Pierre Simon Laplace.

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[6403?] equations [6403?], we obtain fl' = ^i-f6^, as in [6405] ; and when ^ = 0, it becomes

[6403r] 6' = ^L, as in [6406, 6360], corresponding to the epoch 1750.

[6403a]
[64036]

[6403c]
[6403(f|

[6403e]
[6403/]

[6403g]

[6403/1]
[6403f]

[6403A;]

[6403Z]
[6403ni]

[6403n]
[6403o]

[6403p]

Vin. it. ^ 10.] INEQUALITIES IN THE LATITUDES. 83

*A = 0; [6403^] [64031

^ =>^pt; s^'L; [6403A, d] [6404]

V=>p/— ^; 6'^'L-\-b(. [6403o,7] [6405]

$0 that ^L is the inclination of the equator to the orbit of Jupiter, in [6406] 1750 [6403r]. Lastly, if we put y, for the inclination of the orbit of the satellite m, to [6407] the fixed plane [6315]; and 7j for the longitude of its ascending node [6316] ; we shall have, when we consider only the quantities relative to the [6408] displacing of the orbit and equator of Jupiter [6342, 6343], L— / = X. {L—L) ; [6409] and by transposition we easily deduce the following expression of / ; /=(1— X).L + X.L'. [6410] Hence we obtain,* r,. sin. ix = (1 — ^x).«. sin.^-f x.y. sin.7 ; [6411] 7,.cos. 7i = (X — l).^.cos.i'4-^7.cos.7. [6412] Thus, by noticing only the displacing of the equator and orbit of Jupiter, we shall have,t •(3332) Substituting the value of I [6410], in [6332], we get [64116]; and by osing the values [6326, 6330] it becomes as in [6411]. In like manner, the substitution [6411a] of / [6410], in [6333], gives [6411c] ; and by using the values [6327,6331], we obtain [6412]. 7,. sin. 7, = — (1— X).2'.L. sm.{pt-\-A)^\.j! .L .s\n.{pt-\-h) ; [6411i] 7,.cos.7, = (1 — 'K).i.L.cos.{j)t-\-\)-\-'K.7!.L'.cos.{pt-{-\), [6411e] t (3333) If we neglect terms of the order <*, we shall obtain, from the first of the «quations [6404], sin. 4' = ';)<, cos.*=l. Substituting these and 4='L[6404] in [^^^o] the terms of [6411, 6412], whicii liave the factor (1 — X) or (X — 1), we get the terms of [6413, 6414], having the same factor. Lastly substituting the values of 7.sin.7, [6413i] 7.COS.7, [6400, 6401], in the terms of [6411, 6412], containing 7, we get the terras of [6413, 6414], containing a or h. The formulas [6113, 6414] correspond to the satellite [6413c] m ; and by accenting the symbols 7, , 7, , X, we get the following expressions for the satellite m' \ 7/.8in.7/ = (l— X').«L.«p< + X'.ar ; [64I3rf] 7/.COS. 7/ = (X'— 1).'L. + >!.ht . [6413el By adding one more accent we obtain the values corresponding to ml' ; and with another accent they represent the values for ml". 84 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. C^l. [6413] 7i. sin. 7i == (1—^). ^L, ^pt -{->-. at \ [6414] j,,cos.\=l>^\yL -{->-. bt. Relative to the values of p, w^hich depend upon the mutual action of the *■ J satellites, we may put L' equal to nothing ; since the orbit of Jupiter is [6415'] not sensibly displaced by the action of the satellites. We may also, for these values of p, neglect the value of L, in comparison with the corresponding values of /, /', &c. [6417^]. For it follows, from the equation [6341], [6416] that the value of pL is multiplied by the small factor ; therefore it is of the order of the product of the ellipticity of Jupiter, by the masses of the satellites ; and such quantities we have heretofore neglected. Hence [6417] we may neglect pL and (0).Z/,* in comparison with (0)./, (1)./, &c. ; then the equations [6337 — 6340] become, [6417a] [64176] [6417c] [6417rf] [6417e] [6417/] [6417g-] *(3334) We have in [6313m], i = 40n'" nearly; hence we obtain from [6919] 4i~C ^^ Qgo w nearly. Substituting this, together with the values of n, n', w", M, in terms of n!" [6025«, b, c, i], and those of m, m', m\ m!" [7162—7165], in [6337c], and retaining only the first significant figure of the results, for the mere object of ascertaining the order of the terms, we get, pL = ^ . n'". 1 0,00001 . [L—L') + 0,002.(L— Z) + 0,0005.{ L-V) + %QQQ5.{L—l") + 0,00004.(L— r ) } . Now we have, as in [6025A;], —.n'"^ 241830"; substituting this in the preceding expression, and transposing the terms depending on L to the first member, we get, by a rough calculation, neglecting L' as in [6415], {p— 735"J.jL=— 483".?— 121".^— 121"./"— 10".Z'". Substituting the values of /', /", /"' [7226 — 7229], corresponding to the first angle p, we find that L is less than y^ . I ; similar results are obtained by using the values [7233 — 7236], relative to the second angle p^; those in [7238 — 7241], relative to the third angle p^ ; or those in [7245 — 7248], relative to the fourth angle p^. Hence it is evident that we may neglect L, as in [6415']. VIII. iv. ^ 10.] INEQUALITIES IN THE LATITUDES. 85 0= 5;'-(0)-[o]-(0,l)~(0,2)-(0,3) j . / 1 + (0,l)./'-f(0,2).r+(0,3).r; 2 0= 5;>-0)-[T]-(i,o)-(i,2)-(i,3)J./' 1 +(l,0)./4-(l,2).r+(l,3).Z'"; 2 = J ;,-(2)~[T]-(2,0)~(2,l)-(2,3) j . /" 1 +(2,0)./+(2,l)./'+(2,3)./'" ; 2 = |;,-(3)-[T] -(3,0)-(3,l)-(3,2) j . V" 1 -|-(3,0)./+(3,l)./'+(3,2)./". 2 If we suppose, [6418] [6419] [6490] [6431] [6423] I will vanish fr.im the preceding equations, and we shall obtain four equations s/mboit. between the indeterminate quantities ^', ^", ^", and ^ ; whence we may [6423] find p hy an equation of the fourth degree. We shall put p, pi, p^ ps, [q424] for the four roots of the equation, and ^/, li", r; ^.', ^.", la'"; ^3', ^3", r; [6435] for what ?, ^', |"' become, when we change successively p into pi, Pi, P3. W^e 5A«// then suppose that s, s\ s!\ s!", instead of expressing as in [6033, 6036'] nearly the latitudes of the satellites above the fixed plane, [6426] express their latitudes above Jupiter^s orbit, these last latitudes being part.vcularly required in the calculation of their eclipses ; in this case we shall have,* [6426'] * (3335) The latitude of the satellite m is expressed in [6298x line 1], by a series of terms of the form s = lf.l.s\n.(v-{-pt-\-A) ; and we have observed in [6298y] that the '■ ■• terms depending on the angles pt-\-A, p,<-|-A, , p,<-}-A, , Pj^-|-A, , arise from the mutual action of the sun and satellites, and on the ellipticity of Jupiter. These terms are [64196] explicitly retained in the expression of s [6427 lines 2, 3, 4, 5]. We have also observed, in [6296z], that the remaining terras, depending on the angles pjt-\-^4, P»<+a,, &c., and arising from the displacement of Jupiter's equator and orbit, are reduced to the single ,^.,n^, term (X - l).«'.sin.(»-fy) [6362] ; which is the same as that in [6427 line 1], referred 10 the variable orbit of Jupiter [6360—6362] ; so that the expressions [6427—6430] will VOL. IV. 22 86 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cel. [6427] Latitude* above the variable orbit of Jupiter. [6428] [6429} [6430] s == (X — 1). 6', sin. (v 4- y') + /. sin. (v -\- pt -{- A) + li. sin.(^+;7i^+Ai) + 4. sin.(v +Piit + A^) + Z3. sin.(i? +p3t + A^) + ^'. lsin.(v'-\-pt-{-A) +^,'.h.sm.(v'+p,t+A,) -{-^^,h,sm.(v'+p^t-{-A^) 'i'^^'.l3'Sm.(v'i-p,t+A,) 5" = (x^'— I).^'.sin.(/' + ^) +f. /. sin.(v"+p^ + A) +^,".h.sin.(v"+p,t-{-A,) +^3".l2.sm.(v"+p3t+A,) s"'= (\"'—l)j'.sm.(v"'+^') +^"',l.sin.(v'"-{-pt+A) +^/".Zi.sin.(i/"+pi^+Ai) +^^",l^.sin.(v"'+p,t+A,) ' 1 [Satellite m] 2 3 4 5 1 [Satellite m'] 2 3 4 5 1 [Satellit© i«"J 2 3 4 K 1 [Satellite m"'] 2 3 4 6 [6419rf] be very nearly the latitudes referred to the variable orbit, as in [6426']. Hence it appears that the expression of 5 [6427], contains the sensible terms of the proposed forms v-\-pt-]-A, v-\-pjt-\-A^ , he, depending on the mutual action of the bodies and the secular equations of the orbit and equator of Jupiter. In like manner we obtain from the expressions of 5', s", s'" [6298a; lines 2, 3, 4], the corresponding values [6428, 6429, 6430], We may incidentally remark, that if we neglect the first term in each of the expressions of s, s', /', s'" [6427—6430], depending on *', we shall find that the four remaining terms of each of these expressions become of similar forms to those in the values of 6v, W, y, Sv'" [6241—6244], changing g, T, h, ^, 8ic. into —p, —a, I, ^, &c. respectively, and retaining, in every instance, the same number of accents on the corresponding symbols. Hence we shall have a table of symbols, similar to that in [6229d]f and corresponding to the four roots p, p^, p^, ps ; and we may form, in like manner, a table of the values of these parts of s, /, /, /", similar to that in [6240^-M?,or 6241jg - A:]. From these we deduce analogous results to those in [6241m, &;c.] ; namely, that the first [6419fc] satellite has a peculiar inclination Z, and longitude of the node — pt — a ; the second satellite has a peculiar inclination V, and longitude of the node — p^t — a^ ; the third [6419c] [6419/] [6419g] VIII. iv. «^11.] INEQUALITIES IN THE LATITUDES. 87 The expressions /, /,, l^, ^; a, Ai, Ao, A3, are the eight arbitrary constant quantities, which can be determined only by observation. Jf we wish to obtain the latitudes of the satellites, above the fixed plane, we must add to the preceding expressions of s, s\ s", 5"', their values, upon the supposition [6431] that the satellites move in the plane of the orbit of Jupiter. 11. JVe shall nmo consider those inequalities of the motions of the satellites in latitude, depending on their mutual configuration, which acquire very small divisors by the integrations. It is evident that the terms of the differential equation [6295], depending upon an angle which differs but little from* v, [6432] will acquire such divisors. Now if we consider only the first power of the inclinations of the orbits, we shall find that all the angles of the different terms of this equation will be included in the formf i'(v — t/) zkv' ; and t/ [6433] satellite has a peculiar inclination /", and longitude of the node — p^t — a, ; and that the fourth satellite has a peculiar inclination /'", and longitude of the node — pj — a,. l"4lin] Moreover we see, as in [624 W — u], that each of these latitudes may be considered as the sum of four distinct latitudes, computed for four different orbits, passing through the places [64191:] of the nodes of the four satellites respectively. ♦ (3336) The equation [6295], for the determination of s, can be reduced to a form like that in [6446] ; in which the coefficient of s is nearly equal to unity, as in [6447J. This equation is integrated like that in [6049A:, /], changing t into v ; observing that the coefficient «' is nearly equal to unity, so that the divisor mj^ — aj^ [6049/], becomes ■i^' — 1, nearly; which is very small when m, is nearly equal to unity, or when the angle m^v-\-t^ becomes nearly equal to t> + e, , as in [6432]. [6433a] [6432i] f (3337) The action of the satellites upon each other being similar to that of the planets, it is evident that the forms of the terms of ^s, relative to the satellites, will be similar to those which we have computed for the planets in [1030]. Now the general term of Ss, [1030 line 3), depends on the angle i.{n't — ti/-|-«' — «)+w'-f"* — ^ » *"d by neglecting terms of the second order in e, 7, we may substitute, in this angle, the values n<-f-» = r, nt-\-/-=v' [953] ; hence it becomes of the form i.{v' — v)-\-v; i being any integer^ positive or ne^ntice, from »= — x to «=x [1028'], inelu/iing also t=0, in the term [1030 lino 2]. Then the corresponding term of [1033 line 3], has the divisor n" — \n — i.{n — w')j", which is easily reduced to the form, jn+[n— i.(n— n')]!. In—in—i.in—n')]] = {2n— t.(n— n') j.t.(n-ii'). Now it is evident, from the d(;finiiion of » [6434</], that we may change 1 into » -|- 1 » and bytbisineaos the angle i.{x/—v)-\-v [6434c], changes into {i-j-l).{v'-v)-\-v=i.{i/-v)-^i/; [&134a] [6434&] [6434«] [Gi34d] [6434e] [643V1 [643lffl [6434A] 68 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cel. [64341 <^^ff®^s but very little from ^v [6150'] ; so that the angle i.(v — v')±v' will differ but very little from v, if ^(izt'i) = 1 ; which gives either i= 1, [6435] or i = 3. In the case of i = l, the proposed angle becomes v; and in the case of i = 3, this angle is reduced to 3v — Av'. We have just examined the first of these cases, upon which the secular variations of the [6436'] orbit depend, in [6298, &c.]. It now remains to consider the inequalities depending on the angle Sv — 4y'. [6437] The expression of R [6297] contains the term R = m'.A'^^Kcos.(^v-Av'); r^ ds /dR\ [6438] therefore the term — li'TT'l^)' ^^ ^^® differential equation [6295], produces the expression [6440], by the substitution of s = Lsm.(v-^pt-\-A) [6439] v y2 [6300], and putting t='- [6439a-c],* Ti" = « [6299a], th ft [6434i] and the second form of tlie divisor [6434/], becomes {2n — (i-\-l).{n — n')].{i-]-l).(n — n'). The form of the angle i.{v' — v)-\-v' [6434A], is evidently equivalent to that in [6433], [643 lA] ygijjg jj^Q values of i [6434<Z]. Again, the divisor [6434i] becomes small when the first [6434^] of its factors is nearly equal to nothing, or 2n — (i-\-l).(n — n') = 0, nearly ; and if we substitute n' = ^n [6151], it becomes n.{2 — |.(«-[-l)| = Jw.(3 — i) ; which vanishes [6434m] ^^^^^ i= 3, as in [6435]. The same divisor [6434i] also vanishes when ^-f-l=0, or i = — 1, as in [6435] ; the difference in the sign being of no importance, considering that [6434ra] i [6434<Z] may be made positive or negative, as well as the term zkv' [6433]. When i=3, the angle i.(v' — v)-\-v' [6434A] becomes 4«' — 3v, as in [6436]; and when ■* i = — 1, it becomes — {v' — v)-{-v' = v, as in [6435]. The terms depending on v are [6434p] noticed in [6298, he] ; the angle 4u' — 3v, or 3v — 4«', is treated of in [6437, &tc.]. We find, in like manner, for the action of the third satelhte m" upon m, that the ^■' angle [6434A] becomes i. {v" — v) -\- v", and the divisor [6434i] changes into [6434r] {2n — [i-\-\).{n — r^')\.(i-\-V).{n — n"). This factor vanishes when i-|-l=Oj and then the angle i.{v" — v)-\-v" [6434$'], becomes v, as in [6435] ; which is treated of in
[6434s] [6298, &c.] The other factor 2w — {i-\-\).{n — ri') [6434r], does not become small,

because rf=:iln, nearly [6151]; hence this factor becomes n.{2 — {i-\-^)'i\^ which
'• -l vanishes when * = i; but i being an integer [6434flf], we may neglect this terra.

For similar reasons we may neglect the terms depending on the action of the satellite
[6434u] rd", and on the sun S ; since they do not produce, by the integrations, terms

having small divisors, like that depending on the angle 3v — 4«' [6434p], which is

computed in [6437, &;c.].

[6439a]

n'v n's
n n
These differ from the expressions in [6439,6441], by the constant parts of t, v' ; but this

* (3338) The values of v, v' [6091], give <=-^ — - ; v'=n'i+s'=*^— ^'+£'.

VUI. iy. ^11] INEQUAUTIES IN THE LATITUDES. 89

4m'. aA^*\ 1 8in.(4ts-4i/). cos/r + •2- r + a V [6440]

n'v
If we substitute t/= — [6439a — c], which may always be done, when [6441]

we neglect the excentricities of the orbits, the function [6440] will give, by
its development, the term,

2m'. aA^*K L sin/Sr— ^.v — ^,v^a\ [«42]

The term 75 •(-j-) of the differential equation [6295], produces the [6443]
following ; *

produces no effect in the subsequent calculations, because the differentials of the expressions [6431)6]
[6439rt], which are used in [6439d, &.C.], are the same as the differentials of the expressions
given by the author in [6439, 6441] ; and the re-substitution of the assumed values of /, t/, r/vjop i
[6455'J corrects for this neglect of the constant terms. The part of R [6437] gives

— - j = 4»i'.^^^\sin.(4»— 4i;') ; and the value of s [6439] gives,
— = /.A -f- ^Vcos/i? -{- L.v -^ a\ = l.cos.fv -{- L.v-{-a\ nearly. [6439rf]

Substituting these and tt = a [6439], in the term [6438], it becomes as in [6440].

[6439e]

hi

Reducing thb by [18] Int. we get the term [6442], using the value of v' [6441]. The

4n' p p

angle 3v .r 1> [6442], is very nearly equal to v, because — is very small, [6439/"]

and — = i nearly [6151]. The other terms of [6090] may be neglected, in computing
the term [6438], because they produce no angle which is nearly equal to v. Thus

2n.' n

the terms depending on A^\ A^^\ &c., produce the angles t; .v — — .t>, [643%]

2p — - — .V .Vf &ic. ; and neither of these angles is equal to v, or nearly equal to it.

♦ (3339) Taking the differential of aR [6297] relative to *, and retaining only the
terms producing the same angle as that in [6442], we find that the part [6297 line 3] gives,

a.f^\ = -iw'.aV.|*'-#.co8.(r'-«)} . { iJJ^0M-^>.co«.(«'-i>)-|-B<3'.co8.(2tr'-2rH-J5<3).co8.(3r'-^H-&c |. [6444a]

Substituting « = t« [6439], in the first member, it becomes the same as the term in

[6443] ; and its second member contains the terms in [6444]. For the term — m',a^(^.J, [64446]

being multiplied by the term jB'^.cos.(3o' — 3r), produces the term of [6444] depending

on /; moreover the terra + m'.oV.t. cos. (t/—r) being multiplied by the terms [6444c]

TOL. IV. 23

90 MOTIONS OF THE SATELLITES OF JUPITER. [M^c. C6I.

[6444] m'.aV. ^ Is.IB'^^^+B^'^].cos.(4^v—4>v')—B^''Is'.co3.(3v—3d') ] ;

which introduces, into the equation [6295], the following term,*

[6445] im',a^a'. { B^'Kr—i.(B^^^+B^'^).ll,sm.(3v —^.v^^.v—a\

Hence the equation [6295] becomes, by noticing only the terms depending
on the angle 3v — 4v',f

^ ddS , , An t

^ = ^ + -^> , ^

Nf being equal to the coefficient of s in [6299 line 1], namely,

[6447] iV^ = 1 + ^-(Prii^ + 1. "^ + 4.2.m'.aV. B'^'K

' an

We have, by means of [996, 1006], J

[6448] 2«>)— i.aV.J^^^'^+jB^^^} = - .2a.6^^^— i.a^6|)— i.a^6|);

and the formulas [966, 971], give,§

[6446]

J5'^3\cos.(3u' — 3u)-}-B^\cos.(5v' — 5y), in the last factor of [6444o], produces, by using
[6444d] [20] Int., the terms m'.aV.|5.{J5^3^+J5^5^}.cos.(4u' — 4?;), as in [6444]. These are the
only quantities which are necessary to be retained in finding terms of the form [6442].

* (3340) Substituting the values of s, s' [6300 lines 1, 2] in [6444], reducing the
[6445a] products by means of [19] Int., it produces terms depending on the angle (3y-4«'-^^-A) ;
and by using the values of /, v' [6439, 6441], it becomes as in [6445].

t (3341) The differential equation [6295] is reduced to the form [6299] ; and the
r6446 1 *^^"^^ ^'^ [6299 line 1], are the same as in [6446 line 1], using for brevity the symbol Nf
[6447]. These terms, being connected with those in [6442, 6445], produce the equation
[6446].

[6448a] X (3342) We have, in [996, 963^'], 2rt. ^^^^ = — ^ . 6^^ = — 2a. i^\ Also from
[1006, 963*^] we obtain,

[64486] — laV.i?(3) == __ i.f^ . J(3) ^ _ , q^s. jw . _^aV.5^5>=-i. -^ . lf=^-\^^. bf,

a^ 2" 5" a^ 2 2"

Substituting these in the first member of [6448], it becomes as in its second member.

<§> (3343) Substituting ^ = 5, s = f in [966], we get [6449]; also i = 4, and
[6449a] ^^^^ ^gj^g substituted in [971], give [6450]. Using the values [6449, 6450] in the
second member of [6448], it becomes by reduction equal to — |a2.6j|\ as in [6451].
[64496] Substituting this and ^ aV. 5^^^ = | a^. 6|> [64486], in [6446], we get [6452].

Vm. iv. ^11.) INEQUALITIES IN THE LATITUDES. 91

8.(l-|-tt«).A^,<>— 9a.iJ)
b^^^ _7 1; [6449]

*^<) = Z _ L . [6450]

Hence we deduce,

-2a.J<^> - ^a».6<')— }a».6^») = io.\bf ; [6451]

therefore the differential equation [6446] becomes,

= ^ -f. AT*. 5 + Jm'.a«.i^^\ (/'_/).sin/3i;— ^'.r— :^. r—A^ . [6452]

This gives, by integration,*

i.)n'.a«.6^3\(/'— /).sin/3y— ^'. t,— | . »— A^

This divisor is equal to | 3— ^ — -^ 4-iV; M 3— ^ — ^ — iV, | .
Now — is very small [6302] ; N^ is nearly equal to 1 [6447], and n is

nearly equal to 2n' [6151] ; therefore the factor 3 — — N^ is very

small, and the factor 3 — + iV^ is very nearly equal to 2. Hence

n V

m
s =

'.a«.6^3).(/'— /).sin.(3»— 4t/— J9^-A)

,.(s-±i-L-N)

It is evident that the different values of p, /, /', give, in the expression of
5, an equal number of terms, similar to the preceding.

These inequalities of 5, by reason of the smallness of the divisors,
considerably exceed the others, which are produced by the action of the
satellites. They are the only ones which require any notice ; and we shall

[6454]

[6455]

we get, by re -substituting the values v' = — .v, < = — , [6441, 6439], [6455^

[6456]

[6457]

• (3344) Integraung the equation [6452] by the method in [6049k, /], we get [6453] ;

and by substituting for T e denominator of [6 .53] its value 2.^3 ""* "^z ) °^^W» C^56a]

it becomes as In [6456].

92

MOTIONS OF THE SATELLITES OF JUPITER.

[Mec. Cel.

hereafter find, in [6931, &c.], that even these are insensible.* The sun's
action produces, in the vahie of s, an inequality which might become
[6457'J sensible, bj reason of the smallness of its divisor. This inequality depends
on the angle v — 2U; and we easily find, -by § 9, that the difierential
equation in 5 becomes, by noticing this term only,t

[6457a]

* (3345) This is evident from the expression of s [6931 line 1], using the values
of mf [7143] ; from which it appears that the coefficient of this inequality is of the
order 0;0008.(Z' — Z), which is very small in comparison with I.

t (3346) The expression of aR [6297 line 4], depending on the sun's action, produces,

in «-(-7~)j the terms given below in [6458&], depending on the angles v — U, 2v — '2U.

[6458al Substituting the values of s, S', [6300], we obtain the terms in [6458c] ; and, by
reduction, those in [6458(/], depending on the angle v — 2U — pt — a. These terms
increase considerably, in consequence of the divisors introduced by the integrations
[6049^, /] ;

[64586]
[6458c]
[6458c/]

[6458*]

JW2

[6458/]
[6458g-]

[6458/i]

[6458i]

= — — .j— 6/.sin.(t;+^^-fA).cos.(2u— 2C/)+12i'.sin.(l/4-p^+A).cos.(«— C/}

= — — . J 3?.sin.(i;— 2 Z7— 2?^— a)— 61.'. sm.{v—2U—pt—A) \ .

The terms in [6458c] produce also, by means of [18, 19] Int., some terms depending on
the angles 3u — 2U-{-jpt-{-A, v-\-pt'{-A. This last angle has already been noticed in
[6300], and the terra depending on the other angle is not increased by the integration
[6049A:,/j.

Again, the same part of the value of aR [6297 line 4] gives

ds

neglecting the other terms. Multiplying this by j- = l.cos.{v-\-pt-\-^) [6439J], then

reducing, by [18] Int., and retaining only the term depending on the angle v — 2U—]}i — a,

we get,

/dR\ ds 3M^ J . , ^^j ^ ,

-''\-^)-d-v=-4^' '• sm.(.-2/7-p^-A).

Adding together the expressions [6458J, A], and substituting, in the first member of the sum,
« = -^ [6439], we get,

Substituting this in [6295], and changing the second term s into JVj^Sf as in [6446a], it

VIII. iv. ^11.] INEQUALITIES IN THE LATITUDES. 98

= ^ + iV;'.s-|- _ . (L'— /).sm.f r— — . ©— ^. r— aJ ; [«58]

whence we obtain, by integration,

-^^.(L-/).sin.(.-~.n - ..-AJ

5 =

^ + ^+-'^,-1

[6459]

Connecting together the parts of 5, which depend upon the configurations
of the satellites and the action of the sun, we obtain,

m'.o.\bf.{l'—l).s\u.{^v-^v'—pt —a) SZfo •'

* 4.(3_il_iL_A;) '

\ n n V

^.(L'-/).,i„.(.-2[/-p<-A)

±!! + Z + JV-l

each of these terms being supposed to represent the sum of the similar terms, [6460']
corresponding to the different values of p.

In the eclipses of the satellite m, U is very nearly equal to v — 200°
[6032,6041] ; hence the inequality [6460 line 2] is reduced to the following
form ; *

[6461]

^^.(i'-4).sin.(., + p( + A)

[6462]

becomes as in [6458]. Integrating this equation, as in [6049A:, /], we get the term of s
[6459] ; observing that the divisor m^ — a^ [6049/j. becomes, in the present case, hy *■ J
using the values of Uy t [6102, 6455'J,

('-^-f^^=('~-l+ - -)•0-^'-^^v,)=..(.-»^-^iv,). ,^,

(2.V p \
}- — + A] — 1 j , as in [6459J. Connecting

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