^K.

Now the measure of the water running in the contracted vein, in the same

time, by proceeding as in Prob. 2, will be — — - X 3r — 2p.

But these two measures are equal ; therefore Zqr'^ rv = up^ (3r — 2p). ,

Further, as the measure of the water running through the hole in the time
T, is Iqmr'^A, the motion of the same, by Prob. 6, is 3q^mr^Av. And the
motion of the water running through the vein in the same time, by proceed-
ing as in Prob. 2, is found ^^, X (6rV — 8Rp^ -|- 3p').

Now these two art equal; and hence Q^V^rV = u^(6rV — 8Rp^ -|- 3p*).

Then these two equations being rightly reduced for exterminating r, we
come to the following equation, p^u'^ + 1q\jvr^f^ = I2q\'^r^f —Qqv'^r*. From
which may be found any of the three quantities p, u, 5 ; viz.
g = ^ ^/^v X \/(u -I- 69V — 2v/(3^uv -f 9^V — 2u^),



,=:^-^X (,- + 2^/3p'^-2r^),

g'"



VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 2Q3

Schol. I. — It was supposed above, that the motion of the water running
through the contracted vein, is equal to the motion of that which runs through
the hole. But this is not true in mathematical strictness. For the motion of
the water running through the hole, is equal both to the motion of the water
through the contracted vein, and to the motion of the ring of air surrounding
the vein, which air is drawn into motion by the water running through the
vein. But the motion of the ring of air is considered as little or nothing, since

its thickness is not greater than ^^^, and its density not greater than the gOOth

part of the density of the water. And thus the equations are rendered much
simpler than otherwise they would be.

Sckol. 2. — By corol. 1, prob. 5, when the water issues into a vacuum, the
same ratio continues between the radius of the hole, and the radius of the con-
tracted vein, whether the motion of the effluent water be in any degree dimi-
nished by resistance, or not. Hence, as to a physical quantity, it is accounted
sufficiently true, that the ratio between those radii be considered as given, even
when the water flows through air, however the motion of the effluent water
may be diminished by resistance, or at least that the said ratio is varied the least
possible. And the same is found to be true by the experiments hitherto made.

Also if the ratio is given between r and f, the ratio between r and r is also
given, or the ratio between the radius of the hole, and the imaginary radius,
by which the velocity u, by gradually decreasing, is reduced to nothing. For
by eliminating u from the two equations above, we come to an equation which

gives R = Aj -f — ■====-^. Besides, from one of the two equations we ob-
tain 3r^R : p^(3r — 2p) :: u :g\: and since the former ratio is given, the latter
ratio is also given, that is, the quantity — is given.

Of two remarkable Caverns, the one Icy, and the other emitting noxious Effluvia.
By Matthias Belius,* F.R.S. N° 452, p. 41. Abridged from the Latin.

The icy cavern opens from the frozen Carpathian mountain, near the village of
Szelicze, the mouth, which faces the north, being 18 fathom high, and Q wide.
When the cold is severe in the country, M. B. says, the air within the cavern is
warm ; but that it freezes within the cavern, when on the outside the sun shines
with the greatest heat. When spring begins, and the snow melts, the water
trickles down into the inside of the cavern, and is there frozen into transparent
ice, by the power of the internal cold, forming large clusters of icicles, as

• Authorof a history of Hungary in 4 vols, folio, with plates, entitled Notitia Hungariae Historico-
geographica. Viennee Austr. 1735 — IZ^S.



294 l»HILOSOPHICAL TRANSACTIONS. [anNO 1739-

thick as casks branching out into many surprizing forms.* Thus not only the
arches, formed by nature in the solid rock, but also the floor of the cavern are
thickly covered with clear ice ; which shines all about within the cavern, as if
it were incrusted with crystal.

M. B. seems chiefly to ascribe the freezing quality of the cavern to the saline
nature of its texture. The nature of the Carpathian mountains, is saline, ni-
trous, aluminous, and vitriolic ; hence he concludes must ensue an almost con-
stant congelation.

As to the cavern at Ribar, a village in the county of Zol, it emits very
noxious vapours. It was formerly a rude copious fountain, and the water
rising to a good height, overflowed on all sides. The water was petrifying;
and generating a tophus, formed it gradually into such a mass, as became a
kind of mound about the mouth of the spring, and dammed it up so as to
prevent it from overflowing.

But afterwards, when subterraneous waters flowed from the interior of the
fountain in the hidden passages, the ground began to give way near the old
foundation, and at length formed a new opening ; when it began to emit
noxious vapours again, destructive to birds and other animals. In this cavern
is heard the murmuring noise of running water ; so that a river probably flows
through the interior passages, and at last loses itself in some kind of swallows.

A very extraordinary Tumour in the Knee of a Person, whose Leg was taken
off. By Mr. Jer. Peirce, Surgeon at Bath. N° 452, p. 56.

William Hedges of Stratton in Somersetshire, 25 years of age, of a mus-
cular healthy habit, had never known any kind of disease ; but about 8 years
before, he first observed a small swelling on his right leg, near the supe-
rior epiphysis of the tibia, which he called a splint, about the size of a
split horse-bean. He was not conscious of any bruise on the part, and was
quite free from pain; yet from its constant increase, which during the first 2

* Dr. Townson, who, in his Travels into Hungary, gives a particular account of this remarkable
cavern, states that he found abundance of icicles in it in the month of July, but in a stale of thaw.
The temperature of the cavern at that time (near Midsummer') was at Oof Reaumur's thermometer, i.e.
at the degree of melting snow. He therefore infers that the masses of ice found in this cavern must
be formed in the winter, and consequently that contrary to this author's account and to vulgar
report, the temperature of the air in the cavern, regulated to a certain extent by the temperature of
the atmosphere without, is lower in winter than in summer. However, as the temperature of the
cavern is but slowly affected by the temperature of the external air. Dr. T. admits that, when a very
warm spring suddenly succeeds to a severe winter, a freezing cold may prevail within the cavern, for
some time after it has begun to thaw without, and vice versa.



VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. IQS

years was very slow, but afterwards very fast, he was rendered quite incapable of
labour from the time of hay-harvest 1735.

Oil taking off the limb in May 1737, it weighed, with the leg and foot,
69 lb. which was 27 more than the leg some years before taken off at St.
Bartholomew's hospital by Mr. Gay, for the like disorder.

On examining this tumour, the adjacent muscles were found destitute of
their fibrous and fleshy appearance, probably from the pressure, and great ex-
tension, which they had suffered, and the little motion which for some years
they had employed on the tarsus and toes ; but the fasciae and common mem-
branes of the muscles, being greatly thickened and callous, adhered to the sub-
jacent tumour; and on removing this callous integument, the tumour appeared
covered with great quantities of blood-vessels, much distended, and of a colour
more intensely red than natural.

The tumour itself was cartilaginous for the space of half an inch from its
external surface ; from whence it formed numberless bony substances of various
forms, colours, and consistences, which, growing more and more numerous
as they lay deeper, at last formed a continual substance completely ossified :
in the centre of this bony substance was found about a quart of mucilaginous
liquor, no ways fetid, though it was then 10 days from the operation, the
colour and consistence of which nearly resembled that of linseed oil ; in which
were observed many little bony substances loose and floating, similar to many
others adhering to the internal surface of the cavity, all which had nearly the
appearance of those irregular incrustations, which in hollow rocks are some-
times made by the dropping of petrifying waters. After the operation, every
circumstance of the cure proceeded well and the stump healed.

Mr. P. thinks it worthy of remark that the parts above the tumour were
very little altered from their natural state. The cartilaginous extremity of the
femur was perfectly smooth ; nor had the rotula suff^ered any other injury, ex-
cept the ossification of the ligament by which it is fixed to the tibia ; but the
superior extremity of the fibula was wholly lost in the tumour.

An Experiment concerning the Spirit of Coals. By the late Rev. John Clayton,

D. D. N° 452, p. 59.

Mention is here made of a ditch, 2 miles from Wigan in Lancashire, the water
in which would seemingly burn like brandy, the flame being so fierce, that
several strangers boiled eggs over it ; the neighbouring people indeed aflirmed,
that about 30 years before it would have boiled a piece of beef; and that whereas
much rain formerly made it burn much fiercer, now after rain it would scarcely
burn at all. It was after a long continued season of rain that Dr. C. went
to see the place, and make some experiments, when he found that a lighted



296 PHILOSOPHIC At, TRANSACTIONS. [aNNO 1739.

paper, though it were waved all over the ditch, would not set the water on fire.
He then had a dam made in the ditch, and the water thrown out, to try whether
the steam which arose from the ditch would then take fire, but he found it
would not. He still however pursued his experiment, and caused it to be dug
deeper ; when at about the depth of half a yard, he found a shelly coal, and
the candle being then put down into the hole, the air caught fire, and con-
tinued burning.

Dr. C. observed that there had formerly been coal pits in the same close of
ground ; and having got some coal from one of the nearest pits, he distilled it
in a retort in an open fire. At first there came over only phlegm, afterwards
a black oil, and then also a spirit arose, which he could noways condense,
but it forced the luting, or broke the glasses. Once, when it had forced
the lute, coming close to it, to try to repair it, he observed that the spirit
which issued out caught fire at the flame of the candle, and continued burning
with violence as it issued out in a stream, which he blew out, and lighted again,
alternately, for several times. He then tried to save some of this spirit, taking
a turbinated receiver, and putting a candle to the pipe of the receiver while the
spirit rose, he observed that it caught flame, and continued burning at the end
of the pipe, though you could not discern what fed the flame : he then blew
it out, and lighted it again several times; after which he fixed a bladder, flatted
and void of air, to the pipe of the receiver. The oil and phlegm descended
into the receiver, but the spirit, still ascending, blew up the bladder. He then
filled a good many bladders with it, and might have filled an inconceivable
number more ; for the spirit continued to rise for several hours, and filled
the bladders almost as fast as a man could have blown them with his mouth ;
and yet the quantity of coals he distilled was inconsiderable.

He kept this spirit in the bladders a considerable time, and endeavoured
several ways to condense it, but in vain. And when he wished to amuse his
friends, he would take one of these bladders, and pricking a hole with a pin,
and compressing gently the bladder near the flame of a candle till it once took
fire, it would then continue flaming till all the spirit was compressed out of the
bladder.

But then he found, that this spirit must be kept in good thick bladders, as in
those of an ox, or the like ; for if he filled calves bladders with it, it would lose
its inflammability in 24 hours, though the bladder became not at all relaxed.*

An Experiment concerning the nitrous Particles in the Air ; by the same.

N" 442, p. 26.
Dr. C. took a small gally-pot, and ground the top of it very smooth and true,

* This so called spirit of coals, was inflammable air.



VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 207

and adapted to it a cover of blue slate, which he had likewise ground with
much care. Into this gally-pot he put equal quantities of nitre and flowers of
sulphur, about 1 dr. of each. He then fixed on the cover, putting it into a
new digester, for 3 or 4 seconds. On opening it the next day, he perceived
something had transpired between the top of the gally-pot and the cover ; the
top edges of the gally-pot, where the glazing was ground off, being discoloured,
though the nitre and sulphur were very little diminished as to their weight ;
only they were melted into one lump, which he took out of the gally-pot.

Having set the empty gally-pot on a shelf, on looking at it the next day, he
found long hoary hairs, very bright and brittle, all around the ground edges of
the pot ; which he gathered, and, tasting them, found them to be pure nitre.
He then set the pot on the shelf again, and in 3 or 4 days, still finding there
were fresh shoots made, as large as at the first, he gathered them a second and
third time ; so that he supposed the pot Would have continued to have shot
fresh nitre much longer, if he had not had urgent use for it, to make other
experiments in. However, it is to be observed, that he had already gathered
more nitre than he put into the pot at first ; though he had taken all, or nearly
all the nitre that he first put in together with the sulphur, out of the pot in a
lump. Hence he infers we may have some conception of the nature of mineral
earths, and how they increase, when once impregnated with the seeds of a
mineral. This is also a proof of the quantity of nitrous particles with which
the air abounds, since the large quantity of nitre which he collected out of the
pot, when left empty on the shelf, could be supplied by the air only.

Concermng the Poison of Laurel-Water. By John Rutty, M.D. N''452, p. 63.

Dr. R. expresses a wish that Dr. Mortimer's experiments with the milk, had
more fully determined and ascertained it to be an antidote, than they have yet
done. He was informed that some apothecaries in England, being used to
sophisticate black-cherry-water with laurel leaves, will not be persuaded, that
this is a poison on human bodies, notwithstanding our few instances ; but Dr.
R. confirms that it really is so by the following case :

At Lisminy in Westmeath, a girl of 18 years old, very well and healthv, took
a quantity, less than 2 spoonfuls, of the first runnings of the simple water of
laurel-leaves ; within half a minute she fell down, was convulsed, foamed at
the mouth, and died in a short time, nor was there any swelling on her body.

VOL. VIII, Q Q



2g8



VHILOSOPHICAL TRANSACTIONS.



[anno 1739.



Essay on the Measure and Motion of Effluent Water. By Dr. James Jurin,
F.R.S. W 453, p. 65. Translated from the Latin. Part 11, being the
Continuation from p. 284 of this volume.



Of the Resistance of the Parts of Water among themselves arising from a Want

of Lubricity.

We must now consider that resistance of fluids whicli arises from the mo-
tion of its parts among themselves, and is called, by Sir I. Newton, a resist-
ance arising from a want of lubricity. He makes this of two sorts ; one aris-
ing from the tenacity of the fluid, the other from the mutual attrition or friction
of its parts.

The former he thinks is uniform in a given surface, or that it produces an
effect proportional to the time ; and this opinion is agreeable to experiments.
The latter he considers as increasing in proportion to the velocity, or but little
less. About this however he determines nothing, for want of experiments.

Hypothesis. — The resistances arising from the want of lubricity in water.
Dr. Jurin considers as in a ratio compounded of the three following. 1. Of
the ratio of the surface of the parts moved ; 2. Of the ratio of the relative
velocity with which the parts of water are moved among themselves ; 3. Of
the subduplicated ratio of the altitude of the fluid. All which are allowed by
Sir I. Newton, and most other philosophers.

Prob. 8. — To explain the Resistance of the Parts of the Cataract which arises

from the Want of Lubricity.

Let r denote the radius of the hole, A the altitude of the cataract, y the
radius of any horizontal section, :r the altitude of the cataract above that section,
z the radius of any circle in that section, and v the velocity of the water in the
centre of the hole.

Then uv'- will be the velocity of the water in the centre of the section

having the radius y ; and



fi/- the velocity in the circumference of the

circle of the radius z; also -Z)/- the relative velocity, and Imzi the surface of
the nascent cylinder, to the radius z and altitude x : then, by the above 3 posi-
tions, the resistance of cylindric surface, is as
9,mzx X -2 V - X yoc = — -—xz.

Now let <v, x, and y be considered as constant quantities, while z flows till



VOL,



XLI.^ PHILOSOPHICAL TRANSACTIONS. igg



it becomes equal toy ; then the fluent of the above expression will be — — i-, or

^!^i', by making z = i/, which will be as the resistance of the nascent cylinder,
of the radius 1/ and altitude x.

But, by the nature of the cataract curve, xy* = Ar*, ory^x = r^A; hence
the resistance of this nascent cylinder will be as ^^.r ; and the resistance of

the whole cataract, will be as the fluent of this fluxion, or as ^mvr^-, or as

^mvr^ ti? by making ar = a. And since, by prob. 4, it is v = S^v, the re-
sistance in the cataract will be as ^^mvrV a?, or qxrV x^. a. e. i.

CoroL- — Since v is as v'a, the resistance in the cataract will be as qiA?.

Schol. — In the above solution, Imzi' has been used instead of 2mzV.f^ +^j
the true quantity ; and if this be used, as also the subtangent and tangent of
the curve as in prob. 4 ; then, by going through the same process as above,
the resistance through the whole cataract will be as

irmvry/A? X I —775^71 + W:\s:Va* ~ sStW + ^^ -^"^ '^ ^'^^ altitude a
be considered as infinite, with respect to the diameter of the hole, all the terms
of the series after the first will vanish, and the resistance will be barely
as ^mvr^A?, the same as before determined.

If A = lOr, the resistance will be as ^mvrt/ a^ X (1 7-) nearly.

If a = Ar, the resistance will be as \mvr*/ a^ X (1 5) nearly.

We may therefore use \mvr^ a^ for the measure of the resistance, without
sensible error, even where the altitude of the water does not exceed 2 diameters
of the hole, and much more in a far greater height.

Prob. 9. — Having Given the Measure of the Water issuing through a Given
Circular Hole, in the Middle of the Bottom of a Circular f^essel, of a Given
Depth ; to determine the Measure of the Water issuing from another Vessel of
Any Given Depth, through Any Given Circular Hole.

Let r denote the radius of the given hole, a the altitude or depth, 2mqr^A
the given measure of the water issuing in that time in which a body would fall
in vacuo through the altitude a.

Then, by prob. 4, S^^mr^AV will be the motion of the water issuing in the
same time ; and, by the cor. to prob. 4, the motion lost in the same time by
the resistance will be mr^Av(i — 3^^). Hence therefore an equal force of re-
sistance can generate this motion in the same time.

But the motions generated in the same time are proportional to the generat-
ing forces. Therefore the motion mr^AV, which the weight of the column of
water mr* a can generate in this time by prob, 1, without any resistance, is to
the motion mr^Av(l — 39^), which the resistance can generate in the same

ao 2



300 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739.

time, as the weight of mr^A, is to that resistance. Hence this resistance is
= mr^ a(1 — 39^).

In like manner, by putting « and e for the radius of the hole, and the altitude
of another vessel, and 2pnis^E for the water issuing in the same time, in which
a body falls in vacuo through the height e, the resistance in this new vessel
will be = ms'^E(l — 3/)^).

But, by the cor. to prob. 8, these two resistances are to each other, as qrx^
to psE*. Making this proportion then gives this equation, jbrE(l — 3^^) = qsx
(1 — 3/>^) ; hen ce p =

\/I^^^7-i^, or, = x/i+(irJ£„)._-V„ by putting
rE := nsA. And hence 2pms'^E is the measure ot the water issuing from the
second vessel, in the time that a body falls freely through the altitude e. q.e.i.

Corol. 1 . If the diameters of the holes be in the ratio of the altitudes of the
water, the ratio of the measures will be the same, as if the water issued with-
out any resistance. For, if r : « :: a : e, or rE = sa, and n = 1, then h p=.q ;
and hence 2qmr^\ : 2pms^E :: 2mr'^A : ttw^e, which is the ratio of the measures
when void of all resistance.

Corol. 1. If E be considered as nothing in respect of the altitude a, then will
n be as nothing also, and hence/) =■/•!-. Therefore the smaller the altitude e
is taken, the nearer p approaches to V -r.

Corol. 3. If s be infinitely great in respect of r, then \s p = V -^. There-
fore the greater s is taken, the nearer p approaches to v^^.
Prob. 10. — The Water Jlmving into the Air; to determine the Ratio between
the Diameter of the Hole and of the Contracted Vein.

This ratio cannot be determined without experiments. By prob. 7, p» =

£Zl! X {v -\- 6q\ — l*^ 3qvv -j- gq^v^—lv^) ; hence p is determined, when q

and V are known.

But no experiments are known, by which q and v may be measured. Poleni's
experiments show the measure of the effluent water, whence q is known :
but they do not show the greatest distance to which the water is carried, when
issuing horizontally from the hole ; nor the distance to which the middle part
of the vein reaches, that issues with the velocity v. And Mariotte's experi-
ments measure the greatest perpendicular height, to which water jets, when
its motion is turned upwards, whence f' is known ; but they do not show the
measure of the effluent water. Therefore, for want of proper experiments,
the ratio can only be determined approximately, as follows.

By schol. 2, prob. 7j it was made probable, that the ratio is constant be-
tween these two radii, or at least that it is very little varied. From Mariotte's
experiments it appears, that the difference between the altitude to which the



I



VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 301

water jets, and the altitude of the vessel, has nearly the duplicate ratio of the
altitude of the vessel. Therefore let a be the height to which the water, in
the axis of the vein, with the velocity v, can jet ; then, by Mariotte's experir
njents, a — a is as a^ and — — ■ is a given quantity.

But in one experiment, which Mariotte esteems a fundamental one, a was
= 60 Paris inches, and he found a = 5Q inches, the diameter of the hole being
half an inch. So that in this case — — — = 3600; and since this is a given quan-
tity, it will be always SSoOa = 3600a — a% or a = — ^g^" "^ = A — ^.

Therefore, if a = 1 inch, or double the diameter of the hole, it will be



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