Pierre Simon Laplace.

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and if we notice only its chief term in [7522 line 1], we shall have nearly

[70Glc] Z = 05 = fl.3^,43267.sin.(D-f 510,3787) nearly. Now dv is to dv,, as n to n— J/, as

VIIL viU.
^ 26] ON THE DURATION OF AN ECLIPSE. 226

therefore we shall have, very nearly,*

i^.sinAr, = «^— (l-j-p^.Z^— 2.(l-}-p7.sm.i?,.Z. -^ . [7062]

Hence we deduce,

(i+r')'.Z-'#

\/j7 + ('+e).fS.Jf-(.+P').fS.

If s he supposed to express the tangent of the latitude of the satellite, above [7064]
Jupiter^ s orbit, at the time of the conjunction [6033], we shall have very
nearly Z =zrs\^ and as r is nearly constant, the preceding equation
becomes.

is manifest from the definitions of », u,, n, M [6023c, </, 6022/, 602 1 z] ; and as ^ is

very small relative to n [6025/], </», may be considered as of the same order as dv ; [7061(f|

consequendy — - is of the same order as — . Substituting in this last expression the

value of Z [7081c], we find that -r- or -j— is of the same order as Z; and the same [7061el

dv ttv^

holds eood relative to -r-:: , &c.

• (3539) Taking the square of z [7061], and neglecting terms of the fourth order in

rfZ [7062a]

Z, »i, it becomes s^=Z^-\-2.sin.v^.Z. - — [- &c. Substituting this in [7060], we get

[7062], which is a quadratic equation in sin.c, , and its root gives sin.©, [7063] ; observing

that if we transpose the last term of the second member of [7062] into the first member, the

rfZ [70e2i]

equation becomes r^.sin.'rj -|-2.( 1 -f-p')*- s'm.«,. Z. — = a' — ( 1 + f')'- 2* 5 which may

evidenUy be put under the form,

+ ^s-^l =la+il+,').Z].\a-(l+',-)Z\; ' '

A-

neglecting terms of the order Z'tiZ*. Dividing this by r*, and taking the square root,
we obtain [7063].

t (3540) The expression of Z [7065] can be deduced from [6036], by neglecting
terms of the order x^. The orbit of the satellite being nearly circular, r is nearly constant ; ['Otwoj

dZ ds

and then the difierential of Z, relative to v^ , gives -r— = r. -— nearly. Substituting

these values in [7063], we get [7066]. The remarks in [7067, 7068], relative to the [70656]
agns, are evidently correct ; taking into consideration that the positive values of «, follow
after the conjunction.

VOL. IV. 67

226 MOTIONS OF THE SATELLITES OF JUPITER. [Mic. Cel.

P0661 sin.«, = -(i+p?. ^ ± v/^ 5 7 +(i+p')-^ \ ■ \^-(i+n-s I .

[7067] This formula, with the sign + prefixed to the radical, denotes the sine of the
arc described by the satellite, by means of its synodical motion, from the

[7068] conjunction to the emersion. With the sign — , it denotes the negative value
of the sine of the similar arc, from the immersion to the conjunction.
We shall put,*

* (3541) If we notice only the chief inequalities in — and y, relative to the satellite
m [6131,6132], and also the chief term of the elliptical values [669], we shall have,

[7071al L ^ i_e.cos.(n< + £— Tzr) "^^^^^ .cos.(2nt—2n't-4-2e—2e) ;

[70716 ] V = nt-\- s -f-2e.sin. {nt-\- s—a) -f ^2,—JV ' sin-C^^^— 2»'<4-2£— S/) ;

and if, for brevity, we put,
[7071c] X' == 2e.cos.(7i^+ £— -5!)+ . ""'•"'/,, .cos.(2n<— 2n'<+26— 2^ ;

we shall find that the expression [7071a] becomes,
[7071rf] L^i_iX'.

Moreover if we take the differential of [70716], and divide it by ndt, substituting X'
[7071c], we shall get [7071e], observing that we have very nearly ndt = {'2n — 2n').dt
[6151] ;
[7071c] ^==1+X'; or dv = ndt+ndt,X'.

[7071/] Now if we suppose, as in [6l01'J, that Mdt represents the sideral motion of Jupiter,
during the time dt, we shall have, from [6023c, <Z], dv= dv^-\-Mdt, nearly. Substituting
the value of c?t; [7071e], we obtain ndt.X' = dv^—{n—M).dt. Dividing by {n-M).dt,

[7071g-] we get — - . X'= — - — 1 ; so that if we put -. X' = X, it becomes as in

n — M {n—M).dt ^ n—M

r707Ul ['^0'71]- Lastly, as n — M is nearly equal to n [6025iJ, we shall have X' = X nearly;

and it is evident, from [7071c], that X', or X, must be a small quantity of the order e, &c.

From what has been said it appears, that if — contain a periodical term of the form
[7071t] ^^

— ^X [7071rf], — - will contain a term of the form +X [7071e]. A similar result
nat

[7071X;] holds good also with the satellites w', m", rn" . This is evident as it respects the elliptical
terms depending on e', e", e'", which are of the same form as those in [7071rt, 6] depending

on e. Moreover by comparing the coefficient of — ^ [6160], with that of -^ , deduced

from [6161], and observing that n — n' = n' nearly [6151], they become of the same
[7071 to] ^/^^/ ^^„

form. Also —^ [6164], and -;;- deduced from [6165], correspond in like manner.

VIII. viii. <§> 26.] ON THE DURATION OF AN ECLIPSE. 227

Symbols

r= the time employed by the satellite in describing ^^the greatest width « T,i,X

of the shadow, by means of the synodical motion ; UOQd]

t = the time of describing the angle Vy , by the synodical motion ; [7070]

^ = {v^J\i)At — ^'^ovdv,= (n-M).dL(li-X) ; [7071]

X being a very small quantity, a is the mean distance of the satellite from [7072]

Jupiter [6079] ; — is the sine of the angle under which the radius « [7073]

appears at the distance a. Putting this angle equal to j3, or,

sin.^ = — , [7074]

a

we shall have, very nearly,*

t = —^ -' . [7075]

If we substitute in this expression for Vi its sine, which differs but very
little from the arc [70616]; and for sin.i/i, the preceding value [7066] ;

also |3 = — nearly [7074] ; we shall have,

.= T.(l_X).J_(l+pr.-i.,^±^{f + (l+p).i-].^f-(l+p').^^5.

If we notice only the equation of the centre of the satellite, we shall have,
as in [7071c, dy e, A], nearly,

r = a.{] — iX}; or —=l-{-iX; [7078]

and it also follows, from the same articles [7071c, d, c, A], that this equation

* (3542) During the eclipse we may suppose the variation of v^ to be proportional to

dv V V [7075a]

t, so that — -i =: -^; hence [70711 becomes X= , b 1 J consequently,

at t ■" (n — M).t * "*

rp.. I -I -tri

or by multiplying the numerator and denominator by T; < = ^' m\t '^ ^"^ since

(n — M).T represents the synodical arc described by the satellite in the time T, which, [7075c]

by [7069, 7074], is equal to p, it becomes t = — '—— , as in [7075] ; or, as it may

p

be expressed very nearly, t = T.(l — X). — '— . Substituting the value of sin.v, [7066], [7075rf]
it becomes as in [7077], observing that — = a nearly [7076].

[7076]

[7077]

228 MOTIONS OF THE SATELLITES OF JUPITER. [M^c. Gel.

,-^««, holds good even when we include the chief inequalities of the satellite ;
[7079] ° 1 *

therefore we shall have, very nearly.

[7080]

t = T.{l-X).UlHr'j'£;±\X\l+'iX-^{lH)-j^^^^

[7081] If we put the whole duration of the eclipse equal to t\ we shall have,t

Symbol

[7^] t'=^2T.(\-X).\/^\+\X+{\+f').~\^.)\+hX-(.'i+t)-j\-
Hence we deduce, t

[7083] S=^^ ^ '

[7084]

This equation will serve to determine the arbitrary constant quantities
which enter into the expression of s, by selecting the observations of the
eclipses in which these quantities have the most influence.

The duration of these eclipses being one of the most important parts of
their theory, we shall examine particularly the preceding formulas for

* (^^'^^) Substituting in the radical expression of [7077], the value —=l-\-^X
nearly [7078], we get [7080].

r7081 1 ^ (3544) We shall suppose the time from the conjunction to the emersion to be t^ ; the
time from the immersion to the conjunction, considered as negative, is of the form — ^j,,

[70815] t^^ being positive. These values may be deduced from [7080], attending to the remarks in
[7067, 7068] ; hence we have,

[7081c] ,^ ==r.(l-X).^-(l+p')^.i-.^ + ^Jl+iX+(l+pO. j |.|l+iX-(l+p').|(|;

[7081rf] -^„=T.(l~X).^-(l+p02.-i-.£-^|l+iX+(l+pO.^S.|l+iX-(l+^^^^

[7081e] Subtracting the second of these expressions from the first, and putting t^-\-t^^ =t' [7081],
we obtain [7082].

X [3545] Connecting the two factors in the radical [7082], and neglecting X^, we get

^''^^1 <' = 2T.(1— X).\/l-f X— (l+p')2. ■^.

Squaring this we obtain,
[''0835] <'2^ 4T2.(1— X)— 472. (l_X)2.(l+p')2. ~ ;

\y^ ] hence — == ^^ -1____^ . Extracting the square root, and multiplying by ^, we
obtain [7083].

VIII. viii. ^26]. ON THE DURATION OF AN ECLIPSE. 229

computing them. The radius of the shadow a varies with the distances of

the satellites from Jupiter, and of Jupiter from the sun. Putting

D = D' — 6D for Jupiter's distance from the sun ; D' being the mean [7085]

distance; and supposing, as in [7078], r = «.(I — \X), we shall find that

the variations of « will be represented hy the following expression ;*

(l+p).i2'. \ \X— ^ \ , ^-^^^ = variable part of «.
JX is always very small in comparison with -jyA and this last quantity is [7087]

[7085']
[7086]

[7085a]
[70856]

* (3546) The values of D, r [7035, 7085'J, give,

Substituting this in [7048'], we get,

„ = (.+p).«'.jl-"-^|+(l+p).fi'.|iX-^].'-^.

Of the two terms contained in this value of a, the first is constant, and the second variable j
being the same as in [7086].

t (3547) If we put the values of D [6275, 7085] equal to each other, we shall get,
D'—W = D'—D'H.cos.{Mt-]-E—I)', whence ~ = H.cos.{Mt-{-E—I) ; [7086a]

and as H= 0,048... [6882i], -— is of the order 0,048. On the other hand, the values

[70866]

ly
of ^X, corresponding to the satellites wi, m', m", m'", are found in [7527,7488,7432,7377]
to be respectively of the order 0,0039; 0,0093 ; 0,0013; 0,0072; which are much less [7086c]

W
than the preceding value of — , as is observed in [7087] ; and the author has therefore

neglected the term of [7086], depending on ^X. To estimate roughly the value of this rynggji
neglected term, we may observe, that the chief term of a [7048^], namely (I-f-p)-R', is

to the term of [7086] depending on JT, as 1 to ^X.-^—^.-^^ and by substituting the [7086c]
values ' =9 [7045g], — = ^1^ [7045i, &c.], this ratio becomes nearly as 1 to

^6,01. X Now if we take even the greatest of the preceding values of ^X= 0,0093 [7086/*]
[7086c], we shall have 0,01. X= 0,000186; and its actual value is generally much less.
Multiplying this by the semi-durations of the eclipses [7562 — 7565], we get the greatest [7086^]
effect of this neglected term in the times of these eclipses ; and this correction very rarely
amounts to a second of time of the centesimal division. Hence we see that we may safely
neglect the term depending on X^ in [7086] ; and if we substitute for (l-fp).il' its value [7086/i]

a nearly [7048'j, the remaining term of [7086] becomes — ^a. ~ .—,. — ; and
VOL. IV. 68

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

[7087'] equal to H.cos.(Mt+E—I) [7086a] ; therefore the variation of a is very
nearly represented by the following expression,

_„. ilz^.^. H.cos.{Mt+E—I) = variable part of «.

[7088]

\ 1>
Hence it follows, that in the preceding formulas we must substitute, for

— , the following function,

[7089]

-A I— ^-^"Y^ • ^ • H.COS.(Mt + E—I) ] . fcorrected valueof -1

[70901 ^' *^ ^^^'^ function, corresponds to the mean motions and mean distances of

the satellite from Jupiter, and of Jupiter from the sun.

T denotes the time that the satellite employs in describing half the width
[7090'] ^^ ^^^ shadow a [7069] ; this time increases in consequence of the variations

of «, by the quantity,

[''^^1 —T. ^^^—Sl . « . H.COS.(Mt-tE—I) ; [nrst increment of r]

A, J-f

and it increases, because the synodical motion, in the time dt, is very nearly
equal to,*
[7092] (n-M).dt. ^ l+X— ^^ . H.cos.(Mti-E—I) \ ;

by substituting the value of —7 [7086a], it becomes as in [7088]. So that if we use the
[7086t] mean values of a, p, we ought to change a into a — a. ~ . -^ .H.cos.{Mt-\-E — /) ;

Aj U

therefore — must be changed into the factor [7089]. Now as T [7069] denotes the

[7086A;] i\f^Q of describing the arch a, this lime will be increased in consequence of the variable
part [7088], by a quantity which is proportional to that variable part, as in [7091].

* (.3548) The longitude of the sun, as seen from Jupiter, is Mi-{-E\-2H.sh.(Mt-\-E-I)

[7091a] |-6ioi, 6102, 6882a]. Its differential, divided by dt, is Mi-2HM.cos.{Mi-\-E—I);

which must be substituted for M, in the expression of the synodical motion

dv^ = {n — M).dt.{l-}-X) [7071, 7055], corresponding to the time dt, and by (his means

it becomes,

[7091c] \n-M-2HM.cos.{Mti-E-I)].dt.{l^T) = {n-M).dt. i l-\-X-^^.H.6os.{Mt-{-E-I) I

nearly, as in [7092] ; which is to be used instead of {n—J\l).dt, in the preceding
[7091<fl calculations. Now as the time T [7039] is inversely as the synodical motion, it will be

[7091«] represented by the mean value of T, divided by ] 1-j-X -.H.cos.{Mi-{-E—I) i ,

VIII. vili. § 26.] ON THE DURATION OF AN ECLIPSE. 231

[7093]

which gives for the increment of T, depending upon this cause,

T. } ^^— ^.iy.COS.(M/+i: — /)— Z S ; [709 le] [second increment of r]

Therefore by neglecting Z, » as we have done above, we shall find, that by the r^^gg,,
combined effect of both these causes, T changes into,

T. < 1 "f ( jT, • ^7 ] 'H. COS. (Mt-\-E /) > . [corrected value of tJ

but these two causes have no sensible effect, except upon the eclipses of the
fourth satellite [709 lo-].

In the beginning and end of these eclipses, the quantities of the order s'^,
which we have neglected in the radical part of the expression of t,
[7063,7075, &c.], may become sensible [7383/, &:c.] But the only one which

[7094]

[7094'1

s'h

has any influence is the square of* (1-fp')^ — — , which must be added to [7095]

[7091/]

which augments the value of T, by the two terms given in [7093] nearly. Increasing T

by the terms in [7091, 7093], and neglecting X, as in tlie last note, it becomes as in

[7094]. The effect of this correction is, in general, insensible ; and even in the eclipses of

the fourth satellite it is hardly deserving of notice, as is shown in [7379], where the [709Jg-]

correction of T is found to be of the order 0,0006101. T, at its maximum; and if we

suppose T to be equal to 9890", as in [7565], it will never amount to 6 seconds of the '■ -■

centesimal division, and in general will be much less.

* (3549) In finding sin.v. [7063], we have neglected the term ( ^ ) t7095a]

under the radical, and by substituting — =s [7065], =— nearly, it becomes

** '"'^"^ '''> [7095a']

C,, , ,,„srfs)2 ,.,.,, T.(\—X) , . -.

l(l~rp) • 'j — 1; • I-'J's IS to be multiplied by to produce the correspondmg part

of t [7075i, 7077, or 7080] ; or, in other viords, the function under the radical in

[7077, 7080], must be multiplied by -^ ; by this means the neglected term in [7077,7080] [70956]

1 C sis ) 2

becomes — . ] (1-f-p')^- — ( ? as in [7095] neaily ; and for brevity we may represent it

by (5X \( we neglect the part depending on p', on account of itssmallness, we shall have [7095c]

s^ds^
&JC= „ — . Now the product of the two factors of the radical in [7080, or 7082] is \70U5d]

nearly l-\-X — (l-j-^jS. — . j and if we increase this by the neglected terra 6X, it will be

^ [7095c J

the same as to change X into X-\-SX, under the radical; in [7080, 7082], as is remarked

in 17095,7096].

232 MOTIONS OF THE SATELLITES OF JUPITER. [Mcc. C61.

the quantity contaioed under the radical. We may notice this by
[7096] augmenting X, by the quantity .^ ^ , under the radical in the expression

of t' [7082], and decreasing* X by the same quantity, under the radical of
'■ ^ the expression of s [7083].

We have confounded the arc v^ with its sine [7076] ; now we have very

nearly, as in [70616], f
[7097] i;^ = sin.ri + -i.sin.^??i ;

* (3550) The author, in the original work, says, that in the expression of s [7083]
*■ ^ vie mnsi increase -^ under the radical, by the quantity ^Jf [7095c?J ; but this is not accurate,

and we have corrected the mistake in [7096']. For if we multiply together the two factors
under the radical in [7082], neglecting X^^ and then increase X by ^X, under the
radical, as in [7095e], we get the expression of t' [7096c]. If we now introduce the
external factor 1 — X, under the radical, it becomes as in [7096c?j ; always neglecting
terms of the order X"^, or X^X. Squaring [7096c?], and reducing, we obtain the value
of s [7096e] ;

[70966]

[709ac]
[7096d]
[7096e]

^.s/\AT^.{\ — X-\-8X)-t'^

2r.(i+p').(i-^

If we now compare the values of s [7083, 7096e], we see that — X must be changed
[7096/"] into — X-\-bX, under the radical of the expression of s [7083], to obtain the corrected

value [7096e] ; or, in other words, X must be decreased by — 6X^ in that part of the
[7096g-] expression. This agrees wiih the corrected translation given in [7096'], but differs from the

last paragraph of the original work.

t (3551) Tlie expression of t [7075] is proportional to — ; and if we substitute in it

the value of v^ [7097], and the similar expression of ^ =sin.^-}-i.sin.^p, we get very
nearly, by development,

[709/ i] _L = _ — \ \ \ , . ^ = - — -.n+i.sin.^v^— ^.sin.2^i
3 sin.|3 l-f-^.sin.2^ sin.(3 ' ' ** * " '^ >

^''^^^ 1 = ^3 ' 5 1+tV.(1-cos.2«J-tV.(1-cos.2p) } = "j^- 1 1-^.cos.2^+tVcos.2p j .

.3 t . x^ V 1/ 1^ V -/, gin_p

sin.tj,

[7097t/]

Now in [7076] we have used . ' ^ for — , in the chief term of t, which does not contain

s ; and to correct this, we must multiply it by the factor 1 — T2 - cos.2Di-fY2-cos.2p nearly
[7097e] [7097c], as is observed in [7101] ; we must also use the same factor in the expression of

VIII. viii. <§>26.]

ON THE DURATION OF AN ECLIPSE.

233

therefore the preceding value of t' must be multiplied by 1 +i'Sin.^?;j.
Relative to the first suellite, Vy is about ten degrees [7104, 7554] ; and this
renders the product of /', by -i.sin.^iJj, sensible. But this error is corrected,
in a great measure, by the supposition we .have made in [7076], that

_ = ^. For we have — = sin./3 [7074] ; we ought therefore to have

supposed — = /3 — -i-.|3=' [43] Int. ; which amounts to nearly the same thing

[709,8]

[7099]

as to multiply the value of t' by l-|.sin.^/3, because the term — ^ '^^J ' ^

contained under the radical in the expression of t' [7096^], being a small
fraction in the theory of the first satellite, we may neglect its product by
-i.sin.-|3 [7097O-], The value t', determined by the preceding formula, must
therefore be multiplied by l+-i.sin.^i;i-|.sin.^f3, or by l-y^^.cos.22;i+yi^.cos.2|3
[7097c]. The arc Vi differs but little from |3 relative to the first satellite,
so that the product of i' by yi^.(cos.2i;i — cos.2j3) is insensible.* [7102]

[7100]

[7100']
[7101]

t' [7082]. We may finally observe that the correction relative to /3 [70976], produces a
factor 1 — ^ sin.2|3 without the radical in [7077] ; or very nearly 1 — ^.sin.'*^, under the
radical ; but this ought not to be applied to the term depending on 5, mentioned in
[7097 f/], because the value of ^ [7076] was not introduced into this part of [7077] ; but
as tliis term is very small, we may neglect the effect of the correction arising from its
multiplication by ^.sin.^|3, as in [7100'].

* (3552) If the satellite pass through the centre of the shadow, it will describe the arch
2^ [7076], during its passage; but when the latitude is large, the described arc, which we
may call 2v^, must be less. To estimate roughly the effect of X, ^ [7527, 7529],
corresponding to the first satellite, upon the time t', we shall observe, that X [7527] being
much smaller than (^[7529], we may neglect it; and then the expression of «' [7531]
becomes ^' = 9426"./!=^; but when 5 = 0, the value ^ [7528] vanishes, and t'
becomes 9426". The arch described in the first case corresponds to 2v^ nearly ; and in
the second case, to 2^ [7102«]. Hence 2^:21?^:: I : ^1—t^ nearly; and if we put (3
equal to q=l lo,1780 [7104,7554], we shall get 2^ = 22^,3560, and 2i;,=22°,3560.\/r:>2
nearly. The least value of 2Uj corresponds to the greatest value of 2,, and this is nearly
^=7 [7529]; therefore the least value of 2v^ is 2Uj = 22^,3560./! = 21^,0774.
Hence we get cos.2yj — cos. 2^ =0,0067 and t'2.(cos.2w, — cos. 2^)=: 0,0005 at its
maximum. Multiplying this by 9426" [7102c], we get the greatest error in the value of
i', corresponding lo the first satellite, which will not therefore exceed 5", and is generally
much less. For the second satellite we have, in [7558], 2(3 = 2y== 13^,9692, and

VOL. IV. 59

[7097/]
[7097g]

[7102a]

[71026]

[7102c]
[7102rf]

[7I02e]
[7102/]

234

MOTIONS OF THE SATELLITES OF JUPITER.

[Mec. C61.

[7102']

[7103]

[7104]

?
[7105]

[7106]

[7107]
[7108]
[7109]
[7110]

The value of T, determined by a very great number of eclipses, will give
the mean distance of the satellite from the centre of Jupiter, in parts of the
diameter of the equator of this planet, supposing the satellite to disappear

at the moment when its centre enters the shadow of Jupiter. For — [7073]

is the sine of the angle under which the half width of the shadi w is seen, from
Jupiter^s centre, when this planet is at its mean distance from the sun, and
the satellite at its mean distance from Jupiter. Now if we put this angle
equal to q, we shall have, from [7047, 7103, &c.],*

{1+P).R' C, (l-\)

j 1 0-\) A I

sin.(

The observed value of T ivill give that of the angle q, which is nothing more
than the corresponding arc described by the satellite, by means of its synodical
motion ; therefore ive shall have the four following equations ;

{l±p)^ S a'" (l-\)

a"' ' I a \

(l+p).il' (^fl- 0-\)

'I a' \ ' D' S

{1+P).R'

a" ( a" \

■[•-

a

w

= sm.q ;
sin.q' ;
= sin. 9" ;
= sin.a'".

[7102i]

[7102g-] 2v, = 13°,9692.v/r^; and as the greatest value of ^ [7491], is |, it becomes
2v^ = 13°,969I.v/f = 120,0977. Hence jV- (cos.Sy^— cos.9^) = 0,0005. IVIuhiplying

'■ ■' this by 11951'' [7493], we get the greatest possible error in the value of t', corresponding

to the second satellite, which does not exceed 6", and in general is much less. For the
third and fourth sateihtes, v^ is quite small ; so that if we put cos2yj=l, and use
^ = q", or q'" [7556, 7557], we shall find that the greatest error in t' will be, by

[7102ik] [7437,7385], 14838".yV.(l— cos.2/) = 11", and 19780".tV-(1— cos.2/')==5". Both
these expressions will be decreased by using the actual values of cos.2i;j ; and in general
these errors will be incomparably less than the limits here computed.

* (3553) The value of a [7048'] being divided by ff, putting also D' for D [7085],
[7107o] . „

and a for r, gives — = sin.g' [7104], corresponding to the first satellite, as in [7105].
[71076] ^ °

Changing the symbols «, q, relative to the first satellite, successively into a',q' \ a", q" ;

a'", q'", corresponding respectively to the second, third and fourth satellites, we obtain
[7107c] the equations [7107 — 7110], the forms being altered a httle by the introduction of the

[7111']

VIII. viii. ^^6.] ^ ' ON THE DURATION OF AN ECLIPSE. 235

a'"
Each of these four equations gives a value of 7—- — —-^, , that is, the value [7111]

of a'" in parts of the radius (l-{-f).R' of Jupiter's equator [7048] ; for there

a'"
is but little uncertainty in the ratio — , given by Pound's observations,

a'" a'" a'"
which are quoted by Newton [71076?]; and the ratios — , -— , — - , are

a a a'

well determined in [6797 — 6800]. The differences of these values of a'"
will make known the errors of the supposition that the satellites are eclipsed [7112]
at the moment of the entrance of their centres into the shadow. The
penumbra, the magnitude as well as the brightness of the disc, and the
refraction which the sun's rays may suffer in Jupiter's atmosphere, are [7112]
sources of error whose effect it will be difficult to ascertain with correctness.

quantities a, a', he. between the braces. Before closing this note we may remark, that

we have already mentioned, in [67876, &c.], that observations similar to those of Pound ['1"'"]

[7111'J, have been made with great accuracy by Professor Airy.

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

CHAPTER IX.

DETERMINATION OF THE MASSES OF THE SATELLITES AND THE OBLATENESS OF JUPITER.

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