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Pierre Simon Laplace.

. (page 25 of 114)

together the two terms of s [6456, 6459], it becomes as in [6460] ; taking care to

re-substitute 2C7 for â€” .r, and pi for â€”.v [6455', &c.].

[6458m]

â€¢ (3347) The longitude of the sun, as seen from Jupiter, is U [6041] ; and in
eclipses of the first satellite, the longitude of the satellite v is equal to this quantity ^ ''I
increased, or decreased, by 200Â° ; hence U=sv â€” 200Â°, as in [6461]. Substituting this
in the term [6460 line 2], and reducing, it becomes as in [6462] ; observing that,
8in.|Bâ€” 2(tâ€” 200Â°)â€” p<â€” a| =sin.(iH-2râ€” ^<â€” A)=siQ.(-r-;pr-A)= â€”s'm.{v-^pi-\-A). [64616 J
VOL. IV. 24

94 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. C61.

When the values of p are relative to the motions of the equator and of

the orbit of Jupiter, we maj neglect p [6342] in comparison with M

[6463] [6927, 6025m]. Moreover the sum of all the terms (L'â€” Z).sin.(r-}-jo/+A),

is then equal to â€” (>^â€” l).a'.sin.(t5-f^') [6357/]; and the preceding

inequality becomes,*

3JW2
[6464] -(X-l)-,-;;^.^'.s.n.(.+V)

- + iV,-l

n '

[6462a] * (3348) Prefixing the symbol 2' [6324'J to the function [6462], so as to include all
the terms of that form, it becomes as in the first member of [646"2c] ; substituting the value

[6357/], we get its second member, neglecting â€” . This second member is the same as

[64626'! ^^^ function [6464] ; the sign being changed from what it is in the original work, to correct
for a mistake.

m&^c^ â€” ^.{ L'-r).sm.(v+pt+A) 3m^ (x-i).^.sin.(r+T')

n ' n ' n ' '

Adding this to the chief term of the expression of the latitude of the satellite m above
[6462<ri Jupiter's orbit (Xâ€” l).d'. sin. (v-{-^') [6427 line 1], we get the following expression of the
part of the latitude depending on this argument v-\-^', in eclipses,

The ratio of the general value [6462<i], to the value in eclipses [6462e], is as 1 to
3J
2M

3M^

[6462/"] 1 â€” THirs r , as in [6465]. Hence it is evident that if we determine this

[6462g-] inclination by means of eclipses, we must increase it, in the ratio mentioned in [6465].
We may observe that, in the original work, the word increased [6465] is printed
decreased, and we have changed the sign of the term depending on M^ to correct for the
mistake [6465]. The same phenomenon occurs in the lunar theory. For in eclipses,

[6462f] when 2 3) long. â€” 2olong. = 0Â°, or 400Â°, we find that the two chief terms of the moon's

latitude [5595 lines 1, 3], give for the general inclination 18524'.5.sin.arg.lat. ; and in

J eclipses (18524*.5â€” 528*.4).sin.arg.lat. The ratio of these two expressions is as 1 to

1 â€” ttV nearly. This is nearly the same as the ratio [6465]; for by putting N^=l,
[6462Z] j^^

which is nearly correct in the lunar theory [5049 lines 1-4], it becomes as 1 to 1 â€” f. â€” â€¢

M

In this expression â€” is equivalent to m = 0,0748.. [5117]; hence the preceding ratio

becomes as 1 to 1â€”0,028, or 1 to 1â€” i^, as in [6462/].

Vni. iv. ^ 11] INEQUALITIES IN THE LATITUDES. 95

Hence the inclination (f of the equator to the orbit of Jupiter, deduced
from the eclipses of the satellite m, ought to be increased in the ratio of 1

to 1â€”

[6468]

We shall now consider, in the same manner, the periodical inequalities of
the motions of the second satellite in latitude. For this purpose, we shall
resume the differential equation [6295], which becomes, relatively to the
second satellite,*

R' being what R becomes relative to this satellite. The terms of this
differential equation, which depend on the angle 2v â€” 3v', acquire a small '^

divisor, because v differs but little from 2v' [6151], so that the coefficient
of i/, in the angle 2v â€” 3v\ differs but very little from unity ; therefore
it is important to consider these terms. If we notice these terms only, it
will be evident from ^ 9, that the preceding differential equation becomes,t

â€¢ (3349) Substituting 75 =Â« [6439], in the third term of [6295], and then changing

'' [6466a]

reciprocally the elements of m into those of m', it becomes as in [6466] ; R' being

the value of R deduced from [6297], by changing, in the same manner, the elements of [64666]

m into those of m', and the contrary ; by this means we obtain,

*^'*^"^''"-) 4- Ja'^<'>^+a'^^'>.cos.(Â»'â€” t>)-j-a'^Â«'*>.cos.(2i/â€” 2Â»)+&c. (2

. ^-aa'^, \ Â»Â»'-^ .(5Â»-f Â»'3).cos. {y'-v) \ .{ Jfi^o^+fi^^.cos .(tj'-i;)+5Â«Â»\cos.(2t?-2i/)+&c } )3 ^ ^

_- ^ -_ ^ . I l>3*'Â«-3S'3+3.( l-yÂ»-*S"Â«).cos.(2p'-2i;)+ 1 ^'S'. cos. (1/- 17) } 4

it being evident from [6089, 6090, 6296] that the values u4<'^ R'\ are not changed,

except in the term A^*\ as it observed in [997, IOCS'] ; but the tertn ^<'> is not med in [6466rf]

the subsequent part of the calculation ; therefore this difference is not noticed,

t (3350) We have found, by developing the equation [6295], as in [6299], that the

dds [GiG9a]

two first terras become -r-r -f- ^*'' ; '^,^ being used for brevity, as in [6447]. The same

process being performed with the similar equation [6466], we obtain, for its two first terms,

M^ [64696]

~-f JV;*.Â«', as in [6469] ; A*;Â« being deduced from iV," [6447], by changing

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

[6469] Â«v

reciprocally the elements of m into those of ml ; hence we have,
[6469c] iV'2= 1+2.^-^^^4-1. ~+i.2.m.aa'2.B^i\

The remaining terms of the equation [6466] depend on /?', and in substituting the value

[6469rf] of R' [6466c], it will only be necessary to retain the terms depending on angles which
differ but little from v', because they increase considerably by integration, as in the
siniilar case, relative to the first satellite, treated of in [6432fl â€” 6]. Now by proceeding as

[6469c] in [6434a â€” m], we find, that by changing the elements of m, into those of m', and the
contrary, the form of the angle [6434A], %.{y! â€” v)-\-v , will become i.{y â€” 1?')-]-^; and
the corresponding divisor [6434iJ will change into {2u' â€” {i-\-\').{r^â€”n)\.{i-\-\).{}{ â€” n).

r6469 1 '^^'^ becomes 0, when t-|-l=0; and then the angle i.{vi â€” i'')-f-v [6469/J, is equal
to v' , producing terms analogous to those of v [6436'J ; it also becomes when the

[6469/1] factor 2Â«'â€” (i+I).(n'â€” w) = 0, [6469/]. Substituting w = 2n' [6151], in this last
expression, it becomes n'. (2-{- i-{-l) ; which vanishes when Â«' = â€” 3, and then the

r6469'l ^"S^^ *-(^ â€” ^')"f"'^ [^'169/] becomes 3v' â€” 2t;. or 2i; â€” 3y', as in [6468]. The satellite
rd' produces in / a similar term. For the factor [6469^] changes into
2w' â€” {i-\-\).{id â€” n") ; and by putting n"^^w', it becomes nearly w'.{2 â€” \i â€” \\.
This factor is very small when i = 3, and then the angle [6469i] changes into
3.(u" â€” '/)-!-Â«" = 4Â«" â€” 3v', or 3i;' â€” A.v" , being of the same form as that which is
computed in [6476]. The satellite w!" produces nothing of importance, because the
factor [6469A] changes into 2n' â€” (t-|-l).(n' â€” n"') ; and by putting n'" = |^n', it becomes
nearly w'.|2 â€” |i â€” 1| ; being small only with fractional values of ^, which is contrary to

[6649m] the supposition in [6434^] ; so that these terms may be neglected. We shall now proceed
to compute the terms of [6466], connected with the angle 2Â« â€” 3v' [6469t].

[6469A]
[6469Z ]

[6469n] From [6300 line 2] we have â€” = Z'.cos.(i;'-{~F^+a) ; observing that pt = â€”;^ nearly,
[6439, 6441], and â€”^ is so small that it may be neglected, as in [6439c?]. Moreover,

/f]Tf'\

[6469o] from [6466c] we get â€” a'.f â€” ] =2w.a'-4'^^\sin.(2v' â€” 2y) ; neglecting the other quantities,

which do not produce the required angle 2v â€” 3v', connected with sensible terms.
Multiplying these two expressions, and reducing, by [19] Int., we obtain the following terra;

[6469p] â€”a'. '^Â£,.(~\ = â€”m.r.a'A^^Ks\n.{2v-^3v'â€”ptâ€”A).

Again, the expression of a'R' [6466c] produces, in Â«'-("T7)> the terms in the second
[6469^] member of [6469/-, s]. Reducing them successively, by means of [18 â€” 20] Int. ; then

VIII iv. <^ U.] INEQUALITIES IN THE LATITUDES. 97

[6469^

iV/* [6469c] being what N^ [6447] becomes relative to m', and the
values of A^\ B^'\ B^\ &c. [996, 1006], being the same for R' as for
R [6466rf]. Therefore we shall have, by means of [996, I006],Â» [6470]

o = ;^+iv,".y

â€¢ (3351) From [996J we get the first of the following equations. In like manner

a

a

[6471]

We have, by using [971, 966],

*f â€” i ""' ^l'â€” i^'bf==^i a. bf ; [6472]

hence [6471] becomes,t

substituting the values of s, / [6300], and retaining only those terras which produce the
proposed angle 2v â€” 3d' â€” pi â€” a; we get [6469^, u].

a'Y^=-m.fla'Â«.|Â»-Â»'.cos.(wU)|.{iB^Â«^+B(>\cos.(t/-t>)+5Â»>.cos.(2Â»'-2t))+&c.} [6469r]

â€” -in-aa^^ .B^3).cog.(3r'_3u)4-m^rt'a.a'K;os.(v'-o). 1 5^^>^os-(t/-t;)+i?^3'K;os-(3r'-3t>) J [6469*]

= { \m.aa'^. l. m^-\m.a(^\ I'. {m^+W^) \ .sin.(2tf-3r'-p/-A). [6469u]

Substituting the values [6469/?, m] in [6466], and the value of pt [6469n], we obtain the

terms of [6469] depending on the angle now under consideration.

[6470o]

we obtain the second and third, from [1006], by substituting -7 =a [963''].

Substituting these expressions in [6469] we get [6471] ; observing that 2t>= -7- nearly [6470e]
[6439a, kc.] ; neglecting the constant parts depending on s, /, as in [6439, 8a;.].

t(3352) Putting t=2 and Â« = i, in [971], we get i|'=Â§a.6|)â€” i.(14-aÂ«).W>. ^5473^^

Substituting this in the first member of [6472] it becomes, by reduction,

TV*.^^'-i.(l-KÂ«).^-ia.6|>. [6472^3

Now putting Â» = 3 and Â« ==| in [966], and transposing the terms into the first member,

4
we get fcl'â€”g^. (l+aÂ«).&|)-f^.6^> = 0. Multiplying this by â€” ^a, and adding the [6472c]

product to [64725], we find that the terms depending on 6y\ b^ mutually destroy each

r6472</]
other, and by reduction the sum becomes â€” ia. ij', as in [6472]. Substitutbg this last

expression in [6471], we get [6473].
VOL. IV. 26

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

[6473] ^=^, + ^/'â€¢^'+ 4 ^- ^- bf'G-n- sm.(^.v'â€”3v' - ^ . v'-'iS .

whence we get, by integration,*

72n
m.^.b'i'.^l â€” rj.sir

[6474] 5' ==

4.f:_3-4-iv;)

The action of the third satellite adds also to the expression of 5', a term

which may become sensible by its small divisor. This term is analogous to

that which the action of w! upon m, produces in s ; therefore, by

[6475] putting y^f for what Ul^ becomes, relative to the second satellite,

compared with the third ; we shall have, for the part of 5' depending on
the action of m", f

w".a'2.6'f.(Z''â€” r).sin.(3i;'â€” 4i;"^i?<â€” A)
[6476] 5' = ^â€” ^, .

V n n 'J

We may therefore unite, into one term, the two terms of the expression of

* (3353) Tlie equation [6473] being integrated as in [6049Ar, /], acquires the divisor
which is given in the first member of the following expression, and this is easily separated
into two factors, as in the second member ;

[6474a]

|:-3-â€ž^y-^;==(^"-3-5-^:')-(|-3-^+iv;).

/2n

2n
Now we have nearly â€” = 4, iV/= 1 [6151, 6469c] ; therefore the factor

n'

2n

3-^ + iV; = 2 nearly;

[64746]

hence the divisor [6474o] becomes 2. ( â€” â€” 3 â€” 4- â€” ^/) â€¢ Dividing the last term of

[6473] by this quantity, according to the directions in [6049ZJ, we get the expression of
s' [6474].

t (3354) If we change, in [6456], the elements of w, m', into those of m', ni'

respectively, the expression 01=-^ [6470Â«], will change into ^; A*^ will become JV/

[6447,6469c], &c.; and the value of s [6456], will change into / [6476] ; using 6f>

[6475]. IMultiplying [6477] by â€” 2, and adding Su â€” 3w' to both members of the
[6476 ] profjuct, we get 3u' â€” 4Â«" = 2v â€” 3i/ â€” 400^ ; substituting this in the first member of
[6478], we get its second member.

TTII. iv. ^11.] INEQUALITIES IN THE LATITUDES. 99

y, which depend on the action of the first and third satellites. For we
have ver^ nearly, as we have seen in [6155],

V â€” 3i/4-2Â©'' = 200Â°; [6477]

which gives, as in [64766],

sm.(3v'â€”A,v"^pt'-A) = sin.(2râ€” 3i/â€” ;i<â€” A). [6478]

If we connect this term with that which depends on the sun's action,
observing also that the equation n â€” 3w' + 2n" = [6152], gives [64781

â€”, 3 = 3 -; we shall have, for the expression of the inequalities of [6479]

% n

the second satellite in latitude, relative to the mutual configurations of the
satellites and the sun, the following expression ; *

[Â«4801
3Af Â«. (L^-r).sin. (t/-2 U-pt-A)

We shall find in the same manner, for the expression of the corresponding
inequalities of the third satellite in latitude,!

[6479al

2 4n"

â€¢(3355) Multiplying the equation [6478', or 6152] by -7, and adding 3 ;-,

n n

in"
to both members, we get the equation [6479]. Substituting this value of 3 ;-, in the

divisor of the expression [6476] we get 3 -, ^ â€” JV/ = -^ â€” 3 â€” -^ â€” iV/ , [64796]

which is the same as that in the divisor of [6474]. Substituting also in [6476] the
expression [6478], we get the term of [6480] depending on m" ; that depending on m'
being the same as in [6474], using the value of a [6476a]. Lastly, the term of [6-180] ^ ^
depending on the sun's action, is easily obtained from [6459], by changing the elements of [6479i<
m into those of m'; by this means it becomes as in [6480 line 2].

t (3356) The satellite m" is situated, relative to m', in the same manner as m' is
relative to m ; hence it is evident tliat we may deduce the value of a", arising from the ^ "^
action of the satellite m', and the sun ; by adding an accent to each of the symbols m, a,
t^f It /', t>, n, n'f JV*/, 6J\ In this way we find that the terms depending on m, [64816]

[6480 line 1], produce those in [6481 line 1] ; and those in [6480 line 2] give the terms
depending on the sun's action in [6481 line 2] ; observing that the angle 2v â€” 3v' â€” pt â€” a, [6481*]
[6480 line 1], b the correct form under which this argument appears, in [6474], after

[6484]

100 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cel.

Periodical ,. * ' 4 . . _, , a ii ^ a\

terms of 5"= -â€” ^ , Sm.(2v' 3v"â€”pt A) J

^^^^^^ _ SM^{L'â€”r).sm.(v"â€”2U-r^â€”A) 2

Periodical Lastlj the same expression becomes, relatively to s'"*

'"/'"' ,, SM^Uâ€”r).sm.(v'"â€”2Uâ€”ptâ€”^) .

[6482] - 4n-(^ + Z^ + ^^;._l) '

[6483] iV/' and Nl" being what iV [6447] becomes, relative to the third and
fourth satellites. We must apply to the terms of 5', 5", 5'", contained in
[6480 line 2, 6481 line 2, 6482], what we have said upon the corresponding
term of s ; namely, that in eclipses each of these terms is confounded with
the corresponding term, depending upon the inclination of the equator to the
orbit of Jupiter ; and that, on this account, it increases^ the inclination

re-substituting the assumed value of â€” = t, Sic, in the terms where it had been introduced.

[6481tf| It is not necessary to notice the action of the two satellites m and m!" upon m" ; for
the factors similar to 2n â€” (t-|-l).(n â€” n') [6434i], corresponding to the action of the two
satellites m, rd", become respectively 2n" â€” (*4-l)- (w" â€” n) ; 2n" â€” {i-\-\).{n" â€” n'").

[6481e] Now we have very nearly, in [61516], n = 4n", n'"=fn"; hence the preceding
factors become n".(5-f-3i), and n".(^- â€” f i) ; which do not vanish with the integral

[6481/j yj^iygg Qf I [6434rf] J therefore we may neglect these terms.

* (3357) Changing the elements of m" into those of m'", in the terms depending
[6482o] Qjj ^|jg gyjj5g action [6481 line 2], we obtain the terms of /' [6482], corresponding to the

sun's action on the satellite m'". We may neglect the action of the three satellites
[64825] 'm,m',rri' upon m'" ; because the factor 2n â€” {i-{-\).{n â€” n') [6434tJ becomes, for these

three satellites respectively,
[6482c] 2n'"-(z+l).(n'"_n) ; Sti'"â€” (i+l).K-n') ; 2Â«'"â€” (t+l).K-n").

[6482d] Now, by [61516], we have nearly n = -^w'"; ?t' = J/-n'"; vS'^lvl"-, hence the
[6482e] factors [6482c] become respectively f (31+25i).7i'", ^.(17-i-lli).n'", ^.(10+4i).n'";

and neither of them vanish with the integral values of i [64346?] ; therefore these terms

may be neglected.

t (3358) In the original work this is said to decrease the inclination ; we have altered
^ "^ it to correct for the mistake of the author, as in [6462/, &Â£c.].

Vm. IT. ^ 11.] INEQUALITIES IN THE LATITUDES. 101

Approsi-

which is deduced from observations of eclipses. We may also observe, that ^"^ ^
we can put, in all these expressions, without any sensible error,* N^ N',

*(3359) Wc have, in [6796,6797], 2. ^i^^= 2. ^^^^ = 0,001 nearly, and [6485Â«]

JMÂ«

from [6025Â»] J. -^=}- (0,0004)' is a very small fraction in comparison with the
preceding; therefore it may be neglected in [6447]. Multiplying the second of the
equations [64706], by m'. â€” = m'.a [6470a], we get

[64856]

a:
m'

I'-aV-B*" = m'.aÂ«.W) = -Sm'/^V. 6^ [992] ; [6485c]

and by substituting the values of a, 6"\ [6801, 6802], it becomes of the order 2m' or

"^ [6485rf]

0,00005 [7163] ; so that this term and the similar ones depending on m", fn"\ are very

small in comparison with that in [6485a] ; therefore if we neglect these terms of iV,' [6485e]

(P â€” ^9)
[6447], we shall have A'/=l-{-2. , whose square root gives very nearly

A'^= 1 -j â€” , as in the first of the equations [6485]. Changing successively a into [6485/']

df fl*, t^'f we get the other expressions in [6485, 6486].

VOL. ir. 26

102

MOTIONS OF THE SATELLITES OF JUPITER.

[Mec. Cel.

CHAPTER V.

INEaUALITIES DEPENDING ON THE SQUARES AND PRODUCTS OF THE EXCENTEICITIES AND

INCLINATIONS OF THE ORBITS.

[6487]
[6487']

Action
of the

satellite.

[6488]
[6489]

12. It will suffice, in calculating these inequalities, to notice the secular
variations analogous to those which we have determined for the planets, in Â§ 5,
Book VI. It follows, from this article, that if we notice only the action of
m! upon m, the part of an.R, depending wholly upon the secular
inequalities, is*

an.il = -J.(0,l).|e2+e'2|-f[^l.eer(^'-^)+i.(0,l).{7/2_27,.7/.cos.(7;-'70+7i'l;

7i and 7/ being the inclinations of the orbits of m, m' to the fixed plane ,
\ , '7/, the longitudes of their ascending nodes upon this plane.

The part of an.R, depending on the sun's action, corresponding to the

[6488a]

[64886]
[6488c]

[6488d]

[6488e]
[6488/]
[6488g]
[6488ft]

* (3360) That part of an.R, depending on the squares and products of the
excentricities and inclinations, and which is independent of the configuration of the bodies,

affects the secular inequalities, and is as in [3765]

an.R =-|.(0,l) ! h^-\- Z24-A'2^Z'2 5 + 0,1 .(M'+^Z0+i(o,i). ! {p'-pfM^-qY\ .

Now we have, in [3756&, e], the equations [6488e]. From [1032] we easily deduce the
expressions [6488/], by using [24] Int. ; observing also that cp, 9', &, &' [1030', 1030"],
are changed into 7^ , 7/, 7^ , ?/, respectively, in [6489] ; and that on account of the
smallness of 7^ , 7/, we may take these arcs instead of their tangents. Substituting the
values [6488/] in the development of the first member of [6488^], we get successively its
sfecond member [6488^, or 6488/i]. Finally, substituting [6488e, A] in [64886], we get
[6488].

A2 _{_ ^2 = e2 ; A'2 _}- /'2 = e'2 ; hh! -f //' = ee'.cos.(^-ra) ;

P' + ?^

P'^ + ?'" = 7/' ; PP' + ??' = 7x.7/-cos.(7/~70 ;

{p'-pf-\- W-qy = ip'' + q'') - ^{fp'+ qq') + {p'+ q')

= 7.'"â€” 27,. 7i'.cos.( %'â€”\) + 7,2.

VIII. V. Â§ 12.] INEQUALITIES DEPENDING ON eÂ», e'Â«, &c. ; 7', 7^, &tc. 103

secular inequalities, is by ^1,* AeikMor

^___^ the tan.

an./J==â€” J.[Tj.{f'+//Â«-.yiÂ«+27,.y.cos.(ij - 7) - -/Â»}. [6490]
Lastly, the part of an./?, depending on the ellipticity of the spheroid of

Cllbct of

Jupiter, is by [6052, &c.],t S.?!"'^

an./J= J.(0).{aÂ» + 2tf.7,.cos.(>i'-f i,)-f ^,Â«â€” e''}. lÂ«9i]

â€¢(3361) These terms of /i may be deduced from those in [6042], neglecting â€” ,

as in [6277'J, and substituting the elliptical values of r, D, together with those of s, S',

&c. [63326, 6328c, S:c.], retaining only the terms independent of the configurations. ^ '

We may also derive [6190] from [6488], by changing the elements of the satellite m',

into those of the sun's relative orbit about Jupiter. This is done by changing m', e', xs', â€¢â€¢ J

7/1 '^11 *Â», a, into S, H, /, 7, 1, M, â€” , respectively, as appears by the comparison [6490c]

of the definitions of these quantities, in [6021c â€” 6024Â»i]. Now if these changes be made

a? [6490rf]

in the expression m.a', it becomes S. â€” =JW.<r' [6105]; and by substituting the

â–¼alue o' = n~' [6110], we find that w'.a' changes into -^ ; so that this quantity, [6490e]

multiplied by a or jr,j must be very small [64856], and may be neglected ; or, in other

words, we may neglect terms of the order a*. In this case it will only be necessary to

take the first terms of the developments in [989], which give 6^]1 = 2 ; 6^^\ = â€” a. [6490/"]

Substituting these in [1082], developing according to the powers of a, and neglecting

tt*, we get 0Â»1 = â€” foum'.n.^ â€” a4-a.} = 0j moreover [1076] becomes, by [6490^-]

neglecting quantities of the same order, (0,l) = f.m'.a'.n. Now changing, as in [6490e],

J|f9 I 1

m'.aÂ» into M^.tt^, it becomes (0,1 ) = f. â€” = [6216]. Hence it appears [G490A]

that the change in the symbols, mentioned in [6490c], makes 0,1 vanish, and r^vjq^-.

(0,1)= . Substituting these and the values [6490c] in [6488], changing also

I 1 [6490*]

cos.(7â€” li) into cos.(7i â€” 7), we get the expression [6490], corresponding to the part

of an.Rj which depends on the sun's action.

t (3362) The part of R, depending on the ellipticity, is given in [6052]. We
must substitute, in it, the values M = 1, ^=1 [6082,6282']; and the part of the ^ "^
elliptical value of f^ [3702c], which is independent of the configuration, namely r649i&]
r-3x=s<r^.(l -{- JÂ«") ; then multiplying by oiif and using the value of (0) [6216], we

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

[649rj Therefore we shall have,*

obtain the first and second expressions of an.R [649 lc(]. Multiplying together the factors
of this last expression, and neglecting terms of the fourth order in e, s, s^ , we get

^ ^^ [6491e]. Developing this expression and neglecting the term â€” ^-(0), which produces
nothing in an.dR [6492], we obtain [6491/].

[6491rf] an.i? = -Â«n.(p-J9).{^-(.-50^].Â«-3.(l+3e2) = _(o).{i_(,_5j2j.;i_|.3e2j

[6491e] = _ (0) [ ^ 4- J e^-{s-s,y \

[6491/] = (0)\s,^^2s,s-\-s^'-ie^.

Now we have, as in [6323<Z, 63326],
[6491g-] Sj = â€” 5. sin.(i;+'i') ; s = %.s\n.{v â€” 7j.

Taking successively the squares and product of these values of 5, 5^, and reducing, by
'â–  â– ' means of [17] Int., we get the following expressions, by retaining only those terms which

are independent of the configurations ;
[649H] 5,2=i^^; s2^^7,2. s^s = â€”ldy,.cos.{-f + '],).

Substituting these last values in [6491/], we get [6491].

[6492al

* (3363) Adding together the three functions [6488, 6490, 6491], we obtain the

complete value of that part of an.R, which is taken into consideration in this article.

The difierential of this expression, relative to the characteristic d, is taken in [6492],
[64926] considering the elements of the satellite m as the only variable quantities, as in [6055] ;

so that we must suppose e, zs, 7^, 7^ [6061, &z;c. ; 6315, &ic.] to be the variable quantities,

a being constant as in [1044"] ; also e', zs', 7/, 7/, H, 7, 7, 6, ^, are considered as
[64926'] constant, relative to the characteristic d. As e is connected with sin.w, or cos.w, in

[6493, 6494]; and 7^ with sin.7i or cos.7i, in [6499, 6500] ; the quantities (e.sin.'ci),

* (e.cos.-Bj), (7j.sin.7j, (7j.cos.7i), are considered as the variable quantities in [6492];

[6492d] and the parts of an.R [6488, 6490, 6491] may be made to contain these terms, by the

substitution of the following expressions, which are easily deduced from [23, 24] Int. ;
[6492c] e2==(e.cos.z3)2-j-(e.sin.'w)2; e'^=:{e'.co5.z/)^-\-{e.sm.z/f; 7i^={7i'0os.1^y+{y^.sm.l^Y;
[6492/"] ee'.cos.('ia' â€” zs) = (e.cos.'Zj).(e'.cos.'cj')+(e.sin.'cj).(e'.sin.'3/) ;
[6492g-] 7,7.cos.(7,â€” 7) = (7,.cos.7,).(7.cos.7)+(7i.sin.7,)-(7.sin.7) j
[6492/1] ^ 7i-cos.(^ + 7i) = (71- cos. 7 J.(^.cos.^)â€” (71- sin. 7i).(^.sin.^).