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Pierre Simon Laplace.

# Mécanique céleste (Volume 4)

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which is to be substituted, for the longitude of the node, in [6555/*], and then it becomes,

6v^ — X^. (0). ^ . sin.(/)<-f-A— «r'). [6555m]

The greatest value of t [6300 line 1 ; 6298</, &c.] is /, which may therefore be taken for

the inclination of the orbit of the satellite, instead of 7 [4813] nearly. Substituting this *- ^^

value of 7, in [6555m], it becomes as in [6555].

TOL. IV. 32

126 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Gel.

[6556]

CHAPTER VJ.

ON THE INEaUALITIES DEPENDING ON THE SaUARE OP THE DISTCJEBING FOECE.

14. We have already considered, in [1215 — 1242'], one of the most
remarkable of these inequalities. It depends, as we have seen, upon the
circumstance, that, at the origin of the motion, the m,ean longitude of the
first satellite, minus three times that of the second, plus twice that of the third,
differed but very little from the semi-circumference ; and then the mutual
attraction of these three satellites has caused this difference to vanish. We
shall now resume, by another method, the abstruse theory of this inequality,
in order to give it a further development, and to determine its influence
upon the various inequalities of these satellites.

[65571 ff ^^^ consider the orbits as variable ellipses, ^ representing the mean
longitude of the satellite m, we shall have, as in [1195, 6110c],

[6558] dd^ = Sandt.dR,

We shall notice, in the expression of the motion of the satellites, only those
terms which depend on the angle nt — 3n't-\-2n"t-\-s — 3s'-{-2s", and have
also the divisor (n — 3n'-{-2n"y; since these terms may become sensible,
on account of the extreme smallness of this divisor. It is plain that
Sandt.dR contains terms depending on the proposed angle, which may
acquire this divisor by the double integrations. Moreover such terms cannot
be introduced into the value of v, except through the expression of ^ ; for
it is evident, by the inspection of the values of de, d^^, ds, de', &c., given

[6561] in [1258, 1258«, &c.], that they cannot produce such terms in v, at least
when we carry on the approximation no further than to include the terms
depending on the square of the disturbing force.* Therefore, by noticing

[6559]

[6560]

[6561a]

* (3400) We shall see, in [6567—6569], that R, R', R", dR, dR', dR", do not
contain terms depending on the angle v — 3i>'+2y", connected with coefficients having the

VUI. vi. ^ 14.] INEQUALITIES OF THE ORDER m«, mm', m'«, &c. 127

only such of these terms as can acquire this divisor, by integration, we
shall have,

ddv = 3andt.dR, [6562]

In like manner we shall have,

ddi/ = Sa'n'.dt.d'R ; [6563]

ddi/' = 3af'n".dt.d"R' ; [6564]

R' and R" being what R becomes relative to the satellites m' and m" ; [65G5]
and the characteristics d, d', d", correspond respectively to the co-ordinates [6566]
of the bodies m, m', m". We must now determine the terms of d/2, d'R, j'^*^'
d"R', which depend on the angle nt—Sn't-\-2n"t+e - .3^-{-2^'. ^'j^'

The expressions of R, /2', R', do not contain angles depending on [6567]
V — 3i7'-r2y"; they give, by development, only terms depending on the
radii vectores, the latitudes, the elongations, v — i/, v — t/', v' — v", and the
multiples of these elongations.* But by the substitution of the parts of [6568]
r, V, r', v', &c., which depend upon the disturbing forces, there may arise,
in dR, d'R, d"R", some terms of the order of the square of the disturbing
forces, depending on the angle ni—3n'( + 2n"t+s — 3s'-|-2£". We have
determined, in [6116, 6119], the perturbations of r, v, /, r', r", v", and
we have seen in [6131, 6132], that the principal inequalities of r, v, arising
from the disturbing forces, depend upon the angle 2nt — 2n't ; that those of
r' and xl [6139,6140,6148,6149], depend upon the angles nt — n't and

[6569]

[65616]

divisor n — 3n'-|-2nf', or its square. For similar reasons it will follow, that the terms

r.(^), {r^), dr, Sr [6094, 6057, &c. ; 6049A:, /, &c.], in Sv [6060], do not contain

terms depending on that angle, and having that divisor, even when we notice quantities [6561c]
of the order of the square of the disturbing force. Moreover if this angle be found in

r/— j, it will only produce in fndt.r.(~\ which occurs in 6v [6060J, a term [6561cn

having the first power of the divisor n — 3n'-j-2n", noticing the same order of terms.
Hence it appears that the only term of 6v [6060], which can produce this divisor, is that
depending on Jf3andt.6R, corresponding to dd^ [6558] ; or, as it may be written, [6561«]
ddv sss 3andt.dR, corresponding to [6562], or to the similar values of ddi/, ddi/' [6561/]
[6563, 6564].

• (3401) This is evident by comparing [949 or 951] with its development
[957 or 101 1], or with the expressions in [1226, 1227], which produce, as in [1230], some ^ ^
tmnt of d/?, depending on the angle nt — 3n'<-f-2ii"/-}-i— Si'-f 2i".

128 MOTIONS OF THE SATELLITES OF JUPITER. [M6c. C61.

'2,n't — 2n"t ; lastly that those of r", v" [6164, 6165], depend on the angle

'■ ■' n't — n"t. These inequalities acquire^ by integration, very small divisors,
which render them much greater than the other inequalities, so that we need
only notice them in the present question. Some of the arguments of these
inequalities, when combined with the elongations of the satellites and their

[6572] multiples, by addition or subtraction, may produce the angle nt — 3n't-\-2n"t.
The first of these arguments 2nt — 2n't, and the last, n't — n"t, cannot
form it* by their combination with the angles v — v', v — v", v' — v", and
their multiples, as is evident by using the following mean values of v, v', v"
[6091, &c.], namely,

[6572'] v = nt-\-z', v' = n't-\-^', v" = n"t + s".

Therefore in the expressions of dR, d'R', d"R", ive may dispense ivith the
consideration of the perturbations of the satellites m, m" ; it will suffice to
notice those of the satellite m'. The inequality of the satellite m', relative
to the angle 2n't — 2n"t, being combined by subtraction with the angle
V — v', and its inequality relative to the angle nt — n't being combined, in
like manner, with the angle 2v' — 2v", will produce some terms depending

[6574'] ontheanglet nt—3n't-\-2n"t^s—o^'-\-2s".

[6572a]

* (3402) Thus the angle '2nt — 2n'^, being combined with a mnhiple of
V — v' = nt — n't-\-z — /, will not contain n"t, and will not therefore be of the form [6572].

[6572a'] The sanne angle 27?^ — 2/i'^, when combined with a multiple of v — v"=ni — n"i-\-e — s",
will form an angle containing — 2n'^, instead of — dn't [6572]. Lastly, if tlie angle

[65726] 2nt — 2n'^ be combined with a multiple of v' — v'\ it will contain 'Hut instead of nt, as
is required in [6572]. In like manner n't — n"<, being combined with a multiple of
V — «', contains — n% instead of -\-'in"t [6572] ; if it be combined with a multi|)le of

[6572c] V — v", it will contain n't instead of — Zn'l ; and if it be combined with a multiple of
v' — v", it will not contain nt [6572]. Hence it appears that in noticing only the terms
depending on the angle nt — 3n't-{-2n"t [6572], we may neglect the inequalities depending

[6572e] on the angles 2nt — 2n't, n't — n"t ; or, in other words, we mry neglect the terms 6r, 6v
[6131,6132], also Si", 5v" [6164, 6165], with thnr diffirentials d5r, d(h, dd/', div" ;
and notice ^/, dV, d5/, rf(5i;' [6148, 6149, 61 5S, 6159], depending on 2n't-2n"t, nt-n't.

[6572/]

t (3403) The angle -2{n't-n"t), which occurs in [6148, 6149], being combined with
[ °J (v — 2j')j or nt — n't, produces the angle nt — 3«'/-f2n"^. In like manner, if we

combine the angle {nt-n't) [6158,6159] witli -2(i;-i"), it produces {nt-3n't-\-'2n"t).
[65746] The terra of R depending on v—v', is noticed in [6575, &,c.] ; and the term of R

depending on (2©' — 2v") is noticed in [6595, &;c.].

VHL vi. ^ U.] INEQUALITIES OF THE ORDER «•,««', t^, &c 129

We shall now consider the second term of R [6090], which we shall
represent by iJ = m'.A^^Kcos.(v — t/). If we notice only this term we shall [6575]
hife/

dR = — iii'.^(»>.rfr.sin.(t?~i/)+m'/^\(ir.cos.(tJ— v'). ^6576]

If we neglect the perturbations of in, and the eccentricities of the orbits,
we shall have dr = 0, and dv == ndt [6572'] ; therefore,

dR = — m'. A^^K ndLsin.(v—v') ; [d578]

which gives, in d/?, the following terms of the order of the square of the
disturbing forces ; f

dJK = m',A^^Kndt.6vf.Q0s.(v-^')'^m'J—j,ndLir'.s\n.(v^}f). [6579]

We have, in [6148, 6149], by noticing only the perturbations depending on
the angle 2n'i— 2n"/,

^ ^ "*"-'*'-^' . cos.('2n'<— 2n"^4-2s'— 2/') : [6580]

^ = J'li'ZjN' • sin.(2n'/-2n"^+2.'-2e"). [«»1]

Substituting these values in [6579], retaining only the terms depending
on nt — Sn't-\-2n"if and observing that n is very nearly equal to 2n' [6151],

• (3404) We have already remarked, in [60896], that A^^^ is a function of r, r^, and r^cy^ ,
does not contain v. Then taking the differential of R [6575], relative to the characteristic
d, which affects only r, v, we get the expression of dR [6576]. Substituting in it the [65766]
values of dr^ dv [6577], we get [6578] ; observing that the variations relative to the
characteristic S, of the second terra of [6576], which contains rfr, produces nothing of [6576c]
the required form or order in [6583] ; dSr, d6vy being neglected, as in [6572/].

[6579a]

t (3405) Supposing / to be increased by the quantity &/ [6148], and t/ by ^
[6149], we shall find, by Taylor's theorem [617, &ic.], that A^^^ will become

•^ +("1 — j'^^) nearly; and sin.(i>— r') will become sin.(t> — t/)— V.C08.(i> — i/).

Substituting these in [6578], we obtain the additional terms of dR [6579], depending on
ir'y 6i/, The value of h' is given in [6149], being the same as in [6581] ; that of

— is deduced from [6148], by putting, in the first member, ^=1 [6297c]. The terms [6579e]

of i/^, itf [6139,6140], are neglected, because they do not produce, by combination, the [6579<n
angle [6574'] ; as we have seen in [6572a, &ic.].

TOL. IV. 33

130

MOTIONS OF THE SATELLITES OF JUPITER.

[Mec. Gel.

1:6583]

[6584]

[6584']

[6585]

[6586]
[6587]

we shall have,*

Sandt.dR = -

. dt^.sm.{nt-dn:t-^2n"t-{- s-SZ-f- Ss") .

S.{2n'—2n"—JV')
We have very nearly, as in [6145],

Moreover 2n' — 2n" is equal to n — n' [6154] ; or at least, their difference
is so small that it has hitherto been insensible by observation. Therefore if
we substitute, in [6583], the values [6572'],

which can be done in the present instance ; and also the value
ddv = Sandt.dR [6562], we shall obtain,

ddv
'd^

^m'.m".n^.^.F'G

s\n.{v—Sv'-{-2v").

S,(^n—n'—N')

The part of R, relative to the action of m" upon m, contains only
terms depending on the angle v — v" and its multiples ; therefore it adds no

term to this value of -j^ [6572a', &c.].

[6583a]

[65836]
[6583c]
[6583(Z]
[6583e]

[6583/]

* (3406) Substituting n' = ln [6151] in the numerators of the coefficients of
[6580, 6581], and then their results in [6579], we get,

^ ( —4A^^).cos.(v—v').s\n.(2n't—2n"t4-2^—2e'') )

8.(2n'-2n"-JV0 ^ _2a'.(-^Vsin.(«— t)';.cos.(2n'^— 2n"<+2£'— Ss") (

Putting now v — v' = nt — n'^-}-£ — s' [6572'], and reducing the expressions by means of
[18, 19] Int., retaining only the terms depending on the angle [6574'J, we get,

IVTuliiplying this by Sandi, it produces the expression [6583] ; observing, that by

neglecting the excentricity we may change (-77) into i~T~,)) ^s in [62026, &;c.].

Substituting in [6583] the values G [6584], 2«'— 2n" [6584'J, and ddv [6562], also the
expressions [6585], we obtain the equation [6586]. IVIuhiplying this by dt^, and
substituting [6582], it becomes,

Sa.ndtAR

dm'.m".n^^.F'G.dt^
a

8.(n— n'— iV')

.sin.(i;— 3w'+2i;").

VUl. vi. ^ 14.] INEQUALITIES OF THE ORDER m«, mm', m'*, &c. 181

We shall noxo consider the term /?' = m. i4/'\cos.(t? — vf)^ being a part of [C588]
the expression of R\ arising from the action of m upon m\ as we have
seen in [6134'].* If we notice onlj this term, we shall have,

-^ j . COS.(r— t/). [6589]

This function, being developed, contains the following terms ;

^ \ . n'J^<Jr'.sin.(r— i;')+m. 4«'>.rf«5i/.sin.(v— r') 1

— OT.^/".n'J/.6t/.cos.(i;— v') + w.f -T7- j . d6r',cos,{v—vy 2

Substituting, in [6590], the preceding values of <5r', 6v' [6580, 6581] ; and
observing that 7i" = Jn' [6151] nearly, also that we have very nearly, as ^ ^
in [6137, 6145],

G = 2a'. 4(')— o'^.f ^ j ; [6592]

we shall obtain, f

3a'. n'rf/. d'/r. = ^"!f/'''''T^Jf ' sin.(r— 3i;'+2i/'). [6593]

16. (n — n — A) ^

* (3407) Comparing [6090, 6134'j with the definition of R' [6565], we obtain the
term of R' [6588J. Its differential relative to the characteristic d', which affects only ["^®^<»]
r', t/ [6566], gives [6589]. Supposing now / to be augmented by 5/, and v' by [65886]
M\ then developing the second member of [6589], according to the powers of ^/, 5r' [65886'j
[617], retaining only the terms depending on the first power, we shall get [6590]. For [6588c]
the variations of the three quantities «4^'\ dv\ sin.(« — v'), which enter as factors into
the first term of the second member of [6589], produce respectively the three terras of [6588rf]
[6590], containing 5/, dM, 6v' ; observing that we may substitute n'dt for dv [6585].
Moreover the variation of <//, in the last term of [6589], produces the last term of [6590]. [6588e]
The variations of djiy\ cos.(r — v'), in the last term of [6589] , are neglected, because
they are multiplied by rf/, which is of the order e', and may therefore be neglected, as ["^^'^i/J
in [6577], since they produce nothing of the required form and order.

t (3408) Taking the differential of [6580], and comparing the result with [6581], after
iubsliluting in the coefficient 2/i' — 2/»" = n — n' = n' [6151, 6151], we get the first of l**^^^l
the expressions [6593c]. Then taking the differential of [6581] and comparing it with [65936]
[6580], after making similar substitutions, we get the second of the expressions [6593r],

^ = J n'dtM ; d.y = — 2n'rf/. ^ . [qs93c]

Substituting these expressions in the two lines of the second member of [6590], they

132 . MOTIONS OF THE SATELLITES OF JUPITER. [M6c. Cel.

We may here observe, that, by comparing the two expressions of 3andt.dRi

Sa'n'dt.d'R' [6sSS, 6593], we shall obtain,*
[6594] m.dR + m'.d'R'^O;

which is conformable to what we have found in [1202].
[6595] The part of R' relative to the action of m" upon m', contains the term

R'^m",A'^''Kcos.(2v'—2v") [6090,6146]. Noticing only this term, and

[6593rf] produce the two lines of the value of d'R' [6593e] respectively ; and by using the value
of G [6592], it becomes as in [6593/].

d'i. = -!!!:^/fsin.(.-.').f-a'f^') + 2«'.^<»|

[6593e]

[6593/] =—^^.G.y-^.s\a.{v—D') + iSt/.cos.{v—<,')'i.

m". n'. F'
Now if we put for brevity B= _ , ' ' — rz, and use the values of v, v\ v" r6572'1,
[6593g] '^ 2n'-2n"—JV •■ ■"

we shall find that the expressions [6580, 6581] may be put under the following forms ;

[6593A] ^=— JJ5.cos.(2t;'— 2u") ; 8v' = B.s\n.{Qv'—2v").

Multiplying the first of these equations by sin.(« — o'), and the second by |cos.(r — r');
then taking the sura of the products, and reducing successively, by means of [22] Int.,
we get,

[6593Jk] — . sin. (»-«)') -|-J5t)'.cos.(v-D') = -^B. { sin. (r-i;') .cos. (2r'-2t>")-cos. (»-©') . sin.(2» -2»") }
[6593Z] = -^jB.sin. (w-3i;'+2t;").

Substituting this in [6593/], and then multiplying by 3a'.n'dt, we get the first of the
[6593m] expressions [6593n]. Re-substituting B [6593^], n'=^n [6151], and 2n-2n"—n-n!
[6593a], we finally get [6593o], which is the same as [6593].

[6593n] Ba'.n'dt.d'R = lm.n'KBG.dt^sm.{v—Sv'-i-^v") = |"Jj^^^^^^ . sin. («— 3v'+2t/')
[65930] = ie,^^_^._^.^ . sm.{v-3v'+2v").

[6594al *(3409) Multiplying [6583/] by g-^, and [6593] by ~t = ^t ""^''^^
[6151] ; we find that the second members of these products are,

[6594^] T 8.(„-n - ;V0.a- " ^■n-(»-3'^+2«") 5

and as they have different signs, their sum vanishes ; hence the sum of the terms in the first
[6594c] jjjgjjjijgfg Qf tjje game products, becomes m.diZ -j- m'. d'jR' = 0, as in [6594].

[65*7]

VIII. vi. ^ 14.] INEQUALITIES OF THE ORDER m« mm', m^, &c. 133

taking its differential relative to d', which only afiectf r', tf [6566], we
shall have,

d'/r = — 2m".^'<^\rfi/.sin.(2r'— 2t?")+i»".<ff^/^jJ^^ . cos.(2t/— 2»"). [6596]

This function, being developed, contains the following terms ;*

d'R' = -2m"/ ^^j . n'J/.ir'.sin.(2t/— 2i;")— 2m".^'<^<for'.sin.(2t;'— 2»")

—4m". n'di. A^^'Kn/, cos.(2r'— 2'/) + m".(—-j . (/ar'.cos.(2o'— 2tj").

e have, by noticing only the action of the satellite m upon m',

6139, 6140], t

a/ m.n'.G J

7 = - 2.(n-n-~.JV-) • COS.(n^-n^4-e-0 ; [6598]

_ _/ /J

^ = n-r^L-N' • sin.(n/— n'/+£ — £')• [6599]

'hen, by observing that n' = 2n" nearly [6151], we shall have, with a
losiderable degree of accuracy, by [6147],

F' = -4a'. J'<=> —a". ("^^ ; [6600]

nee we obtain, J

[6597o]

• (3410) The expression [6597] is deduced from [6596], in the same manner as
[6590] is obtained from [6589], in [6588a—/]; namely, by Bnding the increment of
[6596] from the change of r^, tf, into /+^> v'-\-tv', respectively; neglecting, as in [6597i]
[6S88/^, the two terras having the factor dr^.

it
t (3411) Substituting — = 1 [6579c] in the first member of [6139], it becomes as

in [6598]. The value of hv' [6140] is the same as in [6599]. Substituting T<"==Jn'- ^^^^'^
[6151] in [6147], it becomes as in [6600].

X (3412) The computation of [6601] from [6597], is similar to that of finding [6593]
firoro [6590], as in [6593a— o]. In the first place, the differentials of y, W [6598, 6599], t^^^*]
being compared with [6599, 6598] respectively, give the same results as in [6593c].
Substituting these values in the two lines of the second member of [6597], they produce [66016]
respectively the two lines in the second member of the following expression ;

d'iJ' = m".n'</<. 3.8in.(2i/— 2p"). \ —2a'.('^')-{-4jT'^ I
+m''.nV/.air'.cos.(2r'— 2t;"). \ —4A^^^-^W-C^^) ] -
VOL. IT. 34

134

[6601]

MOTIONS OF THE SATELLITES OF JUPITER.

3a'.n'dt.d'R' = -— — — - . sm.(v—3v'-\-2v").

32.(71—11— N') ^ ' '^

[Mec. Cel.

.(n— n'— iV')

Connecting these two terms of 3a'.n'dt,d'R' [6593, 6601], we get, as in
[6601m],

[6602]

ddv'

9m.m".n\F'G

- . sm.{v—Sv'-]-2v"),

S2.{n—n'—N')

It now remains to consider the value of d"R". The part of R", depending
[6603] on 2i;'— 2i;" [6574], is as in [6595, &c.], R" = m',A"^''Kcos,(2v'—2v");
and by noticing only this term, we shall have,

[6604] ^"ji» ^ 2m'.A'^^\ dv'\sm.(2v'—2v'')+m',dr''/~^ . cos.(2i?'— 2y"). •

This function contains the following terms ; *

m. n'. G

[6601d] If vve now put, for brevity, B'= '""''^j^, , and use the values of v, v', v" [6572'], we
shall find that the expressions [6598, 6599], may be put under the following forms ;

[6601e]
[6601/]

[6601g]

[66OI/1]
[6601t]
[660U]

[6601Z]
J660lm]

[6605a]
[6605&]

— = — iB'.cos.(u— r') ; 6v' = B'.sm.{v—v').

Multiplying these values by sin.(2y' — 2u"), cos.(2«' — 2v"), respectively; reducing the
products by [18, 19] Int., and retaining only the terms depending on the angle v-Sv'-f-Si;",
■we get,

^ . sin . (2t;'-2i''0 =iB'. sin . {v-3v'-\-2v") ; Sv'.cos.{2i/-2v") = | ^,sin. (y-3v'-\-2v") .

Substituting these last expressions in [6601c], and making successive reductions, using F'
[6600], we get,

d'R'=^m\n\dt.'^i.[-2a\(^-^) + AA'^^^^^

=:m".n'.dt. -^ . B\sm.(v—3v'-}^2v").
4a' ^ '

Multiplying this last expression by 3a'n'.dt, and re-substituting the value of B' [660 l^j,

also n'^= — .n^ nearly [6151], we get [6601]. The sum of the two parts of

8

3af.n'dLd'R' [6593, 6601], being taken and substituted in [6563], gives,

9m.m".n3. F' G.'dt^

ddv' = 3a'n'.dt.d'R' =
Dividing this by dl'^, we get [6602].

32.(n-n'— A*0

. sin.(i;— 3u'+2«").

* (3413) We have, in [6595], R' = m".A'^^\cos.(2v'—2v"); and as A'^^> is the
same for both satellites w', m" [6466£/], we shall have R" = m'.A"-^\cos.(2v'—2v"), as
in [6603], The differential of this expression, relative to the characteristic d", which

Vni. vi. \$ 14.] INEQUALITIES OF THE ORDER m«, mm', m'», kc, 136

d"/?" = 2m'.n'dt.6r'/^\ . sm,(2v'—2i/')i-4>m\A^'*\n"dt.6v'.cos.(2v'—2i/'), [6605]

Substituting the parts of 6v' and i/, depending on the angle nt — n't
[6598, 6599], observing also that we have very nearly n' = ^n, n" = ^n ^

[6151], we obtain,

Sa'\n"dtA"R" = — e^^^_^._^.^ • ^ . sin.(iJ-3i/4-2t/'). [6606]

Hence we easily find, that, by noticing only the reciprocal action of m'
upon m", we shall have,*

m'. d'R' + m". d"R" = ; [WOT]

which agrees with [1202]. Therefore we have,t

rfiT ^ 64.(n-n - A-) ' 7 ' Sin.(t^-3t/+2. ). [6608]

affects only /', r", gives [6604]. Supposing / to be increased by 5/, and p' by [6605c]
V, we find that the expression [6604] will be increased, by the terms in the second
member of [6605], as is evident by proceeding as in [6588a, &tc.], and neglecting the '• ^
terms multiplied by d/', as in [6588/] ; also those depending on d6/', d6v'\ as in
[6572/J ; and finally putting dv" = n"dt [6572']. Now if we substitute, in [6605], the
values [660le], we shall get [6605/]. Reducing the products by [18, 19] Int., and t^^^^l
retaining only the terms depending on the angle v — 3u'-}-2t;", we get [6605^] j and by
using r, B' [6600, 6601J], we obtain [6605A].

d"/r' = m'.i^'dt.B.\—a'. (^) . cos.(t;-u').sin.(2t?'-2i;")+4^'^».sin.(r-i,').cos.(2t;'-2t/')? [6605/]
= m'.vl'dt.B.S^l '^'(^da')'^'^'^'''\ ' sin(^-3'^+2«;") [6605^]

= -.m'.n"dt.B. ^ . s\n.{v-3v'+2v") = - ;;;';;^X^^!;;^' . sin.(t;-3«'+2t/'). [6605*]

Multiplying this by 3a".WJt, and substituting, in the second member of the product, the

values n' = \n, n" = in [6605'], we obtain [6606]. ^^^^^

• (3414) Muliiplying [6601] by ^-^ = ^-^ ; and [6606] by ^^r^ = 5^^ ; (ggOT,]
we find that the second members of the products become,

tnd being of different signs, their sum vanishes in the second member, and the first member
of the sum becomes as in [6607].

t (3415) Substituting [6606] in [6564], and dividing by dfi, we get [6608]. [660ea)

136 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cd.

Now if we use the following values of k, 9,

[6610] ' <p = v—3v' + 2v";

we shall obtain, by connecting together the expressions of — ~, —,

^^ [6586, 6602, 6608], the following equation ; *

[6611] — ^ = k. n^. sin. <p .

15. We may suppose k and n^ to be constant in this equation,
because their variations are very small ; then, by integration, we obtain,t

[««12] "^^ - v/ c-2k.n^.cos., '

[6612'] c being an arbitrary constant quantity. The different values which c

the value may have, gives rise to the three following cases ;

* (3416) Multiplying [6602] by —3, also [6608] by 2, then adding the products

ddv—3ddv'-]-2ddv"

[66II0] to the equation [6586], we find that the first member of the sum is

which is easily reduced to the form — , by using (p [6610] ; being the same
[66116]

which is easily reduced to the form — ■ , by using 9 [6610] ; being the same as the first

dt

member of [6611]. Moreover the second member of this sum, by using the value of k
[6609], becomes equal to k7i^.sin.(i; — 3v'-{-2v")^kn^.sm.(p, as in [6611].

t (3417) The equation [661 1] is the same as [1235], changing V into 9, and /3
[6612a] jj^jQ j^ jjg integral, found as in [1236a], is the same as in [6612]. From this equation
[66126] we may deduce the same results as in [1236', &tc.] ; and the three cases relative to the
value of c, given in [6613, 6615, 6617], correspond respectively to [1236', 1237"", 1237^'].
We may observe that the symbols c, k, are printed in the Roman, instead of the Italic
type used by the author, to distinguish them from c, 7c [602U', x]. For the same reasons
we have placed a mark below his symbol tu, in [6620, he], to distinguish it from zs
[6024^]. We may also remark that the equation [661 I] will not be altered, if we change,

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