as 1 to 6, as in [9036].



VOL. IV. 162



646 SUPPLEMENT TO THE THEORY OF THE [M^c. Cel.



[9037]



[9038]



[9039]



CHAPTER VIII.

«
SUPPLEMENT TO THE THEORIES OF THE PLANETS AND SATELLITES,

23. We have given, in the sixth book, the numerical expressions of the
inequalities of the planets. The care which had been taken not to omit
any sensible inequality, authorized the belief that the tables of the planets'
motions would be improved by the use of these formulas ; and it became
an object of interest with astronomers to apply them to this purpose.
These hopes have been realized by the labors of Delambre, Bouvard,
Lefrancais Lalande and Burckhardt, who have compared the theory with
a very great number of observations, in order to deduce from them the
elliptical elements of the orbits of the planets ; and I have also reviewed,
with great care, the theory of their perturbations ; so that by these united
efforts we have obtained very accurate tables of the motions of the planets.
This new examination of the theoretical results, has not indicated any
sensible inequalities to be added to those which had been before
determined, except in the motions of Jupiter and Saturn. The nearly
commensurable ratio of their mean motions gives rise, as we have seen in
the second and sixth books, to some very great variations in the elements
of the orbits of these two planets, depending on inequalities whose periods
exceed nine centuries. The variations of the excentricity and perihelion
of Jupiter's orbit, depending on this cause, produce in the motion of Jupiter
a very sensible inequality [4394], whose argument is three times the mean
motion of Jupiter minus five times that of Saturn. The similar variations
in the excentricity and perihelion of Saturn, produce in the motion of
Saturn a great inequality [4468], whose argument is twice the mean motion
of Jupiter minus four times that of Saturn. These two inequalities may
in fact be considered as real equations of the centre, whose excentricity and
perihelion vary with extreme slowness. Now the two great equations of



[9040]



X.viii.Ȥ.23.] PERTURBATIONS OF JUPITER AND SATURN. 647

the centre of these two planets give rise to some very sensible inequalities ;
therefore, by substituting in the expressions of these inequalities, instead
of these great equations of the centre, those which we have just mentioned,
there will be produced some small similar inequalities, which may be of
sufficient importance to be noticed ; we shall therefore consider, in this
point of view, the chief inequalities of Jupiter and Saturn, depending on
the excentricities.

We have seen, in [4392], that the expression of &v'^ contains the
following inequalities ;

. — 138^373.sin.(2w"^— w'"^ + 2£^— s'^— t^'') \ [904i]

+ 56%6S^.sm.(2n''t— n'U-{-2B'^— s'^— ^M f [9042]

— 44^461. sin.(3?i"/—2«'"^ + 3a'— 2s'^—t3"^) [' ^^^ [9043]

+ 84%942.sin.(3n"^— 2/i^'^4-3s"— 2a''— ^') I [9044]

they are the most important ones arising from the first power of the
excentricities. The first and third depend on the equation of the centre of
Jupiter, +2e''.sin.(?r^ + s''— ta'') [3834, 4390^, &;c.]. We have seen, in [9045]
[4394], that Jupiter's motion is subjected to the following inequality;

169*,266.sin.(w''^ + a'' -f 55' 40"" 49*— Sn'i + 2/r i— 5a' + 2a''). [9046]

This inequality may be considered as a second equation of the centre of
Jupiterh orbit ; whose excentricity and perihelion vary with extreme
slowness, their variations depending upon those of the angle on^t — 2w''^.
This being premised, we shall put the inequality [9041 ] under the following ^

form ;

1 38 * 373

— ^J.^ ' . 2e".sin.(w''^ + a" - n'' + 2n''t—2n'U + 2s'— 2a''). [9048]

If we substitute in it, instead of * 2e''.sin.(n'''^ + s'"' — w"), the expression
[9046],



[9046']



[9049a]



* (4090) The term 2e''.sin.(n''i-}-s'' — -sj'') is the most important part of the
equation of the centre [3834] ; and if we substitute instead of it the term [9046 or 9049],
it will be equivalent to changing 2e'' into 169',266, and — xs*' into,

55" 40'" 49^—571'^ -f 27i''<— 5£'+ 2s'' ; [90496]

and by this means [9048] changes into [9050]. In like manner we obtain [9053].

This short but indirect method of computing the small inequalities
[9050, 9053, 9058, 9060 or 9061, Sec], of the order of the square of the disturbing '^^^^^''^



648 SUPPLEMENT TO THE THEORY OF THE [Mec. Cel.

[9049] 1 69%266.sm.(3n}U—5nU + Ss''^— 5e" + 65'' 40« 49^,

we shall obtain the inequality,

[9050] __ i^^^ . l69%266.sm.(n}U—3nU + £-— 3 £^+ 55'* 40'" 49^.

[9051] Putting in like manner the inequality of the expression [9043] under

the form,

[9052] __ l^^l ,2e^sin.(w''^ + £-_^- + 3»-f— 3w'^^ + 3s'— 3s'''),

we shall obtain, by substituting the same term [9046], the following
inequality ;



[9053] _ 44%461

[9054]



[904Grf]



^ . . 169^266.sin.(— 27i'i— 2s'' + 55HO"'490.



The terms [9042, 9044] depend on the equation of the centre of Saturn,
2e\sin.(»''^+ s'— Z3''), and we shall put [9042] under the form,

[9055] ^^^. 2e\(n'^^ + £'—«'' + n''i—n'^^ + s'^—s").

We find, in [4468], that Saturn's motion is subjected to the inequality,

[9056] — 669^682.sin.(n'^ + s''-{-56^ 10"* bT—brCt + 2n}''t—5s'' + 2s-).

This may be considered as a second equation of the centre of Saturn, whose
[9056'] . . . . . *' , . .

excentricity and perihelion vary with extreme slowness, these variations

being dependent upon that of the angle Sn^t — 2/1'"^. Therefore by

[9057] substituting it for* 2e\sin.(n'^+ s'^ — to')^ in [9055], we shall obtain the

following inequality ;



masses, must be considered as nothing more than a tolerably near approximation for
obtaining their values, since several of the small parts of the general expressions of these
terms are neglected ; those parts only being retained which are derived from the variation
of the first term of the equation of the centre of Jupiter, 2e*''.sin.(n'''-{- ^''^ — '^^^) [9045],
or that of Saturn, '2e^ .sm.{Tf t-\-s'' — -sj") [9054]. We have already mentioned a

[9049e] somewhat similar defect in the abridged method of computing the small inequality in the
motion of Mercury [3872, &tc.]. Plana has noticed this imperfection in vol. 2, page
406, Sic. of the Memoirs of the Astronomical Society of London, to which we may refer ;

._ since it is not necessary to go into any particular detail on the subject, taking into view the

smallness of these inequalities [9061, &;c.], and that the corrections to be made in them

rr.r..r. , ^^^ ^f '^^^Y h^tlc importauce. Similar remarks may be made relative to the other
[90492'] ....

mequalities, computed in this section of the work.

* (4091) If we proceed in this case as in [9049a, 6], we shall find, by comparing the
expressions [9056, 9057J, that we must change 26^^ into — 669*,682, and — w'




[9061]



X. viii. § 23.] PERTURBATIONS OF JUPITER AND SATURN. 649

— ^-i^ ' 669%682.sin.(n''i— 3/1^^ + a-— 3s- -f 56' 10™ 570- ^9058]

In like manner, bj putting [9044] under the form,

84* 942

-^^ . 2e\sin.(n''t + s^— t^i^ + 2n''t—2n}''t + 2s-— 2s'-), [9059]

we shall obtain, by the same substitution, the following inequality ;

— —7^ • 669^682.sin.(— 2/1-/— 2s- + 56'' 10" 570- [9060]

Jubstituting in [9050, 9053, 9058, 9060] the values of e'\ e' [4080],
we find that the four expressions [9041 — 9044] produce the following
inequalities ; *

P,1809.sin.(3w"/— 71'-/ + 3s-— s»-— 55HO'"490 ; 1

+ 0%3794.sin.(2ri-/ + 2s-— 55'^ 40"* 490 ; 2

+ r,6352.sin.(3ri"/— w'"/-f-3s-— s'-— 56'' 10^570 ; 3

+ 2%4525.sin.(27i-/ + 2s-— 56" 10™ 57*). 4

These inequalities are very small ; but as they may be connected with others
of similar forms, they will not render the tables more complicated, and will
make them more accurate.

We have seen, in [3916, &c.], that the inequality of Jupiter [4394],

169%266.sin.(3/i'-/— 5w"/ +3s'-— 5s- + 55'' 40^490, [9062]

is the result of the variations in the equation of the centre and the
^ perihelion, depending upon the angle 5iVt — 2n'^t. If we represent these
variations by 6s'- and 6^'-, the preceding inequality may be put, as in .
[3916], under the form,

'2.6e\sm.{n'''t + s'-—- n'-)— 2c'-.<57:3'-.cos.(yi'-/ + s'-— ^•-). [9064]

The expression of Jupiter's true longitude in terms of its mean longitude,t
[668], contains the two terms,

A.e'-2.sin.(2w'^/ + 2s'-— 2:.'-) + J-|.e'-^(37i'-/ + 3s'-— 3..'-), [9065]



[9063]



into 56'^10"'57' — 5n-< + 2n'-^ — 56-4-2gi-; substituting these in [9055], we 1-90576]
get [9058].



* (4092) After making the substitution of e'-, e- [4080], we must divide by the
radius in seconds 206265% and we shall obtain the numerical values [9061].

t (4093) Changing e into c'-, and nt into 7i'-<4"^'''— '^'^ to conform to the
notation here used.

VOL. ir. 163



[9061o]
[9064a]



650 SUPPLEMENT TO THE THEORY OF THE [Mec. Gel.

which gives the following expression ; *



[9066] ^ ^ ^ )' (Q)

. . ( 5e'\sin.r3w'^^ + 3s'-— 3tji'0> i ^ ' 3

The two first of these terms give the inequality depending uponf

4/1^^^— 5ri^^ + 4s'"— 5£" + 45''21'"44% which we have determined in
[9068] [4440]. If we represent by p.s\n.(n}H -{- s''' — tn'^-f/), the inequality of
Jupiter [9046], depending upon Sn'^'t — 5rft, we shall have,

[9069] f == .2re^ _5^v^ _l_ 2 3iv_5 ^v j^ ^. _j_ ^^d 49â„¢ 49. .

[9070] 266'" == p.cos.f; — 2e'\(Jt3'^ =^.sin/; p = 169*,226.

Hence the terms [9066 lines 3, 4] become,!
[907]] i_3 .e\e\p.sm.(Sn'''t + 3£'" - 373'^ +/) ;

therefore by substituting /, p [9069, 9070], we get,
[9072] L3.e'^e'M69^266.sin.(5n'7— 5w"^ + 5 s'"— 5-="— 2z3'' + 55^ 40"* 49^.

Reducing the coefficient of this expression to numbers, we obtain the



[9065al * (4094) Taking the variation of [9065], considering e'^, zs^", as variable, it becomes
as in [90661.

[9067al ^ (4095) If we put p = 169%266, and use the value of / [9069], the term [9046]
will become p.sin.{n}''t -\- s'"" — to'^-j-/), as in [9068]. Developing it by [21] Int. we get,

[90676] p.cos./.sin.(?i'^<-|- £>^ — w^^) +p.sin./.cos.(n'^^+£'^-— cj'^).

rQOfir Putting this equal to the expression [9064], we obtain 25e^''= p.cos.f; -2e^'^.5'!s^''=p.sin.f,
as in [9070]. Substituting these values in [9066 lines 1,2], they become,

|^.e'^|cos./.sin.(2n'^< + 2£>"— 2^^^) + sin/.cos.(2ni^^ + 2s'"— 2zn'^)}
= fj9.e>\sin.(2w'^^ + 2£'"— 2to»^+/), [21] Int.
Re-substituting the value of / [9069], it becomes,

[9067e ] ||).e'\sin. (4ni^ t — 5n^ ^+4si^— 5s^ — xs'^-f- 55"* 40"* 49'),

being of the same form as that in [9067], which is computed in [4440 or 4439] by a
similar process, changing K [3827] into — p [9068], and using -ra'^ [4081] nearly.

f (4096) Substituting the values [9070] in the two terms [9066 lines 3, 4], we obtain,
[9072a] -V-e'".e'\;?.{cos./.sin.(3n''^ t + 3£'^— 3i!j'^) + sin./.cos.(3»'^i + 3s'"— 3z3'^)},

which is easily reduced to the form [9071], by using [21] Int.



[9075]



X. viii. -^ 23.] PERTURBATIONS OF JUPITER AND SATURN. 651

following inequality ; *

0%6358.sin.(5w'"^ — 5w^^ + 5^''—5s'' + 34'^ 58"* 4P). [9073]

The following inequality is given in [4438, 90736],

— 4^0247.sin.(5?i'7 — lOri + 5s-— lOs^ + 51''21'^550. [9074]

We have seen, in [4006', &c.], that in all the arguments of Jupiter and
Saturn, where the coefficient of t is neither 5rf — 27i'% nor differs from it
by n'"" for Jupiter, or n'' for Saturn, we must increase the mean longitudes
w'7 + =''', ^"^ + 5% counted from the fixed equinox of 1750, by their great
inequalities depending upon 5rft — 2w"'/. If we wish to use the mean
longitudes thus increased in the inequality of Jupiter [9062],

1 69^266.sin.(3w'7 — 5/i^^ + 3 s''— 5 s' + 55*^ 40™ 49^, [9076]

we may put q'\ q\ for these longitudes thus augmented, and then put this [9076']
inequality under the following form ;

169^266.sin.;37^'— 5g^— (3p^'^+ 5p') -f55H0'" 49^} ; [9077]

p" being the great inequality of Jupiter, and — p^ that of Saturn. If we [9078]
develop the preceding function, we shall get,t

169',266.sin.(3g'^— 59' + 55*^ 40'" 49^ 1

—(Sf + 5/).I69s266.cos.(3^''— 59^ + 55'^ 40™ 49*). 2

Now we have very nearly, J



[9079]



.| * (4097) Substituting the values of e'^ [4080], also ^'^== 10''2I'"04* [4081], in rgQ^g^-.

'^ [9072], it becomes as in [9073]. The inequality [9074] is the same as [4438]; it is

';^ printed with a different sign in the original work, but it is corrected in [4438]. This is [90736]

"i hereafter combined with the term which is computed in [9083].

f (4098) This development is made as in [60] Int., by putting,

z = 3^'^— 5^^-1-55'' 40"' 49% a = — (.3^«^+5p^), [9076o]

neglecting the square and higher powers of a ; then multiplying by the coefficient
169^,266, we reduce the expression [9077] to the form [9079].

J (4099) If we put for brevity T= 5n^^—2nW + 55^—28^^-1-4" 2 1"* 20% and use
the symbols p'", p" [9078], we shall have very nearly,

p'^=1265%3.sin.T [4434] ; p^ = 2939%6.sin.T [4492]; [9080a]

hence 3p'^-}-5p^= 18493',9.sin.T, as in [9080]; the argument being taken the same
as for Saturn in [4492], because this produces by far the greatest part of the coefficient in
[9080]. IMoreover the difference of the arguments in [4492, 4434], is not of much [^ J
importance in the small inequalities which are computed in [9085, &z;c.]. Dividing the
coefficient 18493%9 by the radius in seconds 206265% and multiplying the result by [9080c]



[9086]



652 SUPPLEMENT TO THE THEORY OF THE [Mec. Cel.

[9080] 3pi^4-5/?"= 18493%834.sin.(6w^^— 2n'"^ + 5s^— 2s''+4''21"'2(y) ;

which gives,

—(Sp'' + 5p'')A69%266.cos.(Sf^5q'' + 55' 40'" 49^ 1

[9081] C sin.(Sq''—5q'' +5n''t—'2rVH ^5 s''— 2 ^'4-60' 02« 09^ ) 2

' I _sin.(39'^— 5q^— 5n7H-2n'^^— 5s' + 2a^^+61'^ 19™ 29') ) 3

We may substitute in these two last terms, without any sensible error, 5'^
[9082] and (/% for n'H -\- s'"" and rft -\- s^. The first term will then be confounded

with Jupiter's equation of the centre. The second becomes very nearly

equal to,
[9083] 7%5882.sin.(59^^— 10^^ + 51", 19"* 29^) ;

and by connecting it with that in [9074], namely,
[9084] — 4^0247.sin.(59'^— 10^^ + 51^9^290,

we obtain the following result,
[9085] 3^5635.sin.(55>^— 10^^ + 51'^ 19"* 29^.

Thus we may use q'" and 5% instead of n'^'t + s''' and nH-{-s'', in all

Jupiter's inequalities except the great inequality.

We shall now consider the analogous inequalities in the motion of Saturn,
which are much more sensible than those of Jupiter. To determine them
we shall observe that we have found, in [4466], that the motion of Saturn
contains the two following large inequalities, depending on the first power
of the excentricities ;
^9087] — 182%069.sin.(2?i7— «+2a-— a''^— ^')) 1

+ 4>n%05S.sm.(2n''t—n'H+2s''—s'^—i^'')\' ^ ^ 2

The first of these inequalities depends on the equation of the centre of



169%266, it becomes 15*,1765; lience the first member of [9081] is,

[9080rf] — 15%1765.sin.(5n^^— 2n'^^-}-5s^— 2ai^+4''21"» 20*)Xcos.(3^>^— 5^'+55M0'"49*) ;

reducing this by means of [19] Int., we get the second member of [9081]; observing
that the author has given the angle 51'^19'"29* [9081 line 3] equal to
[9080e] 57°,0725=51''21'"55% being too great by 2^26% from the data he has used. This
is corrected in the formulas [9083 — 9085]. Now substituting §-'% y% according to the
directions in [9082], we find that the second member of [9081] becomes,

[9080/] — 7^5882.sin.(j'^-f 60" O2'"O90+7',5882.sin.(5^'^— 10^^-f Sl'^ 19"* 29*) ;

of which the first may be combined with the equation of the centre, and the second is as
in [9083].



X.viii.Ǥ>23.] PERTURBATIONS OF JUPITER AND SATURN. 653

[9088]



Saturn +2e\sin.(n''/-j-£'' — zi"), and it may be put under the following
form ; *

182 ',069



The inequality in the motion of Saturn [9056],

— 669',682.sin.(2/i^"i — 4/i"^ 4-2s>^— 4a^+56'^ 1 0™ 57*), [9090]

which, as we have observed in [9056'], may be considered as a second
equation of the centre, will therefore produce, by its substitution in [9089],
the following term ;
182* 069

\ . 669^682.sin.(n'7 —Sn^'t + s-^^'^Ss" +56'' lO*" 570- [9091]

The equation [9087 line 2] arises from the equation of the centre of Jupiter,
and it may be put under the following form ;

417* 0*58

-— ^ — . 2e'\sin.rn'"^ -{- s'^— :.'' + 2nrt—2n''t 4- 2s'— 2a'0- ^9092]

The inequality of Jupiter,

-fl69^266.sin.(3ri''^— 5w7+3£"'— 5s'-f 55'^ 40"* 49^, [9046] [9093]

which, as we have seen in [9046'], is a second equation of the centre of
Jupiter, will therefore produce, by its substitution in [9092], the following
term ;

417* 0'i8

^Vl . 169%266.sin.(w''^^— 3n'i + s''—3e' + 55' 40™ 49^. [9094]

p Therefore the expressions [9087] produce, as in [9089c], the following

inequalities ;

— 5*,2568.sin.(3w7— w'7+3£'— s»^— 56'^ lO*" 57*) [9095]

— 3*,5594.sin.(3n"^— n'^^+Ss'— s^''— 55M0'" 49*). [9096]

The expression of Saturn's true longitude in terms of the mean longitude,

contains the inequality,!



[90S9a]



* (4100) The calculation is here made in the same manner as for Jupiter,
[9049a, 6, he], by putting the inequality [9090] under the form,

— 669%682.sin.(7i'^+£^-f-56'' 10"* 57^4-2n''^— 5n^^ -|_2£i^— Ss'), [90896]

which is similar to [9046] ; and changing, as in [9049a, 6], 2e'' into — 669*,6S2j also
— ^^ into 56-^ 10*" 57*+2n'' <— 5n^^+2£>''— 5£' ; and then [9089] becomes as in [9091].
In like manner we get [9094] from [9092]. Reducing the expressions [9091 , 9094] to [9089c]
numbers, they become as in [9095, 9096] respectively.

f (4101) This calculation is made in the same manner as for Jupiter, in [9065, &ic.].
The second term of [9065] is similar to that in [9097] ; those in [9066 lines 3, 4] are
VOL. IV. 164



654 SUPPLEMENT TO THE THEORY OF THE [M6c. C61.

[9097] ^.(ey.(3nH + 3s^— 3^").

Therefore by putting ie"" and 6-^^ for the variations of the excentricity and
perihelion, depending upon Bti^t — 2n'''t, we shall have the function,

[9098] y.e\c\{(J€\sin.(3w"^+3£"— 3sj^)— c\(J^\cos.(3n7+3£^— 3ra'')j ; (0)

To obtain Se"" and <5to% we shall consider this inequality of Saturn,

[9099] - 669%682.sin.(2?i"7— 4w7 +2£'^— 4s^ -^56' 10" 570, [9090]

and we shall suppose it to be produced by the variation of the equation of
[9100] the centre and perihelion, in the term 2e''.sin.(?i7 -j- s" — to") ; and we shall
then have for the expression of this inequality,

[9101] 2(5e\sin.(w"^ + £'— ^0— 2e\(5TO\cos.(?i"i + £"— TO^.

Hence it is evident that the function [9098] will become,
[9102] — y .e\e\669%682.sin.(2/i'"— 2/1^^ + 2£'^— 2£"— 2^-+ 56'' 10™ 570-

This inequality, reduced to numbers, is equal to,
[9103] — 3^4402.sin.(2?i'7— 2w^^ + 2£''— 2s"— 120'' 7'" 1 7").

[9104] We have in [4496] the following inequality, corrected as in [4495a — d] ;
[9105] 8s2645.sin.(4/i^"^ — 9n7 +4£'"— 9 £" + 51'' 49™ 37').

We have seen, as in [9075, 9076'], that we must change, in all the
inequalities of Saturn, /i'7 + £'" into g% and n't-\-s'' into ^% excepting
in the great inequality, and in the following ;
[9107] — 699^682.sin.(2w'7— 4717 -f 2s'"— 4£" + 56" 1 0" 57'). [9099]



similar to [9098] j the term [9057] is the same as [9100] ; the term [9062] is similar to
[9099] ; the terms [9064] are similar to [9101]. Now if we put,

[90976 ] / = 2n'"^ —bri't + 2£«" — 5s" + to" + 56"^ 1 0"» 57%

the expression [9090] will become, by using [21] Int.,
'^ — 669',682.sin.(n"^+£v— ro"-f/) = — 669%682.cos/sin.(n"< -]-£v_^v)

i^^'^<^'^ — 669%682.sin/.cos.(w"i+£"— TO").

rnnn^^. CompaHng tliis with [9101], we get 25e"=— 669%82.cos/; 2e"(5«"=669',82.sin./.

[90'J7o] ....

Substituting these in [9098], it becomes,

[9097c] — i/e".e".669%682. {sin.(3w"i+3£"— 3TO").cos./+cos.(3n"<+3s"— 3£").sin./} ;

and by means of [21] Int. it is reduced to,
[9097e'] — J/e".e".669%682.sin.(3n"<-f3£"— 3ttf"+/).

Now re-substituting the value of / [90976], it becomes as in [9102] ; and by using the
[9097 /l value of e" [4080], it changes into [9103]; observing that the value of cj" = 88^ 9"* 07',

[4081], gives — 2ts"+56''10"*57* = :— 120''7"' 17^



[9109]



X. viii.<^23.] PERTURBATIONS OF JUPITER AND SATURN. 655

If we wish to use (f" and ([ [9076'] in this last inequality, we must put it
under the form,

— 669^682.sin.(29'^— 4(/^— 2/^— V + ^^' 10™ 57^ ; [^1^8]

f^^ and — p' [9078] being the two great inequalities of Jupiter and Saturn.
The inequality [9108] becomes by development,*

— 669^682.sin.(29'''— 4^" + m^ lO"* 57^ 1

+ 669*,682.(2/'^+4p^).cos.(29''^— 4^^ + 56" lO" 570; 2!

and we have very nearly,

669',682.(2y^+4j90-cos.(29'^— 2^^+56'^ 10™ 57^ 1

_ i sin.(29'^— 4g^+5/i7— 2ri'7+5£^— 2£'^+60''32'»170) 2 t^"^^

~ ' '\ — sin.(25'^— 47^— 5w^^+2w'7— 5£^+2£'^+51H9'^ 37^ j * 3

We may, in these two last inequalities, change 7i'7-f-s''' and w7+s' into
(f" and (i' ; then the first will be confounded with the equation of the
centre of Saturn, and the second becomes,

— 23^1960.sin.(4g'^— 9^^ +51"^ 49"" 37^. [9"2]

Connecting it with that in [9105], namely,

8%2645.sin.(49'"— 9^" +51"^ 49" 37"), [9ii3]

we obtain the following inequality,

—14',9315.sin.(49'^— 99^ + 51" 49" 37^. [9n4]

We may thus use (^^ and 7% instead of ti'V+s"" and 717 + £% in all the
inequalities of Saturn, except in its great inequality.



[9111]



[9115]



* (4102) This development is made as in [9076a], by using [60] Int., putting

2; = 2^i^— 45'^-f56'', 10'"57% and a = — (2j9'^+4p0 ; then multiplying the developed t^^^^^l

value of sin.(2r-}-a), by — 669S682 [9108] ; hence we get [9109]. Now the values

of ^'% _p^ [9080a], give 2p'^+4p^= 14289*. sin. T; substituting it in the term rgj^^jj-,

[9109 line 2], and dividing by the radius in seconds, which gives

669',682 X 14289* ,^, „^^^ . ,

-^j^^^, =46*,3920, It becomes,

46%3920.sin. r.cos.(2j'^— 4^^+56'^ 10"* 57^
= 23%1960.sin.(25«^-4^^+56''10'»57*+T)— 23%1960.sin.(29'^-4^^+56''10'»57'-T). '■ '^^
Re-substituting the value of T [9080a], it takes the same form as in [9110 lines 2, 3] ;
and by using the values of §-•% (f [9111], it becomes,

23',1960.sin.((?^4-60''32"' 17*)— 23%1960.sin.(42'^— 9^^+51* 49"*37'). [Qinrf]

The first of these terms can be connected with the equation of the centre, and the second
is as in [9112].



656 SUPPLEMENT TO THE THEORY OF THE [M^c. Cel.

[9116] We must, for greater accuracy, increase q^ by the inequality [4468 line 6],

namely,
[9117] S\%02b.sm.{SrfH — rCt + Ss^' ^£^—85'' 34"' 12*),

[91181 arising from the action of Uranus, and which must be applied to Saturn's
mean motion, as we have seen in [4472, Sic.].

If we connect the preceding inequalities with those which have been
determined in the sixth book, we shall obtain the formulas of the true
longitudes of Jupiter and Saturn. Bouvard has compared these formulas

[91191 ^^^'-^ observation, by means of the oppositions of Jupiter and Saturn, which
he has collected chiefly from those of Bradley and Maskelyne at Greenwich,
and those made at the observatory in Paris, in late years. These
observations were made with excellent transit instruments, and with the
best mural quadrants, during an interval of more than fifty years. They
furnish, by their accuracy, as well as by the greatness of the number of
observations, the most accurate method of correcting the elements of the
elliptical motion. From these sources were obtained, from 1747 to 1804

191201 inclusively, fifty oppositions of Jupiter and fifty-four oppositions of Saturn,
[9122c]. These have given one hundred and four equations of condition,
between the corrections of the elliptical elements of the motions of the two
planets ; and as the value of Saturn's mass is somewhat uncertain, the
correction depending upon it was introduced into these equations. It was
soon discovered that the value of this mass, given in [4061], must be

[9121] decreased by , which reduces it to ^rrj-r^ , that of the sun being

taken for unity [9122^, 4061c]. This important correction, which is
evidently indicated by the preceding observations, and by those of
Flamsteed, is one of the principal results of this improved theory. The