Pierre Simon Laplace.

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depends upon the oblateness of Saturn. For this purpose we shall put t j.^^^^^
for the time of Saturn^ s rotation upon its axis ; and we shall have,*

We have, by observation,

t = 0'"ys428 ; a = 59, 1 54 ; [7676]

hence we deduce,

9 = 0,165970. [7677]

We shall suppose that the oblateness of the earth is to the value of <p, which
corresponds to it, as the oblateness of Saturn is to the corresponding value

[7678]

* (3706) We may deduce, from [7665^], the expression of the centrifugal force of a ry/^-c ,
particle on the surface of the spheroid of Saturn, by changing a into 1 [7598'], and T into

43r2

t [7674] ; by this means it becomes -— . Dividing this by the expression of the gravity [76756 ]

of the same particle towards Saturn, [7665Zc], we get the expression of <p

[6044', 75876], as in [7675]; substituting the values [7670,7676], we obtain [7677J.
VOL. IV. 84

334 MOTIONS OF THE SATELLITES OF SATURN. [Mec. C^L

of (p.* This principle being found, in [2069], to hold good very nearly for

Jupiter compared with the earth. Now for the earth we have (p = —

[7679] , 1

[1594«] ; and if we suppose the oblateness of the earth to be — r, in

conformity with experiments on the pendulum [2048'], we shall have, as

in [7674c],

243 T2

[7680] p — l(p = — - . — — .

[7680'] Then by noticing only the part of K' [7621], depending upon this quantity,

Assumed i ii r* l i

value of we snail iind,t

[7681] K' = K.^^.^=0,m9.K.

* (3707) Using 9=-—, p = —- [7679], relative to the earth, we have,

^ ^^ 335 ^ 289 670x289 670 ^'

243

[76795] and if we suppose, as in [2069], that the same ratio p — ^9= — .9, holds good for

243 T^

[7679c] Saturn, we shall have, from [7675], p — 1? = ^i;;; X ^5-^ , as in [7680]. If we suppose

the oblateness of the earth to be — :, as in [2056^;], instead of —^ [7679], the expression

[7679a, c] will become,

._-„„,, , 1.1 278 278 278 T^

[7679i] p — iqj = i . — - = = — . = — . ,

^ •" '^ '^ ^ 300 2 289 000x289 600 ^ 600 Pm^ '

[7679e]

which is about a quarter part more than the expression [7680] ; and the value of K'
[7681] will be increased in the same ratio.

f (3708) Dividing [7680] by a^, we get the first of the following expressions
[7680a], which, by successive reductions, and the substitution of the second value of K
[7668], becomes as in [76806] ;

[7680a] P:zA:?_243 T^^ _-243 1_ / T^\ / T^^X _ 243 Jl^ A T^X

a2 670 • f2.a5 670 * a^ \** Ty ' y' t^ J ^ 670 * a^ ' V* t^ J

[76806] = — . J- K —

335 a5 i3 •

This last expression is the same as the first of those in [7681], and by the substitution of
[7680c] the values of a, t, T [7676, 7670], it becomes 0,4219.iSr, as in [7681]. This would

be increased about one fourth part [7679e], by using the corrected ellipticity — - [7679^?].

0\)\)

VIII. xvii. ^36.] THEORY OF THE OUTER SATELLITE. 335

We cannot suppose K' to be less than this value, because it is increased r-r/^Q,,,

l_7ool J

by the action of the inner satellites and by that of the ring.*

We have by observation A = SS'^^SSSS ; hence this value of K gives,t [7682]

d = 24°,0083; ^— ^ = 9°,3250; [7683]

9= 0,03926.Z"; p = 1,30412.Z; [7684]
The observations of Bernard in Marseilles, in 1787, give,

X = 25°,222 ; [7588] [7685]

7 = 13^,593. [7599] [7686]

* (3709) Putting for brevity, for a moment, h = — 2.sin.7.cos.7.sin.(v — ^), we find ry^g, ,
/fjjj \ ^ ■'

itiaX the part of «-(-^) [7602], depending on the ellipticity of Saturn's mass, is

— —-hj and that depending on the interior satellite [7617], is rr^T^**^* Dividing [76816]

these by A, we get the corresponding parts of K' [7621, 76216, 7617], namely,

p ^(p Sm'.d^a'^

— r~ ^^^ A I » , /o^s > ^^^ as this last term is positive, it follows that the sum of the [7681c ]

similar terms, corresponding to the action of all the satellites, or ^B [7621], must be
positive, or of the same sign as — — ; so that IC ]>• [7621]. Now the expression [7681rf]

of K' = 0,4219. K [7681], is computed upon the supposition that B is nothing ; hence
we have IC^0,i2\9.K, as in [7681'] ; observing also that this coefficient 0,4219, will [7681e]

be increased about a quarter part, by using the corrected ellipticity — [7680c].

t (3710) If we put c = 0,4219, we shall have X' = c^[7681]; substituting this [7682a]

in [7646], we get tang.2a= ^"'^1^ . The value of A = 33° i [1682], gives

c-\-co3.^ [76826]

cos.2.^ = J, sin.2v2 = |\/3 ; hence we obtain tang.2^= r— - ; and from this we get

6 [7683]. Now if we substitute the preceding values K'=cK, cos.2A==^, in [7682c]
[7649, 7650], we shall get,

q = 1 Z".! 1+c— v/i+c+r2j ; p = l K{ l+c+v/i+H=^|— ? ; [7682rf]

and by using the value of c [7682a], we obtain the expressions of p, q [7684]. As the
expression of c ought to be increased above a quarter part [7681 e, 7682a], we shall make
a rough estimate of the effect of this correction upon the values of 6, p, q, supposing [7682e]
merely for an example that c is increased to c = 0,6. This gives,

tang.2^=-^ = 0,787 = tang.42°,46 [76826]; or d = 21°,23 ; [7682/]

4,4

and y/i-^c+c2 = v/T^ = lj4 ; therefore we have, from [76826?], q:=0,05.K, [7682g-]
^=l,45.ir, instead of the values [7684].

336 MOTIONS OF THE SATELLITES OF SATURN. [Mec. C^l.

Hence we deduce,*

[7687] Y=71°,354

[7688] ts = 16°,96l

[7689] n=:37°,789

consequently,!
[7690] IP = 0,00000364437.

We have then very nearly, by the preceding formulas, J

* (3711) In the triangle JS'Hl, fig. 98, page 324, we have, by comparing the symbols
[7687a] |-^g28c-/] with their numerical values [7682-7686], the three angles JVf/J=7=13°,593,
[76876 ] HNI= X = 25^222, NIH = 200°— .4 = 166° ,6667 ; to find, by the common rules

of spherical trigonometry [1345^, 1345^^], the sides JVfl = * =71°,354 [7687], and
[7687c ] JV7'= 24°,926. Then in the spherical triangle JVIK we have the preceding value of

JV/=24°,926; the angle J5riV/=X = 25°,222 [7685] ; and the angle,
[7687d] KIN=200°—{^— 6) = 190°, 61 50, [7683];

to find, by the formula [134539], the side Kl= 200°— n = 162°,211, or n = 37°,789

[7689]; and then, from [1345^5-|^ the angle JVJr/=TO = 16°,961 [7688].

[7687c ]
[7690a]

f (3712) From [7654] we get b^ = s\n.^-a.(p — g'.cos.2n) ; and by substituting the
values of p, q, -us, n [7684, 7688, 7689], it becomes as in [7690].

X (3713) If we substitute the value of K [7671] in [76S4], we shall see that p is
[7691a] 2 yip

much larger than q, or 62 [7690] ; so that the fractions ~ , — |- , must be quite small.

Again, by multiplying together the two radical expressions in the denominator of [7658],

and putting, for brevity, p^ — q^ — h^p = m^, we get dv = ■ / ^ -^ . Developing

[7691c 1 ^1^® radical, and multiplying the whole expression by m, we obtain nearly

mdv = dn'A- ^— - . </n'.cos.2n'. Integrating, and adding the constant C, we have
2m"

[7691rf] C-\- mv = n'-j- -7— 2 ' sin.2n' ; or, by transposition, n' = C-\-mv — — . sin.2n' ; and by

subtracting p.sin.2n' from both sides of the equation

( 62(7 ■)

[7691e] n— 3.sin.2n' = C+mu— ) P+ 4 2 ( ' sin.2n'.

]VIoreover p [7660] is of the order — ; and if we neglect terms of the order — , we shall
get, from [7659], n = n' — ^.sin.2n'; hence the equation [7691e] becomes,
[7691/-] n = C + mv— ^ |3+ ^J • sin.2n'.

If we neglect terms of the order ^, in the value of n' [769le], we shall get n'= C-\-mv ;

VIII.XTO.'§.36.] THEORY OP THE OUTER SATELLITE. 337

n = C +v.\/f-f-ty'p - J ^ + 4(^!^lt.^) \ ■ sin.2. { C+v.,y^^If^p } ; [7691]

Reducing this expression to numbers, and determining the arbitrary constant r769i/i
quantity C, so that n may be equal to 37°,789 in 1787, we obtain,* Motion of

the node.

n = 38°,721+t.944",805— 9937",7.sin.2.(38°,721+t.944",805) ; [7692]

i being the number of Julian years elapsed since 1787. [7692^

These results depend upon the accuracy of the observations above quoted,
and particularly upon the ratio of K' to K [7681]. This last quantity
depends upon so many and such different elements, which are so difficult to
ascertain, that it is almost impossible to determine it a priori. We may find
it a posteriori, after we have determined exactly, by observation, the annual [76931
motion of the orbit of the satellite upon Saturn's orbit. For if we suppose

and by substituting it in the last term of n [7691/], we obtain,

n= C+mu— I ^ + ^ ] .sin.2.(C+nj«) ; [7691^]

which is reduced to the form [7691], by re-substituting the value of m [7691 i].

* (3714) The revolution of the outer satellite of Saturn is completed in 79^*y%3296,
[7669] ; therefore the mean value of the arc v, described in i Julian years, is

tJ = i.-r^^r— -.4000000" = 18416830".i; moreover, by using the values of K, p, [7692a]

q, h^ [7671, 7684, 7690], we get /^a _ g2 _ 52^ == 0,000051294 ; hence [76926]
i?.y/p2_^2 — b'Hp = 944",7.i. Again, the same values of p, q, l^, give

— —^^1-—— = 0,000554 ; also, — = sin.l°,9168=sin.i? : whence, [7692c]

62(7

p = tang.J5 = 0,01 5055 [7658c?], and p-f- _-^ - i_p— = 0,015609 ; [7692rf]

multiplying this by the radius in seconds, it becomes 9937",7 ; substituting these
numerical values in [7691], we obtain,

U = C +944",7.i — 9937",7.sin.2.( C +944",7.i). [7692e ]

To find C, we must put, as in [7691', 7692^], i = 0, and n=37°,789; and then [7692/]
the equation [7692e] becomes,

37%789 = C— 9937",7.sin.2C ; or C= 37°,789+9937",7.sin.2C. [7692^1

From this last expression we easily perceive that C must be nearly equal to 38°, and
9937",7.sin.2C nearly equal to 9937",7.sin.76° = 9240" ; so that

C=37°,789 + 0°,924==38°,713 nearly; [7692^]

and by repeating the calculation with this new value, it becomes C = 38°,721. Hence ,^^0^.,
the expression [7692e] becomes very nearly as in [7692].
VOL. IV. 85

338

MOTIONS OF THE SATELLITES OF SATURN.

[Mec. Cel.

[7693'] that the fixed plane, to which we have referred the orbit of the satellite, is
[7694] the orbit of Saturn itself, we shall have * zs = \ r = ; and the preceding
analysis will give,

* (3715) If we suppose the
point K' in fig. 98, page 324, to
fall in JV, or the point K to be
200° distant from the point JV, the
arc IK will coincide with the sun's
orbit JVIJV^, and we shall have the
angle flJ?7 = angle HJVI, or
TO = X [7628e]. IVToreover the
expression of r = 200° — NK
[1628d] becomes r=200°-200°=0
[7695a], as in [7694], corresponding
to the annexed fig. 99. Substituting
«=X, r = 0, in [7637, 7637'J,
[7695c] and dividing the last of these
expressions by sin.X, we get

[7695a]

[76956]

[7695d]
[7695e]

[7695/ ]

[7695g-]
[7695^]

[7695, 7696] respectively ; which may be put under the following forms ;
^ = — ;^ . { A.sm.7.cos.7.sm.T} ;

dn _

-r- = jDl.cos.X-

dv

K ( sin.X )

IVIultiplving these equations by dv, integrating and neglecting the variations of the
coefficients of dv, we get the following values of the variations of X, n, which we shall
represent by 6\ Jn, respectively ;

K'

"5^ = — -^ • {K.sm.y.cos.y.swx.irl.Vj

in = 5 ^.cos.X- ^ . \k. '^^!;^^ .cos.^1 I .«.
C ^ L sm.X J 5

Substituting in these expressions the values K, X, 7, y, v [7671,7685,7686,7687,7692a],

K'

and retaining the factor -— , they become,

K

[7695t]
[7695fc]
[7695n

K'

5X = — 140",03.^.r

-5n = ^ — 692",76+175",27. ^l-i]

K'

as in [7697, 7699]; and by using the value of — = 0,4219 [7681], they become

6X= -59",074.i, — 5n = — 618^81.^, as in [7698, 7700].

Some objection having been made to the accuracy of the equation [7696], by M. Plana,
in a paper published in the second volume of the^ Memoirs of the Astronomical Society of
London, page 346, etc., another demonstration was given by La Place, in the Conmisance

VIII. xvii. ^ 36.] THEORY OF THE OUTER SATELLITE. 339

— - = — ^'.sin.r.cos.y.sin.^ ; [7695]
av

du -_ ^y., sin.y.cos.r r'r«ofli

-r- = K.cos. X — K', — T . cos.Y. [7696]

av sin.X

des terns for the year 1829, page 248, by means of the formulas [5786A, i] ; which, by nQg^jfi]
neglecting terms of the order e^, and dividing by ndt, become,

ndt sin.y' \dd'/ ' ndt sin.y \dyj

As it will serve for an example for illustrating the use of these formulas, we shall here give

the substance of his demonstration. In these formulas, y represents the inclination of the

satellite's disturbed orbit to the fixed plane, and ^ the longitude of the ascending node of r«^q^ -■

the same orbit upon the same plane, and counted from a fixed point in that plane, as the

origin of the longitudes ; as is evident from the definitions in [5786c — d]. Now if we

neglect the secular motions of the equator and orbit of Saturn, taking the sun's relative orbit [7695p]

about Saturn FJVl, fig. 99, page 338, for the fixed plane, and F for the fixed point,

from which the longitudes are counted, we shall have, according to the notation which is

used in [7695n], / = angle Ji^Y/, arcFJV=^. But we have supposed, in [7628e], [7695g]

that the angle HJVI = X ; hence y = X. Moreover, if we suppose the longitude of [7 695r]

the ascending node of the equator upon the fixed plane to be represented by i^/=a, we ryggs-i

shall have FN = FI—JVI=^ a — JVI. Now in fig. 99 we have supposed that the points

K'j iV" coincide, therefore N1=K'1=ti [7628c]; and by substituting the values of [7695* ]

FN, NI [1695q,t] in [7695^], we get ^ = cl — n, whose differential is d6f = — dn; [7695m]

a being considered as a constant quantity, because the secular equations of the orbit are not

here taken into consideration [7695/?]. Substituting the values /=X, d6f = — dn, ^ ^J

[7695r, u] in [7695n], we obtain,

d\_ a_ /dR\ dn a_ /dR\ nm^

ndt sin.X * \dn J ' ndt ein.X ' \dX / * ■'

If we neglect the terms of R depending on the action of the interior satellites and rings,
we may put B=:0 [7619] ; and then we shall have, in [7621],

^ 3S.a^ ^, p — i9

•^=4^' ^=^"^'' ^^^^^^'^

and the parts of R which are to be noticed are, that in [7587] depending on the sun's
action, and that in [7597] depending on the ellipticity of Saturn. After substituting [7596a] [7695^1
in [7587], and reducing, we, must neglect all the constant quantities became they produce
nothing in [7695mj] ; we must also neglect the terms containing U or its multiples, because
these periodical quantities are not here noticed ; and if we also neglect the terms of the
order s, we shall find that the only term of [7596a], which requires notice, is the square
of the second term of the second member, namely, «^i)^.sin.^C7.cos.^X.sin.^«, which [7696al
can be reduced to the form i.a^D^.cos.^X-|-&tc. , as is evident by substituting
s'm.^U=l — i.cos.2C7; sin.^v = ^ — ^.cos.2t; ; hence we get from the last term of

[76952]

[7696d]

340 MOTIONS OF THE SATELLITES OF SATURN. [Mec. C61.

K'

[7697] Substituting the preceding values of r, * and \ we find 140",03. -— , for

[7698] the annual decrease of x in 1787; which becomes — 59",074, by adopting

[76966] R [7587], the following expression, flJR = — ^^.cos.2x = — Jjf.cos.^x [7695a?],

[7696c ] depending upon the sun's action. Again, by substituting M = l, JB=1, r = a
[7602a], and K' [7695a;] in [7597], and neglecting, as in [7695y], the constant part,
which produces nothing in [7695i^>] , we get, in aR, the term aR = K'.v^. Substituting

[7696e] the value of v^ [7601], and neglecting the terms of the order *, then putting
sin.2(« — y) =z^ — J.cos.2.(« — *), we get, by retaining as before only the terms which are

[7696/] independent of t>— t, aR = ^K'.sm.^'y = ^K' — IK'.cos.^y. Adding this to the other
term of aR [76966], and neglecting as above the constant term ^K\ we finally obtain,

[7696g'] aJ? == — I Jf.cos.2 X— ^ K'.cos.^y.

Substituting [7696^] in the formulas [7695ty], we get,

not sm.X V rfn /

[7696i ] - — = ^.cos.X — -^ . ( — -^ ) .

ndt ein.X \ d\ J

In the spherical triangle NHI, fig. 99, page 338, we have, by means of the formula

[1345^], the equation [7696^] ; which, by using the symbols [7628c—-/], becomes as

in [7696Z]; t

[7696;^] eos.NHI = s'm.JVIH.sin.HNLcos.Nl—cos.NIH.cos.HNI'j

[7696n cos.y = sin..^.sin.X.cos.n-|-cos..^.cos.X.

Substituting the value of <?.cos.7 [7696Z] in [1696h, i], and putting ndt = dv, we get,

[7696m] — = — J?'.cos.r.sin../2.sin.n ;

dv

[7696n] _— = A.cos.X — —. . Jsm.^.cos. X.cos.n — cos.^.sm.XJ.

dv sin.X

[7696o]

[7696p]

Now if we suppose the perpendicular arc JVP to be let fall upon the arc HIP, so as to
form the two rectangular spherical triangles JVPI, NPH, we shall have in the first
triangle NFI, sin.iVP = sin.A'IP.sin.JVi= sin..^.sin.n [1345^^]; and in like manner
in the second triangle NPH, sin. NP = sin.NH.s]n,NHP = sin.^.sin.'y ; hence we
have sin../5.sin.n =sin.^.sin.7 ; substituting this in [7696w], it becomes as in [7695].
If we now suppose the arc NP to be drawn perpendicular to HN, so that the angle
PNI= 100° — X, we shall have, in the rectangular spherical triangle HNP,
'■ ^■' cos.iVP/=sin.iVHP.cos.iVH = sin.7.cos.Y [1345^^] j and in the oblique spherical
triangle NPI, we have cos.NPl=sm.NIP.sm.PNLcos.NI-cos.NIP.cos.PNI [134522];
or in symbols cos.iVPJ= sin.^^.cos.X.cos.n — cos../3.sin.X. Putting these two expressions
of cos.iVPJ equal to each other, we obtain sin*5.cos.X.cos.n — cos..^.sin.X=sin.7.cos.Y;
substituting this in [7696n], it becomes as in [7696]. Hence we see that the formulas

[7696r]

[7700]

VIII. xvii. § 36.] THEORY OF THE OUTER SATELLITE. 341

the preceding ratio of K' to K. Then we find, for the annual motion of
the node upon the orbit,

— 692",76+175",27. ^'; [7695t] ,^ t7699]

which gives — 6 18", 81 for this motion in the same hypothesis. The
observations we now have are not sufficiently accurate to determine, by

K'

means of the preceding formula, the ratio of — ; they serve only to show

that the motion of the node of the orbit, upon the orbit of the planet, is

K'
The ratio -^ , so far as it depends upon the action of the planet Saturn,

as we have seen in [7681], is inversely proportional to the fifth power of the [7701]
semi-axis of the orbit of the satellite,* or of its mean distance from Saturn.

given by La Place, in [7695, 7696], agree with those which are deduced from [5786A, i\ ;

and it will be found, upon examination, that M. Plana's calculations lead to the same

results, after correcting for two small mistakes in his calculation. Now without entering

into a minute discussion and explanation of the method used by M. Plana, we shall merely '■ ■'

observe, that he deduces his results from the formulas [13376], with the value of R [7597],

and that of v or f* [12860], which is similar to [5344]. But in making the reductions in

page 346, line 7, of his memoir, I have found that he accidentally omits a term, connected

with the factor 1 — f .sin.^t^; ; the corrected value being \-^.sm.^w-l.sm?w.cos.{'il6-\-'2^), [7696m]

according to his notation. This first mistake is not particularly noticed, either by La Place

or M. Poisson, in their remarks on this subject, in the Connaisance des terns for the years

1829, 1831. The second mistake of M. Plana was pointed out by M. Poisson, in the

Connaisance des terns for the year 1831, page 38. It arises from M. Plana's having

neglected the effect of the reduction of the longitude of the satellite to the plane of the orbit

of the planet ; and it is rather singular that this second mistake produces a correction in the

same factor of exactly the same form and magnitude as the first ; so that the true value of

this factor becomes,

1 — l.sin.^z^— J.sin.2zf?.cos.(2^-j-2*), instead of 1— f.sin.^M;. [7696x]

Finally we may remark, that the calculations of M. Poisson, in the Connaisance des terns
for 1831, agree with those given by La Place in [7695, 7696], and also with those
corrected formulas of M. Plana [7696z^?, &c.].

[7696j/]

*(3716) We have, in [7681], ^ = (i|^. — ). ^ ; and as T', ^^670, 7676]

K' 1

are the same for all the satellites, we shall have — proportional to — . Now for the

outer satellite a = 59,154 [7676], and for the next inferior satellite a = 20,295 [7702] ;
VOL. IV. 86

[7701a]

342 MOTIONS OF THE SATELLITES OF SATURN. [Mec. Cel.

Therefore, for the sixth or the outer satellite except one, we must multiply
[7702] the preceding value of — by (gQ^) > to obtain the value of —

corresponding to it. Hence we have,
[7703] ^ = 88,754 ;

which gives,
[7704] 6 = 3088",.

Therefore the inclination of the intermediate plane [7625] to the equator
™q5, of Saturn, is insensible to us ; and as the satellite will move very nearly

upon this plane, if b be nothing or very small,* we see that Saturn's

K'

[77016] therefore the value of — , corresponding to this inferior satellite, will be found by

multiplying the value -^ = 0,4219.. [7681], corresponding to the outer satellite, by

(59 154 \ ^
^-^—z ) ; and by this means it becomes as in [7703]. Substituting this in [7645/], also
^ ^ 20,290/ ^

the values of cos.2^ = |, sin.2.4 = ^.v/3 [76826], we get,

[7701d] tang.24 = ,,f.^^, , = 0,009703 = tang.6177" ;

hence 24 = 6177", or 5 = 3088", which differs a little from the calculation of the
[7701e] author, who gives in [7704] 5 = 2933",6, which we have altered to 3088"^ ; the mark c
being placed as usual to denote this alteration.

* (3717) The angle formed by the fixed plane and the orbit of the satellite, is

TS^HKI, fig. 97, page 324 [7628e], and we have b =y^p—q.co3.2n.sm.zi [7654];

hence if 6=0, we shall have «=:0; and if 6 be small, we shall have zs small;
[77056] observing that for the inner satellites q is much smaller than p, as is evident from

[7649, 7650]. For in this case K is very small in comparison with K' [7703, &c.] ;
[7705c] therefore we have very nearly ^{K^-\-'iiKK'.cos.^l+K'^) = K'-\-K.cos.2A = K'-{-iK

[7701c]; substituting this in [7649,7650], we get q = iK, p=^K'-{-^K; whence

[7705<f] i_ = ^, — - nearly ; and if we use the value of -— for the sixth satellite, given in [7703],

it becomes — = -— nearly. This must be still further decreased for the inner satellites,
[7705c] ^ 710 >

in the ratio mentioned in [7701] nearly. Hence we see that for these satellites if b be
small, « will also be small [7654] ; therefore they must move very nearly upon their
intermediate planes [7624, 7626], coinciding almost with the plane of the equator.

If we neglect the forces arising firom the action of the satellites and from the oblateness of
Saturn, the expression of K' [7621, 7620, 7597] will vanish, and we shall get, from [7646],

VIII.xvii.<§.36.] THEORY OF THE OUTER SATELLITE. 343

[TTOST

[7706]

action can retain this satellite in very nearly the same plane ; and much
more so those satellites which are inferior to it, as well as the rings of Saturn.
This is conformable to what we have demonstrated in [3689].

However, if the mass of the outer satellite be a two hundredth part of thai
of Saturn, the fixed plane, upon which the orbit of the next inner or sixth
satellite moves, will be so much inclined to the plane of the ring that the
satellite will vary from this last plane by a sensible quantity. To prove this,
we shall observe that the fixed plane, upon which we suppose the orbit of
the satellite to move, may be determined by supposing the satellite to move
in that plane, and to be retained in it, by the mutual destruction of the
forces which tend to draw it from the plane. To prove this we shall resume
le expression of s [7623] ;

s = Kv.sui,\.cos.'>^,cos.v — ^r.sin.7.cos.7.cos.(© — y). [7707J

'he fixed plane, upon which the orbit of the outer satellite moves, being [7707']
iclined to the equator of Saturn by the angle 6 [7625], if we suppose the [7708]
rbit of the satellite to coincide with this plane, we shall have,*

y=:6, X = A — &, ^ = ; [7709]

K sin 2j?
ig.2d= — ^ — ^ = tang.2^, or ^ = .4; hence it follovro that the fixed plane IK, [7705/^

fig. 98, page 324, upon which the orbit of the satellite moves, will then be the orbit of

Saturn; and it will have an annual retrograde motion of — 692";76 [7699], arising fit)m

K' [770%]

sun's disturbing force. As -=- increases, the fixed plane, which is determined by the

K'

angle 6 [7646], approaches nearer to the equator ; and when -— is very large, these two

. [7705A]

planes nearly coincide. Hence we see that the oblateness of Saturn, upon which IT

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