# Electronic library

Pierre Simon Laplace.

# Mécanique céleste (Volume 4)

. (page 72 of 114)
Font size

r \TrdvJ r*

hence we deduce,!

[8904] j_ _ r^dv.\\-\-2e.cos.{v—Ts)-\-e^\^ ^

^^~ «.(i-e2) '

d^ _ r^dv.{l-\-^e.cos.{v—zi) + e^}t .
dT^ ~ a^l—e^f

[8905]

therefore,

[8906]

2K.(p(y\ r2 dv. \ l+2e.cos.(tJ— «)+e2 J

*(4052) Comparing [372, 1057], and putting f/- = l [8892], we get [8901]. The

[8902a] ^ „^

value of r [8902] is the same as that in [378]. We have in [8894a] ds = \/rfxa -j-df ;

and by substituting its value, deduced from the first equation [372], we get the first

expression in [8903] ; which is easily reduced to the second form in [8903].

t (4053) The expression of r [8902] gives, as in [1256],

r8904al JL. = e.sm.jv — 'us) ^ J_ _ l+c.cos.(t?— w)

r^dv a.(l— c2) ' r a.(l— eS)

Substituting these in the last value of ds [8903], we get [8904] ; observing that the

numerator of this expression may be reduced, by putting,

[89046] €2.sin.2(i;— "5^)+ {14-c.cos.(v— ro)}2 = l+2e.cos.(u— w)4-c2.

T^dv
The cube of ds [8904], being divided by the square of dt = y====. [8901], gives

[8905] ; and by substituting this last value in [8894], we get [8306].

X. vii. <§. 18,] EFFECTS OF A RESISTING MEDIUM. 613

We shall suppose that the function,* [8906']

[8908a]

[8908/]

* (4054) When the form of the function (?(—) is known, we may make the

developments of the first member of [8908], by processes similar to those which are
employed in [964, &c.] ; and by this means we can ascertain the values of A, B, C, &c.
We may also use the method of definite integrals, in the following manner. Putting for
brevity V equal to the first member of [8908], also v — '5J = v; and, for the sake of t^^^^^]
symmetry, j1 = A^; Be = jl^; Ce^ = A^, he, we obtain the following expression [89G8c]
of the function V ;

V = K.J-\ r2. J l+Se.cos.v+e^Ji = ^„ -f ^..cos.v +A^.co%SIy^A^.cos.Zv-{-Uc. [8908rf]

Multiplying this hy dv and integrating, we get,

fydw = A^.v -f ^,.sin. v-f I ^j.sin.Sv +&c. [8908e ]

This vanishes when v^O, and when v = '?r it becomes /^Vrfv = .^o.*; dividing this
by *, we get A^ [8908A:J. Multiplying [8908rf] by Jv.cos.v, and reducing the second
member by means of [20] Int., we obtain by integration,

/V.cos.v.Jv=/|.^o.cos.v+^.^i.(l-|-cos.2v)-f-|.^2-(cos.v-f-cos.3v)-|-&;c.}.£Zv [8908g-]

= .^o.sin.v+|^i.(v-j-^.sin.2v)+J^2.(sin.v + |.sin.3v)-{-&tc. [8908A]

The second member vanishes when v = 0, and when v = * it becomes ^A^.'K', hence
we have /^V.cos.v.<Zv= J^^.* ; which gives A^^ as in [8908/]. In like manner, [8908i]
multiplying [8908c?] by cos.nv, n being a whole number, we get by integration the
expression of |.^„.tf; whence we deduce .^„ [8908m].

A,= ^.f-Ydv; [8908A:]

A = 7 -/r V.cos.v.c? V ; [8908Z ]

An = — ./* V.cos.n v.d v. [8908m]

If the integrals of the second members of these equations can be obtained in finite terms,

or with the assistance of circular arcs, logarithms, elliptical functions, or converging

series, we shall have the values of A„, A^, A^, &;c. If neither of these methods can [8908n]

be advantageously employed, we must ascertain the integrals by means of quadratures, as

in [7929a;, &;c.] ; supposing di to be changed into dv, and y'-^ into V.cos.nv, in [8908o]

finding the general value of A„ [8908m].

As an example of this method, we shall put ?(~) = "^; this being the hypothesis [8908p]

used by Encke [56676], in calculating the perturbations of the comet which bears his
name, in vol. 9, page 333, of the Astronomische Nachrichten. We shall investigate the
formulas for the determination of the values of A^, A^, or A, B; these being the ^^

VOL. IV. 154

614 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.

[8907] K,Jj\r\{l+2e.cos.{v'-vs)+e'\^,

[8908r]

only coefficients of the series [8908c?] which are required, when we restrict ourselves to
the computalion of the secular inequalities, as in [8909', 8915, &,c.] ; and these secular
inequalities are all which deserve attention, in the present imperfect state of our knowledge
[89085] relative to the nature and density of the resisting medium; the periodical inequalities
depending on this cause being comparatively unimportant. Now substituting

<p(r) = — [8908^] in [89086/], we get the following general expression of V;

[8908u] V = K.\i-{-2e.cos.v-\-e^i = ,5„-f-^^.cos.v+^2-cos.2v+&c.

This last development may be obtained by the formulas [975", 976, Sec], putting a = v,
a = — e, s= — i; hence we get,

[8908u;] X =l4-2e.cos.v+c2; [975"J

[8908x] X* = 1 14-2e.cos.v-f e2 ji = ^(^J + 6y.cos.v-f-6i!j.cos.2v-|-&c. ; [976]

[8908y] V = K.\l'\-2e.cos.v+e^^ = K.^ b^ + A:.&ilJ.cos.v -f &c. [8908w, a;]

Comparing together the two last of the developments of V in [8908mj y], and resuming
the values of A^, B^ [8908c], we easily deduce,
[89082] .2 = ^0 == JT. 1 6^5 ; 5e = ^, == ^.6^ .

The values of ^6^, 615 , ^"^^ S^^^" ^" [989] ; and by putting a = — c, as in [8908t>],
we obtain,

„ . „,,,,, C . 1.1 - 1 1.1.3 , 1.3 1.1.3.5 . . >

If we neglect the square of e, as in [8921], these expressions will become A:=K,
[8909c] B = K; being the same as those which are deduced from [8922,8923], by putting

When e is large, the values of A, B, can be more easily obtained by the method of

elHptical functions, with the tables published by Legendre, which give, by mere inspection,

I oyuy w I , .111

the values of .^, B ; and as this example shows the importance of these functions, we shall

compute the formulas which are necessary in this calculation. For this purpose we shall
[8909e] put c= fj;-} and shall use the symbols given in [82a, b], being the same as the values

[8909/] of A(c, (p), F(c, (p), E(c, (p), F\c)y E^c) [8910t— A]. Substituting the first value
of V [8908w] in [8908^], we get the first form of A„ [8909A] ; its second form is

^ ^■' obtained by putting v = 2(p ; its third by making cos.2(p=l — 2.sin.'^(p ; then substituting
c, E^ [8909e,/], we successively obtain,

X.vii.«^18.] EFFECTS OF A RESISTING MEDIUM. 615

is developed in a series, arranged according to the cosines of (v — w) and [8907']

^0 = - 'f^l l+2e.cos.v+e2 j^.rfv = ^ .f^^\ l+2e.cos.2(p+e2}i.2rf(p [890%]

tytr Qir

= — .(l+e)./i'^\/l_c2.8in2cp.(^= — .(l+e).Ei(c)=^. [8908c]. [8909ft]

X—l €2

Putting for a moment X= l-j-2e.cos.v -f- ^^j we get cos.v = — ^ , and [8909f]

V = KX^ [8908m]. Then from [8909A, 1c] we have,

yj^{l4-2e.cos.v+e2}i.f/v=y;^Xi.e/v==2.(l-|-e).Ei(c); [8909Z']

and [8908/] gives, by successive reductions,

^^=^'fo''iX-l-e^)'XUv==f^ .f,^X^.dv^~.{l+e^).frXUy [8909m]

= fe 'fr^'^'dv- ~ . (l+e).(l+e«).EXc). [8909n]

The term in [8909w], under the sign /, may be simplified by substituting, as in [8909g-],
v = 2(p; whence we have, .

X^=5l+2e.cos.29+e2p = ^l4-e)»— 4c.sin.2(p}i = (l+e).v/l-c2.sin.2(p = (l+e).A ; [8909p]

writing for brevity A, instead of A(c, cp) = y/l— c2.sin.29 [8910i]. Substituting this, [8909^']
and dv = ^d(p [8909o], in [8909n], we get,

A = — - {l+e)\fl^AU<p^ ^ . (1+c). (l+e2).EXc). [8909g]

Now it is easy to prove, by differentiation, that we have generally, for any value of cp,

/A3.£?9 = ^c2.sin.(p.cos.9.A-{-i(l— Jc2).E(c,(p)— i.(l— c2).F(c,(p). [8909r]

For by taking its differential, dividing by d(pj and using the values E(c, (p), F(c, (p),
[8909/ J, then writing the terms down in the order in which they occur, without any
reduction, we obtain,

a3 = ^c2.(cos.2 9— sin.2cp).A — Jc^sin.^ip.cos.^ip.A-* +f.(l— |c2).A — ^.(1— c2).A-». [8909^]

Multiplying this by 3A, and substituting cos.2(p = l — sin.^9, it becomes as in [8909^].
Connecting together the terms depending on A^, we get [8909m] ; substituting
A^= 1 — c^.sm.^cp, we obtain [8909v] ; and by successive reductions, it changes into SA"*,
[8909ifj], as in the first member of [8909^] ; therefore the diflferentials of both members of
[8909r], are equal to each other;
3A4 = c2.(l— 2.sin.2(p).A24-c4.(_siu.2<p-fsin.4(p)+(4— 2c2).a2— (1— c2) [8909/]

= (4_c2_2c2.sin.2(p) .A^Jr{(^—[—c^.sm.\-{-c*.sm*(p) [8909u]

= (4_c2— 2c2.sin.2(p)4-c2.sin.2(p.(— 4+c2+2c2.sin.2(p)+(c2— 1— c*.sin.2(p4-c^sin.V) [8909u]
= 3— 6c2.sin.2(p-f 3c'».sin.V = 3.(1— c^.sin.^fp)^ = 3A4 ; [8909m>]

616 MOTIONS OF THE HEAVENLY BODIES. [Mec.Ca

[8907''] its multiples, in the following form ;

observing that the expressions ±c^, ±c^.sin.^(p, in [8909w], mutually destroy each
[8909a:] other. The first term of the second member of the integral [8909r], vanishes at the two
limits 9 = 0, <p = |ff, and the general integrals E(c, 9), F(c, 9), become E^(c),
F^(c) [89]0m,n], respectively; therefore we have,

t^^^^^ ft^''df> = H^-ic^) ' E^(c)-i.(l-c2).Fi (c).

Substituting this in [89095-], ^"^ making a slight reduction, we obtain,

01?" Ql(^

[8909Z] ^^= -^.5(l-|-e)3.4(l_|,2)_(i_^,).(l_^,2;)|.E^(,)__^^.(l+,)3.(i_c2).p(,) .

[8910a] and since c2 = -ii- [8909e], gives 1— ic2 = [^±^, l-c2 = 5-J^, it becomes,

[89106] A = £.(l+6).|(l+e2).EXc)-(l-e)^Fi(c)l ;

substituting this in B = A^.e~^ [89095], we obtain,
[8910c] 5 = £^^.(l+e).{(l+e^).E'(c)-(l-e)2.F(c)S.

The process we have here used in finding A,^ or A^ by a series [8909fl, h], or by
[8910</] elliptical functions [8909Z(r, 89106], may be used when 9(— ) is supposed to be of a

different form from that which is assumed in [8908p] ; as, for example, we may suppose,
[8910e] ,(i)=£, + ^.+^. + 5 + . ... + «»;

and by re-substituting the value of r [8902], we can reduce the expression of V to the
last form in [8908(/J, and then Ji^, A.^, may evidently be obtained by means of elliptical
functions. This calculation has been made by Plana, in a valuable memoir on the
resistance of the ethereal medium, published in Zach's Correspond ance Astronomique,
he, vol. 13, page 341, &,c. Instead of finding the values of the differentials de, da, dn,
in terms of the differential of the true anomaly dv, as La Place has done in
[8909^,8915, &c.]. Plana gives them in terms of the differential of the excentric anomaly
[8910/"] iJ,u, according to the method which is used by La Place, in computing the perturbations of
a comet, in [7872, &;c.]. We may remark that the values of A^, A^, &c. [8908rf], may
be derived from Aq, A^, by a similar method to that which is used in [966, &c.].

The expressions of .^, B [8909A:, 8910c], contain the elliptical functions F\c), E^c),

18910^] . . 2t/c

which require the computation of c=— j— [8909e], from the given value of e. This

may be avoided by reducing these functions to others, depending on F^(e), E^(e). In
^ ^J making these reductions we shall use the method and notation of Legendre, putting as in
[82a, 5, &c.].

[8910e'J

X. vii. ^ 18.] EFFECTS OF A RESISTING MEDIUM. 45 J 7

K.U^\r\{\+2e.cos.(v-7;s)+e'li^A-^Be.cos,(v - ^)+Ce\^^^

[8908]

(^=^~; sin.(29'-(p) =c.sin.<p; [8910/^]

l-\-c

A (c, (p) = v/l— c2.sin.2(p ; A (c', <p') = v/1— c'2.sin.2(p' ; [8910i ]

F(c, 9) = r-^=J^=== ; F(c', (?') = /*, ^"^ ; [8910A]

E(c, (p) =yH(p.v/l— c2.sin.2(p ; E ( c', 9') =/<:i(p'y 1— c'^-sin.s <?/ ; • [8910/ ]

^ J •/ /i_c2.sin.2ip' ^ -' -^ y/i_c'2.sin.2(|/' *• -"

Ei(c) =yj'^ «?9.v/l— c2.sin.2(p ; E^(c') =y|'^ <^(p'.v/l-c'2.sin.2 cf/. [8910n]

Having found c', 9', from c, 9, by means of the assumed equations [8910A], we shall
have the following integral formulas ;

F (c', <pO = i (IH-c).F(c, 9) ; [89100]

^^"^'^P') = j:^.E(c,9)-|(l~c).F(c,9)+j^^.sin.9; [89iqp]

Fi(c') ==(l+c).Fi(c); ' [8910g]

^'(^) =i:^-EXO-(l-0-FX^)- t8910r]

For by using the symbol A for v/1— c2.sin.29, as in [8909/], we shall have, from the [sgiOr^
second equation [89I0A], and [23] Int.,

A = v/l— {c-sin.9)2 = v/l-sin.2(29'— 9) = COS. (89' — 9) ; [89105 ]

COS.29' = COS. { (29' — 9)4-9} =cos.(29' — 9).cos.9 — sin. (29' — 9).sin.9 = COS.9.A — c.sin.^9. [8910< ]

Substituting this last value of cos.29', in 2.cos.^9'=:l-|-cos.29V 2.sin.^9'=l— cos.29',
[1,6] Int., we obtain,

2.cos.^9' = 1 — c.sin.^9 -|- COS.9.A ; [8910m]

2.sin.^9' ^l-f-c.sin.^9 — C0S.9.A. [8910i>]

We have, by developing as in [21, 31] Int., and using the values [8910A, s],

sin.29' =sin.(29' — 9-I-9) =sin.(29' — 9) .cos.9-f 003,(29' — 9).sin.9; or,
2.sin.9'.cos.9' = c.sin.9.cos.9-}-A.sin.9 = sin.9.(c.cos.94-A). [8910m)]

Substituting the value of c' [8910A], and that of 2.sin.29' [8910y], in the first member of
[8910a?], altering the arrangement of the terras and making successive reductions, it
becomes as in [8910y] ;

(l-f-c)./!— c'2.sin.2 9' = ^(l_l-c)2_4c.sin.2 9' = \/{\-\-cf — 2c.(l-|-c.sin.29— COS.9.A) [8910a;]

= v/^(l — c2.sin.29)+2c.cos.9.A-|-c^.(l — sin.29) \

= \/{a2-]-2c.cos.9.A-j-c2.cos.^9| =A4-c.cos.9. '• ^-'

VOL. IV. 155

618 MOTIONS OF THE HEAVENLY BODIES. [Mec. Cel.

[8908'] A, B, C, being functions of e^, we shall have, by neglecting the

*d(p

The differential of sin.(2^' — ^)=c.sin.^ [8910A], gives, by using [89105],

dcp
[8910z] (^dp' — d<p).A=zcd<p.cos.<p; hence df>' ==—- .{A-{-c.cos.<p).

Substituting [89i0y, z] in the expression of F{c',<p') [8910A:], we get,
[89U.] F(</, /) =/-^j=^ = J.(l+c)/^ = i.(l+c). F(c, ^). .

If we substitute the same expressions [89I0y, z] in E(c', cp') [89 lOZ], and multiply it by
2(I4-c), using c2.cos.2^ = c2— c2.sin.2^ = c2— 1+A2 [8910/], we get,

[89116 ] 2.(l+c).E(c', <p') = {2+2c).fd<p'. v/l-c'2.8in.2(p' =/'-^ . (A-j-c.cos.'p)^

[8911c] = /"-^ .{ a2+2c.cos.^.a+(c2— 1+A2) J

[8911rf] = /*^ . 5 2a2+2c.cos.?).A+c2— 1 }

= 2j'd<p.A-\-2cJcl<p.cos.<p — {l—(?).r-

[8911c] =2E(c,^)+2c.sin.<p— (1— c2).F(c, <f>).

Dividing this by ^.(l+c)} we get,
[8911/j E(c',^') = ji-^ .E(c,^)-i.(l-c).F(c, ^)+ j-^^ .sin.^.

,^^,, If we put sin.5 = c.sin.'?, we shall have generally sin. 5, independent of its sifijn, less
[8911g"] .

than c; and as c<^l, 5 must be less than a right angle ; and we shall have generally

2^' — ^ = 5, as is evident from [8910A] ; hence we get 2ip' = ^-j-5. Now when

[89llfc] ^ = 0, or <z> = 180^^, 5 vanishes [8911^], and we have ^ = 2^'; so that if we take the

integrals relative to <|>', from <p' = to <p' = 90'^, we must take those relative to <p

from 9=0 to (PsztISO"*; or what is equivalent, we must take them from <p = to

<p = 90**, and double the results ; since it is evident, from a mere inspection of the formulas

[8911i'] [8910fc— n], that when q> = ISO'', we have ¥{c, 9) = 2Y^{c) and E(c, <p) = 2Ei(c) ;

observing that the expression of A, and the elements of the integrals, are the same for

[89n)fc] 9=: 90*^ — <pi, or 9 = 90'^+<p, ; <Pi being any angle less than 90'^. Putting therefore at

first 9 = 0, 9' = 0, in [8911a,/], and then 9=180'', 9' = 90^ we get,

[891.in Y^{d)^{l-\-c),¥\c)',

[8911m] W{d) = -^ .E^(c)— (1— c).F(4

rftQii 1 If we now change c into e, the expression of d [8910A] becomes equal to that of c,
[8909e], in the notation used by La Place in this chapter. Making these changes in
[891lZ,m], we get.

X. vii. § 18.] EFFECTS OF A RESISTING MEDIUM. 619

periodical functions * [8908"]

Fi(c)=(l+e).Fi(e);

[891 In]

Ei(c) = -^ .Ei(e)— (l-e).Fi(e). [8911o]

l-\-e

Substituting the expressions {891ln, o] in [8909A:, 8910c], we obtain, after making some

slight reductions,

2/r

^ = _.{2Ei(c)— (1— e=).Fi(e)i ; C^^"?!

sJ£.^?^.Wie)-'i^.FHe)]. [8911,]

If we take for an example the value of e = 0,8446862 = sin.57''38'« 18* [4079»t], [8911r]
corresponding to Encke's comet, we shall have, from Legendre's tables,

Fi(e) =2,09575; Ei(e) = 1,23357. [8911a]

Substituting these in [8911^,9-], we get,

^ = ^Xl,18838; 5 = ^X0,90015; [891K]

which will be used hereafter, together with the value of a = 2,224346 [4079ffi], and [8911m]
e [891 Ir].

Finally, if we substitute the values of A, B [891 Ip, q\ in [89082:], we get, by rejecting
the common factor jBT,

U^ = ^ .|2Ei(e)— (1— e2).FXe) | ; [8911«]

*iii = i.^<i±^.EV)-'i^.FHe)|, [S91H

SO that we shall have 6i?^, 6i!|, by means of elliptical functions; and by a similar [^39113.1
process we may obtain the general value of the coefficient 6^'\ by means of such functions.
The values of ¥J^ may also be derived from those in [8911y, ii?], using the formulas
[966, 975, &c.], and putting a = — e [890Sw]. Thus from [992] we obtain,

f == ;7I3^-^2E^(^)-(l-^')-FHe)} ; [89112/]

^f = ;;7:(^.- {(i+^^)-EKe)-(i-e^)-FK«)l- [8911/]

In like manner the formulas [990, 991] give, by substituting the values [891 lu, w]^

6(o)=i.Fi(e); [89112]

6y = — . |Ei(e)— Fi(e) } ; [89112^]

and so on for other cases.

* (4055) Substituting v— z3 = v [89086], in [8908], and then multiplying by
2rfu.(l+2e.cos.v+e2) • r u j v r r [8912a]

a2.{l— e2)8

we get the expression of the second member of [8906] ; hence we have,

620 MOTIONS OF THE HEAVENLY BODIES. [Mec.Cel.

r89091 rf i - l2A.{l+e^+^Be-ldv ^
[8909] rf.-_ a^l-e^)^ ' ""'

[8909Q <fo=- (l-e^^ ^•

Then we have, as in [371 or 927],
[8910] X = r.cos.t? J y = r.sin.t> ;

hence we get,*

[89126 ] <?• 7 = a^a-e^)ii ' I (^ +e^)'{'^-hBe.cos.y-\-hc.)+2e,co5.y.{Jl+Be.cos.y-\-hc.) ] .

If we neglect the periodical functions, as in [8908^], and put 2e.cos.vXBe.cos.v= B^-l-kc,
[8912c]

[6] Int., we get [8909]; developing the differential, we obtain [8909^]. From [8919]

[8912c?] we have log.n = — |.log.a, whose differential is — = — §. — ; substituting da

[8909'], and multiplying by n, we obtain,

[8912c ] ^^ ^ n.\3A.{lJre^) + SBe^.dv ^

a.(l — e^) 2 '

which will be used hereafter.

* (4056) The differential of x [8910] is as in the first of the expressions [8912/],
which is easily reduced to its second form ; then substituting the values [8904a], we
get [8912^, A];

[8912/*] dx ^ dr. cos.v — rdv. sm.v = — T^dv. } — —r- . cos.i; -] . sin.t; i

T^dv
[8912g] = . I — e.sin.(v — •cj).cos.r-j-[l+e.cos.(« — w)].sin.v?

[8912^] = 71 . ^sin.iJ-j-e.[sin.v.cos.(t> — ©) — cos.'i>.sin.(v — w)] }•

The coefficient of e between the parenthesis in this last expression, being reduced by
[22] Int., becomes sin.fu — [v — to) | = sin.-ci ; hence this value of dx becomes as in
[J [8911]. We may find in a similar manner the differential of y [8910], which gives
successively [8912^, Z, m\. The last of these values is easily reduced to the form [8912],
by using [24] Int. ;

c dr . 1 ')

[8912A;] ciy = dr.sin.v -\- rdv.cos.v = r^dv. < -^-r • sin.i;-| — . cos.v i

T^dv
[8912Z] ^ "Ti i" * l^'Si"*(^' — TO).sin.i'-{-[l+e.cos.(i; — c:f)].cos.«|

[8912m] = — . I cos.i - {-e.[cos.r.cos.(i? — w)+sin.v.sin.(r — w)J \ .

This value of dy might also have been easily derived from that of dx, by decreasing

'1

X.vii.<§>18.] EFFECTS OF A RESISTING MEDIUM. 621

[8911]

dx = r, . Csin.v + e.sin.ttf) ;

«.(1 — e-*) ^

T dv
dy = — - — ^fT.fcos.i^+c.cos.w). [8912]

^ a.(l — e-^) ^ ^

From these we easily deduce,*

[8912n]

the angles v, ■a, by a right angle ; for this does not alter the angle v — ■&, or the value in
i [8904a] ; but it changes the first expression of dx [8912/], into the first expression of dy,
[89I2A:] ; therefore by making the same changes in the values of v, «, in dx [8911], we
shall get dy [8912].

* (4057) Multiplying [8904] by 2K.(p(-\ and substituting the developed value
[8908], we get,

2KJ^\ds = ^^^. . {A-{-Be.cos.{v—'a)-\- Ce^cos,{<iiv—2zs)-\-hc. ] . [8913a]

Multiplying this by xdy — ydx = r^dv, and by t^ = ' , ^ [8901], we obtain,

2K.<p(^\ ^.(xdy—ydx)==^ . {Ai-Be.cos.{v—vi)+Ce^.cos.{2v—2':,)-\-hc.]. [89136]

Theproduct of [89136] by dx [891 1], being substituted in [8898], gives [8913c]; and
in like manner the product of [89135] by — dy [8912], being substituted in [8899],
gives [8913J] ;

d.(e.sin.zi ) = — - . {sin.u ~\- g.sin.w} . \jl-\-Be.cos.{v — w)-|- Ce^.cos.(2u — 2w)-}-&£C. } ; [8913c ]

^.(e.cos.'cj) = — ^ .|cos.t;-j-c.cos.w|.{.^-^-J5e.cos.(« — ts)-\-Ce^.cos.(2v—2'&)-{-hc.}. [8913cri

Multiplying together the factors in the second member of each of these expressions,
substituting sin.«.cos.(« — 'K)=^.sin.'5j+&;c. ; cos.t>.cos.(t; — •s3)=J.cos.w-|-Sic. [18,20]
Int., and retaining only the terms which are independent of v, we obtain, from [8913c],
the expression [8913]; also from [8913d!], the expression [8914]. If we develop the
differentials in the first members of [8913, 8914], they will become of the following forms

respectively, usmg for brevity the symbol C = — - — ^— ; [8913/^]

de.shiM -{-edzs.cos.'a = — C.sin.xif ; [8913g']

fZe.cos.-ss — ed-a.s'm.vs=^ — C.cos.-a. [8913A]

Multiplying [8913^] by sin.w, and [8913^] by cos.w, then adding the products, and
using sin.^5i-|-cos.^'5i= 1, we get de = — C, as in [8915, 8913/]. Again multiplying [8913t]
[8913^] by cos.trf, and [8913A] by — sin.zs, adding the products and making the same rsgisji
reduction, we get dzi = 0, as in [8916]. Finally as dy, d6, vanish [8886A], the
secular variations are reduced to the finding of those of a, e [8909^, 8915], as in ^ -■
[8916']; those of n being derived from a [8912d].

VOL. IV. 156

[8913c]

622 MOTIONS OF THE HEAVENLY BODIES. [Mec. Gel.

,^^,„, , . . . ( 2 A 4- B).edv. sin. -a
8913] d,(e.sm.^) = — ^ — ^,/ 3, ;

^ '^ a.(l — e-*)

[8914] c?.(e.COS.-ra) = — ^ -I— ^ — .

0.(1 — e'*)

Hence we obtain, as in [8913i, A;],

[8915] de = — - — J- — ^ ;

^ ■• a.(l — e^)

[8916] d^ = 0.

Therefore the perihelion is immovable, and there is no variation except in
the greater axis, and in the excentricity of the orbit [8913/J.

v?[j^ations Dividing the expression of de [8915] by that of da [8909'], and
multiplying the quotient by da, we get,

, _ {2A-i-B).e.{l—ee).da
~ 'a.Y2A.{l+ee)-}-2Be^l '

If we integrate* this differential equation, we shall obtain e in terms of a ;

* (4058) Instead of proceeding in this general manner, it will be found sufficient, for
[8917a] all practical purposes, to integrate the expressions of da, de, dn [8909', 8915, 891 2e],
for one complete revolution of the comet, or from v=0 to v = 2'f; supposing the
elements of the orbit in the second members of these equations to be constant during this
period, on account of the smallness of their variations. By this means we shall have, for
the corresponding variations 6a, Se, Sn, respectively, the following expressions ;

{2^.(1 +e2) + 2Be2}.2*^

of the
elements

[8917]

[89176]

[8917c] Sa = —

[8917rf] ^e = —

{l-er
{2A-{-Bl.e.2ir ^

a.(l— e2) '

[8917e ] Sn = n.f3^.(l+e^)+3ge^j.2. -

a.(l— e2)2

Substituting in these formulas the values of A, B, a, e [8911^, M,r], corresponding to
Encke's comet, and to the hypothesis which he has assumed for the resistance, we obtain,

[8917g] 5a = — 4 1 0,057 . ^ ;

[89m] 5e=— 27,290. iBT;

[8917i] 6n= 1,434.^.

r8917t'l ^^^^ substitute the values of A, B [Q9\\p,q\ in [8917c, fi?,e], and then put t^ = l—h^,
we shall obtain, by some slight reductions.

X. vli. <§> 18.]

EFFECTS OF A RESISTING MEDIUM.

623

and by substituting this value of e in the expression of da [8909'], we shall [8918]
have, by integration, 'o in terms of a ; or a in terms of v. [8918']

I*.'

fe = -4. ^ .if.(2-3-|).F' (e) + -3/ . ^ . Ji:.(i - |).E>(e) ;

being the same as were found by Plana, in vol. 13, page 352, of Zach's Journal [8910e'],
and used in Pontecoulant's Theorie Anahjtique, he, vol. 3, page 288.

Plana, in the memoir we have just referred to, makes the numerical calculations of the
values of Sa, de, in two particular cases. First; where the density is constant, or all the
terms of the series [8910e] vanish except B^. Second ; where all the terms of the same
series vanish except B^ , which is the same as Encke's hypothesis [8908/?] ; the numerical
results of Plana, in this last case, agree very nearly with those in [8917^ — i]. Encke
uses the common method of quadratures [7929a?, Sic], in finding the values of Sa, 8e,
considering t as the independent variable quantity, and computing the co-ordinates at
equal intervals of time. He found this method to be convenient, because these co-ordinates
had been previously ascertained by him, in making the calculations of the perturbations of
the comet by the action of the planets. The intervals he used were 4 days, when the
comet was near its perihelion, and within the sphere of the attraction of Mercury;

 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.