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Pierre Simon Laplace.

. (page 22 of 114)

which is used in the planetary theory [1133, 1137"], where we have found,

q=^.N.cos.{gt-\-^) ; p = l..N.sm.{gt-\-fi) ; s = q.sm.{nt-\-s)^p.cos.{nt-{-t). [6398*1

Substituting these values of y, p^ in that of s [6298A] and reducing, by [22] Int., we
get $= l..:S.s\n.{nt-\-zâ€”gtâ€”^). Now changing JV, â€”g, â€” p, into /, ^, A, [62961] respectively; also n/ + s, which is the mean value of t), into v [6298a]; it becomes of the same form as in [6300 line 1]. In like manner we may deduce the forms of [62984] /, #", &ic. [6300] from those of q, /, &tc. [1 133] ; observing also that, in this notation, the quantity g, or the equivalent expression â€” p [6298Â»], is of the order of the L"*^^*J disturbing forces^ as in [1097c]. We shall now make a few additional remarks on the process of integration, which is used 64 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cel. [6298'] h^ = a [6110^], we shall find that the preceding differential equation by the author in this chapter. We shall see, in the next note, that by retaining only the r6298 1 ^^*â„¢^ mentioned in [6298], the equation [6295] can be reduced to the form [6299]; which is linear in terms of s, s', s", s'", S', St. This differeniial equation corresponds to the satellite m ; and there are three similar linear equations, corresj^onding to the satellites m', rrb', m'" ; also an equation relative to the motion of Jupiter in its orbit ; and another relative to that of the equator of Jupiter. Now if we suppose s to contain a term of the form s = Z.sin.(u-f-i'^-| - '^)) it will produce, in [6299 line 1], some terms of the form l- ^J l^.sin.(v-^f-pt-\-A) ; which must be destroyed by similar terms in [6299 line 2], so that the terms in this last line must produce the quantity â€” l^.s\n.[v-\-pt-yA), which depends on the same angle v-\-pt-\-A, but has a coefficient â€” /^, with a different sign irom that [6298p] which is produced by the terras in [6299 line 1]. We must therefore assume such forms for /, /, /', S', s^, as will, by substitution -n [6299 line 2], produce terms depending on sin.("-[-p/-[-A), and this 'is obtained by means of the fomis assumed ui j 6300 lines 2-6], [6298g] as is easily proved in the manner pointed out in ihe note in [63026, c. &:c.J There is a perfect symmetry in all these expressions, since the argument, or angle, under tiie sign sin. , is found, in every case, by adding the same angle pt-\-A, to ihe longitude of the object, whose latitude is to be computed by that formula. Thus, the longitude of the I- J second satellite, v\ is added to j?^-}-A, In [6300 line 2], to find the argument of the [6298s] latitude / of the second sateUiie ; the sun's longitude U, is used in [6300 line 5], in finding the sun's latitude S' ; and ihe longitude v, of the first satellite, is used in [6300 line 6], in findiog the latitude s^ of that satellite; supposing it to move in the plane of the equator of Juplier [6051]. In the same manner as we have found the six expressions [6300], corresponding; to the ande pt-l-A, we m?y obtain similar ones, relative to any [6298u] â– - -" ' o or,} j ' j oiher angle pZ-'r^i vvhich fcatisfies the proposed diflerential equations, enumerated m [629Sn] ; and by changing the coefficients I, V, I", k.c. Into l^, //, &Â£c., we obtain another system of equa lions s = /^.sin.(Â«Â»-l-/?i^f-Aj, s'=li.s]a.(;o'-\-pJ-{-A^),kxi., similar to [6300]. A third angle, pJ-f-A^, will in like manner give s=l2.i,m.(v-\-pJ-\-A2) ; [6298w] s''==//.sin.(//-[-p2/-{-A2), &c. ; and so on for other values of p. Finally, it is plain, from the linear form of the equations [6298m, &;c.], that we may add together all these values of s, for the complete value of s ; all the values of /, for the complete value of /, he. ; so that we shall finally have, by using the symbol 2' as in [6324'], s =1. sin. [v-[-pt-{-A)-^l^. sin. {')-}-;) f+Aj-|-Z .a\o.{n-\-p^l-\-A^)-\-SLC. = 'E'.l.sm.{v-{-pt-{-A) ; 1 s' = v. sm.(,v'-\-pt-\-A)-\-l^.8m. {v'-\-p^t-\-A^)-\-l^'. i^in.{o'-\-p^t-\-A^)-\-SLC. = 2'. I'. sm.{v' -^pt -\-a) Â«'' = Z^sin.K-[-;jf + A) -H ^sifl.(Â«''+p^i + A J +Z;^ si a.( y''-l^^f +Ag) +&C. ^ [6298x] ^,â€ž ^ p,gin.( â€ž'"_|-pf-|-A)-|-Z^'".sin.(Â«'"^-p^f+A^) + l^'".sm.{v'"-\-pJ-\-A^ )t &c. = S'.l'".8m.{v'"-^pt-\- a ) S' = I.'.sin.( U-{-pt-^A)-\-L\sm.{ U-\~p^t-i-A^)-^L^.Bm.{ U-\-pt-\-A^)-{-&,c. = 2'.i'.sin.(L7"-f;)f-f-A) VUI. iv. ^ 9] INEQUALITIES IN THE LATITUDES. 65 becomes, â€¢ [6298^ These forms agree with those in [6427 â€” 6430] ; the terms depending on the angles jp'-j-A, PJ-h^t* PÂ»'~f"^j> PÂ»^'{'^9i arising from the mutual action of the sun, satellites, and the [G296y] ellipticity of Jupiter, are explicitly retained, in [6427 â€” 6430]. The remaining terras depending on the angles pJ-\-A^, pj~\~^i> ^c., arising from the displacement of Jupiter's equator and orbit, are reduced, in [6342 â€” 6414], to a single term of the form [6362], â€¢â€¢ *â€¢â€¢ corresponding to those in the first line of each of the expressions [6427 â€” 6430]. â€¢(3299) Substituting r=a [6296'], and h* = a [6298^, in [6295], it becomes, [G299a] dds /dR\ da /dR\ If we introduce into this expression the value of aR [6297], and retain only the terms depending on s'm.{v-\-pt-\-A)y as in [6301], it will become as in [6299]. For the terms L^Â«99c] dd$

VT -{- * [62996], are the same as the two first terms in [6299 line 1] ; and we shall now

proceed to prove that the last term of [62996] may be neglected, and that the remaining

(dR\
â€” j, produces the rest of the terms of [6299]. The chief terms of ail [6297]

containing Â», are of the form ail = 2.^.cos.Â»(Â»' â€” v); which gives, [62904]

/dR \ /....//

a. ( â€” -j=2.Â«5t.sm.i(t> â€” V).

ds [6299Â«]

Multiplying this by j- = Lcos.{v-{-pt-\-A) nearly, [6300 line 1], it produces terms

[6299/]

depending on the angles tv'â€”(iztl)v^pt^A; which cannot be reduced to the form
v-\-pt-]-A [6299c], by using, for i, any of the integral numbers 1, 2, 3, &Â«:. ; therefore
it may be neglected. The other terms of o.(-â€”j [6297] are of the order s, or of the

forms which are neglected in [6297']. We shall now compute the terms of [6299]

/dti\ r6299^1

depending on the remaining part Â«â€¢ (-^J of the equation [62996] ; which, by substituting

[6297], becomes,

a. (â€” j =l.m.~.{ Â«'-*.cos.(t;'-i;) } -2.m'.a V. { /-Â».cos. (Â»'-Â») | . | i B'^^-\-R^\cos.(v''v)-\-kc } 1

â€” ^.{â€”6*â€” 6*.cos.(2t>-2i7)+12S'.cos.(t;-^)| 4-(f^^|2,â€” 2*,|. 2 ^^^*^

As we retain only the terms depending on the angle v-}-pt-]-A^ after substituting the

values [6300], we may neglect all the terms of the first line of [6299A], except that

depending on B^'^ ; and this term produces 2.iii'.flV.*.cos.(t/â€” Â»).J?>*\cos.(o' â€” Â»), or by [6299i1

reduction, using [20] Int., J Z.m'.oV. I?''. * , which is the last term in [6299 line 1].

It also produces the last term of [6299 line 2], namely, â€” 2.m'.aV.^Â»./.co3.(t;' â€” Â»).

We may also neglect the term depending on cos.(2j; â€” 2U) [6299A], since it does not [62994J

VOL. IV. 17

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

[6299] av (^ a n )

To integrate this equation, we shall suppose, as in [6298c-/],

s =1. sin.(v +pt-\-^); 1

5' = /'. sin.(t)' -f-^^ + A) ; 2

^'=l".sin.(^'+pt + A); 3

s"'^r'.sm.(v"' + pt + A); 4

ASf' = L'.sin.(t/ + p^ + A) ; [6329] 5

Si =L.sm.(v +pt + A) . [6323] 6

[6300'] Putting pdt = ^.dv [6089/], and substituting the assumed values [6300]

[6301] in [6299] ; then comparing together the coefficients of sm.(y-\-pt-\-A),
neglecting the square of ^ ; p being a very small quantity of the same order
as the disturbing forces [6298/] ; we shall obtain,*

[6302]

^'-i.5_..xÂ».'.â€žV.5<'>5

[6303] / 1 \ Tl^r

If we put, as in [963'% 964],

produce a term of the form [6299c]. The two terms in the second line of [6299A],
[6299/:] depending on â€” 6s, -}-12*S", produce the terms in [6299 lines 1, 2], which are
multiplied by M^ ; and the terras of [6299A], depending on (p â€” J9), produce those in
[6299] having the same factor.

* (3300) Substituting the values of S', s' [6300 lines 5, 2] in the first members of
[6302a] [6302 6, c], and then reducing, by means of [18] Int., retaining only the terms depending

on the angle v-\-'pt-\-A, we get the second members of these equations 5
[63026] *S'.cos. ( Uâ€”v) = L'.s\n.{ U-j-pi-\-A) .cos. ( C7â€” v) = | L'.sin. (v+pt+A.)

[6302c] /.cos. (v'â€”v)=r.sm.(v'-\-pt-\-A).cos.{v'â€” v) = ^Z'.sin. (v-\-jpt-}-A).

Again, the second differential of s [6300 line 1], talcen relative to v, using pdt [6300']

dds / p\^

[6302i] gives t^ = â€” Ul -{- ^j .sm.(v-\-pt-{-A) ; adding this to the same value of 5, and

[6302c] neglecting the term of the order p^, we get -^-j-5 = â€” 2?. â€” . sin.(Â«-|-p^+A) .

Substituting the values [63026, c,e; 6300 lines 1, 6], in [6299], and then dividing by
â€” 2.sin.(u+p<+A), we get [6303].

VIII. iv. <^ 9.J INEQUALITIES IN THE LATITUDES. 67

~ = a ; [C304]

(1â€” 2a.cos.Â«+aÂ«)- = J.6/^+6;Â»>.cos.Â«-f V.COS.20-I-&C. ; [6305]

we shall obtain, as in [1128],

aV.Bt'> = aÂ».6^Â»>. [6306]

Now we have, as in [1076, 1 130],*

(0,1 ) = â€” ^f:!"^*fj = i.m'.n.aÂ«.6(;) = ;^.m'.n.aV.B('> ; [6307]

hence we get,t

0=/.^H0)-[T]-(04)-(0,2)-(0,3)|+(0).I^[7]l'4-(0,1).Z'+(0,^ [6308]

We shall find, in the same manner,

0=^^Hl)-[T]-(l,0Hl,2)-(l,3)^+(l).L+[7]x'+(l,0)i+(l^ [6309]

0==r^|p^(2)-[T]-(2,0)-(2,l)-(2,3)|+(2)X+[7]X'+(2,0).Z+(2,^ [6310]

0=r^M3)-Q-(3,0)-(3,l)-(3,2)^+(3)X4-|T].i.'+(3,0)./+^ [6311]

There is, between the quantities p^ /, /', /", /'", L, L', an equation,
which depends on the displacing of Jupiter^s equator, by the combined
actions of the sun and the satellites. To obtain this equation, we must [6311*]
determine the precession of Jupiter's equinoxes and the nutation of its axis
relative to the fixed plane, using, as in ^5 of the fifth book, the following
symbols ; J

â€¢ (3301) The first and third values of (0,1) [6307], are the same as in [1130] ; and
by substituting the values of a^af.B^^ [6306], in the last expression, we obtain the second â– '

form [6307].

t(3302) Multiplying [6303] by n, and substituting fm'.n.aV.B<'^ = (0,1) [6307],

with the similar values (0,2), (0,3), &c. ; also the values of (0), I I [6216] ; we get [6a06a]

[6308]. Changing successively the elements relative to m, into those relative to
m', m", &tc., we get [6309 â€” 63 11], corresponding to the other satellites.

X (3303) These symbols are similar to those which are used in figure 58, page 803, of
the second volume, in computing the precession and nutation of the earth's axis. The '* â– '
symbol 7 [6314] represents the longitude of the ascending node of Jupiter's orbit, relative [6313&)

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

Symbols.

[6312] 6 = the inclifiation of Jupiter's equator to the fixed plane ;

[6313] "0- = the retrograde motion, or precession, of the descending node of

Jupiter's equator upon the fixed plane, and counted from the fixed

axis of X ;
[6313'] 7 = the inclination of the orbit of Jupiter to the fixed plane, [being the

same as that of the relative orbit of the sun about Jupiter] ;
[6314] 7 = the longitude of the ascending node of Jupiter's orbit, counted from

the fixed axis of x ; [being the same as the longitude of the

ascending node of the relative orbit of the sun about Jupiter]

[6313^0-];

[6315] 7i= the inclination of the orbit of the satellite m, upon the fixed plane ;

[6316] 7i = the longitude of the ascending node of the satellite m, upon the

fixed plane, and counted from the fixed axis of x ;
[6316'] it = the rotatory motion of Jupiter in the time t.

Then we shall have, as in Â§ 5 of the fifth book, bj neglecting the square

[6316"J

[6313c]

[6Sl3d]
[6313e]
[6313/]
[6Sl3g]

[6313/i]

of 6, *

to the fixed plane, and counted from the axis of x ; observing that this longitude is the
same, whether we consider the actual orbit of Jupiter in its motion about the sun, or the
relative motion of the sun about Jupiter, supposing this planet to be at rest. It being
evident, that when the sun is considered as at rest, the heliocentric longitude of Jupiter, at

[6313e] the moment of passing the ascending node of its actual orbit, will be represented by "7 ;
and that if the planet be considered as at rest, the longitude of the sun, at the time of
passing the ascending node in its apparent orbit about Jupiter, will also be represented

[6313jg-] by 1, as seen from the planet, and counted from the same axis x, or point in the heavens.
This symbol 1 is equivalent to r^, in the notation which is used for the earth in
[3067, &ic.]. The longitude 7^ [6316] is counted from the fixed axis of x; and if we
add to it the precession ^ [6313], from the same axis, we shall get Ti-f-"^? /or the

[6313t] longitude of the ascending node of the satellite m, upon the fixed plane, and counted
from the moveable vernal equinox of Jupiter.

[6313A;] We may observe, that the planet Jupiter revolves about its axis in 0'^Â«y%41377 [6920] ;
hence its motion in a Julian year is i = 353094Â° 71' nearly, [6021 w]. Comparing this
with the values of n, n', n", n'" [6025A:], we get the following results, which will be of
use hereafter ;

[6313;Â»] i = n. 4,27565 == n'. 8,58250 = n". 17,2911 3 = n'". 40,33405.

[6313Z]

* (3304) The expression of â€” , corresponding to Jupiter and its satellites, is easily
[6317a] deduced from the value of 6 [3089], relative to the earth and moon. For */ we neglect

VIII. iv. ^ 9.] INEQUAUTIES IN THE LATITUDES.

dt

4Â».(;

.{ 3P. r.sin.(7 -|-i')-}-2.ifi.n*. 7|. sin. ( ii -[-'Â»') I â€¢

69

[6317]

the terms of [3089] depending on cos.2v, cos.2p', which are very small [3377] ; then
take the differential of the remaining tenns relative to t, and divide by dt^ placing an
accent below the letters m, n, X, to distinguish them from the same letters m, n, \ which
are used in [6022e?,/; 60246, iic] for other purposes, we shall get,

^^3_ (p=Â±Z?J . {(m,Â»+/m;).cos.^.2.c.sin.(yi4.p)->m^c'.cos.^.sin.(/'<+^){.

To obtain the similar expression for Jupiter and its first satellite, we must change, as
in [3060],

â€¢an*c mass . snn's mass

mto mr =

= JWÂ« [6105] nearly.

(disU 8un aud earUi)3 "' (dist. sun and Jupiterp

We have also, as in [3079], by changing what relates to the earth into the corresponding
terms relative to Jupiter ; and using the values of a, m, n^ [6021c, 6022rf, 6110] ;
2 mass of the satellite m

' ' (distance of satellite and planetp o3

The rotatory motion of the planet nt [3015], is changed Into tV, In [6316'] ; so that
n^ = I. We have in [3073f/], by changing tang, y into 7, on account of its smallness,
2.c.sin.(//-|-^) =7'Sin.A. Substituting A=r^-}-'*' [3069], and then changing r^
Into 7, as in [6313^], to conform to the system now under consideration, we get
2.c.sin.(//-|-p) =7Â«sln.(7 + *). The quantity c' [3086], is changed into tang.7,
[6315], or 7j nearly, so that for the satellite m, we shall have c' = y^. The
expression â€” ft â€” p' [3086] represents the longitude of the ascending node of the orbit of
the satellite, counted from the moveable vernal equinox, and this is changed Into li-\-^
in [6313Â»].

Now substituting the values [6317^, c, /, A, Ic] in [6317c], and putting as in [6316"],

dd

COS. ^:^ 1, we get the following expression of â€” , corresponding to the system of Jupiter
and its first satellite ;

^a^3.(2C-^-B) J (^M^^m.n^y^.sm.{n+^)^rn.nyr,.^^n.{\+^)\,

If we take, as in [6398], the orbit of Jupiter in 1750 for the fixed plane, y will be of the
order of the disturbing force [6313'] ; and by neglecting quantities of the order of the square
of this force m.y, we may reject the term m.n*.7.sln.(7 -j-'r) [6317fn], and we shall have,
di 3.(2Câ€” ^â€” i?)

If in the last term of this expression, depending on the satellite m, we change successively
the elements of m, into those corresponding to m', m", m"\ we shall get the parts
depending on these satellites ; and by adding them to those in [6317o], by means of the

symbol 2, we shall get the complete value of â€”, as In [6317].

VOL. IV. 18

[63176]
[631761

[6317e]
[6317rf]

[6317e]

[6317/]
[6317^]

[6317A]
[6317t]

[6317A]

[6317/]
[63I7JII]

[6317fi]
[63l7o]

[6317p]

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

[6318] We shall also have,*

[6319] ^- J = ^'^^^7c~^^ 'K^'+^-^-^')-<>+i^V.COS.(7+-p)+XW.W^7i.COS.(7,+i-)}.

Miiltipljing [6317] by sin.^, and [6319] by cos.^; then taking the sum
of these products, we get,t

[6320] â€” ^-^- â€” ^ =â€”=â€”â€”â€” â€” ' ,\(M^+i.m.n^)Xcos.^-\-M^r.cos.i -\-i.m.'nr,ri'Cos.'} i].
In like manner we shall have, J

* (3305) Neglecting the terms of [3096] depending on sin.2w, sin.St;', as in [6317&],
'â–  "'^ then putting as in [6316"J, cosJ = l, sin. 6^=6, we get, by multiplying by 6, and
accenting m, n, as in [63176'] ;

[63196] ^.^ = ?^^?|^M (^/'+/^.')-^+K"^ (/'<+P') !â€¢

Substituting the values of mj*, ^mf', n^, c', â€”ft â€” ^' [6317rf, c,/, i, Tc] ; also putting
successively, as in [3075, 3069, 6317^],

[6319c] 2. c.cos.(/if4-^) = 7.cos.A= 7.cos.(r^4"*) = 7.003.(74-*) ;

we obtain,

[6319(/] ^ - ^ = ^r " .j (JW'Â»+m.n8).5+(JW8_j_^.â€žs).7.cos.( 7 + Y)+Â»i.n3.7,.cos.(7,+Y) j .

Neglecting the term ni.n^.y.cos.( 7 + '*')> as in [6317n] ; and prefixing the symbol 2 to
I- ^1 the terms depending on m, so as to obtain the sum corresponding to all the satellites, we
get [6319].

t (3306) The first member of this sum is â€” , which is evidently

r6320al equal to the development of the differential expression in the first member of [6320]. The
second member of this sum may be reduced, by means of tlie following expressions, which
are easily obtained from [24] Int. ;

[63206] sin. (7 -j-^).sin.*-4-cos.(7 -|-^).cos.Y = cos.{(7 -j-*^) â€” *} s=cos.7 ;

[6320cT sin.(7i-[-*)-sin.Y+cos.(7i-f-*).cOS.* = cos.{(7j+>i') â€” y} =cos.7, ;

hence the second member of this sum becomes as in the second member of [6320].

X (3307) Multiplying the equation [6317] by cos.^f, and [6319] by â€” sin.^, then
taking the sum of the products, we find that the first member of this sum is

r6321al â€” '' â€” '' 1 ^ â€” ~' wbich is easily reduced to the form in the first member of [6321].

Substituting in the second member the two following expressions, which are easily deduced
from [22] Int.,

VIII. iT. ^ 9] INEQUALITIES IN THE LATITUDES. 71

â€” L_ â€” r s= ^.{-(iT-f 2.W.II*) J.iiii.r-f-lfV.Â»in.7 4-2.Â»i.Â»*.-/j.fiii.7 ,}. (6881)

To integrate these two equations, we shall obsenre, that the latitude of the
jirtt tateliite, supposing it to mote in the plane of Jupiter^s equator^ is r^MMi
â€” i.siii.(v-|-r) above the fixed plane.* But we have already shown that this
latitude is equal to a series of terms of the form L,Bin,(v-\-pt-\-A), and we [6393]
shall denote this series by

â€” *.sin.(t;+*) =2:'.L.3in.(r+p/+ii). [6394]

The characteristic if signifies that the sum of (dl the terms of the proposed [6384^
function^ which are similar to those under this sign, is to be taken as ^'
in [6298a:].

The characteristic 2 [6118/'], denotes that the sum of the terms relative [6385]
to the different satellites is to be taken, ^

Therefore we shall hare,t

w dbtam, far the second aieinber of the sum, tlie ssune leault as in the second member
of [6321].

* (3306) The motion of the mellite m is direct, and reprcMMed by v [6092],
eoonted from the axis of x. The retrograde raotioo of the d e se ^ n din g node of Jupiter's l*****'
eqwuor is y [6313], counted iirom the same axis x. Therefxe the dina i ifn of the
sMeOke, from that deteerndtng node, b (v-f-r) ; conseqneatlj its di s tsnee , from the
a t emdiw g node of the equator abore the fixed ptane, is v-f-r 4-200^. Moltifrfyiog its
wme, by the inrftwtion I of the equator to the 6xed pbne [6312J, we get rerj nearly, .,.
as in [533a], the eleration or (fistanee of the satellite from the fixed pbae, sopposbg it to ^ '
ibM CfMlor ; beaoe this distanee is tjm4^9-^^-{''2O0r^), or â€” ijm^w-j-r), as VOXUX
[6322]. llovlbiilMgblisicpmcMedby Â«J6051]; therelbre Â«, Â» ^^.s^n^v+r) ;
ias *, isftpwantediÂ»[6g98>lme6] byaseriesoftcnnsof theferm JLam^w-^pfj-A), [9Uae]
sum is represented by prefitbg the symbol 2' [6324'J, ve sfasU have
.(â€¢-f*') = t.LMuJ(w'^'\'A), as in [6324]. ^*

[6aÂ»]

t(39O0) Devcfepmc the first members or[Â«396c,^], by MMi of [21] Int., we obtain,

sin.( V + r ) as sio.rxos. * -|-eoa.v.sin.T ; [MMn]

mm^9^'\-A)^9ia.Â»xm(pt'^-A)+ctmjrMm.(jt-\'A). [Â«aÂ»]

SnhrtiMing these in [6324],and pottmg the tanas dspindiwg on eos.v, ain.e, separately

aqHl lo caell other, we g<!t, by ehangpg the iipM of sB the mow, iIm two

[Â«ns,e327].

IÂ«>fc]

72 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. C6I.

[6326] ^ . sin. * = â€” 2'. L. sin. {pt -{- a) ;

[6327] fl. cos.^ = â€” 's!.L.cos.{pt-\-A).

[6328] In like manner the sun's latitude, above the fixed plane, is r.sin.(f7 â€” i);*

[6329] but this latitude is equal to 'l' .LI .s\n.{U-\-pt-\-A) ; hence we get,

[6330] 7. sin. 1 = â€” 2'. L. sm.{pt + a) ;

[6331] r.cos. 7= 2'.I/'.cos.(j?^ + a).

We have likevs^ise,t
[6332] 7i. sin. -Ji = â€” 2'.;. sin. {pt -\- a) ;

[6333] 7l.cos.^^= 2'./.cos. (;?^ + A) .

If we substitute these values in [6320, 6321], w^e shall obtain, by comparing

separately the coefficients of the same sines, %

[6328a]

[63286]
[6328c]
[6328rf]

* (3310) The projection of the radius vector of the sun's relative orbit about Jupiter,
makes the angle U with the axis x [6041] ; and "7 [6314] is the longitude of the
ascending node of the same orbit, counted from the same axis of x. Hence U â€” 7 is
the argument of latitude of the sun's relative orbit ; and as 7 [6313'] is the inclination of

[6328c] that orbit to the fixed plane, the sun's latitude above that fixed plane will be found, as in
[533a, Sic. ], to be 7.sin.(C/ â€” 7) nearly; being the same as in [6328]. Therefore we
have very nearly, for the sun's latitude S' [6040], the following expression;
S' = 7.sin.(C7â€” 7). But by using the symbol 2' [6324'J, we get, from [629ac line 5],
S' = "^ .LI .svc\.{U-\'pt-\-A) ; hence we obtain,

[6328e] S' â€” 7. sin.( C7â€” 7) = 2'. L'. sin.( U^'pt-\-A).

Now by development, using [22, 21] Int., we have,

[6328/] sin.( C7 â€” 7) =sin.f.cos.7 â€” cos. L/". sin. 7 ;

[63285-] sin.(Â£/-|-p^-|-A)=sin.C/.cos.(p^+A)-fcos.C7.sin.(p^-}-A).

Substituting these last expressions in [6328e], and then putting separately the coefficients
of â€” COS.C7, sin.C/, equal to each other, we get [6330,6331].

f (3311) The equations [6332, 6333] are obtained in the same manner as we have

'â–  "-' found [6330, 6331] in the preceding note ; changing the elements 7, 7, S', LJ , U, of

the sun's relative orbit, into 7, , 7j , s, I, v, respectively ; so as to conform to the orbit

of the satelHte m [6313' â€” 6316]. By this means the expression of S' [6328e] changes

[63326] into s = y^.sin.(v â€” 1j) = lf.Ls'm.{v-\-pt-{-A)', which represents the latitude of the

satellite m above the fixed plane [6298jc line I] ; moreover [6330] changes into [6332],

and [6331] into [6333].

X (3312) Substituting the values [6326, 6327, 6331, 6333] in [6320], then putting the

terms depending on cos.(^^-|-a) equal to nothing, and dividing by cos.(j?^+a), also

[6334a] transposing the terms to the second member, we get [6334]. In liiie manner, by substituting

[6326, 6327, 6330, 6332] in [6321], transposing and dividing by sin.(p^+A), we get the

same equation [6334].

Vni. iv. ^ 10.] INEQUAUTIES IN THE LATITUDES. 75

0^pL+ ^^^^^^^ . { Af Â».(L'-.L)+2.Â».nÂ».(/â€” L) } . [6334]

We may here observe that, if we suppose Jupiter to be an ellipsoid, we shall
have, as in [3408, &c.],*

C "" f,^n,H\dH
n^ being the density of the elliptical stratum, whose radius is R, [6336]

10. We shall now consider particularly the equations [6308-631 1, 6334],
putting them under the following forms ; f

0= \ ;,^(0)-|T|-(0,l)-(0,2)-(0,3) \ .(L^l) 1

^ ' â– > [6337]

+(O,l).(I^/0+(O,2).(Z^/'0+(O,3).(Z^/''0+|T].(I^L') - pL ; 2
0= j;>-(l)^|T]-(l,0)-(l,2)-(l,3) \ .(L-/') 1

^ ^ [6388]

4-(l,O).(I^O4-(l,2)<I^/'0+(l,3).(L-/''04-|T].(L-Z.')-P^; 2

(H)
0= j;, - (2)-[7]-(2,0)~(2,l)-(2,3) j ,(L-l") 1

[6339]

-f (2,0).(L~/)+(2,l ).(L-/')+(2,3).(L-/'") -f \V\ (L^L')^pL ; 2

0= |i>-(3)-[r|-(3,0)-(3,l)-(3,2) I .(I^/'") 1

+(3,0).(L-/)-f (3,l).(L-Z')H-(3,2) . (L-Z") + [T|(L-LO~/?L ; 2

â€¢ (3313) If we change a(p into 9, and aA into p, as in [6046a]; also the
density p [2947], into n^, [6336]; and the radius a into 7?, [2942,6336]; we