# Electronic library

Pierre Simon Laplace.

# Mécanique céleste (Volume 4)

. (page 29 of 114)
Font size

[6612c] in [6610], (pinto — 9, putting 9 = — (v — 3v'-\-'2v") ; because this will merely change
the signs of both members of [6611], without altering the form or value of the equation ;
so that instead of the expression [6610], we may put, more generally,

9 = ± (v— 3r'-|-2t;") , or ± 9 = v—3v'-{-2v".
Hence we see that the sign of 9 is indeterminate in the differential equation [6611], and
'■ ^ the same occurs in its integral [6612, Sic]

[6612c]

[6612rf]

[6612c]
[6612/]

[6612g-]

VIII. vi. ^ 15.] INEQUAUTIES OF THE ORDER m>, mm', m'«, &c. 137

First, If the constant quantity c, neglecting its sign, exceed 2kn', it [6613]
must necessarily be />o*i/ire. Then the angle ±9 will increase indefinitely, -„,^

[ool3j

and may become equal to one, two, three, &c. semi-circumferences.

Second. If the constant quantity c, neglecting its sign, be less than [6614]
2kfi', k being positive, the radical y/c— 2kna.cos.9 will become imaginary, 2^
when ±9 is equal to nothing, or to one, two, &c. circumferences. In this ^■'^oo*
case the angle 9 mil merely oscillate about the semi-circumference, which [6615]
will represent its mean value.

Third. U the constant quantity c, neglecting its sign, be less than [66I6]
2kii', k being negative, the radical /c— ^kn^.cos.? will become imaginary, '"»|^«^
when db? is equal to an uneven number of semi-circumferences. In this
case the angle 9 will oscillate about zero, so that its mean value is nothing. ^

The case of c = ± 2kn^, may be supposed to be included in the
preceding forms. We may, moreover, consider the probability of this case
as infinitely small. We shall now see which of these cases really obtains,
in the system of these satellites.

We shall find, in [7270, 7272], that k is positive; therefore the third
ease [6616] must be excluded; and the angle 9 must either increase
indefinitely [6613'], or oscillate about the semi-circumference [6615]. If

Symbol

we put, w

9 = «'±«; [6(»0]

t being the semi-circumference whose radius is 1 ; we shall have,*

dt-= . ,'^ [6621]

[6618]

[6619]

[GHaUb]
[6633c]

♦ (3418) The equation [6621] is easily deduced from [6612], by the substitution of [6633o]
[6620]. Now if the angle -a^ increase indefinitely, the limits of c -f 2k/i'.cos.«^ will be
c-f-2kn*, c — 2kn'; and it is evident that c -|- 2kn'*.cos.w^ , will sometimes become
negative, unless c be positive and greater than 2kn'; but if c — 2kn'.cos.ra^ be
negative, the denominator of the expression of dt [6621], will become imaginary, and w^
will not increase with t. Hence we evidently see that, if vt^ increase indefinitely with
t, we shall have c positive and greater than 2kn^, as in [6621'] ; consequently the
radical v^c-f-3kni).co«.f3, , will exceed v/3kn« or n.^/Sic, while «^ varies from 0° to [6633</]

KXF; and within these limits we shall always have, from [6621], dt<^—y^, whose

intesnl gives <<^j;^ [6622]. Putting w, = i«', we get <<^jj^, asin[6623];
bcoee we infer, as in [6624], that «^ or 9 [6620], does not indefinitely increase, so that
TOL. IT. 35

[6633e]

138

MOTIONS OF THE SATELLITES OF JUPITER.

[Mec. Cel.

[6621'1

[6622]

[6623]
[6624]

[6625]

Remarka-
ble proof
of the
mutual
attraction

of
Jupiter 'a
latellites.

[6626J
[6627]

If the angles ±9 and «^ increase indefinitely, c will he positive, and
greater than 2kn^ ; hence we shall have, in the interval comprised between

7tf^ = 0, and OT equal to a quarter of the circumference, dt<i — y^,

consequently t < — ^ . Therefore the time t, which is required for the

angle «^ to increase to a quarter of the circumference, will be less than

— y=. We shall see, in [7274], that this time is less than two years.

Now since the discovery of the satellites, the angle zs^ has always appeared
to be insensible, or extremely small ; so that it does not increase indefinitely
[6613']. Therefore it must oscillate about zero, making its mean value equal
to nothing [6622g-]. This is confirmed by observation, and furnishes a new
and remarkable proof of the mutual attraction of Jupiter^ s satellites.

Hence we may deduce several important consequences. The following
equation, which is deduced from [6610, 6620],

v—3v'-ir2v" = -^ziz^^,
gives, by putting the quantities, which are not periodical, separately equal
to nothing,*

nt—3n't-{-2n"t-{- s_3s'+2s" = *.

[6622/] it does not come under the form in case 1, [6613'] ; and we have seen that it does not
come under case 3, [6619] ; therefore we are restricted to case 2, [6615], in which the
^ ^■' mean value of cp is equal to * ; or, in other words, zi^ [6620] is equal to zero, as in
[6622/1] [6624], and c<2kn2 [6614].

[6628a]
[6628&]

[66286']

[6628c]
[6628d]
[6628e]

[6628/]

* (3419) The longitudes v, v', v", are composed of the mean values [6572'], the
terms depending on the libration [6652 — 6654], and the periodical terms. Now if we
neglect these periodical terms, and substitute the others, in the first member of [6626] ;
then put the terms depending on the libration in the first member equal ± «^ , as in
[6620], we shall get, from the remaining terms, the equation [6627]. This equation may
be put under the form [nt—v^t-^-z — /) — 2.{n't — n"^ + e' — s") = ir, which is used
hereafter. This equation holds good for all values of t ; and if we put ^ = 0, it becomes
g — 3s'-j-2s"=='r, as in [6639]. Subtracting this from [6627], we get, for all values of
t, nt — 3n'i-{-'2n"t = 0; dividing this by t, we obtain the equation [6628]. The
equation [6628d] may be put under the form ni-{-'2n"t = Sn't, as in [6629] ; and the
equation [6627] corresponds to the theorem in [6630]. The first member of [6627] may
be put under the form 2.{7i"t — n^+s" — s) — 3.{n't — nt-\-^ — s) ; and by substituting the
values [6240/?], it becomes 2.(0" — 0) — 3.(0' — 0) =0— 3e'+20" ; hence the equation

VUI. vi. ^ 15.] INEQUAUTIES OF THE ORDER m», mm', m'», &c. 139

Hence we deduce,

n-^Sn' 4- 2n" = 0. [6628]

From these equations we obtain the following important results. First. Jj'jr^
The mean motion of the first satellite^ plus twice that of the third, is exactly
equal to three times that of the second. Second. The mean longitude of the
first, minus three times that of the second, plus twice that of the third, is always ^ ^
exactly equal to two right angles. The same result holds good relatively to t^^^^l
the mean synodical longitudes. For in the equation,

nt^Sn't + 2n"t + s —3/ + 2s" = * [6627] , [6631]

we may refer the angles to an axis moving according to any law, since the
position of that axis vanishes from this equation ; therefore we may suppose ^

that nt-\-s, n't-\-^, n"t-\-^', denote the mean synodical longitudes.*

Hence it follows that the three first satellites cannot all be eclipsed at the The thm
same time. For nt-{-s, n't-\-fl, n"t-\-^', being supposed to express the

fim
MtellitM
cannot be
eclipted
At tba

[6627] may be put under the following form, which is used hereafter, in [7391, &c.] ;

e— 3e'-f 2e" = * = 200°. [6628^]

We may remark that, in substituting the numerical values of e, e', e", in the first
member of [6628jf], it may be necessary to add or subtract a circumference 400°, or a L"®*^!
multiple of 400°, to make it equal to the second member 200°. This will be evident
by using the values of e, e, e", given in [7496, 7440, 7386], which, by neglecting the [6628t]
ninutes and seconds, are nearly expressed by

© = 16° + /.82583° ; e' = 346° + ^41141° ; e" = ll°-f ^ 20420° ; [6628A]

being the number of Julian years elapsed since the epoch of 1750 [7284]. Now if
= 0, we shall have e=16°, e' = 346°, e"=ll°; substituting these in the first [^>®28Z]
mber of [6628^], it becomes 16°— 1038° + 22° = —1000° ; which, by adding three
circumferences, or 1200°, becomes equal to the second member of [6628jgf] . Again, if
we suppose ^ = 0,002, the preceding values [6628it] become e = 181°,16; [(jcasml
^ = 28°,28; e" = 5l°,84; hence e-3e'-f 2e"= 181°,16-84°,84-|-103°,68=200o,
as io [6628^]. Lastly, if < = 0,005, the expressions [6628A:] are very nearly represented ^ "^
by e=28S9j e'=151°,7; e"==113°,l ; consequently,

o— 3e'+ 2e" = 28°,9— 455°,1 -|- 226°,2 = —200°. [6628o]

Hence it is evident that the second member of [6628^] may have the sign dc ; or if wo
put t for an integral number, positive or negative, including zero, the equation [662dg']
may be more generally expressed in the following manner ;

e— 3e'-l-2e'' = 200°-f 400°. t . [6628j]

[6628p]

•(3420) This is fully demonstrated in [1240", 1240a, 6], and it is unnecessary to r^j^jj^^-i
make any further remarks on this subject.

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

[6633] mean synodical motions, we shall have, in the simultaneous eclipses of the

first and second satellites, nt + s and n't -f- s' equal to * ; so that the

equation [6631] becomes,
[6634] 2/1"^ + 2s" = 3*;

and then the mean longitude of the third satellite will be equal to ^tf.
In the simultaneous eclipses of the first and third satellites, nt + s and

n"t-\-z" are equal to *; and [6631] becomes,
[6635] 3?i'^ + 3£' = 2* ;

so that the mean synodical longitude of the second satellite is f ■«'. Lastly,

in the simultaneous eclipses of the second and third satellites, n't + s' and

nl't-^-z" are equal to •»• ; hence [6631] gives,
[6636] nt-\-s==2'!f.

The mean synodical longitude of the first satellite is then nothing, and

[6636'] instead of being eclipsed, it may produce upon Jupiter an eclipse of the sun.

We have seen, in [6160, 6161], that the two principal inequalities of the

second satellite, which are produced by the action of the first and third
[6637] satellites, are, by means of the preceding theorems, reduced to one single

term, producing the great inequality, discovered by observation, in the motion
- of the second satellite. Therefore these two inequalities will always be

united, and there is no fear that they will be separated after the lapse of

many ages.

Were it not for the mutual action of the satellites, the tivo equations

[6628, 6628c],
[6638] n—3n'+2n" == ;

[6639] £ — 3/ + 2s" = * ;

would have no connection with each other. We must therefore suppose, at the
origin of the motion, that the epochs and mean motions of the satellites have
been so arranged as to satisfy these equations, ivhich is extremely improbable ;
and even in this case, the smallest force, such as the attraction of the planets

[6639'] ' . .

and comets, would finally produce a change in these relations. But the
reciprocal action of the satellites removes this objection, and gives stability to
the preceding relations. For, by what has been said, we have, at the origin
of the motion,*

[6640a]

*(3421) The differentials of [6620,6610] give ±d7S, = d(p== dv-^dv'-\-2dv" ;
and from [6621, 6626], we have,

VIII. vi. ^ 15.] INEQUALITIES OF THE ORDER •••, «»', «", iic. Ul

^ ^^ 'iidt'^^'^^^'^V^ -2k.cos.(.-3.'+20 ; [6640]

c being less than 2kn' [6622A] ; therefore, to render the preceding [6641]
theorems accurate, it is only necessary to have, at the origin of the motion,

the function — r r H t- » comprised between the limits,* *- *

ndt ndt ndt '^

+ 2k*.sin.(Ji— |s'+«") [6649]

— 2k*.sin.(Je— Is' + O; ' [6643]

and it will suffice to ensure the stability of the system, if the foreign
attractions should always leave the preceding function within these limits.

We know, by observation, that the angle n is very small ; therefore we
may suppose cos.a = 1 — ^-a^ [44] Int. We shall now put.

[6644]

p^ being arbitrary, because it contains the arbitrary constant quantity c ; [6645']
then the diflferential equation [6621] gives,t

dtrf«, = ±</t\/{c-j-2kn».ros.(r— 3»'+2t/'— *)! =±dt.^{c—2knKcos.{v^3i/-\-2v")\, [66404]
Putting these two expressions of ± dra^ equal to each other, and then dividing by ndt^
we get,

^- ^+^='^^U-^^'''^'('-^'-^^' ) \ ' [6640c]

and at the origin of the motion, when < =s 0, we have c = s, t)' = /, v" = s" [eSTS'J ; [6640<f]
hence it becomes as in [6640].

• (3422) Putting, for brevity, the function [6641'], or the first member of [6640]
equal to <ff, we shall obtain [6641c], by taking the square of the equation [6640]. Now *■ •'
we have cos.(s— 3s'4-2«") = 1 — 2.sin."(Js — Js'-f-s")* f'^^"™ [IJ ^"'- Substituting this in [66416]
[6641c], we obtain [664U].

9" = -^ — 2k.cos.(^-3«'4-20 [6641c]

n3
c— 2kna

f 4k.sin.«(Ji— |i'4-i") ; [6641rf]

and since c — 2kn* is negative [6641], we shall have 9'*<^4k.sin."(Ji — J«'-f-«")» or [6641e]
^'<2k».sin.(Ji — ^Z+i")* " »n [6642, 6643].

t (3423) The equation [6646] is deduced from [6621], in the same manner as [1239] [6646a]
b derived from [1238]; observing, that by changing ^ into k, and « into «^, in [66466]
[1838], it becomes as in [6621] ; and by making the same changes in [1239], we get

VOL. IV. 36

142 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Ca.

[6646] vs^==^^.sm.(nt.^k + A);

A being another arbitrary constant quantity.

The motions of the four satellites of Jupiter being determined by twelve

[6647] differential equations of the second order,* their theory ought to contain
twenty-four arbitrary constant quantities. Four of these correspond to the
mean motions of the satellites, or, in other words, to their mean distances ;

[6647'J four correspond to the epochs of the mean longitudes ; eight depend on the
excentricities and the aphelia ; and eight others upon the inclinations and
nodes of the orbits. The preceding theorems [6638, 6639] establish two
relations between the mean motions and the epochs of the mean longitudes

[6648] of the three first satellites, which reduces these twenty-four arbitrary
quantities to twenty-two. To supply this deficiency the two new arbitrary
quantities |3^, A, are introduced into the expression of «^ [6646].

If we resume the equation [6586] and substitute in it the following
expression, which is deduced from [6610, 6620, 6646],t

[6649] v—Sv'-{-2v" = ^ + ^^ ,sm.(nt.\/k + A) ;

[6646c]

'a^ = 'k.sm.(nt.\/k-\-y) } which is of the same form as [6646] ; the constant quantities
X, 7j being changed into ^^, A, respectively. We may incidentally remark, that the
symbols ^^, w^, [6646] are not accented in the original work ; we have accented them to
distinguish them from p, w [6023n, 6024g].

* (3424) Each of the four satellites furnishes three equations of the second order,
^ "^ similar to those in [6057, 6060, 6077] ; making in all twelve equations of the second order ;
and as the complete integral of an equation of the second order, introduces two arbitrary
constant quantities, we shall have in all twenty-four arbitrary constant quantities, as
in [6647].

[66476]

t (3425) Substituting the computed vakie of -a^ [6646], in the expression of 9

[6610, 6620], using also ^^ for zh^,, we get, as in [6649],
[6650a] (p = v — 3t?'-{- 2u" = ir -j- p^. sm.{nt.\/k. + A) .

The sine of this expression gives,
[66506] sin.(«— 3y'4-2tj")=— sin.{^^.sin.(n^.V/k-|-^)} =— ^,.sin.(n<.\/k + ^) [43] Int.,

neglecting terms of the order ^^, on account of their smallness [6658]. Substituting

this in [6586], we get [6650] ; multiplying this by dt, integrating and then adding a

(6650c] constant quantity n^, we obtain the value of — . Again, multiplying by di, and

integrating, adding the constant quantity e^, we obtain v = n^t-\-s^-\-([inciion [6651].
[6650d] ]>fow we may suppose e^ to be included in the epoch ?, and n^t in the mean motion

VHI vl. ^ 15] mEQUALlTlES OP THE ORDER m», mm', m'», &c. 143

we shaJl obtain,

^ = 8.(|,^n - .^-) • 7 • ft-«»n-KV^ + ^)- t6650]

integrating this expression, and neglecting the arbitrary constant quantities
which form part of the epoch and the mean longitude, we get,

Sm'.m'.n.F'G a ^ - , . j- , ^.
^ = — 8k.(n-» - A0 * ? * '''•^'"•(''^•V^ + ^) ; [6651]

and by substituting the value of 1^, we shall have,

^ ^ fi^,s\n.(nt^ -{- A)
" 1 4- ?flf! -I- —:I!L ' [6658]

In like manner we shall find,*

3o .III .

^' — ov„ — ^ — ; [«««)

1 , va.tn . a.m

4a.TO' 4o .m"
a".m

„ Ba.ni

77.^,.sin.(n^.v/k4-^)

- 9tf'.TO n".in

[6654]

4a.»» 4a.in"

Therefore the three first satellites are subjected to an inequality depending on
the angle nt.\/v. + A, It is by observation alone that we can determine the
limit of the arbitrary quantity ^^, and the time when this inequality

«< ; and we may therefore neglect the constant quantities n^, s^, and we shall have

• = function [6651]. Now from [6609] we have, ^^^^'^

Bubstituting this in [6651], we get [6652].

• (3426) Multiplying the second member of [6586] by — ~-, , it becomes equal to

ddi/ 3af.m ddv - [6654a]

the second member of [6602] ; hence we gel — =» — -z — , . ^ . Integratmg and

"I'.m t66546)

neglecting the constant quantities as in [6650e] , we get r' = — t , . v ; and by substituting

the value of r [6652], we obtain [6653]. In like manner, by multiplying [6586]

tTj^ ddv" a^.m ddv

by ui^ , it becomes equal to [6608] ; hence — = ^^, . — ; whose integral gives [6e54e]

»*-» r-^,. V ; and by substituting [6652], we get [6654].

[6657]

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

vanishes, which depends upon the value of A. This inequality deserves the

particular attention of astronomers. JVe may consider it as a lihration of

the mean motions of the three first satellites. By means of this libration the

[6656] difference of the mean motions of the first and second satellites, minus twice

Libration ^f^^ diffcrencc of the mean motions of the second and third satellites, oscillates

■aumtet. always about two right angles [66286']. For this reason we shall designate

this inequality by the name of the libration of Jupiter\s satellites. It has

considerable analogy with the libration of the lunar spheroid, whose analysis

we have given in [3460 — 3470] ; and in like manner as in the lunar theory

it replaces two arbitrary constant terms of the mean longitudes. Moreover,

[6658] this libration, as in the lunar theory, is insensible ; and this arises from

circumstances depending upon the primitive motions of the satellites.

The radii vectores of the three satellites are subjected to the same

[6658'] inequality ; for they produce, in the expression of the mean motion fndt,

the inequality,

^^.sin.fo/.y/k-F^) . r^^K^-,
[6659] • . 9a'.m . a".m ' L^^^^J

[6659q

I -I- +

^4a.m' ^ 4a. m"

which gives in n, considered as variable, the following inequality, which
we shall represent by 6n ;*

^^_n.^,.y/k.cos.(nt.j/k±A) .

[6660] ^ 9a'.m a".m

Aa.ruk 4a.m'^

__2

[6661] and since a= n"^ [6110], we shall have,

[6662c]

* (3427) Putting nj for the mean value of fndt, and v for the quantity [6652],
we shall have, for the mean motion [6658'J, including the libration, the expression

dv
[66626] fndt^=n,t-\-v. Its differential being divided by dt, gives n =: n,-\- — ; the quantity

— [6652], being the same as the second member of [6660], or the value of Sn ; hence
dt

_2 _2 _5

n=n^-|-^»' Substituting this in [6661], we get a={n^-{-8n) ^=n^ ^ — f.w^ ^.8n,

5

neglecting the higher powers of §n, so that a contains the inequality — f.n ^.5n,
which we shall represent by Sa ; and we shall have very nearly,

Sa = — f .n"^. 5n = — ^.rT^. — = — f .a. — ,

as in [6662]. In like manner we shall have, for the satellites mf, m",

[6662J]
[6662«1

oftlM

rkdioa
vactor.

[6663]

[6665]

VIII. vi. ^ 16.] INEQUALITIES OF THE ORDER m«, mm\ m'\ &c. 145

to = — |.a.-; [«662]

M LibfaUon

which is the variation of the radius vector r, depending on the expression
[6659]. We may obtain, in the same way, the corresponding variations of
f' and r" [6662<f].

16. The libration of the three first satellites of Jupiter, modifies all their

inequalities of a long period. It gives to their expressions a particular form , ^^^

which connects them together, and is a singular case of the analysis of 1"^^^

perturbations. If we suppose that x^ .sin. (ii -[-<')> is an inequality of a long "«'|*»wm.

period of the satellite *w, which would take place if it were not modified ^ ^

by the action of the two satellites m\ wi", &c. ; and that x'.sin.(i^-f o)

and x".sin.(i^-f o)» are the corresponding inequalities of the satellites m'

and m" ; we shall have, by noticing only these inequalities,*

ddv .„ • /'rf t \

-^ = — r.\.sm.(i^-fo) ; [6666]

ddv -a % / • /'4 , \

_ = — l". X .sin. (it + O) ; [6667]

ddv" 'O ^ // • /'J I \

— = — l^X .sm. (1/ -f- o) . [6668]

But by noticing only the inequality of libration, we have as in [6586, 6650/^ ,t

ddv k.n«.sin.(t;— 3t;^-f2i;^0
"dF ~~ I . 9a'. m a".m ' [6669]

Connecting together the two expressions [6666, 6669], we shall have,
ddv kn«.sin.(»— 3«;'-f2c") ._ , . ,. . v
^ ^ 1 , 9«->» , -"'^ -i'.\.«n.(,l+o). ^^^j

We shall suppose that

»= Q.sin.(i/-f o) ; t/ = Q'.sin.(i/+o) ; v"= Q".sin.(i/-f-o) ; [6671]

represent the inequalities of r, \/, v'\ depending on i/-f o, and modified

• (3428) Taking the second differentials of the assumed values of the terms of r, t>', c", r6666ol
[6664, 6665], and dividing them by rfl*, we obtain the expressions {^56G& — QG6&\.

t (3429) Dividing the expression [6586] by that in [6650/], and multiplying the result
by k, we get [6669]. Adding this to [6666], we obtain [6670J. {(X,7(ki}

▼OL. IV. 37

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

by the reciprocal action of the satellites ; then putting,*

[6672] ^_3i,'_|_2zj" = ^ + (Q_3g+2Q").sin.(i^-|-o) ;

we shall have,

[6673] ^ ) 1 •^+ ^ 9a-.m a'^nT \ ' Sm-O^ + o).

( 4a.m' 4a.»»" )

Substituting, in the first member, the value of v [6671], we get,t

'a 4a.m' ~^ 4a.m")

In like manner we shall find,t

6675] q = V—

2- ■ »«'•- • «'-"»N '

1

a".7n.

[6676] Q" = X/' +

(9a'.m a .m \

"^ 4a. w' "T" 4o.mV

V ~ 4a.m' ' 4o.m7

[6672a]

* (3430) The object of the present calculation is to notice only angles of the form
if+o, where i is very small, as in [6664, 6672] ; and this expression [6672] is found,
by substituting in v — 3w'-}-2v" the values [6671], and adding to the result the mean
value * of the function v — 3«'+2v" [6626]. The sine of this expression of v — ^v'-\-2v"
gives,
[66726] sin.(«— 3i;'+2«'')=— sin.{(Q— 3Q'+2Q").sin.(i<+o)| =— (Q-3Q'4-2Q^).sin.(i^+o) ;
neglecting the higher powers of (Q — 3Q'-f~'^^') on account of its smallness.
Substituting this in [6670], it becomes as in [6673].

f (3431) The assumed value of v [6671] gives, by taking its second differential,
- = — i^Q.sin.(i^+o) j substituting
by — i^.sin.(ii-j-o)5 we get [6674].

[6674a] -— = — i^Q.sin.(i^+o) j substituting this in the first member of [6673], then dividing

[6675a] X (3432) We see in [6654a, c] that the parts of — , —^ , containing k, are found

[66756] by changing successively, in the similar expression of — [6669], k into — ' '^ , and

[6675c] -^ — TT • l^o these we must add the corresponding terms in [6667, 6668], and we shall

[6675<ri obtain the expression of -^ , — , in forms like that of — [6673] ; and which may

Vm vi. "^ 16]. INEQUALITIES OF THE ORDER in', mm\ m'«, &ic. 147

These three equations give,*

i«— kn«

coDsequently,

V =

(i.-kn.).(.+^^, + i^)^
?^,.kn«.(\-3x; + 2x;) , . ,.

(6877)

[0878]

[6679]

^..kn^(X-3x; + 2x;)
„" =: </x;'-f. __: ^^ \. sin.(i/ + 0). [6680]

It is important here to remark^ that in the preceding analysis we have supposed [668(K]
i to be much less than n — 2n', or n' — 2n". For in changing, as we have
done in [6385], nt-{-e, n't-]-^, n"t-\-^' into v, t/, r", respectively, in [6681]
the angle nt-'3n't + 2n"t + s—S/-\'2s'\ [6583], it is necessary that the

same changes may be permitted in the values of -r^ and 6v'

[6580, 6581, 6598, 6599], which we have used in computing [6586, &c.] ; [G681']
observing that these expressions depend upon the angles 2n't-2n"t-{-2^-2s",

and n't — nt-]-e — f. We shall first consider the part of — - depending

be derived from it, by changing, as in [6666 — 6668], X^ into X^' or \" respectively,
tnd k as in [66756, c] ; moreover, we must change, as in [6671], Q into Q', or Q",

 Using the text of ebook Mécanique céleste (Volume 4) by Pierre Simon Laplace active link like:read the ebook Mécanique céleste (Volume 4) is obligatory. Leave us your feedback.