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follows from thence, that if we subtract the weight of the column cb, from
the aggregate of the attractions of the column on, the residue must be equal
to the sum of the centrifugal forces of the column cn. Now to endue our
spheroids with this property, we will resume the expression of the centrifugal
force in e, which we found art. 14, which will give

( ^^^ — ^ + z==^-z= — ) - , for that part of the centrifugal force which

3 + px5+/) 3 + ?x5 + ?e
acts according to cm, in any place m, by expunging the terms in which aa

would be found. This value being multiplied by r, and by the density, will
give, when we have taken the fluent, ^^ ' =^



+



8c/ge'+^ + ?A



+



2+px3+/)X5 + p Z+p X 3-\-q X 5^ q

cge^ £qj. jj^g gm^ Qf jjjg centrifugal



2+?x3+px5+p 2+qx3+qx5+q

forces of the column cn, still expunging those terms in which either met or M
are found.

mu , • .u- • W I+ipcA^+'^A , 8 + lpcfge^+f + f>.

Then makmg this expression equal to =rrt — ~ — ' —-



l+pX3 + pX 5+p 2+p + qx3+px5+p

+ __EI} ^<fg'"'^L^!^_ + rr7gcg^e^+^_^ ^j^j^j^ jg ^^g difference of the
2 +p + 9 X 3 + ? X 5 + 9 1+9x3+9X5+9

weight of the column at the pole cb, from the sum of the attractions of the
column CN, we shall have the equation

PPff_ , 2pg/g

1 +px 2+p X 3"+7 X T'+p T+P+ qx 3 +p X 5+p X 2 + q



VOL. XL.] » PHILOSOPHICAL TRANSACTIONS '2\Q

, _JpgJg , gggg ^^ Q

2+P + 9 X 3 + 9 X + 7 X 2 +;» \ + q y. 2 + q X 3 + q y. 5+ q '

where we have put e = 1 , for the greater simplicity of calculation.
Determination of such spheroids, as make the principle of the equilibrium of

the columns, and that of gravity perpendicular to the surface, to coincide with

each other.

32. This equation informs us, that when out of all the infinite varieties,
which will be supplied by the equation of the densities D ^frt + gr^ + hr",
&c. we shall have taken at pleasure all the coefficients, and all the exponents,
one only excepted; if this last be such in respect of the others, that it may
fulfil the conditions of the foregoing equation, the spheroid, being supposed
in a state of fluidity, will be in equilibrio, because it will unite as well the prin-
ciple of a perpendicular tendency to the surface, as that of an equipoise of the
several columns.

33. Before concluding this paper, I shall make a few reflections on the prin-
ciples we have now made use of, for determining the figure of a spheroid
revolving about its axe.

The first principle which, after Mr. Huygens, we have had recourse to, and
which consists in making bodies gravitate perpendicularly to the surface, seems
to me of absolute necessity. For if there were never so little water on the
surface of the earth, it could not be at rest, if it had a tendency any how in-
clined to the surface.

The second principle, made use of by Sir Isaac Newton, and which consists
in an equilibrium of the columns ce, cn, cf, could be thought necessary I
think, only for these two reasons ; the first is that which is usually assigned,
that at the first formation of the earth, it was probably in a state of perfect
fluidity; in which case it must acquire such a figure, as will result from the
equilibrium of the columns, and from the gravitation acting perpendicularly to
the surface. Indeed though this reason has a degree of plausibility, yet there
are many who think it to be of small force. Perhaps, say they, the earth has
never been in this fluid condition.

ITie second reason, which I believe will have a greater weight with every
body, is this: considering the earth as it is at present, and without carrying
our thoughts so far back as to its formation, if the ocean, which is now on its
surface, has any considerable depth, and if its parts preserve a communication
with each other, from region to region, by subterraneous canals ; it can only
keep an equilibrium by this means, because its superficies is the same as it
would have, were the whole a fluid.

34. This second reason has suggested a reflection to my mind, concerning

p p 2



220 rHILOSOPHICAL TKANSACTIONS. x [aNNO 1738.

the equipoise of the columns now calculated, art. 31 and 32. Let us first
suppose, that the earth is our fluid spheroid, composed of layers of different
densities; and that afterwards this fluid hardens into a solid, so that the difl^erent
strata, of which it is made up, are of no other use but to cause a gravity by
their attractions. Then let us suppose, that the seas and great waters about
the earth have a communication with each other, by means of some subterra-
neous canals. As the waters of the sea, which unite with each other, are pro-
bably homogeneous, the foregoing calculation, in which we have considered the
spheroid as a fluid, can no longer take place, because we have there supposed,
that the fluid contained in the canal bcn is of a density, that varies from the
centre to the circumference. From hence it seems to me, we must undertake
the computation of the equilibrium of the columns after another manner, thus:
We must examine whether two canals, as on and bc, which are filled with a
homogeneous fluid, will be in equilibrio, all the other parts of the spheroid
continuing as above.

35. To do this, we will begin with finding the gravity of any column on,
fig. 15, arising from attraction alone. First then, we must resume the expres-
sion of the attraction in any point m, art. 7- Then we must multiply it by
f -\- \r, which will give

2c/r' + r . sT 4.pcf>^r^ +^r , i cfrr +^r , 2cgr' + ?r ^

3+p 3 + P X 5 + p 3 +p X 5 +p 3+J

And taking the fluent of this quantity, we shall have

^e= + =^CL. + ==Jt^:i^-= + ^^li=, &c. for the

Z + py.i+p 3 + p y. S + p 2+px3 + pxS+p 3 + ?x2 + 5

gravity of the whole column on.

36. If in this value we make x = 0, we shall have the gravity of the column
at the pole.

37. And if we subtract the gravity of the column at the pole, from the
whole sum of the attractions of the column cn, we shall have

*^'^' _. ._ + —^ ^— , which must be equal to the sum of the centri-

3+px5+p 3+9x5+9

fugal forces of the column cn, in order that the columns cb and on may be in
equilibrio. (

But we shall find this really to obtain, if we resume the quantity

( — gSffl— — -I ^Ji ^Jl^) r ^ which,, by art. 31, expresses that part of

3+px5+p 3+9x5+? e

the centrifugal force in m, which acts according to cm. Then multiplying this
expression by r, and seeking the fluent, we shall have



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 221

*5^' "^ - — + ^^ ' — =^-=1 for the agsjregate of the centrifugal forces of the

3 + px j +p 3 + 6x5+6

column CN. And this being the same as the foregoing, shows that the columns
CB and CN are in equilibrio, supposing them to be homogeneous; nor are we
here obliged, as in art. 32, where we consider them as heterogeneous, to sup-
pose the coefficientsy, p, &c. to have any certain relation among one another.

38. Perhaps it may be urged, that the foregoing calculus agrees only to a
canal, as bcn, which passes through the centre; and that we ought to prove
in the same manner, that the water included in any other canal pgr, would ob-
serve an equilibrium. But it appears to me, that this property may be derived
from the former; for it follows, from the foregoing calculation, that if we
might be allowed to make this hypothesis, viz. that independently of the attrac-
tion of any matter, the gravity at any distance cn from the centre, see fig. 15,

would be proportional to -^ -|- - ^ = + - — , &c. it is plain

3+;» 3+PX5+P 3-j-px5+p

from thence, that a mass of the homogeneous fluid, which should turn about
the axis cb, would assume the same form as that of our heterogeneous fluids.
But if this spheroid should then put on a fixed state, except only some canal
pqr, the water in this canal would be in equilibrio; for without this, the sphe-
roid could not be esteemed as having arrived to its fixed state. But this suppo-
sition comes to the same as that of our heterogeneous spheroid, composed of
elliptical layers, in which should be found a canal pgr of a homogeneous fluid;
provided that the space, which this canal possesses in the globe, be not of so
large an extent, as to change the law of attraction.

The only three planets, in which we can be assured of gravitation, and the
centrifugal force, are the sun, Jupiter, and the earth. As to the sun, the cen-
trifugal forc^ is there so small, in respect of its gravity, that his poles must be
very little depressed, so that we cannot be sensible of it by observation. Then
as to Jupiter, observations make him something less flat than according to Sir
Isaac Newton ; that is, than if he were composed of matter of a uniform den-
sity. Therefore by the foregoing theory, he must be a little more dense to-
wards the centre, than at the parts near the superficies. We might make a
thousand hypotheses about the manner of distributing the inequality of density,
proceeding from the centre towards the circumference, which would all agree
with the figure observed, and which are very easy to calculate by the principles
here laid down.

As to what concerns the earth, I shall wait till we receive the observations
which must have been lately made in Peru; that by comparing those with what



222 fHILOSOFHICAL TRANSACTIONS. [ANNO 1738.

observations we have made under the arctic circle, and with those of Mr.
Picart in France, we may have the true difference of the earth's diameters at
the equator and at the poles. Then our theory may be applied, to determine
whether the earth is more or less dense at the central parts than at the surface,*
or whether it be every where of a uniform density, as it ought to be, if, with-
out admitting very gross errors in the observations, it may be concluded, that
the earth is really the spheroid of Sir Isaac Newton; and this case would be
the simplest and the most natural of all.

I am here obliged to acknowledge, that if the observations we have made in
the north may be relied on,-|- and if we must admit as incontestable as well the
measure of a degree as the length of the pendulum, the foregoing theory could
not be reconciled to the phenomena. For it follows from our observa-
tions, that the diameter of the equator must exceed the earth's axis by more
than the 230th part; and that the gravity at the pole must be greater than that
at the equator by more than the 230th part also; which will by no means agree
with what we have deduced in art. 23.

As to what concerns the measure of gravity in Lapland, as being not so
liable to error as the measuring a degree; the earth may be not quite so flat as
Sir Isaac's spheroid requires. By the table of the length of the pendulum,
exhibited in the treatise concerning the figure of the earth, published this year
by Mr. de Maupertuis, and by art. 22 of the present discourse, the earth may
be more elevated at the equator than at the pole by the 266th part, or there-
abouts. After the true quantity of the earth's flatness shall be fully settled, if
it should be found to have this figure, I should be apt to think it is a little more
dense at the centre than towards the superficies. But if on the contrary we
should be well ascertained, that the earth is raised higher at the equator than at
the pole, by above the 230th part: and if, for any sufficient reason, we may
something shorten the length of the pendulum that beats seconds in the north;
there would be some grounds to allow, that the earth is not so dense at the
central regions as at those near the surface. But if it shall happen, that we
can neither diminish the length of the pendulum, nor the excess of the equa-
torial diameter above the axe, I must then give up my hypothesis. Yet I shall
think it may be of some use to have thus discussed it, because possibly no one
would have imagined what might have been the result of it. It appears that

• It has since been determined, by Dr. Hutton's calculations on the observations made at mount
Schehallien, in Scotland, that the mean density of the earth, is about 2 times more than at the sur-
face ; and that it is therefore probable the central parts of the earth are still much more dense. See
Philos. Trans, vol. 68, part 33.

+ It has since been found very erroneous. See p. 287, of this volume of Abridgments.



I

!



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 223

even Sir Isaac Newton was of opinion, that it was necessary the earth should
be more dense towards the centre, in order to be so much the flatter at the
poles; and that it followed from this greater flatness, that gravity increased so
much the more from the equator towards the pole.

New Experiments on Ice. By the Abb^ Nollet* F. R. S. at Paris. N° 449, p. 307 .

1. Ice that begins to melt, and water that begins to freeze, have always the
same degree of cold.

2. Thiit cold may be increased by a mixture of salts.

3. It has been thought for a long time, that saltpetre was most fit to increase
the cold of ice; but experiments have shown, that few salts increase cold so
little as that salt. Mix one part of fine saltpetre with two parts of beaten ice,
and Mons. Reaumur's thermometer will descend in it but 3-f degrees below the
freezing point.

What had caused this mistake, is, that people generally made use of salt-
petre of the first or second melting, as being the cheapest; but that salt-
petre, not being purified, contains a great deal of sea- salt ; and it was in pro-
portion to the quantity of the sea-salt that the effect was the greater.

From this last observation, one may deduce an advantageous method for
trying gunpowder; for as of the three ingredients of which it is made up,
salt-petre is the only one that can increase the cold of ice; if one pari of gun-
powder, or a little more, be mixed with two parts of ice; and it increases its
cold more than 3i degrees, it is a sign that the salt-petre contained in it is not
well purified; and the best powder will be that which least increases the cold of ice.

4. Sea-salt, that is the Bay-salt, which is commonly used at table in France,
and that which is immediately taken from the mines, called sal gemmae, usually
gives the greatest degree of cold; for pot-ash gives sometimes a little more,
but generally less. Sea-salt mixed with ice in the abovesaid proportion, gives
15 degrees of cold on Mons. Reaumur's thermometer, and sal-gem. 17.

5. Ashes of green wood 3 deg.

6 Sea-coal

7 Vitriol 2

8 Tartar 10

9 Common pot-ash (in French called sonde ordinaire) 3

10 Pot-ash made of se^ weed 11

This last pot-ash may be substituted instead of sea-salt, for making ice-

* The Abbd Nollet acquired a great degree of celebrity by his lectures and writings on experi-
mental philosophy, and particularly by his experiments and observations on electricity. He was a
member of the French Academy of Sciences, of the Royal Society of London, and of various other
ficientiflc institutions. He died at Paris in 1770, at the advanced age of 70.



224 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738.

creams, in places where salt is dear, as in France, where it is sold for lO sols
a pound. 1st, Because in France this pot-ash is sold only for 2-i- sols a pound.
2dly, Because, not freezing so fast, it does not spoil the creams by reducing
them to icicles. 3dly, Because ice-creams made this way, will keep longer in
a condition fit to serve at table.

11. Sugar J 4 deg.

12. Alum IJL

13. Salt of glass 10

14. Sal ammoniac 12f

] 5. Quick-lime 1^

1 6. Glaubers salt 2

17. The cold of ice may still be considerably increased by a mixture of spirit
of wine ; about a drinking-glass full of spirit of wine to a pound of beaten ice.

18. The cold of ice will not increase, unless the ice melts.
Experiments. — Put into one vessel 4 ounces of ice, beaten very small, and

into another vessel 2 ounces of sea-salt ; set the two vessels in a mixture of
ice and salt, which is to be renewed still, till by means of the thermometer
you find, that the salt and the ice of the first two vessels have acquired each of
them 10 or 12 degrees of cold; then mix the salt with ice, and this mixture
will not increase the degree of cold that the ingredients had acquired, because
the mixtcire does not melt. But if, instead of salt, you mix with the ice
spirit of nitre, cooled to the same degree as the ice, as this last is liquid, it
will melt the ice, and considerably increase its cold.

19. Salt mixed with water, increases its cold.

20. Of all salts, sal ammoniac gives the greatest degree of cold;* so that if
that salt has been cooled in ice, and then one part of it be thrown into 2 parts
of water, cooled to the same degree in ice, that water will become colder than
ice, and will freeze other water thrown into it in a small quantity.

This last observation may be applied to the cooling of liquors where no ice
is to be had ; for there is hardly any place, but what has wells : now the water
of a well moderately deep, wants about 8 or 10 degrees of the cold of ice;
and sal ammoniac being cooled beforehand in the well, will, by mixing with
some of the water of that well, come very near to the cold of ice.

An Observation of the Magnetic Needle being so affected by Cold, that it
would not traverse; by Capt. Christopher Middleton, F.R.S. N°449, p. 310.

In the Philos. Trans. N°418, Capt. M. mentioned a strange phenomenon
relating to the sea-compass, which he had frequently observed, when among

* Mr. "Walker of Oxford has shown that there are other salts which are preferable to sal ammoniac
for producing artificial cold, and particularly muriate of lime. See Phil. Trans, for 1795 and 1801.



VOL. XL.] PHILOSOPHICAL TRANSACTIOH8. 225

the ice in Hudson's Bay, viz. that the magnetic virtue of the needle was so
far lost or destroyed, that it would not traverse as usual, even when the ship
was in a considerable motion: and in his voyage thither last year, he observed
the compass would not move at all, any longer than the quarter-master kept
touching it. There was then much snow on the land, and many isles of ice
around them, and the sea not very smooth: he ordered one of the compasses
to be brought into the cabin, but did not find it any better, till it had stood
near the fire about -^ of an hour, and then it began to traverse very well; he
then ordered it to be placed in the binnacle, and another to be brought into the
cabin, changing them alternately thus every half hour, and found by this means
he could make them traverse as well as in any other part of the world. He
was obliged to continue this practice, till near 100 leagues from the coast ; but
afterwards he had no occasion for that trouble What should be the cause of
this wonderful phenomenon he could not conjecture, being certain the com-
passes, as to their mechanical structure, were very perfect, and answered very
well both before and after, during the whole voyage.

Concerning an uncommon Palsy of the Eye-lids. By Dr. jindrew Cantivell

of Monlpelier. N" 449, P- 3 1 1 .

A lady, about 30 years of age, was troubled with a very singular disease.
It was an intermitting periodical palsy of the eye-lids, which began every
evening about 6 o'clock, with a defluxion from the great canthus, of a whitish
matter of some consistence ; so that she remained blind till next morning, and
then recovered the use of her eye-lids as before. This disorder held her
for 4 months ; from which lime all remedies ordered by her physician proving
ineffectual, she was sent to Baleruc for the benefit of the waters.

As Dr. C. lodged in the same house, he had a fair opportunity of observing
the efFects the waters had on her. She was pumped on the back part of her
head and neck 7 times, without receiving any sensible benefit : the Qth time
her disorder seized her an hour later than usual, and the defluxion was less
and thinner. The next evening it retarded 1 hours, and the following night
she had as much command of her eye-lids as ever. She took the douche, for
so they call that way of pumping, the next morning and evening, and was
entirely cured. Dr. C. sat with her an hour that evening, carefully observed
her eye-lids by candle-light, and asked her several questions on her disorder.
She opened her eyes as well as he did his, and set out the next day for
Montpelier.

VOL. via. G G



226 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738.

An Account of the Man whose Arm with the Shoulder-blade was torn off by
a Mill, Aug. 15, 1737. By Mr. John Belchier, F. R.S. N° 449, p. 313.

Samuel Wood, about 26 years of age, being at work in one of the mills
near the Isle of Dogs, over-against Deptford, and going to fetch a sack
of corn from the farther part of the mill, in order to convey it up into the
hopper, carelessly took with him a rope, at the end of which was a slip-knot,
which he had put round his wrist ; and passing by one of the large wheels, the
cogs of it caught hold of the rope, and he not being able to disengage his
hand instantly, was drawn towards the wheel, and raised off the ground, till
his body being checked by the beam which supports the axis of the wheel, his
arm with the shoulder-blade was separated from it.

At the time the accident happened, he says he was not sensible of any pain,
but only felt a tingling about the wound, and being a good deal surprized, did
not know that his arm was torn off, till he saw it in the wheel: when he was a
little recovered, he came down a narrow ladder to the first floor of the mill,
where his brother was, who seeing his condition, ran down stairs immediately
out of the mill to a house adjacent to the next mill, which is about 100 yards
distant from the place where the accident happened, and alarmed the inhabit-
ants with what had happened to his brother; but before they could get out of
the house to his assistance, the poor man had walked by himself to within
about 10 yards of the house, where, being quite spent by the great effusion of
blood, he fainted away, and lay on the ground; they immediately took him
up, and carried him into the house, and strewed a large quantity of loaf-sugar
powdered into the wound, in order to stop the blood, till they could have the
assistance of a surgeon, whom they sent instantly for to Limehouse; but the
messenger being very much frighted, could not give the surgeon a clear idea of
the accident, so that when he came to see the condition the man was in, he
had no dressings with him for an accident of that kind ; but had brought with
him an apparatus for a broken arm, which he understood by what he could
learn from the messenger to be the case. However, he sent home for proper
dressings, and when he came to examine particularly into the wound, in order
to secure the large blood-vessels, there was not the least appearance of any,
nor any effusion of blood; so having first brought the fleshy parts of the
wound as near together as he could by means of a needle and ligature, he
dressed him up with a warm digestive, and applied a proper bandage : the next
morning he opened the wound again, in company with 2 surgeons more; and



VOL. XL.] VHILOSOFHICAL TRANSACTIONS. 127

not perceiving an effusion of blood at that time, he dressed him as before, and
sent him in the afterneon to St. Thomas's Hospital, where he was admitted a
patient under the care of Mr. Feme; from which time he was constantly at-
tended, in expectation of a hemorrhage of blood from the subclavian artery;
but there being no appearance of fresh bleeding, it was not thought proper to
remove the dressings during the space of 4 days, when Mr. Feme opened the
wound, at which time likewise there was not the least appearance of any blood-
vessels ; so he dressed him up again, and in about 1 months time the cure was
entirely completed.

On examining the arm, within a day or two after it was separated from the
body, they found the scapula fractured transversely, as were likewise the radius
and ulna in 2 places: but whether these bones were fractured before the arm
was torn off, the man cannot possibly judge. The muscles inserted into the
scapula were broken off near their insertions, but the muscles arising from the
scapula came away with it entire. The latissimus dorsi and pectoralis, were
likewise broken off near their insertions into the os humeri. The integuments
of the scapula, and upper part of the arm, were left on the body, as also the
clavicle.

But it is very surprising, that the subclavian artery, which could never be
got at to be secured by art, should not bleed at all after the first dressing; the
artery being separated so happily, that when the coats of it were contracted,
the fleshy parts pressed against the mouth of it, and prevented any effusion



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