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Pierre Simon Laplace.

. (page 41 of 114)

Among the other terms contained in the expression,

s'" = 0,001447815.(Z'"â€” L').sin.(i;"'â€” 2C/â€” p^â€” a) ; [6934] [7348]

the only one which is sensible, is that which corresponds to the peculiar
inclination of the orbit of the satellite to its fixed plane. In this case
L' = [6415], since the position of Jupiter's orbit is not sensibly altered by [7349]
the action of the satellites. IVIoreover we have, by what has been said in
[7324', 7329],

* (3597) The expression of s"' [7344] is the same as that in [7323/], using the
symbol 2' as in [6324'J. The terms under this symbol in [732.3/], are reduced to one
term of the form [7323^], which is the same as that in [7345]. Comparing the first of [73445]
the expressions [7323/] with the last of [7323^], we find tliat we may change ^.(l"'â€”L') [7344c]
into 2Â°,9805l, and yt-^-x into 51Â°,3787 â€” ^.153",8. Making the same changes in [7344(i]
[7341], after having prefixed the symbol 1/ to the terms in the second member, we get
[7346] ; which is easily reduced to the form [7347].

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

[7350] l"',sin.(v"'-\-p.,t+AO = â€” 2771",6.sin.(i;"'+83Â°,29861+^.7528",01) ;

hence the preceding term [7348] becomes,*
[7351] 5'" = â€” 4",01.sin.(v"' - 2t7â€” 83Â°,29861â€” ^7528",01).

Connecting together these different terms of the latitude 5"' of the fourth
satellite above Jupiter's orbit, we obtain,t

5"' = 2Â°,98051 .sin.(y'"H-5P,3787â€” ^.153",8) 1

â€” .2771",6 . sin.(i;"'+83^29861+^.7528",01) 2

+ 448",93.sin.(v'"+208Â°,32562+^.28220",85) 3

t''^^^ ^ 4",80.sin.(2;"'+285Â°,20, + /.133715",77) 4

+ 43",15.sin.(i;"'â€” 2C7â€” 5P,3787+M53",8) 5

â€” 4",01.sin.(?j'"â€” 2(7â€” 83^29861â€” i.7528",01). 6

In the eclipses of the fourth satellite, and in the eclipses of Jupiter by the
same satellite, these expressions of v'" and s'" become more simple ; for we
[7353^ may supposed 2n = 2Â©'", 2U= 2v"' ; and then we shall have, at the times
of these- eclipses,^

r7351al *(3598) We obtain from [7350] ^^+ a, or rather pJ-{- a ^=83", 29861 +t.l5QS", Oh

and Z'"=â€” 277l",6, or tâ€”L' = â€”211l",6 [7349]; hence,
[73516] 0,001447815.(Z'"â€” i')=â€” 4",01.

Substituting this in [7348], it becomes as in [7351] ; using the value o[ pt-^A, or rather
[7351c] Pst-^'As [7351a]. It is unnecessary to notice the other values of p, depending on the

terms in [7352 lines 3, 4, &z.c.], as the coefBcients are too small to be noticed ; the greatest
[7351cr| of them is 448",93 [7352 line 3], being hardly a sixth part of the preceding term; aad,

of course, the coefficients will not be one sixth part of that in [7351].

f (3599) The expression of /' [7352] is the sum of the terms which are contained in
[7324, 7329, 7335, 7340, 7347, 7351] respectively.

X (3600) It is evident from the definitions in [6023^?, 6024/], that, in these eclipses,
'â€¢ "-' 0'", n are equal, or differ by 200Â°; so that we may put generally 2n = 20'". Also
[7353&] from [6022y, 6023c], v'", [7 are equal, or differ by 200Â° ; therefore we may put 2 C7=2Â«'".
[7353c] In like manner, in the eclipses of the third satellite, we may put 2 11=20", 2C^=2t?";
[7353cri in those of the second satellite, 2n= 2Â©', 2C7= 2v' ; and in those of the^rs^ satellite,
[7353â‚¬] 2n = 20, 2Uâ€”2v.

^(3601) Substituting 2n== 2Â©'" [7353'] in [7318 line 10], it vanishes. The same
'â–  â€¢' substitution being made in the term [7318 line 11], it becomes,

[73546 ] +66",94.sin.(^"'â€” 0'") = â€” 66",94.sin. (Â©'"â€”â€¢ sj'") ;

connecting this with the term in [7318 line 1], we obtain 4-9198",62.sin.(0'"â€” in"'), as in
[7354 line 1]; and the whole expression of 1/" becomes as in [7354J. Again, if we

VIlLxii. ^30.]

THEORY OF THE FOURTH SATELLITE.

269

-Z3

)

ij'" = 0'"-|.9198",62.sin. (e'
+ 42",14.sin.2.(0"
+ 0",27.sin.3.(e'"_^'")

â€” 28",36,.sin. (e" â€” e'")

â€” 14",12.sm.2.(e"â€” e'")

â€” 2",95.siii.3.(e"â€” e'")

â€” O",9O.sin.4.(e"_0"')

â€” O",33.siri.5.(e"_0'")

â€” 220",73.sin. (0'"â€”^")

â€” 353",69,.sin.F

â€” 49",51.sin.(i.7681",81+31o,9199).

1

2
3
4
5
6
7
8
9
10
11

Values of

J" J"
U , S ,

in eclipses;

the
longitude
being
counted
from the
moveable
vernal
equinox of
the earth.

[7354]

5'" = 2^,97620 .siii.(2?"'H-51Â°,3787â€” M53",8) 1

â€” 2767",6 .sin.(y"'+83Â°,29861+/.7528",01) 2

+ 448",93.sin.(2;"'+208Â°,32562+^.28220",85) 3

-f- 4",80.sin.(2;'"+285Â°,20, + ^.133715",77). 4

This expression of s'" gives the explanation of a singular phenomenon, which
has been observed in the inclination of the orbit of the fourth satellite, and in
the motion of its nodes. The inclination of the orbit of this satellite to the
orbit of Jupiter, appeared to be nearly constant, and equal to 2Â°,7, from the

[7355]

[7356]

subsiitute 2C7= 2Â«'" [7353'J in [7352 line 5], it becomes,

43",15.sin.(â€” Â«"â€” 51Â°,3787-f-M53",8)=â€” 43",15.sin.(i>'"-f-51Â°,3787â€” M53",8); [7354rf]

connecting this with the term in [7352 hne 1], we obtain,

2Â°,976195.sin.(i;'"+5r,3787â€” <.153",8), [7354e]

as in [7355 line 1], nearly. Making the same substitution of 2?7=2v'", in the term

[7352 line 6], it becomes +4",01.sin.(t;'"-j-83Â°,29S61+f.7528",01) ; and by connecting

it with the term in [7352 line 2], we obtain â€” 2767",6.sin.(w'"+83Â°,298614-^.7528",01), [7354/]

nearly, as in [7355 line 2J. We may observe that the arguments of the inequalities in

[7355 lines 1,2,3,4], are represented by Delambre, for brevity, by H, /, K, L, at the [7354g-]

times of the middle of the eclipse or opposition. The same arguments occur in the values

of s", s, s [7427, 7482, 7522, he] ; and by using this notation, and putting V" equal to

the mean longitude of Jupiter at the conjunction, corrected for the great inequality, we have,

H=V"'-{- 51Â°,3787â€” ^.153",8; [7354t]

I = V"'-\- 83Â°,29861-f^7528",01 ; [7354A]

K == F'"4-208Â°,32562-|-<.28220",85 ; [7354^ ]

L = F'"+285Â°,20c + M337 1 5",77. [7354m]

VOL. IV. 68

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

year 1680 till about 1760. During this interval the nodes had a direct
motion, upon the orbit of the planet, of eight minutes in a year. Since
1760, the inclination has very sensibly increased. We may obtain the
inclination of the orbit, and the position of its nodes, at any given epoch, by

[73r)7] giving to.t the value which agrees with that epoch. We shall put the
preceding expression of 5'" under the form,*

[7358] s'" = A.sm.v'" â€” B.cos.v'".

We can determine A and B, by putting successively! i;'"= 100Â°, and
[7359] ^

v'" = 200^, in the expression of s'". â€” will be the tangent of the longitude
of the node, and \/A^-{-B^ the inclination of the orbit, f This being

[7357a]

* (3602) Any one of the lines of the second member of [7355], may be put under the
form A'.sin.[v"' â€” B') ; A' being the constant coefficient of that line, and â€” B' the
constant angle, connected with v"\ If we suppose the sign 2 of finite differences to

[73576] include the terms in the four lines of [7355], we shall have s'" = ^.A'.s'm.{v'" â€” B').
Developing this by [22] Int., we obtain,

[7357c ] // ^ sm.v'". ^.A\cos.B'â€”cos.v"\ 2. A'.s'm.B'.

[7357d] Now putting 2.^'. cos.i?'=^, ^.A'.sm.B=B, it becomes as in [7358].

[7359a]

t (3603) Putting v'"= 100Â° in [7358], we get s"' = A; and if we put v'" = 200Â°,
it gives s"'=B. Hence it appears that we can compute the value of A, by substituting
t;'"=100Â° in [7355]; and the value of B, by substituting y'"==200Â° in [7355J.

X (3604) It appears, from [533a], that if 7 be the inclination of the orbit, and 6^ the

â– ' longitude of the ascending node, we shall have very nearly, for the expression of the

[73626] latitude /''=7.sin.(t;'" â€” ^^)=7.cos.^^.sin.t;'" â€” 7.sin.^^.cos.i;'" [22] Int. Comparing this last

expression of s'" with that in [7358], we get,
[7362c ] A = 7.cos.^^ ; B = y. sinJ^ .

Dividing this value of B by thai of A, we get the following expression of tang.^^, and the
sum of their squares gives the value of 7^; hence we have, as in [7359, 7360],

B ,

[7362rf] tang.^^ = "TT 5 7 = V^-^^+B^.

With these formulas, and those in [7359], we may compute the values of A, B, 7, 6'
^ ^J [7362] ; for t = â€” 70, t = â€” 30, and t=lO; which correspond respectively to the

years 1680, 1720, 1760 ; the epoch being 1750 [7281'j. The values of A [7359, 7355],
r7362n ^Â°''^^sponding to these times, are iwsitive ; therefore the values of 6^ must be in the Jirst

or fourth quadrant, as is evident from the expression of ^4 = 7.cos.^^ [7362c], 7 being

positive ; and to know which of these must be selected, we must refer to the value of B,
[7362g-] which is ne^Â«<?'t;e ; so that 7.sinJ^ [7362c] must be negative ; consequently/)^ must be in

VIII. xii.'^ao.] THEORY OF THE FOURTH SATELLITE. 271

L7361]

premised, if we put successively t= â€” 70, t = â€” 30, ^=10, which
correspond to the years 1680, 1720 and 1760, we shall have,

Inclination.

Longitude of the node.

1680,

2^7515;

346^,0191 ;

1720,

2V210;

348^,1186;

1760,

2^,7123 ;

352^,3238.

[7362]

1

2
3

If we represent the inclination by the formula 2Â°,7515-f M-f Pf^, ^ being [7363]
the number of Julian years elapsed since 1680, we shall have, by comparing
this formula with the three preceding inclinations,*

N = â€” 0Â°,001035 ; P = 0^0000068125. [7364]

The minimum of the formula corresponds tof ^ := 75'"'"',963, or to the [7365]
year 1756. The mean of the three preceding inclinations is 2*^,7283 ; the

the fourth quadrant of the circle. Tlie calculations of the author in [7362] have been
verified, and found to be very nearly correct; though some slight discrepancies were found [7362/i]
in the fourth decimal place of the inclination, and in the third decimal place of the longitude
of the node.

* (3605) We shall suppose the general expression of the inclination of the orbit y, to ryqf.4 -i
be represented, as in [7363], by

7 = 2Â°,7515+AV + Pi2. [7ZMb]

t being the time elapsed since the year 1680. Putting < = 0, it becomes 7 = 2Â°,7515,

as in [7362 line 1]; then putting ^ = 40, and 7 =2Â° ,7210 [7362 line 2], we get [7364c]

[7364rfJ; lastly, putting ^ = 80 and 7 = 2%7I23 [7362 line 3], we get [73o4e] ;

2Â°,75154-40.V+1600P = 2%7210, or 0Â° , 0305+40 JV-f 1600P = ; [7364</]

2Â°,7515+80iV-|-6400P=2Â°,7123, or 0Â°,0392+80iV+6400P = 0. [7364c]

Subtracting a quarter part of the second of these equations from the first, we obtain
0Â°,0207+20iV=0; whence JV=â€” 0Â°,001035, and 40iV=â€” 0Â°,0414. Substituting
this in [7364^^], we obtain 1600P =0Â°,0414-0Â°,0305 =0Â°,0109 j consequently [7364g-]
P = 0,0000068125. Hence the expression of 7 [7364^*] becomes, as in [7364, &c.],

7 = 2Â°,7515â€” 0Â°,001035.< + 0Â°,0000068125.f^ [7364A]

I (3606) The minimum value of 7 [6364A], is found by putting its differential equal to
nothing, which gives = â€” 0Â°,001035+0Â°,000013625.^ ; whence f = 75,963, as in '^''^^^"^
[7365] ; and this value would be somewhat augmented, by correcting the calculation for ^ J

the small mistakes which are spoken of in [7362A]. Substituting this value of t in [7364A], [7365c]
we get 7 = 2Â°,71 nearly, for tlie minimum value of 7.

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

r7365'l ^^'^^ annual motion of the node, from 1680 to 1760, is 7',88.* These
results agree perfectly with those which have been found by astronomers,

[7366] from the observed eclipses during this interval. Since the year 1760 the
inclination has varied by a very sensible quantity. The preceding value of

[7367J s'" [7355], makes the inclination, in 1800, equal to 2Â°,8657,t and the
longitude of the node equal to 355Â°, 881 7. Observations, by conlirming
these results, compel us to give up the hypothesis of a constant inclination;

[7368] and we should have found it difficult to discover the law of these variations,
without the assistance of the theory.

To obtain the duration of an eclipse of the fourth satellite, we shall
resume the formula [7080] ;

[7369]
[7370]

t=.T.{l-X).^~{li-py,'j/Â£^Â±:\^/[i+hX+^^^

in which T represents the half of the mean duration of an eclipse of the
satellite in its nodes [7069]. Delambre has found, by the mean of all the
observations which he has used, that this semi-duration is equal to 9942" ;

[7371] and since the discovery and use of achromatic telescopes, he finds this
semi-duration to be diminished about 52", by the discussion of all the

[7372] eclipses observed since that epoch. Therefore we shall suppose

[7373] T =9890" [=2",22'",25^ sex.], as in [7565]. The symbol ^ [7073]
represents the mean synodical motion of the satellite, during the time T;

[7374] and we havet |3 = 23613". The value of p' [7049] is

* (3607) The longitudes of the node in 1680, 1760, are 346Â°,0191, and 352Â°,3238
[7362 lines 1, 3] respectively ; so that the motion in 80 years is
[73666 ] 352Â°,3-238â€” 346Â°,0] 91 = 6Â°,3047 ;

dividing this by 80, we get the annual motion 7',88, as in [7365'].

f (3608) Putting ^=50 in [7355], and then successively i;'"=100Â°, v"' = 200Â°,

[7367a] ^g obtain the values of s"\ which are denoted respectively by A, B, in [7359]. With

[73676] these values of A, B, we compute, by means of the formulas [7362^?], the corresponding

inclination 7, and the longitude of &^ ; which are found to be very nearly the same as those

[7367c] given by the author in [7367]. We may observe that the formula [7364/t] is not used for

this purpose, because the value of <=1800 â€” 1680 = 120, is so great that it becomes

necessary to notice terms of the order t^, which are neglected in that formula.

X (3609) Dividing the expression of T=9890" [7373], by the number of seconds in
[7374a] ^ j^^.^^ ^^^^^ 36525000", we get T= 0,0002707735, expressed in parts of a Julian

VIII.xii.<^30.] THEORY OF THE FOURTH SATELLITE. 273

P
P' = -

[7375]

^_(1-X0 a'

and we have found, in [7159], p = 0,0713008 ; hence we deduce,* [7375']

p' = 0,0729603. [7376]

dv'"
The value of X [7071] is very nearly represented by f X= -jjrr â€” 1; 17376^]

therefore if we notice only the greatest term of v'", we shall have,

X = O,O145543.cos.(0"'â€” :.'")â€¢ [7377]

We have also seen, in [7094], that the value of T must be multiplied by
the factor, J

' + 1 ^^M - ^^ -W]- â– ^â€¢'=Â°'- ^ = *'="=*'"â–  Â°*' ^ l^^^^J '- '"*''

year; multiplying this by the annual sy nodical motion v!" â€” iW = 87205380" [6025n],
we get the synodical motion in the time T equal to (n'" â€” JW).T= 23613" =:p, as in
[7374].

a'"
*(3610) We have â€” = sin. 1530",864 [70451]: \ = 0,105469 [7547]. Substituting

2)' â€¢- â– â€¢ Â° [7375a]

these and p = 0,0713008 [7375'], in p' [7375], it becomes as in [7376].

dv '"

t(3611) We have, as in [7071], X = ^-;;;-^^ - 1 ; moreover, rft;'" is to <" ^^3^^^^

as vl" to n"'â€” M, nearly [6023c, of, 73746]; hence X becomes very nearly as in [7376'J.
Substituting, in this expression of X, the chief term of v'" [7318 line 1], it produces, by [73766]
using [7290], the expression,

9265",56. "â– -â– '!: - ':' = 9265'',56 X B^53Â°,46322Uos.(e - .Â»0 _
n at n

Substituting n"' = 8754Â°,2591 [6025^], and dividing by the radius in seconds, 636620",

it becomes X= O,O]4553.cos.(0"'â€” to"'), as in [7377] nearly.

X (3612) If we change n into w"', artd a into a'", in the expression [7094], we shall
obtain the factor of T corresponding to the fourth satellite, namely,

and by using the symbol F= Mt-^Eâ€”I [60236], it becomes as in [7378]. Finally,

a'" [73786]

substituting the values of -~ , \ [7375a], M [6840], and 2fl = 61213",l [68826],

or fl=30606",5, it becomes 1â€” 0,0006087.cos. F; being nearly the same as in [7379]. [7378c]
voLÂ» IV. 69

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

H being the excentricity of Jupiter's orbit. Hence this factor becomes,
[7379] 1â€” 0,0006101. cos.r= factor of T [7369].

We shall put E, = ^-^â– ^- , and we shall have,*

[7381]

[7380]

Symbol P

^ ^==1,352380 .sin.(i;"'+ 5P,3787â€” M53",8) 1

â€” 0,125759.sin.(v'"+ 83^,29861 + ^.7528",01) 2

+ 0,020399.sin.(i;'"+208Â°,32562+^.28220",85) 3

+ 0,000218.sin.(i;'"+285Â°,20, + ^.133715",77). ^ 4

This being premised, if we neglect the square of X, the quantity under the
[7382] radical, in the expression of t [7369], will become 1+Zâ€” ^^ ; and if we

Time of a^so neglect the products of X and ^ by yj, we shall obtain,!

the emer- (iV

sion or

j;~ t = â€” 366",832. p Â± 9890".(1â€” Xâ€” 0,0006101.cos.F).v/T+X-?.

[7381a]

* (3613) From the values of ^, / [7374, 7376], we obtain ^^ = ^^^^5
multiplying this by s'" [7355], it becomes as in [7381], using the abridged symbol [7380].

[7383a]

t (3614) The factor T.(lâ€” X) = 9890".(1â€” X) [7373], which occurs in the
expression of t [7369], being multiplied by the factor of T [7379], neglecting terms of the
order IfX, becomes,

[73836] 9890".(1â€”Xâ€” 0,0006101. cos. F);

and this is to be substituted for T.(l â€” X) in [7369]. Now the radical expression in

[7383c] [7369] may be put under the form Â±v/l+Xâ€” ^2, as is evident by multiplying together
the factors composing this radical, neglecting X'^, and substituting ^ [7380] ; hence we
get the term depending on the radical in [7383]. In the term of [7369] without the
radical, which is very small, we may suppose the factor [7383&] to be simply equal to

[7383ci] 9890"; so that the term itself becomes â€” -. (l+p')^.^ . But from [7380] we

have (l-|-p')2.r2^^2^2^ whose differential, divided by 2, is (l+p')2.s"W'= p^.^rf^;
[7383e ] 0,0

hence the differential expression [7383Â£ZJ becomes â€” 9890".i3. ^-| 5 and by substituting the

value of ^ [7374], then dividing by the radius in seconds, it becomes as in the first term of
[7383/] [7383]. The whole duration of the eclipse t' [7082], is derived from that of i [7080], by

retaining only the radical part of the expression, and multiplying it by Â±2. The same

[7383^] process, being performed in the expression of t [7383], gives the whole duration of the

, eclipse t , as in [7385]. If we now examine into some of the largest terms, which have

been neglected in computing the value of t, or t' [7383, 7385], we shall find that they are

not in general of any importance. The most noted term is that spoken of in [7096],

[7383i ] requiring that X should be increased under the radical, by the quantity Â± , which

VIII. xii. Â§ 30.] THEORY OF THE FOURTH SATELLITE. 275

It is easy to deduce, from this expression, the times of the immersion and
emersion of the satellite ; observing that t expresses the time elapsed from [7384]
the conjunction of the satellite [7070, 7055] ; the conjunction being
estimated by means of its projected place upon the orbit of Jupiter ; and the
time from the conjunction can be determined by means of the tables of
Jupiter, and the preceding expressions of v"\ s'" [7354, 7355]. The whole
duration t' of the eclipse is represented by orthe""

eclipse.

t' == 19780".(1 â€” Zâ€” 0,0006101.cos.F).v/l+Xâ€” f = duration of the eclipse. [7385]

IS

[7383^]
[7383i ]

[7383m]

nearly of the order ^'-f^-lj j ^s is evident from [TSSSe] ; observing that p' [7376] is so

small that it may be neglected. Now \%-w-) niay be considered as of the order l, at

its maximum, and is generally much less. For if we notice only the chief term of ^
[7381 line 1], and represent it, for brevity, by ^=A.sm.[v"'-\-B)^ we shall have, very

nearly, â€” 4: = A.cos.{v"'-\-B) ; whence,

lif = AKsm.{v"'^B).cos.{v"'^B) = \AKsm.{2v"'-^2B), [7383^]

dv

which we shall put equal to U, for brevity, so that h = \^A â€” :^^^.sin.^(2Â«'"-|-2J5) ;

and this varies from to its maximum ^^^ = i.(l,35)^ which is less than unity.
Moreover |3 [7374] is about ^V of the radius ; hence p~ = (^V)^ = t?^ nearly ; so that
if X â€” ^ be very small in comparison with unity, the radical \/\-\-Xâ€”^ [7383] will be [7383n]

varied by ^^^6 = â€¢ nearly, in consequence of this variation in the value of X.

Multiplying this by the factor T=9890" [7383], it becomes lTfb = T.h nearly; [7383o]
which, on account of the smallness of 6, will not be in general of anyjmportance. This
term however must be noticed when the radical y/l-f-Xâ€” ^ becomes very small in
consequence of the great latitude of the satellite, as is observed in [7094' â€” 7096]. We [7383pJ

may find, by a similar process, that the terms of the order sin.^Vi.f â€” j ? &^c., which are

neglected in [7062, 7059, Sec], produce nothing of importance in the values of t, i' ;

and the same is to be observed relative to the other quantities, which are neglected in these l ^J

calculations.

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

CHAPTER XIII.

THEORY OF THE THIRD SATELLITE.

31. We have found, in the preceding chapter [7301, 7296], that

[7386] e" = 11 Â°,39349 -f ^.20420Â°,579040 ;

[7387] .."= 343Â°,82067 + ^.29164",43.

We have also seen, in [7303], that the equation of the centre, corresponding
to this satellite, is

[7388] 6V" = 1 709",05.sin.(e"â€” z.").

This satellite has another equation of the centre corresponding to the
perijove of the fourth satellite [7127], and represented by

[7389] 6v" == 756",6] .sin.(e"â€” :;;"').

The expression of 6v" [6846] becomes, by the substitution of the values of
m, m', m'" [7142â€”7145],*

* (3615) We have, as in [6240n, p, q, s%

[7390a] ntâ€”n"t-\-sâ€”s" = e â€” e" ; n'tâ€”ift-{-s'â€”s" = e'â€” e" ;

[7390i ] n'"tâ€”n"ti-s'"â€”s" = e'"â€” e" ; 2n"tâ€”2Mt-\-2s"â€”2E = 2q"â€” 2n ;

[7390c ]^ ntâ€”2n't-\-Â£â€”2s'-{-g^t-\-r^ = eâ€” Se'-f^" j ntâ€”2n'i-\-sâ€”2/+g,t-\-r^ = eâ€” Se'-fw " ;

[7390rf] n"tâ€”2Mt-\-s"â€”2E-\-gj-{-r^=e"-2n-\-i^" ; 7i"t-2Mt+s"-2E-{-g,t-\-r,=e"-2n-j-!;i"'.

Substituting die values [7390cf, 6] in [6846], also the valuesof m, m',n/' [7142,7143,7145],
we get, by a very easy numerical calculation, the values [7390] .

[7392]

Vni. xiii. pi.] THEORY OF THE THIRD SATELLITE. 277

6v"= 4",21.sin. (0â€”0") 1

â€” ll",84.sin.2.(0' â€” e") 3

â€” 2",37.sin.3.(0'â€” 0") 4

The theorem on the epochs of the three inner satellites [6628^], becomes,*

0â€”0" = 200Â° + 3q'â€”Sq" ; [7391]

therefore the two terms,

4",21.sin. (eâ€” Â©"); [7390 line 1] 1

â€” 2",37.sin.3.(0'â€” Â©") ; [7390 line 4] 2

lay be connected together in one single term, 6v" = â€” 6",58.sin.(30' â€” 3Â©"). [7393]

Substituting in the expression of Q" [6862], the value of 2- = 29009",8

^/ [7393']

[7190], relative to the apsides of the third satellite, also â€” = 0,2152920

[7192], corresponding to this value of g, and â€”2h" = 1709",05 [7388], we [7394]
shall find, that the inequality 6v" = Q".sm.(ntâ€”2n't-{-sâ€”2s'-]-gt+r) [6852], [7394^
becomes,!

this in [7392 line 1], it becomes â€” 4"521.sin.3(0'â€” 0"). The sum of this, and that in [7.3926]
[7392 line 2], becomes as in [7393].

t (3617) The part of the equation of the centre of the third satellite 6v" [6243],
depending on the angle g or ^2=29009",8 [7393' or 7190], has for its coefficient the ['''^^^"l
quantity â€” 2^^" h^, which is represented by â€” 2h" in [6229c? line 5]; and the coefficient
of this term of Sv" is 1709",5 [7388]; hence we have â€” 2A"= 1709",05, or [73946]
h" ~ â€” 854",525. Now for this value of g we have, in [7192],

A' = 0,2152920.A" = â€”0,2 152920 X854",525 =â€” 183",97. [7394c]

Substituting these values of g^, h', h", and that of m' = 0,232355 [7143], in Q" [6862], [7394rf]
we get Q" = â€” 95",18 ; hence the inequality [6852 or 7394'] becomes,

(h"=zâ€”95",l8.sm.(ntâ€”2n't-\-sâ€”2e'-\'g^t-{-r^) ; [7394c]

which is easily reduced to the form [7395], by using the first of the expressions [7390c] ;
and we finally obtain the form [7396], by substituting the value of Â© â€” 20' [7395']. 'â–  â€¢'^

VOL. IV. 70

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

[7395] hv" = â€” 95",1 8.sin.(0â€” 20'+^").

[7395'] ^Â°^ ^Â® ^^^Â® Â®"^Â®' = 2OO^+0'-2e" [7391] ; hence the preceding inequality
becomes,

[7396] ^v" = 95",1 8.sin.(0'~20"+^")-

Substituting in Q" [6862] the value of g or Â£-3 = 7959", 105 [7195], relative
[7396'] ^ ^^ J,,

to the apsides of the fourth satellite, and for -â€” , â€” -, the quantities

[7197, 7198], depending upon this value of g, observing also that we have