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VOL. XL.J PHILOSOPHICAL TRANSACTIONS. 20Q

which may be supposed because of the smaUness of nm; we shall have
c* X PM* X P/> X — J for the attraction of any one of these elements ; put-
ting c for the ratio of the circumference to the radius, and «-for the given
ratio of mn to pm, that is, of de to cd.

Now if we make ca =: e, cb = r, am = z; and for pm, ap, p/», if we sub-
stitute their values expressed by z, and then seek the fluent of the foregoing

quantity ; we shall have — —^ for the value of the whole attraction of

the solid generated by the revolution of BDbsB : to which if we add - — , the

attraction of the sphere, we shall have — 1 — ^^ for the required

attraction of the spheroid on the corpuscle a.

Problem II. Supposing now tfie Spheroid Bsbe, fig. 10, to be no longer of a
homogeneous Matter, but to be composed of an infinite Number of Elliptical
Strata, all similar to BEb, the Densities of which are represented by the Ordi-
nates kt of any Curve ivhatever vx, oj which we have the Equation between
CK and KT ; the Attraction is required tvhich this Spheroid exerts on a Corpuscle
placed at the Pole b. — 2. Making bc ^ e, CK = r, by the foregoing proposi-
tion, we should have — 1 — -— — - for the attraction of the spheroid

KLK, if it consisted of homogeneous matter ; and the fluxion of this quantity
1 — would be the element or moment of the orb klk^M.



But because the density is variable, we must multiply this value of the attrac-
tion of the orb by kt, and tiie fluent of this quantity will be the value of the
attraction of the spheroid klk.

As to the value of kt, which expresses the density of the stratum KLKklk,
we shall take only fr'' -\- g?-^, because we shall see afterwards, that a value
more compounded, as _/r* -)- g-;' -}- hr' + ir', &c. which by the property of
series may express all curves, would not produce any variety in the calculation.

Therefore multiplying the foregoing equation by fr -\- gr^, we shall have

2cfxl + 2»xr^'^f _ ic»fr ^'''^ , 2c gx l + 2«xr ^+? _ 4c«gr^ + ? r ., .. r

eex3+p e*x5+p eex3 + q e^x5 + q

attraction of the spheroid klk, exerted on a corpuscle placed at b.
3. In this value making r = e, we shall have

~ h =^ h -■ + -L. ■ which will express the force of

3+p 3+PX5 + P 3 + q 3+JX5 + 9 '^

attraction of the spheroid be6, exerted on a corpuscle placed at the pole b.
Theorem. ^ Corpuscle being placed in any Point n of the Surface of the
VOL. vm. E E \



'210 PHILOSOPHICAL TRANSACTIONS. [anNO 1738.

foregoing Spheroid BEbe, / say it will undergo the same Attraction from this
Spheroid, as if it were placed at the Pole N of a second Spheroid revolving about
the A.ve no, the second Axe being the Radius of a Circle equal in Superficies
to the Ellipsis fg ; supposing this second Spheroid ngof (fig. 11) to be com-
posed of the Strata Minqa, whose Densities are the same as those of the Strata
KkLlKk, of the first Spheroid. — 4, In the discourse I had the honour of com-
municating to the R. S. being then at Toriieo, printed in the Philos. Trans.
N° 445, I have demonstrated this proposition as to a homogeneous spheroid ;
and the same reasoning will obtain in this case also.

Problem. III. To find the Attraction which the Spheroid BEbe (fig. 10)
exerts on a Corpuscle placed at any Point n of the Superficies. — 5. We will
make, as above, bc = e, ce = e + e«j and also cn = e + ex, and half the
conjugate diameter of cn will be cg = e + ea — ex \ whence the radius of a
circle, equal in superficies to the ellipsis pg, will be a mean proportional be-
tween CB and CG, that is, e -j- ea — ^x. Therefore the spheroid beZ'^ exerts
the same attraction at n, as would be exerted at the pole of a spheroid ngof.
(fig. 11) of which the principal axis would be no = 2e -|- lex, and the second
would be to the principal, as I + « — f a to 1.

Therefore in the expression of the attraction at the pole, (Art. 3) we must
substitute e -\- ex instead of e, and a — aa instead of a. But if _/ and g must
no longer be the same ; for we may easily perceive by the foregoing Theorem,
that the density must be the same in this spheroid ngof, at the distance r-\-rx
from the centre, as it is in the spheroid BE^e at the distance r. Therefore
f(-—^ -\- gir— y must be put instead oi fe -{■ ge . Thus we shall have

3+p 3 + p X 5 + p 3 + p y. 5+ p 3 + q 3+qx5+q S + q X S+q

for the attraction of the spheroid BEie at n.

6. If we make x = a, the foregoing expression will be reduced to this

2je^ ^cfe'+f» ^ej+9 2cge^ ^^.^^ expresses the attraction of the

3 + p ' 5 + p ' 3 + q ' 5 + q ' ^

equator.

7. If we would have the attraction at any point m within the spheroid, in
the expression of the attraction at n, we must put r instead of e. The proof „
of this is plain from the same reasons that Sir Isaac Newton makes use of, |
(Corol. 3, Prop. Ql, I. 1, Princip. Math.) to show that the attraction of an ''
elliptic orb, at a point within it, is none at all.

Problem IV. Let Rllrn- (fig. 12) be a Circle whose Centre my; it is re-
quired to find the Attraction which this Circle exerts on a Corpuscle at n, ac-



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 21 I

cording to the Direction nx ; supposing the point h, which answers perpendicu-
larly below t/ie Point n, to be at a very small Distance from the Point y. —
8. Let there be drawn riHTr perpendicular to the diameter RYr, and let the
space RllTr be transferred to ttIIz. Then the space Trzllr will be the only part
of RflrTr, which will attract the body n according to hy.

To find the attraction of this little space, we will suppose it to be divided
into the elements tIss, the attractions of which, according to hy, will be

TtSS X QT 2HV X Q? X QT . , n . c . • 1 2HY X HQTZ . , .

-— , or , , the fluent or which j is the attraction

NT* ' Nt' ' NT'

of Tzrs, according to hy. In which if we put llTr for Ha, we shall have
nH.Hx2HY ^^ j HY X riH' X c ^^^ ^j^^ attraction required.

NT' NT' '

Q, It is easy to perceive, that if, instead of a circle, the curve Rllr were an
ellipsis, or any other curve whose axes were but very little different from one
another, the foregoing solution would be still the same.

Problem V. To find the Attraction which an Elliptical Spheroid klIi (fig. 1 3)
exerts on a Corpuscle placed without its Surface at n, according to the Direction
ciL perpendicular to cs. — 10. To perform this, we will begin by drawing the
diameter CfAv, which bisects the lines Rr perpendicular to cn; and the ratio of
CH to HY shall be called n. Then accounting the ellipsis Rr as a circle, see the
foregoing article, we shall have, by the foregoing problem, t^'^xrh xch ^^^
its attraction, according to hy ; which being multiplied by the fluxion of mh,
the fluent of this will be the attraction of the segment of the spheroid RMr.

This calculation being made, and not being substituted for nr, we shall have
-—J- for the attraction of the spheroid in n, according to the direction ex.

Problem VI. To find the Attraction of a Corpuscle n, according to ex,
towards an Ellipsoid BNEbe, composed of Strata, the Densities of which are
defined by the Equation D = fr"" + gf^- — H- Take the fluxion of the quantity
— -, which expresses the attraction of the homogeneous ellipsoid klA, and

you will have — ;— for the attraction of an infinitely little elliptic orb; which,

being multiplied by the density d, gives -^^ ^ -^ — ^^ '-, the fluent of

which -i^ 1- —f^I i— !, is the attraction of the spheroid kl/^, according to

5 + jJXe^ 5+yxe* ^

ex. Therefore the total attraction of the spheroid sNE/jie on the corpuscle n,

according to the direction ex, will be ^^f U ?^— .

^ 5+7' ^ 3 + 9

£ B 2



212 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738

Now if we have regard to the smallness of the line nv, and observe how
little angle vnc will differ from a right one, we may perceive that the diameter
CN contains the same angle with the perpendicular nx in n, as the diameter
CN with the perpendicular at v; that is, that the angle ncv is the same as the
angle cnx; so that instead of n we may take — . Therefore the foregoing ex-
pression of the attraction of the ellipsoid BEbe, acting according to the direc-
tion ex, on a corpuscle placed in n, will be -f * x — + — f^ X — .

' ' ' 5+p CN 5+q CN

Problem VII. To find the Direction of the Attraction of a Corpuscle n
towards the Ellipsoid. — 12. By the second Problem we shall find the attraction

of the spheroid according to on to be —^ \- ^^ — , by expunging what

may be here expunged. Then by taking a 4th proportional to these 3 quanti-
ties, the first of which is the attraction according to cn, the 2d is that accord-
ing to ex, and the third is the right line cn; there will arise

5 + p "*" 5 + q _
rl+P , +o X CX CL.

3+p "^ 3 + 9

Whence we shall have ni for the direction required, of the attraction of the
corpuscle n.

1 3. If we suppose p z= q ■= o, that is, if the spheroid be homogeneous, we
shall have ci = ^cx ; which agrees with what Mr. Stirling has found, in that
curious dissertation he has published in the Philos. Trans. N° 438.

Part II. The Use of the foregoing Problems, in finding the Figure of
Spheroids, which revolve about an Axis. — 14. Let us now suppose, that the
foregoing spheroid bneZ^c, (fig. 13) which is still composed of strata of differ-
ent densities, revolves about its axis Bi, and that it is now arrived at its per-
manent state. It is plain that the particles of the fluid, which are on its sur-
face, must gravitate according to a direction perpendicular to the curvature
bne; for without this condition there could be no equilibrium.

We shall now inquire, whether the elliptic figure we have ascribed to our
spheroids can have this property, and to produce this effect, what must be the
relation between the time of revolution of the spheroid and the difference of
its axes.

Let us then put (p for the centrifugal force at the equator, and the centrifugal

'111 ^XPN ^X CX

force at n will be , or , because 2pn X a, = ex.

CE ' 2CE X a.



VOL. XL.] PHILOSOFHICAL TRANSACTIONS. 213

By resolving this centrifugal force according to the perpendicular to cn, we
shall have ^ ^ ^^ ; to which adding -^^-^-; X — + ~-, X — , found by

2<» X CE ° 5 + p CN ' 5 + J CN •'

by prob. 5, will give the whole force of the body n, according to the direction
ex, when the spheroid is turned about its axis. But because this body, by
virtue of the attraction according to cn, and the force according to ex, ought
to have a perpendicular tendency to the superficies; we shall have this analogy,

CN : ex :: -f- — + -I- : — X \- -^, X §- X — . And

3+P 3 + (j 2» ce' 5 + p cn' 5 + q cn

hence, because cn and ce may be assumed as the same on this occasion, it

Will be ip = — == +



3 + p X 5 + p 3 + q X 5 + q

The Spheroid being supposed ellipical. Bodies will gravitate perpendicularly

to its Surface.

And as in this value of the centrifugal force, no quantity enters but what
will agree to any point n; we may therefore conclude, that when our supposed
elliptical spheroid performs its rotation in a proper time, so that the centrifugal
force at the equator may be as before ; then the centrifugal force in any other
place n will be such as it ought to be, to cause bodies to gravitate in a direction
perpendicular to the surface.

The Expression for the Gravity at any Place on the Spheroid.

15. If we now consider, that ed (fig. 14) being taken for the centrifugal
force in e, then will mn express the centrifugal force in n, and consequently
MI will be such a part of this force as acts according to nc ; we shall have

+ = — == to be subtracted from the attraction at n. Hence



3+px5+p 3+qx5+q

•Zcfe'-^f 2p- Wcfi^e'+f Scfue' + f 2cge' + ^ 2y - lOcg^e'+f

3+p 3+~p X sTp 3+px 5 + p 3 + q "*" 3~+Tx 5TT

' scgxe' "'"^

— ■ - Will be the gravity at n. ~

S + q X 5 + q ^ ^

The Gravity at the Equator.

16. In this value making x = «, we shall have

2c/e'+>* 2F^';/-«e'+^ , 2cg-e' +? , 27-^c?«e'+^ ,

-r—. — + ' ~^^= + + — =— for the gravity at the

3+p ' 3+px S+p ' 3 + q ' 3 + qx 5 + q ^ ■>

equator.

17. If we subtract the value of the gravity in n from the value of the at
traction or gravity at the pole, in art. 3, we shall have



ai4



PHILOSOPHICAL TRANSACTIONS.



[anno 1738i



10 — 2pcfxe



1+^



+



10 — iqcg>e



+ ?



'■=^^=:r. Bat it is easy to perceive, that x is pro-



3 + PX5+P Z-\- qy. S + q

portional to the square of the sine of the arc pm, or of the complement of the
latitude. Whence we may therefore conclude, that the diminution of the
gravity from the pole to the equator, is proportional to the square of the cosine
of the latitude ; or, which is the same thing, that the augmentation of gravity
from the equator to the pole, is as the square of the sine of the latitude, as Sir
Isaac Newton has demonstrated in his hypothesis of a homogeneous spheroid.

]8. From the following calculation it is easy to conclude, that Sir Isaac's
theorem, (Prin. Math. lib. 3, prop. 20) which is this, that the gravity in any
place within, is reciprocally as the distance from the centre, cannot obtain here.
For we may see by the foregoing expression, that the gravity in n cannot be to
the gravity in p, as 1 to 1 + ^> except when p = g = O, which happens only
in Sir Isaac's homogeneous spheroid.

It was for want of considering, that this theorem was demonstrated by Sir
Isaac only in the case of his homogeneous spheroid, that several geometricians
have too hastily concluded, that this theorem might be applied to determine the
ratio of the earth's axes, and the lengths of the pendulum observed in two places
of different latitudes. Dr. Gregory is one of those who have fallen into this
mistake, in his Elements of Astronomy, lib. 3, sect. 8, prop. 52. And in the
Philos. Trans. N° 432, it is concluded, from the proportion of gravity at
Jamaica to that at London, that the diameter of the equator must exceed the
earth's axis by the JQOth part, which computation was founded on this 20th
proposition, lib. 3, of Sir Isaac's Principia, which is true only of his spheroid.
The Manner of priding the ^xes of the Spheroid, the Variation of the Densities
of the Strata being taken at pleasure.

IQ. Let us now suppose, that the centrifugal force at the equator is known
by observation, as also within the earth, &c. and that it is a certain part, as the
mth part of the gravity ; by articles 14 and l6, we shall have this equation :



2c/i



■ +;■



+



2p-



2c/e'+^« ^cge'-^f '2q - Zcge^ + ?» 8cfme' + ?a



+



+



Scmge^ "' ^«
"*" F+qxT+q'



S+p ' 3 + px5+p ■ 3 + q 3 + qx5 + g 3+px5 + p

From hence it will be easy to derive the value of «, because y, g,p, q, will be
given, from the hypothesis that will be chosen, for the variation of the density
in the internal parts of the spheroid.

20. And if on the contrary a. be given, that is, if we know by observation
the ratio of the axes of the planet concerned ; then by the foregoing equation
we may perceive, whether we have assumed an agreeable hypothesis for the
variation of the densities : but we cannot precisely determine what this hypo-



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 2)5

thesis must be, because there is but one equation, in which 4 indeterminate
quantities/, g-,jb, q, are involved. And indeed there might be many (nore than
4 indeterminate quantities, if we should assume more than two terms in the
general equation of the densities d = frf + gri + hr', &c.

21. In order to apply the foregoing theory to the earth, it might seem at first
sight, that by the assistance of observations made for measuring the length of
the pendulum, we might have other equations, which with the foregoing
equation a, would determine the coefficients and exponents now mentioned ;
but we shall soon see the impossibility of this, on two accounts : first, there
need be only two observations, as to what concerns the length of the" pendu-
lum. For because, by art. 17, the augmentation of the gravity from the
equator to the pole, is proportional to the square of the sine of the latitude,
two observations as much determine the problem as an infinite number can do:
so that we could have but one other equation besides the foregoing. This

5—pf* , 5-qg»

■n 1 /■ \ P—P 3+PX5+P 3 + 9x5 + ?
equation will be (b) = -^_ ^ — — = •

P p-'^f' , _/_ 1 _£_ I 1-^S 1_

3 + px5+p~^ 3+p "*" 3 + 9 3 + qx5 + q

The first member of this equation expresses the gravity at the equator sub-
tracted from the gravity at the pole, and divided by the gravity at the equator ;
a quantity which may be known in numbers, by determining the length of the
pendulum at two difl^erent latitudes. The other member of the equation is an
expression of the same quantity, as it is deduced by the preceding calculus.

Secondly, This new equation b cannot be of any service in determining the
coefficients and exponents J", g, p, q, &c. For we shall now show, that the

foregoing ratio ^ — - has such an immediate connection with a, that one of them
being determined, the other will necessarily be so too, independently of the
values o{ f, g, p, q, &c. This may deserve our attention, and the proof is
thus :

The Figure of the Spheroid being known, the Augmentation of Gravity from the
Equator to the Pole will be known also ; and so vice versa,
22. Because the ratio of the gravity to the centrifugal force is very great, and
is expressed by m, in the equation a we may reject the third and fourth
terms ; by which means the equation will be reduced to this,
f g imf» imgit

— — + ^-, — = :r-; — ■ -r-r= + Tt^^'t^t^- ^"° " "■°'" this equation

3+p 3+9 3+PX5+P 3+9X5+9 ^

we deduce the value either off or g, and substitute it in the equation b ; having
first rejected the first and fourth terms of the denominator, as in this case may
be done ; we shall have, after the calculation is made, whatever is the number



2l6 PHILOSOPHICAL TRANSACTIONS. [aNNO 1738.

of terms in the equation of the densities, ^—^ = «, or ^^ = — a.

^ ' p im ' p 115 *>

by putting 288 for m, as has been long known. It is easily seen from this

equation, that when a is determined, will be so too, which was the thing

proposed to be proved.

23. But from this equation there follows a very singular proposition, and
which, in some sort, is contrary to the sentiments of Sir Isaac Newton, page
430 of the 3d edition of his Principles. And this is, that if by observation it
shall be discovered, that the earth is flatter than according to the spheroid of
Sir Isaac, that is, if the diameter of the equator exceeds the axis by more than
the 230th part, the gravity will increase less from the equator towards the pole,
than according to the table which he has given for his spheroid, prop. 20 of the
3d book. And on the contrary, if the spheroid is not so flat, the gravity will
increase more from the equator towards the pole.

24. It is thus that Sir Isaac Newton expresses himself about it, when he re-
lates the experiments made towards the south, concerning the diminution of
gravity, which experiments make it greater than his theory requires. He
affirms, that the earth is denser towards the centre than at the superficies, and
more depressed than his spheroid requires. But by the foregoing theory we
may easily perceive, that if the density of the earth diminishes from the centre
towards the superficies, the diminution of gravity from the pole towards the
equator will be greater than according to Sir Isaac's table; but at the same time
the earth will be not so much depressed as his spheroid requires, instead of be-
ing more so, as he affirms. Yet I would not by any means be understood to
decide against Sir Isaac's determination, because I cannot be assured of his
meaning, when he tells us, that the density of the earth diminishes from the
centre towards the circumference. He does not explain this, and perhaps in-
stead of the earth's being composed of parallel strata, its parts may be con-
ceived to be otherwise arranged and disposed, so as that Sir Isaac's proposition
shall be agreeable to the truth.

25. As to Dr. Gregory, who has attempted to comment on this passage of
Sir Isaac, I think I have demonstrated, that he has committed a paralogism.
He says (Element. Astronom. lib. 3, § 8, prop. 52 schol.) that if the earth is
denser towards the centre, or if, for example, it has a nucleus of greater weight
than the other parts, the diminution of gravity from the pole towards the
equator shall be greater than if the whole were of the same density; and in this
he is right. But he is in the wrong, I think, immediately to conclude from
thence, that the earth has a greater flatness. Whence can he conclude this ?



VOL. XL.] PHILOSOPHICAL TRANSACTIONS. 217

it can be only from that proposition of Sir Isaac which informs us, that gravity
is in a reciprocal ratio of the distances: because he gave us the proposition but
the page before, as a method for determining the figure of the earth. But we
are not allowed to make use of this proposition in this case, because it has been
shown, art, 1 8, that it can take place only on the supposition of a homogeneous
spheroid. Therefore, &c.

26. It will not be very difficult, without any regard had to the foregoing
theory, to find the ratio of the axes of a spheroid, which we may suppose to
have a nucleus at the centre, of greater density than the rest of the planet;
and hence we shall be easily assured of Dr. Gregory's mistake.

27. Setting aside all attraction of the parts of matter, if the action of gra-
vity is directed towards a centre, and is in the reciprocal ratio of the squares of
the distances, the ratio of the axes of the spheroid will then be that of 576 to
577: and the gravity at the pole is greater than at the equator by the 144th
part, or thereabouts. Which may be a confirmation of what is here advanced,
especially to such as will not be at the pains of going through the foregoing
calculations. For we may consider the spheroid now mentioned, in which gra-
vity acts in a reciprocal ratio of the squares of the distances, as composed of
matter of such rarity, in respect of that at the centre, that the gravity is pro-
duced only by the attraction of the centre or nucleus.

28. In the foregoing calculations, in order to find the axes of our spheroids,
and to know whether their figure makes a sensible approach to that of the
conical ellipsis, we have had recourse to this principle, that gravity ought
always to act in a direction perpendicular to the surface. Two reasons have
prevailed with us to make use of this principle, rather than the other, which
consists in the equilibrium of the columns. The first is, because the calcula-
tions founded on it are more simple. The second is, that considering the state
of the actual solidity of the earth, it should seem as if this principle were the
more indispensably necessary. However, because Sir Isaac Newton, and all the
other philosophers, who have treated about the figure of the earth, have taken
it, as it were, at its first formation, at which time they suppose it to have been
fluid: we shall here make the same supposition, and we shall assume no other
ratio for that of the two axes, than that of the spheroid which results from a-
coincidence of these two principles.

We shall begin by inquiring what is the whole weight of any column cn,
fig. 15. To do this, we must resume the expression of the attraction in any
point M of the column cn; then multiply it by r -f- xr, and by the density
frP -\- gr9, and afterwards we must find the fluent. Thus we shall have

VOL. VIII. Pf.



218



c/»e*+='^



+



PHILOSOPHICAL TRANSACTIONS



cg'e* + ^?



+



+



[anno 1738.



\ +p X3+P l + 9X3 + g 2 + J» + ?x3+f 2 + p + gx3+</



+ =-



+



+



1 +p X 3 +;> X 5 +p \-\-qx3 + qxS + q 2 + P + ? X 3 + p x 3 + 5



+



8c/g«c



«+^ + f



+



2 + ;) + ?x3 + gx5 + 9
8 4.4;>cg/'Ae^+^ + ?



+



4 + ^pcfye"- + ^Z"



+



l+px3+px5+p
8 + 4?c^Ae*+''+?



+



4 + 2qeg'^>.e* + ?



3 + ?x5 + ?Xl+/>
for the total gravity of



S + p + q X 3 + px 5 +p i + p -\-q X 3 + q X 5 + q

any column cn, having regard only to the attraction.

29. If in this expression we make a = O, we shall have the gravity of the
column at the pole.

30. And if we make x := «, we shall have the aggregate of the attractions of
the column at the equator.

31. Now because the column on is in equilibrio with the column cb; it



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