[6930c], as in [6933 line 2]. Finally, the coefficient of the term of /" [6482] may, by a

3M [6930r]

similar process, be reduced to the form — . (Z'"— i') = 0,00144. (Z'"—Zr') [6930c], as in

[6934] ; which is sufficiently accurate for all practical purposes.
The same reduction may be made in the terms in the first lines of [6460, 6480, 6481] ;

VOL. IV. 51



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



[6931]

[6932]

[6933]



5 =:m'.0,00349437.(/'— /).sin.(3ij— 4i;'— p^— a) 1

-\-0,OOOlliS\2.(l—L').sin.(v—2U—pt—A) ; 2

s' = {m.0,00276975.(/-/') + m".0,00170910.(Z"-/')l.sin.(22;-3i;'-p^-A) 1
•i-0,000301S6.(r—L').sm.(v'—2U—pt—A) ; 2

s" = m'. 0,00135305.(1'— r').sin.{2v'—2v"-^pt— a) 1

+0,00061925.(/"— L').sin.(ij"— 2C7— j9/— a) ; 2

[6934] s'" = +0,001447815.(/'"— L'). sm.(v"'—2U—pt—A).

24. We shall now compute the inequalities depending upon the square
of the excentricities and inclinations of the orbits, whose analytical values we
have given in [6530, 6552]. Those which finally become the most sensible
are the secular equations of the satellites, depending on the secular variations
[6935] of the orbit and equator of Jupiter. But it is easy to prove that they are
-_, at present insensible, and that they will remain so for a long time to come.
For the greatest is that of the fourth satellite, and its expression is, as in
[6530, &c.],*

[6936] s^'" =3 _2. IT] . ^1. cf + 2.(1 —■k"y. [V] . 'L. bt'—3.{3), -K'"". 'L. bt\



[6930« ] for we have, from [60256, 6930a, b], 3— — = 1 ,00727, ^ + JV; = 1 ,00136 ; so that



n ' ' n



the expression of the divisor 3 — — JV, [6460], will not be very much altered

[6930<]

from its true value, by putting JY^-\-— =1, as in [6930] ; and by making this substitution

[6930m] we find that the expression [6460 line 1] becomes — a v o no^or — '^ ^^ substitute in it

m'
[6930r] the values of a, b'f^ [6801,6805], also m' = [6841c], it becomes nearly as in

[6931 line 1]. In like manner, the terra [6480 line 1] becomes, by using the values

[69301*] [6797—6800,6801,6805,6814,6818], and m" = -j-^j^ [6841<?], very nearly, as m

[6932 line 1]. A similar calculation being made with the terms in [6481 line 1] ; using the
[6930x] ^^j^^^ [6798,6799, 6818, Sic], we get the term in [6933 line 1].

|.„»_ ^ * (3506) The expression [6530] corresponds to the first satellite m, and by changing
reciprocally the elements relative to m, into those corresponding to m", we get Sv'"
[6936]. Now it is evident that the terms of Sv'" [6936] are much larger than those of

[69355] dv [6530]. For if we compare the first term of [6936] with that of [6530], we find that



VIII. vli. <§.24.] INEQUALITIES DEPENDING ON e^ e'^, fee; y\ j'^, &c. 203

Now we have, in [6908, 6906],

6= 0",070350 ; ^L == 3°,4444 ; [6937]

and we also have,*

2c =1 ",9446; or c = 0",9723. [6938]



[6939]



Therefore, by making use of the values of x'", 3 and (3), given in

[6926, 6864], and supposing m- = 1 in (3), we find that the secular
equation of the fourth satellite is represented by

6v"' = —0,000135. t^ ; [6940]

so that it will be for a long time insensible. [6940']

We have also, in [6552], the following inequality in the mean motion of
the fourth satellite, referred to the orbit of Jupiter ; f



their ratio is as 3 to ; and from [6864] we have nearly 3 =9. • [6935c]

The same ratio obtains very nearly between the second terms of [6936] and [6530],

X, X', &c. being small [6923 — 6926]. Again, the third term of [6936] is to the third term

of [6530] as X'"^ (3) to X2 (0) ; and by substituting the values [6926, 6923, 6864 hnes 1,4], [6935rf]

we find that the first term of this ratio is much greater than the second. Hence it appears

that this term of Sv'" [6936] is much greater than the corresponding terms of Sv, 5v\ Sv", ["^"^^J

as is observed in [6935'J .

* (3507) If we compare the expression of the excentricity of Jupiter's orbit [6527]
with that in [4407], we get c = 0*,329487, which represents the value of the coefficient 'â–  "^
of t in e*'' [4407]. This term "is produced chiefly by the action of Saturn, as is evident

from [4404c?, &ic. 4246 line 4, 4403, 4405], using the value m^ = ~^ ^ [4061], and [69386]

oo59,40

putting jxv=0. If we use the value l-{-^'^ == ' [6903&], the preceding expression r6938c]
of c will be decreased in nearly the same ratio, making c = 0',3149 =0",972, or [6938rf]



2c== 1",944, as in [6938]. Now substituting in [6936] the values of (3), 3 , X"', b, ^qqssc]

^L, 2c, [6864,6926,6937,6938], also that of H or 6*^ = 0,0480767 [4080], we get
Sv'" [6940] nearly. This is so very small that it does not amount to two centesimal seconds
in a century ; and it must therefore be for a long time insensible, as in [6940'J.



[6938/]



t (3508) The formula [6941] is easily deduced from the terms of [6552], depending



on the angle p, by changing X, (0), , I, into X'", (3), 3 , /'", respectively;



[6941o]



so as to make them conform to the fourth satellite.



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



6.(3).X'"+4.(1— X'").rTl-i.(l-X"0.pl



[6941] ^^>" = __ 1— _ i . a'. V", sin.(p^+A— ^')-



V

[6941'] The first approximation gives,*

[6942] ;?^ + A— ^' =^.7541" + 31°,9199,

[6943] r = 2112";

hence the preceding inequality becomes,!



[69426]



[6942d]



* (3509) The chief term of s'" [7329], depending on the peculiar inclination of the
orbit of the fourth satellite to the fixed plane, may be put under the form,

[6942a] /' = 2771",6.sin.(t;'"+ 283o,29861 + i.7528",01).

In this expression the coefficient of t is relative to the moveable vernal equinox of the
earth [7327, kc] ; and if we refer it to the fixed equinox of 1750, we must change the
coefficient of t from 7528" ,01, to 7682",64, as in [7327]. IVIaking this change, and
then putting the expression equal to that of s'" in [6300 line 4], we get,

l6M2c] r\sm.{v"'-i-pt-{-A) = 277r',6.sin.(z;'"+283°,29861 + <.7682",64).

Comparing the coefficient of the first member with that of the second, we get Z'"=2771"6,
as in [6943] nearly. JVIoreover, by comparing the angles under the sign sin. , we get,

[6942e] pt-{-A = 283^,29861 + ^.7682",64.

Subtracting, from this last expression, the value of *'= — 148°,62129-|-^-0",8259
[6929c], and neglecting the whole circumference 400°, we obtain,

[6942g-] ^^ + A— y = 31^,9199 + ^.768r,8141.

The constant angle 31°,9199 agrees with that which we have given in [6942] ; observing

[6942A] that the author has inserted, in the original work, — 52875", instead of 4-31°59199, in
both the formulas [6942, 6944]. This appears to be merely a typographical mistake;

[6942i] since the true value +31°,9199 is used in the subsequent parts of the work, as in
[7315, 7317, 7318, &c.]. The coefficient of i, which is used by the author in
[6942, 6944], is 7541"; being the value assumed as a first approximation, in [6941', Stc.];

[6942A:] ^^^ jj(jiffgj.s ^ little from that in [6942^], namely, 768l",8141. Substituting in [6941]

[6942^ ]



the values of X'", (3), 3 , pt + A-^', l'% &c. [6926, 6864, 6942^, 6943], also

a'= 30,43519 [7217], and dividing by the radius in seconds, it becomes very nearly equal

to the following expression,
[6942m] 5«'" = — 49",51.sin.(<.7681",8141+3lO;9199);

being the same as in [7318 line 13] ; the coefficient of t being changed as in [6942^].
[6942n] Remarking, however, that, by a rough calculation, the coefficient of this inequality appeared

to be a fraction of a second less than the value given by the author.

f (3510) We have seen in the last note how the inequality [6944], corresponding to

[6944a] ^j^^ angle, or value of p [6942], is computed. The terms depending on the other larger

values of j?, may be neglected, because they are decreased by the increased value of the



VIII. vii. <^ 25.] INEQUALITIES DEPENDING ON e^, e'^, &c.; f, y'\ kjc. 205

5t,'" = _49",51.sin.(^.7541"-f3I°,9199J. [6944]

The inequalities of this liind are insensible relative to the other satellites, rQaAA^
[6944^— m].

25. It now remains to consider the inequalities depending upon the square
of the disturbing force. We have seen, in [6765], that the second satellite
is subjected to the inequality,

62;'=:_5g..(ii)2.sin.(2/i^ — 2n'i+2s — 2s'). [6945]



divisor p, and also by the decreased value of the coefficient V". This is evident by the [69446]

inspection of the value of /' [7352] ; in which the coefficients of t, corresponding to the [6944c]

moveable equinox [7328], are 7528",01, 28220^85, 1 337 1 5",77 ; and the corresponding [6944rf]
coefficients or values of V", neglecting the signs, are 277I",6, 448",93, 4",80 ; so that

I'" 2771 6 448 93

the values of — , relative to these quantities, will be nearly as ' ■ , ' , [6944e]

—7-^ — — , or as 1, 0,04, 0,0001 ; consequently the two last terms must be so much less

lOo/ J O^/ /

than the first [6944], that they may be neglected. The coefficient 49",51 [6944] is the

maximum of that inequality, and this arc is described by the fourth satellite in about 18 [6944/]

sexagesimal seconds of time [6781] ; and generally the effect in eclipses ivill he much less

than this maximum value, on account of the multiplication by the sine of the angle with

which the coefficient is connected in [6944]. If we change the symbols in the formula r(>qAA i

[6941], so as to make it conform to the first satellite, and compute the value of Sv

corresponding to the expression of I, in [7522 line 3], we find, that the coefficient of this

term of Sv is — 2",7 ; which can be described by this satellite in a tenth of a sexagesimal

second of time, and is tlierefore wholly insensible. In like manner, by changing the [6944A]

symbols in [6941], so as to correspond to the second satellite, and to the value of I' in

[7482 line 4], we find, that the coefficient of this correction is — 128",4, which is [6944i]

considerably greater than that in [6944], neglecting the consideration of the signs ; but as

an arc of 128",4 is described by the second satellite in less than 10 sexagesimal seconds

of time, it may, on that account, be considered as of less importance than that of 5v'

[6944/J ; but it seems to be of sufficient magnitude to be noticed. Lastly, by changing

the symbols in [6941], so as to correspond to the third satellite, and to the value of l" in [6944Z]

[7427 line 3], v^e find, that the coefficient of this correction is — 37''',7 ; which can be

described by this satellite in about 6 sexagesimal seconds of time, being considerably less

than that for the fourth satellite [6944/]. The corrections for other values of I, I', I", [6944m]

are decreased like those for the fourth satellite in [6944e], and may be considered

insensible, as in [6944']. We have not, in this note, taken into consideration the terms rgQ44 -i

depending on the square of the disturbing force, mentioned in [6553, &,c.].

VOL. IV. 52



„, [6944A]



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

We have by observation, very nearly,*
[6946] (ii) = 11923";

hence the preceding inequality becomes,
[6947] 6v' = 69",78. sm.(2ni—2n't + 2s — 20.

The inequalities of the same kind are insensible relative to the other

satellites [694>ld, &c.].



* (3511) We obtain from [61726, 6240g'], by successive reductions,
[6947al s\n.{nt—nft-\-s—e') = sin. {2n"i—2n't-}-2^'—^s') = sin.2.(0"— e') = — sin.2. (e'— e").
[69476] Substituting this last expression in [6173], we get fV= (ii).sin.2.(©' — e"). Comparing
[6947c] this with the corresponding term of [7450 line 2], we get (ii) = 1192(y',67, agreeing
nearly with [6946]. IMultiplying the square of this by -^^, and dividing the quotient by
the radius in seconds, it becomes as in [6947] nearly. In lilce manner, by comparing
[7513 line 5] with [6172], and [7390 line 2] with [6174], we get ( i ) = 5050",59,
(hi) = 808",20. The former being substituted in [6764], gives (5«=12",5.sin.4.(0-0') ;
[6947e] and the latter being substituted in [6766] gives 5u" = 0",3. sin. 2.(0' — 0"). This term of
6v" is insensible, but that of Sv might be noticed, with the other similar inequalities in
[6947/*] [7513] ; though it is hardly sensible in observations in eclipses, on account of the rapidity
of the motions of the first satellite ; since the angle 12%5 is described in about half a
sexagesimal second of time [6778].



[6947(f]



VIII. viii. Ǥ. 26.] ON THE DURATION OF AN ECLIPSE. 207



CHAPTER VIII.



ON THE DURATION OF AN ECLIPSE OF ANY SATELLITE.



26. We do not directly observe the motions of the satellites of Jupiter
about that planet. The elongation of a satellite from Jupiter, as seen from
the earth, is so small, that a very slight error in an observation can produce
a variation of several degrees, in the place of the satellite, when referred to
Jupiter's centre. The eclipses of the satellites furnish an incomparably [(5948]
better method of determining their motions ; and it is to these observations
we are indebted for the knowledge of their perturbations. The shadow of
Jupiter is projected in an opposite direction to that of the sun; and the
satellites are immersed in this shadow when they are nearly in opposition
to the sun. The inclinations of the orbits of the three inner satellites, to [6948']
the orbit of Jupiter, and their distances from that planet are so adapted to
each other, that the satellites are eclipsed at each revolution. But the
fourth satellite passes often beyond the limits of the shadow, and is not
eclipsed ; and for this reason as well as on account of the longer duration
of its revolution, the eclipses of this satellite happen less frequently than
those of the other satellites,

A satellite disappears from our view before it is wholly immersed in
Jupiter's shadow. Its light is diminished by the penumbra, and its disc, as [6948"]
it gradually enters into the shadow of the planet, becomes invisible to us,
before it is totally eclipsed. The limb of the satellite, at the moment when [69491
we cease to perceive it, is therefore at a small distance from Jupiter's shadow ; Exterior
and if we suppose that there is at that distance an exterior surface similar to
that of the shadow, the immersion of the satellite, within this exterior surface,
will be the beginning of the eclipse, as it appears to us ; and its emersion
from this exterior surface will be the end of the eclipse.



fictitious
shadow.



[6949']



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



[6950]



[695rj



[6952]



This exterior or fictitious shadow is not the same for all the satellites. It
depends on their apparent distance from Jupiter, whose brilliancy weakens
their light, and on the greater or less power of their surfaces in reflecting
the light ; it also depends upon the penumbra ; and probably also upon the
refraction, and upon the extinction of the solar rays in Jupiter's atmosphere.
The greatest duration of the eclipses of one of the satellites, cannot therefore

[6950'] give, with precision, that of the other satellites ; but the comparison of these
durations will throw light on the influence of the causes we have just
mentioned. The variation of the distance of Jupiter from the sun or

[6951] from the earth, changes the intensity of the light we receive from the
satellites, and this has an influence upon the duration of their eclipses. The
elevation of Jupiter above the horizon, the clearness of the earth's
atmosphere, and the power of the instruments which are used in the
observation, have also an influence upon this duration. All these causes
produce a degriee of uncertainty in the observations of the eclipses of the
satellites, particularly in those of the third and fourth. Fortunately, we can
observe quite often, with these two satellites, the immersion and emersion in
the same eclipse ; which gives the time of their conjunction with a
considerable degree of accuracy, and independent of most of the causes of
error above enumerated.

hi the first place loe shall determine the figure of Jupiterh shadoiv. If
this planet and the sun were spherical, the shadow would be a cone,
touching the surfaces of the two bodies. But Jupiter is sensibly elliptical ;
therefore its shadow must differ from that of a cone.

We shall consider generally the shadow of an opake body illuminated by a
luminous one, whatever be the figures of these bodies. If we draw through
any point of the surface of the shadow a plane which is a tangent to that

[6954] surface, it will also be a tangent to the surfaces of both these bodies. It is
evident that the three points of contact will be in a right line, which will
also coincide with the surface of the shadow ; therefore this surface is

[6955] formed by the intersections of a series of planes, which touch the surface of
the opake and luminous bodies. We shall suppose, as in [19c?], that

[6956] x=^y-\-hz-\-c,

touching is the general equation of these planes, a, b, c, being quantities which are
variable from one plane to another. We may here apply the considerations
used in [1167", &:c.], relative to the orbits of the planets considered as
variable ellipses. If we vary, by infinitely small quantities, the rectangular



[6953]



the
shadow.



VIII. viii. § 26.] ON THE DURATION OF AN ECLIPSE. 209

co-ordinates a;, y, z, they may be considered as appertaining to the same [6957]
plane. Hence we may take the differential of the equation,

a; = ay + bz + c ; [6958]

considering a, b, c, as constant quantities ; which gives,

dx = ^dy + hdz. [6959]

Taking then its differential, supposing all the quantities to be variable, and

subtracting the first differential from the second, we obtain,

= y.d^-{-z.dh-\-dc', [6960]

so that if we consider b and c as functions of a, we shall have, [6961]

I dh dc
" = 2/ + ^-^+^- [6962]

We shall now put,

|x = 0, [6963]

for the equation of the surface of the luminous body ; and X, Y, Z, for the Equation
three co-ordinates of this surface,* at the point where it is touched by the tZ hirai"/
plane. In order to make this plane a tangent to the surface, it is necessary
not only that the co-ordinates should satisfy the equation of this surface, but
that they should also appertain to its differential,

°=(S)-''^+(S)-''y+(S)-^^-

Substituting for dX its value </Z = ac^F-f 6 JZ [6959], we obtain, [6965]

= .Y.J(S)+a.(£)|+.Z.{(g) + b.(S)J.

It is evident that this last equation ought to be satisfied, whatever be the
values of dY and </Z; hence we get.



[6962a]



* (3512) The equations of surfaces are treated of in [19a—/]. The differential of
the equation of the surface (x= [6963], is evidently as in [6964] ; and if this differential
surface coincide with the plane [6953, Sec], it will satisfy the equation [6959], by changing [6962&]
dx, dy, dz, into dX, dY, dZ, respectively, as in [6965]. Substituting this in [6964],
we get [6966] ; which cannot be satisfied for all values of dX, dY, unless both their 'â–  ^-'
coefficients be put equal to nothing, as in [6967,6968].

VOL. IV. 53



210 MOTIONS OF THE SATELLITES OF JUPITER. [M^c. C«.

Combining these equations with the following,*

fA being a function of X^ F, Z ; and then eliminating Xt Y, Z, t(7e o6^om

/Ac /r*/ Jmdamental equation in a, b, c.
}Ve shall also ptU^

[mi] fi' « 0,

S'ui^ for the equation of the opake body ; and X\ F, Z', for the coordinates,



[6970]



Jjj^« corre^qionding to the points where it is touched by the plane ; f*' being
[m%]



considered as a function of these co-ordinates. Then this equation \\'ill give,
in like uuuuier, tlie four following equations ;t

mi] ^==(^) + ^-(S')'

[699^5] fi^'==0; X' = ay'+bZ'+c.

Hence we get a second Jundamental eq^tation between a, b, c. By means
[GD76] of this, and the first fundamental equation [6970], we obtain b and c in
functions of a. Substituting tliese functions in the two equations
[6956, 6962], nameljr,

we shall ha^e two equations between x^ y, s, &c. ; and by eliminating a,
[6979] we shall finally obtain an equation between or, y, r, which will be that of
8Mk««r the surface of the shadow. Tkis is the general solution of the problem for
^^« the determination of the shadoio of an opake body ; and the same solutifm
gims also the equation of the surface qf the penumbra ; for it is evident that
this surface is formed like that of the shadow, by tlie successive intersections



[cseo]



[6Si6dtt]



* (3513) Tlie equations [6969] ar« the same as those in [6963, 6958], changing

or, y, «, into X, Y, Z^ respectively, so as to correspond to the point of contact.

rfioisai ^""'"**^"5 ^» Y, Z, from the four equations [6967 — 6969]} we obtain the first

fundamental equaticMi in a, b, c ; corresponding to the plane which touches the luminous

body.

t (3514) The equations [6973—6975] are similar to those in [6967—6969], changbg
^ J (i, X, Y, Z, into |fc*, X\ Y, Z\ respectively ; so as to correspond to U^e sur&ce of
the opake body.



Vni. viii. ^26J. ON THE DURATION OP AN ECLIPSE. 211

of the planes which touch the surfaces of the luminous and opake bodies ;
with this difference, that in the case of the shadow^ we must consider the
intersection of the planes which touch these surfaces^ on the same side ; whereas [6081]
in the penumbra^ we must consider the intersections of the planes which touch r»««mbi«.
these surfaces upon the opposite sides* We shall now apply this solution to
the shadow of Jupiter*

In the frst place^ we shall suppose Jupiter and the sun to he spherical ; »t***k*A

putting also, ^"^^^

R s= the sun's senni-diameter ; [6869]

R' s= the semi-diameter of Jupiter ; [6969^

D e= the distance of the centres of the sun and Jupiter. [6989*]

Then the orisf^in of the co-ordinates being placed at the sun^s centre^ we have, [6069*]
for the equation of the sun's surface, the expression,* X^-^-Y^-^-S^ — R^ = ; [6983]
so that we shall get, from [6963],

Hence we shall have the four following equations ;

X«4-y' + ^ — ^"-^O; [6968]

Z-fbX—0; [69871

JC — aY-fbZ + c, [<»88]

The three first equations give,t

^.(l+a»-f b») = iJ«; [8989]

and from the three last equations, we obtain,

X(l-|-a« + b^)-c, [em]

* (3515) The equation [6983] a similar to that in [IIM], changing /, «, y, », into
Bi X, y, Z, respectively. Putting this equal to the assumed value f* =» [6963], wo i^*^**J
get f* = X""|-r« + Z* — ii"; whence,

Substituting these in [6967, 6968], we get [6986, 6987]. The equations [6985, 6988] [6983e]
are the same as [6983, 6969] respectively.

t(3516) From [6986, 6987] we get y = - aX, Zat— bX; substituting these r^^ojj ,
in [6985], we get [6989] ; and by substituting the same values in [6988] we obtain the
value of c [6990].



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

Hence we deduce,*
[6991] 0^ = 72^.(1 +a2+b').

[6992] The equation of Jupiter's surface! is (X'— Z))^+Y'H^''— i^'" = 0; so

Equation i r

jupiL'. "*^t "O"^ [6971] we have,

tSj fsf = (X'—Dy+Y'^+Z''—R'\

From what has been said, we obtain the four following equations,!
[6994] (X'—Dy + Y'2 + Z'^—R'^ = ;

[6995] F + a.(X'— Z)) = 0;

[6996] Z' + b.(Z'— Z)) = ;

[6997] Z'— Z) = aF+bZ'+c— Z).

Hence we deduce, §
[6998] (c—Dy = i2". ( 1 + a^ + b^).



[6991a]
[6992a]



* (3517) Dividing the square of [6990] by [6989], and then multiplying the quotient
by R^, we get [6991].



f (3518) If X^, Y'j Z\ be the rectangular co-ordinates of Jupiter's surface, referred
to the centre of the planet as the origin of the co-ordinates, and R' its radius, we shall

[69926] have, for the equation of its surface, as in [6985], Xf â– \-Y'^ -^ Z'^^R'^^^; the
axes of X^, Y\ Z' , being parallel to those of X, Y, -ZT, respectively. If we refer these
co-ordinates to the centre of the sun, and put X' for the new co-ordinate, or the line
drawn from the sun's centre to the centre of the planet, and continued beyond it by the

[6992c] quantity X^, for positive values of the co-ordinate X^, we shall evidently have
X^=X' — D\ the co-ordinates Y' and Z' remaining unaltered. Substituting this
value of X^ in the preceding equation [69926], we get [6992], corresponding to |x' =
[6971], as in [6993J.

f (3519) The equation [6994] is the same as [6992] ; putting this equal to \ii, we
get [6993]; whose partial differentials, relative to X', Y', Z', are.

Substituting these in [6973, 6974], we get [6995, 6996] respectively. Subtracting D