expression, the value of 7,^ [6528^], it produces the terms in [6528 line 2], as is evident
by inspection. In like manner, by substituting the value of d.7,.cos.(i' + 7,) =X. »i.6<, [6528o]
[6528)n], in the term depending on (0), [6517' line 2], it produces the term depending on
(0), in [6528 line 3] ; also by substituting 77,.cos.(7, — 7)=(X — 1).»jL.6< [6528^], in [G528p]
the term of [6517' line 2], depending on , it produces the term depending on ,
in [6528 lined].
If we change 7, 7, into 7,', 7/, respectively, in the identical equation [649%], we get,
7,.r ,'.cos.(7,— 7,') = (r,.cos.7,).(7,'.cos.7,')-f(7,.sin.7,).(7,'.sin. 7/). [6528^]
Again, if we change the symbols 7,, 7,, X, [6315,6316, 6313], corresponding to the
satellite i», into y,', 7/, X', respectively, corresponding to the satellite m', [6489, 6344], [6528r]
we shall find that the equations [6413, 6414] become,
7/. sin. 7,' = ( 1— X'). ^L. 'pt 4- >!.at ; \eS2»$]
7,'.cos.7,' = (X'— l).>L-f-X^6^ [6528<]
Substituting these values and those of [6413, 6414], in the second memoer of [6528^], and
retaining only the terms of the order t^ we obtain,
7..7.'.cos.(7.— 7.0 = {(X-l).X'-j-(X'-l).X}.>L.6^ [6528«]
Moltiplying this by i(0,l), we get the term of [6517Mine2] depending on (0,1), and
116 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. C^l.
[65083 suppose the excentricity H of Jupiter's orbit to be developed in the
following series,
[65B7] H = H,^ct + hc.\ [4407 line 1]
[6927'] JJ^ being the value of H at the origin of the time i ; we shall have,
^^ ^ = - 4.[T].£r,.d + 4.[T|.(1— x)^ii:.6^_6(0).x*.'L.6^ 1
-I- (l_x).x. I (0)+[T]+(0,l)4-(0,2)+(0,3) I . 'L. ht 2
4-J.(0,l).{(>^l).V+(x'— l).x}.^L.6^ 4
+J.(0,2).{(x— l).x"+(x"— l).x}. ^LM 5
+i.(0,3).{(x_l).v"+(x'"_i).x}. ^LM. 6
Hence it is easy to find, by means of the equations between x, x', x", x**
[6347—6350],*
arising from the action of the satellite mf upon n^ as in [6528 line 4]. We may deduce
â– - ' fix>m it the similar expression in [6523 line 5], arising from the acticm of m" upon m, or
from the term depending on (0,2) in [6517' line 3], by merely changing (0,1) into (0,2),
and X into X". We may also obtain, in the same way, that in [6528 line 6], arising
*â– ' from the action of m'" upon m, or from the term depending oa (0,3) in [6517' line 3],
by changing, in [6528 line 4], (0,1) into (0,3), and x' mto X'". Finally we may
observe, that if we multiply the expression [6528] by dt, and integrate it, we shall obtain
[6538r] tenns of Sv, depending on <®, which are similar to those in the lunar theory [5541, 5543]
depending upon the moon's secular equation, as is ol^erved in [6525].
* (3383) Transposing in [6347] the terms dependmg on (0,1), (0,2), (0,3), we get,
[6529a] X.(0)+(X-1).|T] = (0,1).(X'-X)+(0,2).(X''-X)+(0,3).(X - X) ;
multiplying thb by | — X, we obtain,
[65296] (i-X).^X.(0)+(X-l).jT]^ = (i-X) . {(0,1) . (X'-X) + (0,2) . (x"-X) + (0,3) . (X^-X)}.
Now the terms in the first member of this expression are the same as those of [6528 lines 2,3]
which depend on the functions (0), i L multiplied by the common factor ^L.btj as
is easily seen by connecting the coefficients, and making some slight reductions. We may
[6S9c^ therefore neglect the terms depending on (0), I I [6528 Unes 2, 3], and instead of
them substitute the second memberof [65296], multiphed by the common factor ^L.bt, namely,
VIII. T. ^ 13.] INEQUALITIES DEPENDING ON ««, «^, kc. ; y\ y'», &c. 117
it. = — 4.[7].-«^,.c/+4.[T].(l - >^)«.'L.6/ - 6(0).x».'L.6/ ; [6529]
which produces in 6r, or in the motion of the satellite m, the following
secular equation ;*
ar = — 2.[T].^i.c/»-f 2.[^.(1— x)«JL.6/»— 3(0).x».'L.6/». C««o)
We ma^ obser?e, with respect to the three first satellites, that the ratios
which obtain, between their mean motions, change considerably their secular [6o3i]
inequalities, as we shall see in [6663 — 6711,&c.].
When there is but one satellite, we shall obtain, from [6347], t vtineof
I I «« »*•
'^ = -j=ij [6532]
H+(o)
In the lunar theory o is incomparably greater than (0) , and thus we [6533]
(J— X). «X..6/.| (0,1).(X'-X)4-(0,2).(X"— X)+(0,3). (X"'^X) }. [6589rf]
Hence it appears that the coefficicDt of (0,1). 'i^. 5^, in the three lines
[6528 lines 2, 4 ; 6529rf], are respectively,
(I— X).X; J(X— l).x'+J(X'— 1).X; (J-X).(X'— X). [6529f]
If we add these terms together, we find that they mutually destroy each other. For the
coefficient of X', in the sum, setting the terms in the order in which they occur, without
any reduction, is i(X — l)-f-Jx-f-(i — X)=0; and by neglecting X', in the sum, it
becomes, without reduction, (1 — X).X — Jx — (J — X).X = 0. In like manner we find, ^"®^
that the coefficients of (0,2). ^L.ht, and (0,3). ^L.ht, io [6528 lines 2, 4 ; 6529rfJ,
raoish. Therefore the whole expression [6528 lines 2 — 6] vanishes ; aiid the formula i"**^l
[Oo-^] is reduced to its first line, as in [6529].
• (3384) Multiplying [6529] by rf/, and integrating, we obtain in r, or 3r, the r«eoA - |
expressioD [6530]. ^
t (3385) If there be only one satellite m, we must neglect (0,1), (0,2), (0,3), in
the equation [6347], and it will become,
Dividing this by (0)+| I, we get [6532],
TOL. IT. 30
118 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Gel.
shall have very nearly,*
[6534] '^ — I — -==zr ,
[6535] SO that >- will differ but very little from unity ; w^hich reduces the preceding
[6536] expression of the secular equation to the single termf —2.1 o l.Hi.cf^.
[6534a]
(3386) The values of (0), [o], [6216] give J?L = |. liJi^ ^ ; and if we
change the elements relative to Jupiter and its first satellite, into those of the earth and moon
respectively, we shall have the ellipticity of the earth p = -j^-^ [6044, 5593] ; the
■' centrifugal force cp = ^^ [6044', 1594a]; hence p — |(p = ^^^. Moreover,
M earth's mean motion r,r,<^Aa i r^im/ anccv m tj-i i
— = ; : — =m =0,0748 nearly, [6101, 6062^, 5117] ; also,
n moon's mean motion ''
'*"^'' i= ^"'"''f" , = 0,0165510, [6061,6082,5329].
a moon s dist. from earth â– *
Substituting these values in the expression [6534rt], it becomes,
-^ =*-*-(0,0165510)2. (0,0748)-2 =^^Vt nearly.
H
This quantity being very small we may neglect its square and higher powers, in the
development of X in terms of ^ ^ , and then it becomes as in [6534].
[6534rf]
[6534e]
a
f (3387) Substituting the value of X [6534], in the second term of [6530], it becomes
[6536a] /QN
of the order —^—L. . (0), which may be neglected on account of its smallness [6534e].
The third term is also very small in comparison with the first. For the ratio of these two
[65366] terms is represented by |. ^ ^ . — . — = f . ^^Vt- "77" • ~ [6534^?] ; and we have in
a-
b
[6536c] [6937,6938] — = yV nearly; moreover thfe value of ^£« [6937], expressed in parts of
the radius, is nearly equal to 0,05 ; and the excentricity jH^ [6527', 4080 line 5], is
0,048.., which may be considered as equal to ^L; hence the ratio [65366] becomes
[6536rf] I . ig-g\Y . t'j- nearly ; consequently the term depending on (0) may be neglected in
[6536e] comparison with the first term of [6530], and then this expression of Sv will be reduced
VIII. V. ^ 13] INEQUALITIES DEPENDING ON e«, «'•, &c. ; y«, 7", &c. 119
= f . — [6216], it becomes, as in [6536/^, -|. — .H^.cl"; [6537]
which agrees with what we have found in [5541, 6536 1].
The terms, which we have just had under consideration, will finally
become very sensible. Of the other terms, the greatest are those which
depend upon the products* ^7, 6' I', &c. ; and we shall see hereafter, that
/, /', &c. are small quantities, whose squares may be neglected, without any
sensible error. This being premised, we shall consider the following term
[6537']
of ^ [6515];
dM
iir =2.[T|.{ri'*— 27Ti.COS.(7,-7)+'/}. [fi538]
Now it is easy to prove that,t
10 its first term, 5» = — 2. .!/,.«•, as in [6536]. Substituting the value of T
jip [6536/]
[6216], it becomes Sv=: — ^. — ,H^.ct^, as in [6537]. This is the same as that
which is deduced from the expression [5541] Sv= — f .m'./(e'* — E'^).ndt, by changing
M [653%]
u [5117] into .— [6534c], E' into H^f and tf into H, to conform to the present
n
JLfQ
notation. For by this means it becomes nearly — |. — -/{H^ — H^^).di', and by [6536A]
substituting the value of H [6527], we get the term,
JUS Jl|9
•• in [6537].
•(3338) If we compare [6430 line 2] with [7352 line 2], we see that ^'1 = 1'" [6537a]
[6422J is of the order 2771"; moreover ^ = 34352" [7217]. The squares and
products of tiiese expressions being divided by the radius in seconds, give (f^ =» 1854",
#r=150", r'«=l2". The other values of r' [7352 lines 3, 4] are much less than [6537el
the preceding, so that generally their squares may be neglected, as in [6537^]. The terms
depending on ffl, tfl't produce in ~, sometermsconnected with cos. (|;^-f~^~'*'0>^'>
which do not contain v, as in [6540] ; and by integration they introduce into the
•zpressinn of Jr, the divisor p, which is small in comparison with n [6025/f, p] ; and
00 this account it becomes necessary to notice them, as in [6537', he.].
[6537e]
t (3.389) The function [6538] is part of [6515], the other part being neglected as in
[6S&2</] ; and we shall see, in [6540], that the retained part of the function [6538]
[6539'j
120 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cel.
[6539] j.5y^2__.277i.C0S.(7,— 7) + 7'},
is equal to the part, which is independent of v, in the square of the
expression of the latitude of m, above the plane of the orbit of Jupiter.
We have given the expression of the latitude, in [6427] ; and by developing
its square, in sines and cosines of v and its multiples; neglecting the
squares and products of / and /' ; we shall find, for the double of the
[6539"] part which is independent of these sines and cosines,* the following value ;
( +/2-C0S. (;?2^ + A2 * ) + h' COS. (p3^+A3 *') )
Hence the preceding term of -^ produces the following expression ;
contains 6' I, 6'V, &c. Now the latitude of the satellite m, above the iSxed plane, is
represented by 'y^.sm.(v — 7J [6332^] ; and this latitude would be 7.sin.(i; — 7) [6391],
r6539&l ^^ *^ move in the plane of Jupiter's orbit. Hence the latitude of the satellite m, above
the orbit of Jupiter, is 7i.sin.(t; — q^) — 'y.sm.{v — 7). Squaring this, and reducing the
product by [17] Int., neglecting also the terms containing 2b, we may put,
[6539c] sm,%v — /i) =i ; sin.2(v — 7) =i ; sin.(i; — 7j).sin.(tj— 7) =|.cos.(7i— 7) ;
and then by substitution we get, as in [6539],
[6539f^] Iri-sin-C'^ — ^i)— 7.sin.(v— 7)}2= iy^2 — y^^^cos.(l^ — ']) -{- ^7^.
* (3390) The expression [6427] becomes, by using the symbol if [6324'J, of the
following abridged form, which represents the latitude s, of the satellite m, above the
orbit of Jupiter ;
[6540a] s = {\—l).6'.sm.(v-{-f')-j-^\l.sm.{v-{-pt-}-A).
Squaring this expression, and neglecting terms of the second order in /, /', &c., as in
[6539'J, we shall have,
[65406] s^= (1— X)2J'2,sin.2(v+Y')4-2(X— l).a'.sin.(v4-^').2'.Z.sin.(i; -{- pt +a).
Reducing these products by [17] Int., and then neglecting, as in the preceding note, the
terms depending on 2v, we get,
[6540c] sin.2(«-{-V) = ^; sin. {v-{-^). sin. (v-\-pt-\- a) = ^. cos. {pt-{- A— ^) ; &c.
Substituting these in [65406], and then doubling the product, as in [6539^'], we obtain
[6540]. This represents tlie part of y^^ — 277i.cos.(7j — '')+7^ which is independent
of 2v ; and by multiplying it by 2. , it produces, for the expression of [6538], the
function [6541] ; omitting the quantity,
[6540/] 2(1— X)2.^'2. r^ == 2 (1— X)2. 1^2. fo] + 4 (1— X)2. ^L.bt-{-hc. [6405] ;
because the constant part is neglected, as in [6505] ; and the part depending on t, has
already been noticed in [6529], wliere it was found by a different method.
[6540rf]
[6540«]
Vin y. ^ 13.] INEQUALITIES DEPENDING ON e", e'«, &c. ; r«, /«, &c. 121
<fr ^ ^1 1 ^+/,.COS.(/^^+A,|— V)4-/3.COS.(p3^+A3— 1^)i
We shall now consider the following terms, contained in [6516] ;
— 3.(0).j^-f-2*.y,.cos.(i'+i,)+ri'}. [6542]
The expression of the latitude of the satellite m, above the plane of
Jupiter^s equator, is*
x.d'.sin.(i? + ^') + 1 sin,{v+ ft + a) + 1 1. sm,{v -\-p,t + aJ
+ ia.sin.(t? +p^ + Aa) + h' sin.(r +p^-\- A3) ;
hence we easily find, that this term of -^ produces the following ;t
* (3391) The latitude of the satellite m, if it move in the plane of Jupiter's equator,
is — »'.sin.(r-|-i^) [6392], above the orbit of Jupiter. Subtracting this from s [6427], ^ ^
which denotes the actual latitude above the same orbit [6426'J, we obtain the real latitude, [65436^
counted from the equator of Jupiter, as in [6543].
[6544a]
f (3392) The latitude of the satellite m, above the fixed plane, is 7,.sin.(t> — 7,)
[63326] ; but if it move in the plane of Jupiter's equator, its latitude will be — d.sin.(t>-f-*)
[6322]. Subtracting this last expression from the preceding, we get the latitude counted [6544i|
from the plane of Jupiter's equator, y^.s\n.{v — l^-\-Lsva.{v-\-^). This may be derived [6544c]
from the expression [65396], by changing 7 into — 6, and 1 into — ^ ', and by
making the same changes in [6539f/], omitting also the terms depending on 20, we get,
{7,.sin.(t>— '?J+d.sin.(tJ+>i')]2 = i7,2 + 9;.^.cos.(>r + 7,)+i«2; [6544d]
which represents the square of the latitude of m above the plane of Jupiter's equator [6544e]
[65446, &c.]. Now this latitude may also be found, as in the last note, by subtracting
— ^.sin.(«-|-'*^)) which represents the latitude of the satellite above the orbit of
Jupiter, supposing the motion of the satellite to be in the plane of Jupiter's equator, from [6544/]
the general expression of the latitude s [6427], above the orbit of Jupiter [6426] ; the
difference is the function [6543], or X.((.s,\n.{v-\-^)-\-l!.l.sm.{v-{-pt-\-\). The square [6544g]
of this expression being substituted in the first member of [6544(2], then multiplying by
6(0), and transposing all the terms to the opposite members, we get [6544A]. Reducing
the second member of this expression, as we have done that in [65406], we obtain [6544»].
— 3(0){tf«4-24.y,.cos.(>i'+7j-fy.»j = '^(0).{-K.(f.^M\.{v-\-V)-\-7!.Um.{v-\-pt-\-A)\^ [6544*]
= — 6(0).{iX«.a^-f>^.^.2:/.cos.(pr-j-A— i-')}. [6544»]
The first member of [6544A] is the same as in [6542] ; and the terms under the sign 2^,
in the second member, are the same as those in [6544]. The other term, — 3(0).X'.d^, '• ^
may be neglected, as in the calculation [6540c,/] ; observing, that after substituting the
value of fl' [6405], we may reject the constant term — 3(0).X«.«I,«, as in [6505]; [^^^1
and the term — 6(0).X'.'Z«.6^, has already been noticed in [6528n].
TOL. IV. 31
122 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cii.
[6544] i^ = _6.(0).X.«'. \ '•<=°=-(i'« + ^-*')+ h.COS.(p,t + M-^') ) _
dt ^ ^ (-\-L.cos. (p^t + A2 — ^') + h' cos . (pst + A3 — ^') )
We may also notice the term of -jj—, contained in [6514 line 3], namely,
[6545] im'.n. I a'a'.B^'^ + a'a'.(^^ | .{7i'^-27/.n.cos.(7/— 70+7i^}.
We shall observe that 71'. cos. V — 7i.cos.7i, and y/. sin.i/ — 7i.sin.7i,
[6547c, d], are of the order >^, which is a very small fraction in the theory
of the satellites of Jupiter. The sum of the squares of these two quantities
gives, *
[6547] (7i'.cos. 7/— 7i. COS. i.f-}- (7/. sin. 7/— 7i.sin. n,f = 7i"— 27/. ?;• cos. (7/— 70+7' ;
w^hich is of the order x^, we may therefore neglect it, without any sensible
error.
Lastly, we shall consider the term of -^ , contained in [6517], namely,
r^^^.^-. 1 ( fZ.(7..sin.7,) . ^.(7,.cos.7i) )
[6548] i. ^ ,, COS. n,. ^'^■j, — /!• Sin. 1. "-''^^ j .
It is easy to prove that,t
* (3393) Developing the first member of [6547], it becomes, without reduction,
[6547a] 7/2^ ^(,Qg^2 ^^'_|_sin.2 -^/^ —27/. 7;. {cos. 7/. cos. 7i+sin. -7/. sin. 7^ } +7i^- {cos.^ ^j+sin.^ 7 J ;
and this is easily reduced to the form in the second member of [6547], by using [24, &;c.]
1 J Int. Subtracting the expression [6413] from [64136?], we get [6547c] ; and by subtracting
[6414] from [6413e], we get [6547<?].
[6547c] 7/. sin . 7/— 7^. sin. 7, = (X— X').'i. ^p^-f (X'— X) .at,
[6547d] 7/.COS. 7/— 7i.cos. \ = (X'— X). 'L -f (X'— X). bt
The second members of these expressions being of the order X' — X, the sum of their
[6547e] squares must be of the order (X' — X)^ ; which produces, in [6545], only terms of the
order m'.(X' — X)^ ; and these may be neglected on account of their smallness
[6829, 7206, &c.].
t (3394) The latitude of the satellite m, above the plane of Jupiter's orbit, is given in
[65396], and in another form in [6540a]. Now if we put these two expressions equal to
each other, and transpose 7.sin.(« — 7), we shall get,
[6548a] 7i.sin.(t; — 70 = 2'./.sin.(«-j-p^-f-A)+(X— l).d'.sin.(tJ+Y')-J-7.sin.(t; — 7).
[65486] But from [21] Int. we have sin.(tJ+6) = sin.v.cos.6-f-co3.v.sin.6 ; and if we develop
the four terms of the equation [6548a], by means of this formula, we shall obtain an
[6548c] equation which must be satisfied for all values of v; so that we may put the terms
depending on sin.u separately equal to each other, and we shall get [6550] ; in like
VIII. V. '^ la.] INEQUALITIES DEPENDING ON «•, e^, &c. ; y«, /», &c. 123
y,. sin. i| = — I. sin,(pt + a) — /,. sin.(p,t -f a,)— &c. — (x — 1 ). tf, sin-V-j-y. sin. i ; [6549]
y,.cos.i, = /.cos.(f?/-fA)-f /i.cos.(p,r-f-A,)+&c. + (x — l).^.cos.V4-r.cos.i ; [^MO]
hence the term [6548] produces the following expression ; *
^ = —1 CX—1 ) a' f /''•COS-CpZ-f-A— V)+;>, /,.C08.(p,^+A,— V) ) 1
Now if we add together the terms [6541, 6544, 6531], and integrate the
sum, we shall ha?e, for the corresponding part of <Jt>, the following
expression ; f
manner the terms depending on cos.r, give [6549], by changing the signs of all the terms. [6548</]
We may also observe that each of the expressions of 7,.sin.7i, 7, .cos. 7, [6549,6550],
contain two terms, depending on ^, y, which are not inserted in the values of yj-sin.-?,,
y,xo8.7, , [6499, 6500]. The reason for this omission is, that their differentials produce
only insensible quantities in [6501, 6502], on account of the smallness of the coefficients
of /, in the values d', y, y, 7, [6927—6929, 4246], in comparison with the values of ^^^^-^^
p [6025/?]. In fact the smallness of these coefficients enables us to consider ^, xb', y, 7,
as constant quantities, as in [7220] ; and then their differentials vanish from [6501,6502] j
•o that we may neglect them in using the differentials of [6499, 6500]. We shall then 1"^^*^J
have, by using the sign 2', in like manner as in [6324'], so as to include the terms
depending on p, _p, , ;/, , ^3 ;
rf.(y,.8in.7i) . , / ^ , \ «'-(7,«cos.7,) , 1 . , , v
-^ 1 = ^l'.plcos.{pt-\-A) ; ^^'^ '' = —J/.plsin.ipt+A).
[6548%]
[6551a]
• (3395) Substituting the values [6549, 6550, 6548A] in [6548], and neglecting terms
of the order Pp, lyp, &cc. on account of their smallness, we find that it is only necessary
to retain the terms containing ^, in [6549, 6550] ; hence we obtain, for the value of the
function [6548], the first member of the following expression [6551c]. This is reduced
to the form in the second member, by using [24] Int.
-4.(X-l).d^.i'.p/.[cos.i''.cos.(p/+A)-f sin.V.sin.(p<-j-A)J = -}.(X-1). tf. 2'.j;/.cos.(/)/-f ^-^0 [6551c]
This last expression agrees with that in [6551].
t(3396) The expression [6552J b the sum of the functions [6541,6544,6551],
multiplied by dtj and then integrated. For the terms multiplied by , in
[6552 lines 1,2], are derived from [6541] ; those multiplied by (0), from [6544] ; and
those in [6552 lines 3, 4], from [6551]. It will be seen in [6944] that there is but one
term of the expression i«, or rather 6p'", [6552], which requires notice from its
magnitude, and this produces only — 49",51, in the motion of the fourth satellite ; the
Talues for the other satellites being insensible. We may remark that in finding the i"^***l
[6553a]
[65526]
124 MOTIONS OF THE SATELLITES OF JUPITER. [M6c. Cel.
c â– , , ) ( - . sin.(;7^+A-^') + -^ . sin.(;7i^+Ai-^') ) 1
\ Pa Pa )
I. sm.{pt + A— ^') + ^1- sin.(i?i^ + Ai— ^') | 3
4
[6552]
[6553]
This part of 5i? is hardly sensible, and we need not notice it, except in the
fourth satellite [6944]. it must be modified relative to the other satellites,
in consequence of the terms which depend upon the square of the disturbing
force [6699, &c.].
[6553'] If we apply this expression to the moon,* we shall have x = 1 —
[6554] [6534], and j? = I o |> very nearly [6553 J] ; hence we obtain,t
functions [6538, 6542, 6545], from [6515, 6516, 6514] respectively, the terms depending
[6552a!] on the excentricities e^, e'^, &cc. are neglected ; and it is evident that this can be safely
done, because the excentricities [6057e] are very small in comparison with the factor 4 '
[6552e] [7217], which occurs in the terms [6541, 6544, 6551] ; and these last terms, which have
the factor ^', are hardly sensible, as we have already observed in [6552c, 6553].
[6553a] * (3397) The retrograde motion of the moon's node is f wi^.t; nearly [4800].
M
[65536] Substituting ot = — , [6534c], and t) = n^ [6439], it becomes, according to the present
JJJ2 I [
[6553c] notation, f. — .^= \'t [6216]; so that if we wish to investigate the value of 5u
[6552], corresponding to the retrograde motion of the moon's node, supposing it to be
represented by yt [7133c], we shall have p^= • <, or p= L as in [6554].
[6553rf]
[6554a]
t (3398) Neglecting /j, l^^ l^, in [6552], it becomes, for the moon, as in [65546].
Substituting in this the values 1— X = _LJ_ ; ;?= fo 1, [6553', 6554], it becomes as
in [6554c] ; observing that the preceding value of X for the moon, gives very nearly
X = 1 [6534d], which is used in the first term of [6554c] ;
[65545] 5t! = ^ —6 (0). - —4 (l—x).!"^. -i -}- j(l_x) \ . d'. lsm.{pt-L.A—^')
V y L — I p J
[6554c]
=== {—6—4+1;. J2L.fl^Z.sin.(p<4-A - -y) ==-~V-'-^-^'-?-sin.(p^+A - *')-
Li] H
This last expression is the same as in [6555], using the value of jp [6554].
VIII. V. ^ 13] INEQUALITIES DEPENDING ON «■, e^, &c. ; y«, 7", &c. 125
This expression agrees with that which we have found in [5389] ; supposing, £"!.••
in this case, the obliquity of the ecliptic to be very small.* [655^]
•(3399) Instead of using ihe obliquity of the ecliptic X = 2y 28" 17', 9 [5355], if [6555al
we suppose it to be small and equal to (( [6360], we may change sin.X.cos.X into sio.4' [65556]
or d ; then the expression of ^ [5389] will become,
^ = -1^. — — - — - . -— . 7.^. sin. (long, of the ascending node). [6555c]
Now we have D= 1 [5334, 6082'], g — 1 = Jm^ [5347 9] ; or, in the present notation,
by successive reductions, using [6534c, 6216, 6551],
[6555i]
Substituting these values in [6555r], and changing a 9 into 9, also ap into p,
[5333, 5333^, 6044, 6044^], it becomes as in [6555e] ; and by using (0) [6216], it
changes into [6555/] ;
iv=sJ^. — . — . sin.(long. of the ascending node) [6555e]
y.y
= -^. (0) . ^— . sin. (long, of the ascending node). [6555/]
If we subtract the longitude of the descending node of Jupiter's equator, or, as we may call [655^]
it, the longitude of the vernal equinox of Jupiier, — ^' [6361], from the longitude v of
the first satellite [6023c], both longitudes being counted from the same axis x; we shall ^
get v-}-^' for the distance of the satellite from that node, or equinox, counted according r6555tl
to the order of the signs. Again v-\-pt-\-A represents, as in [6300 line 2, 6298e], the
distance of the satellite from its ascending node on Jupiter's orbit. Subtracting this last [655541
expression from that in [6555t], we get the distance of the ascending node of the orbit of
the satellite from Jupiter's vernal equinox, (v-{-^') — {v-\-pt-\-A) = — {pt-\-A — >*•'); [6555/]