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a= 1 — rir. But v^ :\^ ::a: \: 1 — ^^ : 1 . Therefore when the altitude
of the vessel is double the diameter of the hole, there may be taken v^ = v'',
or t; = v.

Further, by cor. 4, prob. Q, as e decreases, p verges to \/-^. Therefore
wlfen the altitude of the vessel is very small, as about 2 diameters of the hole,
then we may takejb or 9 = -/-J-* ,

But, by prob. 7, j'' = i^ X (v + 6qv — Wzqvv + g^^v'—lli^), and here
instead of v and q substituting their values just found, or v and -v/-^, there results
p'' = rViX (I +2i/3 -2^1 + ^/3), or f^ = r^ X {I -V V ^ — 2\^T+V\)
= 1^ X 0.6687553907 ; and hence p = r X 0.81777466.

Here then is the value of p, when the altitude of the water is double the
diameter of the hole : and since by schol. 2, prob. 7, p obtains a constant ratio
to the radius of the hole, it will have the same value in any altitude of the
water, q. e. i.

Carol. 1. By prob. 7, k = ip + /-j— => hence by the value of p just

found, there arises r = r X 3.98877 150, being the value of r when the altitude
of the water is double the diameter of the hole : and since, by schol. 2 of the
same problem, the ratio between r and r is constant, therefore r will have this
same value whatever the altitude of the water be.

Carol. 2. Because v is nearly = v, and q nearly = y^J- when the altitude of
the water is double the diameter of the hole ; therefore, at this altitude of the
water, — = \/3 very nearly. And since, by schol. 2, prob. 7, the ratio between

V and ^v is constant, therefore — will be = v^3, whatever be the altitude of
the water.

Prob. 1 1 . The Water issuing from a Vessel always full, through a Given
Hole, into the Air ; and having Given any one of the three following quantities.



302 PHILOSOPHICAL TRANSACTtONS. [aNNO J 789.

viz, the Measure of the Effluent Water, the Velocity in the Axis of the Con-
tracted Fein, or the Altitude to which the Middle Part of the Vein can jet up-
wards ; required to determine the rest.

Let A be the height of the vessel, r the radius of the hole, 2qmr'^A the mea-
sure of the effluent water, v the velocity in the axis of the contracted vein, a
the height to which the water can jet ; and first let 2qmr^A be given, whence q
is given.



Now, by cor. 2 prob. 10, — = v^3 ; hence v = qvi/3, and u* = a^^v*

But v^ ?;^: A : a = ^ = S^'a.

Secondly, if v be given ;
then 9 = - /+, and 2qmr^A = —^^ /i, also a = -^.

Lastly, if « be given ; since a = S^^A,
hence q^ =^i and q = v/^— . Also v* = — , and v = vy'-. a. e. i.

^ O A O A A A

Prob. 12. Having given the Height to which Water jets through the Air,
from a Vessel of a Given Height, through a Given Circular Hole ; to determine
the Height to which it will jet from any other Vessel, of any Given Height, and
through any Given Circular Hole.

Let the letters r, s, a, e, q, p denote the same things as in prob. Q ; and let
a and e be the heights to which water can jet, when spouting from vessels of
the respective altitudes a and e.

Then, by prob. 1 1, a = S^^a, e = 3/)^e ; hence
39^ = \, and 1 - 3^^ = ^, and q = -/^, and;, = -/^, and jb^ = ■^.

And since, by prob. 9, /> = V\ + C-^~''^y - ^-g^''^' °^ ^y putting
ra = nsA, p = Vj + C "g/^ ")" ~ ^ ~6g'' ^ ' ^^"<^^ ^7 substituting ^^-^ for

1 — 3q^, and y/ j2 for Q> ^"d for a — a writing «, it is /> = ^vali '

and ;>' = ^^-^ g^^ • But /> = 5^ ; hence e =

2Aa + n'>'-««^^4Aa + »'x^ Hence, writing . for e - e, it is ^ = ;^- X

2ao » 6 2Aa

(/4Aa + nV — n a). Now, t or e — e bring given, e is also given, or the
height to which the water can spout from the new vessel, a. e. i.

Corol. 1. If the holes in both vessels be equal, or * = r ; then e = nA,

or n = -, hence £ = ^ X {^ 4Aa + n'«' — nx).



VOL. XLI.3 VHILOSOPHICAL TRANSACTIONS. 303

Corol. 2. If the altitudes of the vessels be equal, or e = a ; then r = ns,
or n = -, hence « = — x (^ Aka + raV — no).

Carol. 3. If the diameters of the holes be in same ratio as the altutdes; then
the water will spout to heights proportional to those of the vessels. For if

r : s :: a: e, or rE := sx, and n = 1, then t = — , or t : a :: e : a, or e — e :a — a

:: E : a, or e : a :: E : A.



Carol. 4. Since 2p\/3Aa = \^4Aa + w'«' — nx, it will be t = — X
2/)V'3Aa =^^^^; hence, for \/a substituting its value above mentioned
9/3A, and properly reducing, « =^, or i = ^^.

Corol. 5. Hence, by making/) = g, then £ = — -, or t: x:: rz^ : sa". That
is, the defects of the jets, are as the diameters of the holes inversely, and as the
squares of the heights of the vessels directly.

Coral. 6. When s ■= r, then t = ^ nearly in the duplicate ratio of the
heights of the vessel ; which agrees with Mariotte's rule.

Corol. 7. When e = a, then i = — nearly ; that is, the defects of the jets
are nearly as the diameters of the holes reciprocally.

First General Scholium. — In examining the truth of this theory by experi-
ment, it will be proper, 1 . To use a very large vessel, at least in the upper
part, that during the time of making the experiment, the height of the water
may not be sensibly changed. But if the vessel be not so large, but that
during the efflux, the height of the water considerably decreases, then a mean
between the greatest and least height is to be taken for the constant height :
which is better than disturbing the natural motion of the fluid, by pouring
fresh water into it.

2. Let the vessel be of such a depth, that when the water spouts from a hole
in the side, the velocity of the fluid through the centre of the hole, may be
safely taken for the velocity through all the hole, when there is no resistance.

3. Let the plate, in which the hole is made, be so thin, or at least have so
thin an edge at the circumference of the hole, that the thickness there may be
accounted as nothing with respect to the diameter of the hole ; observing to
shave away the thickness of the plate on the outer side, leaving the inner side
plain next the water.

These things being prepared, the following experiments may be made, by

which, as so many criteria, the certainty of the above doctrine may be judged of.

Exper. 1. When the water issues through a hole in the side of the vessel.



/
/



304 VHILOSOPHICAL TRANSACTIONS. [aNNO 173Q.

measure carefully the diameter of the contracted vein, observing whether it re-
mains always the same, however the altitude of the water may vary.

Exper. 1. Observe whether this diameter has always the same ratio to the
diameter of the hole, in using different sizes of apertures.

Exper. 3. Observe the quantity of water issuing in a given time, through the
same hole, either in the bottom or the side of the vessel, with different altitudes
of the water.

Exper. 4. Observe the same, with holes of various sizes, but the same alti-
tudes of the water.

Exper. 5. Observe how much water issues in a given time, in two different
cases, in each of which there is the same ratio of the diameter of the hole to
the height of the water. For if the quantities be found in a ratio compounded
of the duplicate ratio of the diameters, and the simple ratio of the altitudes, as
in cor. 3, prob. 9, it will be a great confirmation of our theory.

Exper. 6. In the same two cases, observe to what altitudes the water will
jet upwards, through a large tube fitted to the side of the vessel, and perforated
in its upper part. For if these altitudes be found proportional to those of the
water in the vessel, as in cor. 3, prob. 12, it will be another sure confirmation
of this theory.

Exper. 7. Observe also the height of the jet, with the same hole, but various
heights of the water.

Exper. 8. Observe also the same, by varying the hole, but with the same
height of the vessel.

Second General Scholium. — Till those experiments be carefully tried, we must
avail ourselves of the experiments hitherto made. These are of three kinds :
for they measure either, 1 . The diameter of the contracted vein ; or, 2. The
measure of the effluent water ; or, 3. The height to which the water jets.
As to the

1st. The radius of the contracted vein, as measured by Sir I. Newton, is
0.84r, when the diameter 2r is ^ of an inch ; and by Poleni, it is O.JSr nearly,
when the diameter of the hole is 2^ Paris inches. But by the calculation in
this theory it isO.BlSr nearly, for any diameter of the hole; being nearly a
medium between those two.

2. As to the 2d, none of the quantities, measured by any one, are of any
use, except those by Poleni, when the water issues through the above size of
hole made in a thin plate, which he informs us is much less than when it
passes through a pipe of the same diameter. And the medium among 10 such
measured quantities, is 2mr'A X 0.57 1, when the height of the vessel is 33
Paris inches



VOL.



XLI.]



HHILOSOFHICAL TRANSACTIONS.



305



But the measure taken to this altitude, by our calculation, fron) Mariotte's
fundamental experiments, is 2ffjr*A X O.5768, exceeding Poleni's measure about
the 98th part only.

3. As Poleni rendered all the experiments of his predecessors useless, con-
cerning the measure of the effluent water, because they took no account of the
thickness of the plate through which the water issued ; it might be suspected
that the like fault enters into the experiments concerning the height of the jet
of water. But Poleni has removed this doubt, by another excellent observa-
tion ; viz. that water issuing through short tubes, or through holes in a thin
plate, spout nearly to the same horizontal distance, or issue with nearly the
same velocity.

Therefore, to try the certainty of our theory, let us make use of Mariotte's
experiments concerning the altitudes of jets ; assuming some one of them, as
a foundation, for trying the heights in the rest of the experiments, by our 1 2th
problem. Taking therefore that experiment, in which the diameter of the hole
was 6 lines, and the depth of water in the vessel 34 feet 11-^ lines, or 4 1 94.
inches, in which case Mariotte found the jet spouted to the height of 31 feet
8 or 9 inches, or 380^ inches. Here then, A = 4194- inches, a = 380-|-, and
ft = 39. But, in another experiment, where e, or the depth of water in the
vessel, was 26 feet 1 inch, the water rose, through the same hole, to the
height of 24 feet '1\ inches ; and by our cor. 1, prob. 12, the height of the jet,
or the value of e, comes out 24 feet 3 inches, which is nearly the same.

Other cases are exhibited in the following tables, where it appears how nearly
the calculated heights of the jets, computed by our theory, agree with those
observed in Mariotte's experiments, when made with holes of various sizes, and
with vessels of different heights.



I. With the Hole of 6 lines Diameter.



Alt. of the


Height of the Jet,


Vessel. By Exper. By (


Calcul.


Ft. Inc. Ft."


Inc. Ft.


Inc.


34 II-I-. ..31


84-. . . 31


H


26 1 ...24


24-. . . 24


3


24 5 ... 22


10 ... 22


10


12 4 ... 12


... 11


11


5 6 ... 5


4^. . . 5


5


5 ... 4


11 . . 4


IH


35 5 ... 32


. . . 32


1


VOL. VIII,







II. The Hole of 4 lines Diameter.
Alt. of the Height of the Jet,



Vessel. By Exper. By Calcul.


Ft. Inc. Ft.


Inc. Ft. Inch.


32 114^. ..30


... 30


24 5 ... 21


84^ ... 21 11


5 6 ... 5


A^... 5 4A


III. The Hole .


of 3 lines Diameter.


34 1 H. . . 28


... 28


26 1 ... 22


... 22 1


24 5 ...21


2 ... 20 11


5 6 ... 5


4A... 5 3tV



Kr



306 FHILOSOPHICAL TRANSACTIONS. j^ANNO 1739.

Hence it appears, that the calculations from the above theory agrees so
well with Mariotte's experiments, of the height to which the jets rise ;
as also with Poleni's measure of the effluent water ; and with the diame-
ter of the contracted vein, measured by Newton and Poleni, that it can hardly
be doubted that the above theory is either true, or at least very near the
truth.

A Collection of the Observations of the Eclipse of the Sun, August 4, i738, which

were sent to the Royal Society. N° 453, p. Ql.
By Mr. George Graham and Mr. Short, FF. R. S. at Mr. Graham's home
in Fleet-street, London, by a Refracting Telescope of \2 feet focus, armed
with a Micrometer, and by a Reflecting Telescope ofQ inches focal length,
p. 91.

Beginning of the eclipse at 9'' 59"" 20' A. M.

End at 11 59 36

Duration 3 O \6

Quantity of obscuration by the micrometer 2 dig. 28. min.

2. At Upsal, by M. Celsius, F.R.S, with a 7-foot tube, and a Graham's Mi-
crometer, p. 92.

12** 18"" 52' ... . True time. Beginning of the eclipse.
12 42 22 ... . The end.
O 23 30 ... . The duration.

3. At Wittemberg, by J. F. fVeidler, F.R.S. p. g'i.
The beginning could not be seen for clouds.

At 1 1*" 30™ increasing, 1 digit, eclipsed.
12 19 2 dig, 30' eclipsed.

4. At Bononia, by S. Manfredi, F.R.S. &c. p. 94.
At 23'' O" 10" eclipsed, 1 digit.

3 o the middle ; 4^ digits.

1 18 1 the end of the eclipse.

Some Electrical Experiments, chiefly respecting the Repulsive Force of Electrical
Bodies.— By Granvile Wlieler, Esq. F.R.S. N" 453, p. 98.

The following experiments were made in the autumn of the year 1732, and
repeated to Mr. Grey the following summer.



VOL



. XLI.] rHILOSOPHlCAL TRANSACTIONS. 307



Prop. I. Bodies made Electrical, by communicating with an Electrical Body

excited by Friction, are in a state of Repulsion with regard to such excited

Bodies.

Exper. 1 . — Mr. W. hung a fine white thread by a loop, to a horizontal blue
silk, line, about 4 feet long, tied at each end, and at about a foot distance from
it, placed a glass tube 2-i- feet long nearly, and one inch and quarter diameter,
fixed in the centre of a circular piece of wood supported on three brass screws, so
that the tube and pendulous thread were parallel to each other. The tube being
rubbed, the thread was attracted and repelled 7 or 8 times J in very good wea-
ther it moved to and from the lube 12 times, at above one foot distance.
He then tied a piece of new smooth packthread to the top of the tube, and to
the loop of the thread hanging down as before, and again excited the tube:
the thread, without coming once towards the tube, went into and continued
in a state of repulsion ; but if he only touched the communicating packthread
with a finger, the white thread immediately hastened to the tube : and on hang-
ing another long piece of packthread, which reached the ground, to the com-
municating packthread, and again rubbing the tube, the pendulous white thread
was so far from going into a state of repulsion, that it became attracted to the
tube, and continued so, without showing the least tendency to a state of re-
pulsion, as long as the virtue of tlie tube lasted.

Exper. 2. He tied a piece of small cane, about l6 inches long, and one
fourth of an inch diameter at one end, and a little more at the other, at right
angles to the top of the tube, fixed in the same pedestal as before, and making
unequal arms with it ; and at the end of the larger arm, a piece of stick trans-
versely, about 6 inches long, so as it might slide backwards and forwards to
and from the tube. This moveable short stick at one end supported a very fine
white thread, at the other a very fine blue silk, by which means now a silk and
a thread both at the same time hanging parallel to the tube. The thread, after
the tube was rubbed, first was attracted, but then immediately repelled, and
continued a considerable time in a state of repulsion ; but on tying to the end
of the shorter arm of cane, a piece of long packthread, which reached down on
the table, and rubbing the tube again, the thread continued in a state of at-
traction, without being once repelled during the whole virtue of the tube, as
in the preceding experiment. Yet the silk, whether the long packthread was
added or not to the shorter arm of the cane, continued constantly attracted
towards the tube ; but on putting a short silk only 6 inches long, in the same
circumstances, it would, after some time rubbing the tube, turn into a state of
repulsion, the upper part first bending from the tube, and the lower part to-

R R 2



^08 PHILOSOPHICAL TKANSACTIONS. [aNNO i 73 Q.

wards it, the upper bending still increasing till the whole was repelled ; and it
was remarkable, that the upper part or bending, on the approach of the finger,
or any body not impregnated with electrical effluvia, flying towards it, and the
under part or bending, rather seeming to fly from it, till the whole was satu-
rated, and in a state of repulsion with regard to the tube, and then any part of
it would come to the finger, or any other body, not made electrical. It is pro-
per to add here one more difference remarkable between the thread and the silk.
The thread in a state of repulsion touched with the finger, would immediately
fly towards the tube ; but the silk in the same state, after touching several
times, still continued in a state of repulsion, and would not be attracted till
squeezed from top to bottom between the finger and thumb, once, and some-
times two or three times. And further, the thread would immediately turn
again into a state of repulsion, whereas the silk, after the violence committed
by the thumb and finger, being attracted to the tube, would not without a
good deal of rubbing the tube, be repelled again.

Carol. 1. From the different state of the pendulous silk and threads at the
same time, under the same circumstances, the former being attracted while
the latter is repelled, it follows, that a mere vibration of the parts of the
tube is not sufficient to account for the electrical phaenomena ; which ap-
pears further from the two contrary states continuing some time, and from the
same piece of silk being at once part in a state of repulsion, part in a state of
attraction.

Corol. 1. That some bodies immediately receive and immediately part with
the electric effluvia ; but that others are some time before they receive it, or
receive enough of it ; and when they have received enough of it, part with it
more unwillingly.

Corol. 3. That any light body, as a feather, after touching, or nearly ap-
proaching the tube, must fly from it : on contact or a near approach, it satu-
rates itself with the electric effluvia, and by this means becomes itself elec-
trical ; and consequently from the foregoing experiments, is in a state of re-
pulsion with regard to the tube. As soon as it touches any other body, it loses
its acquired electricity, and therefore may be attracted as at first.
Prop. II. Tivo or more Bodies made electrical, by communicaiivg with an
Electrical Body excited by Friction, are in a state of Repulsion with re-
gard to one another ; or Bodies made electrical by Communication, repel one
another.

Exper 1. Mr. W. suspended two pieces of white thread, each about one
foot long, by loops, on a horizontal blue silk line, 4 feet long, about half an
inch asunder from each other ; and on holding the excited tube over them at a



VOL. XLI.] PHILOSOPHICAL TRANSACTIONS. 30g

little distance, the two threads immediately receded from each other consi-
derably at the bottom. He then removed one of the threads, and held the
tube over the other, in the same manner as before. The single thread was not
observed to move to either side ; consequently the moving of the threads side-
ways was occasioned neither by the attraction of the cross line, nor that of the
tube, nor by the frame of wood, to which the cross line was tied at each end,
but only by their action on each other.

He then added a third string, at the same distance from the second, that the
second was from the first, and on holding the excited tube over the middle one,
at tlie same distance from the cross silk, as before, when the strings continued
in the same plane, the middle one stood still, and the string on each side of it
receded considerably at the bottom part, which in this case must necessarily
happen, on a supposition that they repel each other equally ; for the two con-
trary forces of the outer threads destroy each other, and consequently the
middle one must remain quiet ; but there was nothing to hinder the middle one
from repelling the two outer on each hand sideways. When, as it often hap-
pened, the three pendulous threads did not remain in the same plane, they
then all receded from one another equally, and formed nearly a triangular
prism ; the three threads being the three edges, or rather a triangular pyramid
with the top cut oft".

On suspending 4 threads at the same distance as before from one another ; if
they continued in the same plane, they all parted, but the two outermost
more from their neighbours, than the two in the middle from each other. If
they moved out of the plane they were first in, they formed two prisms, each
extreme with the two in the middle forming one, or rather a parallelopepid, less
at top than at bottom.

When 5 strings were suspended, either the middlemost continued sta-
tionary when the plane was not altered, or if it was, they formed 3 prisms.

Exper. 1. Mr. W. afterwards placed two cross blue silks, of the same length
as before ; about half an inch asunder from one another horizontally, and tied
at each end ; and on each of these, at different times, hung 2, 3, 4, and 5
threads, at the same distances as before, when every thing succeeded, as it
ought to have done, on a supposition of their mutually repelling one an-
other.

Exper. 3. To each of the ends of two threads, suspended as at first, a fea-
ther being tied, the two feathers manifestly receded from each other ; and when
3 threads had each a feather at their extremities, the middlemost became sta-
tionary, and the two outer went off on each hand.-



310 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739,

Exper. 4. Mr. W. suspended afterwards 2, 3, 4, and 5 blue silk strings by
loops, on one cross blue silk, and found the several experiments succeed in the
same manner as in threads ; except that they remained a longer time before
they appeared in a state of repulsion, receded from one another more slowly,
and continued much longer in the repulsive state, after the tube was re-
moved.

Exper 5. This done, he made several experiments, by mixing silks of dif-
ferent colours, and silks and threads of different colours, and suspended them
by turns on silks of different colours ; whence arose several different phaeno-
mena. On suspending two black silks at the before-mentioned distances from
each other, on a scarlet cross silk, they not only opened and receded from each
other at the bottom considerably, but when the tube was held under, ran or
jumped away from each other, to the very ends of the cross red silk that sup-
ported them, taking 2, 3, or more jumps from each other. The same was
observed of two white silks suspended on red silk, but they did not move away
so briskly as the black.

Exper. 6. Mr. W. tried whether threads hanging parallel as above, from a
cross blue silk line, and joined with one or more transverse threads, so that
the perpendicular threads remained nearly parallel, would mutually repel when
the tube was held over them ; and they seemed to repel each other full as
strongly as before. When they were joined by only one cross thread towards
the top, the lower parts separated considerably ; when joined by two cross
threads, one towards the top, and one towards the bottom, they separated
both in the middle parts between the two cross threads, and at their lower ends
under the second or lowest cross thread. When several were tied together at
the top and bottom, and about a foot long, not by transverse threads, but in a
knot at each end, they all bellied out from one another, describing a figure
generated by an ellipsis, revolved about its greater axis ; approaching nearer to
a sphere, the stronger the repulsive force was. And though it was only a ne-
cessary consequence, he could not without some pleasure observe the knot at
the bottom, as the strings swelled out, sensibly rising up. He could scarcely
forbear imagining the bundle of silks, a bundle of muscular fibres.

Exper. 7. He suspended two brass, and afterwards two iron wires on a
cross blue silk, in the same manner as the threads and silks before mentioned,
and found the experiments succeed as in threads of the same number, except that
they did not recede so far from one another, which must necessarily follow from
their greater weight.

N. B. These experiments were n)ade sometimes with the tube held over.



VOL. XLI.] PHILOSOPHICAL TBANSACTIOKS. 31 J

sometimes held under the cross line ; but they generally succeeded best when
the tube was held under the extremities of the pendulous wires, which in this



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