Pierre Simon Laplace.

# Mécanique céleste (Volume 4) online

. (page 50 of 114)
Online LibraryPierre Simon LaplaceMécanique céleste (Volume 4) → online text (page 50 of 114)
Font size
[7831] same time. For this purpose, it is only necessary for the attractive forces in

* (3753) The comet is supposed to be at a great distance from the centre of gravity of
[7829a] ^i^g gyj^ ^^^ planet ; so that it will be acted upon in nearly the same manner as if the mass

l-\-m', of the sun and planet, were placed in the common centre of gravity of these two
[78296] bodies.
Now this mass 1-j-w' acts upon the mass m of the comet, placed at the distance

r-\-6r, with a force which is represented by the sum of these masses l-\-m-\-mf, divided by

[7829c] (r4-5r)2, as in [7829] ; that is, by the force T'^'T^ ? supposing, as in [7829 or 7834'J,

[7829rf] that r-]-5r is the radius vector of the comet, referred to the centre of gravity of the sun and
planet. If we neglect the planet's mass m', and put r for the radius vector referred to

[7829c] lA-rn

the sun's centre, the force becomes —5- , as in [7832].

IX. i. § 2.]

AT A GREAT DISTANCE FROM THE SUN.

367

the corresponding points to be as r-^Sr to r,* which gives,

1 4-m4-m' 1 -}- m
{r-f-ory H
Hence we deduce,

sr = :^m'.r; [7831/]
consequently the co-ordinates of this last ellipsis are,t

(l+im').x; {l+im').y; (l+im').z.

These co-ordinates are referred to the common centre of gravity of the sun
and planet. To reduce them to the sun's centre, we must add the
co-ordinates of this centre of gravity, referred to the sun's centre. Now
these last co-ordinates are evidently represented by mV, m'l/, m'z' ;
therefore the co-ordinates of the comet, referred to the sun's centre, will be,
{\+±m').x-\-m'x' ; (l+±m').y+my ; (l+±m').zi-m'z' ;

[7831']
[7832]

[7833]

[7834]
[7834']

[7835]
[7836]

* (3754) If two bodies revolve about an attracting point in similar ellipses, and in
similar situations, with the same periodical time of revolution, and corresponding radii
vectores r-\-5r and r, it is evident that their tangential velocities in their orbits will be
as the radii r-\-5r and r ; and the deflections from the tangents, in the direction of the
attracting force, and in a given time dt, will be in the same ratio r-\-8r to r; because
the orbits are supposed to be similar in form and position. Now these deflections are
evidently proportional to the attractive forces ; therefore these forces must be to each other
as r-f-5r to r, as in [7831'] ; and if we take, for the forces, the values given in [7829c, e],
we shall obtain the analogy in [7832], From this we easily deduce the equation,

(r+5r)3:==r3.(^±^')=r3.(l+^-^) = ^^^^ nearly.

Extracting the cube root of this last expression, we get, very nearly, r-\-Sr=r.{l-\-^m,') ;
whence 6r = ^m'.r, as in [7833].

f (3755) From the similarity of the forms and positions of the orbits [7831c], it follows
that the co-ordinates x, y, z, must have the same relation to the co-ordinates x-{-Sx,
y-{-6t/, ^-f"^^:, respectively, as the radii r to r-\-5r, or 1 to 1+^w' [7833]; hence we
easily obtain the expressions of x-\-8x, y-\-Sy, z-{-5z [7834]. To these we must add the
parts m'x', m'y', mV [7835], respectively corresponding to X, Y, Z [126, 127] ; observing

or m'x' nearly ; because the value of

l-f-m''

that A= ri261, becomes m this case

x relative to the sun's centre is 0, and the sun's mass is 1 [7804]. In like manner we get
Y-=m'y', Z=m'z', from [127]. These parts being added respectively to those in
[7834], we obtain the co-ordinates x-\-Sx, y-\-^y, z-\-^z [7836], referred to the sun's
centre.

[783lo]
[78316]

[7831c ]
[7831rf]

[7831c]
[7831/-]

[7834a]

[78345]

[7834c]
[7834<i]

368 PERTURBATIONS OF A COMET, [Mec. Cil.

which agree with those in [7825 — 7827] ; and as these co-ordinates contain
six arbitrary quantities [7813'], they satisfy completely the differential
[7836'] equations of the comet's motion, when we suppose R to be equal to the
following function R^ ; *

[7837] J?^ = _- ^ _ m'.(xx'+yy'+zz').(^^ — ^^ .

This being premised we shall suppose,

[7838] R'^Rj^^^ rn',(x:^+yi/+zz'),(^^ - i) ;

[7839] a .r = -i- m'.a: + m V+ 6 x^ ;

[7840] 6y = ±m'.y-\-m'y'i- 6y, ;

[7841] sz = ^m'.z-\-m'z'-{-SZi ;

and the differential equations in 6Xy Sy, 6z [7810 — 7812] will give,t

(7837„] *^''''^ From r=4Aq^^ [78176], we obtain (£)=^^= = f;

substituting this in the partial differential of the assumed value of R [7837] ; which, for the
[7837a'] saJce of distinctness, we shall denote by R^, though it is not so accented by the author;

we get,

[78376] (fr) = !^^ _ «V.(i _ 1) + 3m'.(,xx'+y,y+z^). ^ ,

(tin \
— J, which is used in [7817, 7818]; producing the

parts of the co-ordinates of the comet's orbit contained in [7825 — 7827], or those in [7836].
[7837c] The expression of 7?^ [7837] does not contain the whole of the function i2 [7802], on
[7837rf] account of the terms which are neglected in [7816'] ; and if we suppose that these small

and neglected terms of R are equal to R', we shall have,
[7837e] R = R,-{-R', or R' = R—R,,

Substituting R^ [7837] in this value of R', we get [7838]. If we now suppose that the
[7837/*] quantity R' has the effect to augment the values of 6x, 8y, Sz, by the terms

6x^, (Jy,, Sz^, respectively, their complete values will be as in [7839 — 7841].

t(3757) Substituting R=R,-{-R' [7837e] in [7810], we get,

dd.6x Sx Sx.Sr /dR\ /dR\
[78420] ■ " = ~rf^+;5-— -^r' + l^-rf^y + V-^J-

If we substitute the values of 6x, 5y, 5z [7839-7841], also 5r-\-5r^ [7818a, 7845] for 6r,
we shall find that the parts which are independent of Sx^ , Sy^ , Sz^ , are destroyed by the

[78425] terms arising from (■t-')» as is evident from the calculations in [7818 — 7827J ; therefore m
these terms may be neglected ; and the remaining quantities, depending on Sx^, 6y^, 8z^,

IX. i. ^^3.] FORMULAS FOR COMPUTING ^ ^ If |? ^ ^ £f . ggg

du du du du du du du

dd.Sxi Sec I Sx.Sr^ , /^dR\

dt^ + TT -4- + y^J

Funda-
mental

^ ^ u^u^^^ ,^__ uu,.u.^ 4. r!:fL\' \ [7842]

equations.

Third

form.

= ^-^-,|-^ + (-),.)(B)

_o = '^+\$^-^ + (f>)

In these equations Sti is what 6r [7818a] becomes, when we change
dx, 6y, 5Zf into 6X1, 8yi, 6Zi. These equations differ from those in
[7810 — 7812] only by the change of R into R'. They may be used with
advantage in the calculation of the perturbations of the comet, in the superior
part of its orbit; because R' is then very small [7816', 7837c?]. variations

[7845]

of the
elements
of the

3. We shall now consider the variations of the elements of the orbit. We comet's

•^ *^ ^ ^ orbit.

shall take for the fixed plane the primitive orbit of the comet, ivhich permits us [7846]

to neglect the square of z, being of the order of the square of the disturbing v^q^q^

force. If we suppose, as in [1022, 1010'],

h = c.sin.TO ; I = ccos-^s ; [7847]

e = the ratio of the excentricity of the orbit to the semi-major axis ; [7848]

zs = the longitude of the perihelion, counted from the axis of x ; [7848']

we shall have, as in [1176, &c.],*

dh = dx. \ X. (f )-2/.(f ) \ + Qcdy-ydx).(^-^^ ; [7849]

Sr^, R', will produce the equation [7842]. In like manner we may deduce [7843, 7844]

from [7811, 7812]; or they may be more easily derived from [7842], by changing '■ ^■'

reciprocally x into y, or x into z.

[7849a]

* (3758) Neglecting the square of the disturbing forces, as in [1253'], we shall have,
from [1254], ^e.smM=f', (xe.cos.z* =/; substituting the values [7847], we get
^h=f', i^l=f', whose differentials are ixdh = df', [Ldl=df\ and since df, df [78496]
[1257], or dh, dl, are of the order of the disturbing forces R, we may, by neglecting the
square of these forces, put fji.= l [7804]; and then the differential equations [78496] [73490]
become dh-=df', dl=df. Substituting now the values of df, df [1116], and
neglecting the terms containing z or dz, multiplied by the partial differentials of i?,
because they are of the order of the square of the disturbing masses, they become as in [7849e]
[7849, 7850].

VOL. IV. 93

370 PERTURBATIONS OF A COMET. [Mec. Cel.

These two equations give the values of de, dis ; for we have,*
[7851] de = fZ^.sin.«4"^^«cos.w ;

[7852] ed^ = dh.cos.zs — dLsin.'a ;

and if, for greater simplicity, we take the line of apsides for the axis of x,

[/oS*]

we shall obtain,
[7853] de= dl; edzi = dh.

The equations of the elliptical motion give, as in [605', 606, &c.],t

[7854] fndt-{- £ — ra = u — c.sin.w ;

[7855] r = a.(l — e.cos.u) ;

Elliptical y- — j 1

;;- tang.i.(.-) = \/l±| .tang.it. ; [ (O)

[7857] ^ = ^-

Supposing the symbols to be represented by.

* (3759) The differentials of [7847] give,
[7851a] dh = <?e.sin.w-|-e«?«.cos.t3 ; dl = de.cos.zi — edvi.sin.zi.

Multiplying this value of dh by sin.w, and that of dl by cos.'zar, then taking the sura of
[78516] ^jjg products, we get, by a slight reduction, the value of de [7851]. Again, multiplying

dhf dl [7851a], by cos.w, — sm.zi, respectively, and taking the sum of the products,
[7851c] we get the expression of edixs [7852]. Now in [1188'J the longitude of the perihelion is

-a, and the longitude of the disturbed body m is v ; both being counted from the same axis of
'■ •' a:, as in [500", &ic. 7846] ; so that if we suppose this axis to coincide with the perihelion,
[7851c] as in [7852'], we shall have « = 0; then the expressions [7851,7852] become as in

[7853]; and [7847] changes into h=0, l=e. Lastly, the integrals of [7853] give
^'^^^^•^^ 8e=z6I, e57a=5A; which are used in [7978].

[7854a]

t (3760) The equations [7854 — 7857] are the same as in [605', 606], changing v into
V — rgj and nt into fndt-{-s — «, to conform to the present notation ; as is evident by
comparing the notation in [602'", Sic] with that in [7858, 7859] ; observing that the

[78546] suijstitution of the value of n [605'] in [601], multiplied by n, gives ndt=du.{l-e.cos.u),
whose integral, considering n as variable, is as in [7854] ; s — w being added to complete
the integral, being equivalent to nT in the equation [602], where n is considered as

[7854c] ^jQjjgtant. Comparing the results of the present notation with those in [606, &c.], we
perceive the correctness of the definitions [7859, &c.].

da dn ds de diti dy dd ^

IX.i.^3.] FORMULAS FOR COMPUTING -, -, ^-, ^, ^, f^, -. 371

fndt-{- £ = the mean longitude of the comet ; [7858]

/wJ^-f s-* = the mean anomaly of the comet, counted from the [7858'j

perihelion = V [7903'] ;

1) — -a = the true anomaly of the comet, counted from the perihelion ; [7859]

u = the excentric anomaly of the comet, counted from the perihelion ; [7860]

X, y, are the co-ordinates of the comet x ; the axis of x being the line [786(y]

drawn from the focus to the perihelion ;

then we shall have,*

X = r.cos,(v — w) ; y = r.s\n.(v — w). [7861]

Hence the equations [7855, 7856] give,t

x=:a.(cos,u — e), or cos.'W=e-| — ; ^ [7862]

y = ayi—e^.sin.w, or sin.w = -7r~~i^' [7862'(

We shall suppose that,

X = the inclination of the orbit of the planet m', to that of the comet ; [7863]

y = the longitude of the ascending node of the orbit of the planet m' upon [7864]

the comet's orbit, counted from the perihelion ;
v' = the angle which the radius vector r' of the disturbing planet makes [7865]

with the line of nodes ;
then we have,t

* (3761) The radius vector r forms the angle v — « with the axis of a?, as is evident
from [7859, 7860'] ; hence we easily deduce the values of x, y [7861].

t (3762) The equations [7855, 7856] are equivalent to those of the elliptical motion ryggoai
in [603, 604, 605] ; from which we have deduced the values of cos.v, sin.v [735] ; and
if we change v into v — zs, to conform to the present notation [7854a], we shall have, from
[735], by multiplying by r,

r.cos.(t; — w) = a.cos.w — ae ; r.sin.(i; — «) = a.^i— e^.sin.M ; [78626]

substituting these expressions in [7861], we get [7862, 7862'].

X (3763) The expressions of x' i/, s/ [7866—7868], are found like those in
[7742 — 7744], and they may be derived from these last formulas by changing [7866a]
V, r, X, Y, Zj A, into v', r', a?', y', «', y, respectively; X, X', T', Z', being
unchanged.

372 PERTURBATIONS OF A COMET. [Mec. Gel.

[7866] a/ = /.COS. ^.cos.z?' — /.cos.x.sin. y.s'm.i/ ;

[7867] y' = r'.sin. y.cos.tj'+r'.cos.x.cos. y.sin,v' ;

[7868] zf = r'.sin.x.sin.v'.

The value of R [7802] gives,

[7870] putting,* /= {/{x'—xy-{-{i/—y)^-\-{2f—zy. In like manner we have,

Variations

efeSents. This beiug premised, we find that the value of dl [7850] gives,t

^^ de = — m'.a. [/i — e^.du.cos.u.{xy' — yoif}. J — — — i

[7872] ^ / (i/—v)'>

— m'.a^\/r=r?.cZw.(l— c.cos.w). } ^ — ^^-^ S ;

in like manner, the value of dh [7849] gives,

ed^ = — m'Mdu.sm.u.(xy' — yx').l — — — j

-\-m'.a^^l—e^»du.(l—'€.cos.u). < ^ — Z" S .

We have as in [1177], in the variable ellipsis, observing that fA = l very
nearly [7804],
[7875] d. — — 2AR ;

the differential characteristic d refers only to the co-ordinates of m [916'].

* (3764) It is evident, from [949', 7801, 7801'], that / = ^(:i^—x^^{i/-yf + {zf-zf
expresses the distance of the comet m from the disturbing planet m'.

t (3765) Multiplying [7869] by —y, and [7871] by x, then taking the sum of the
products, we get, by neglecting the terras which destroy each other,

[7872a] ^.(^) -y{^) = m'-{^y'-y^'h l^-jh]-

The differentials of a?, y [7862, 7862'J, give dx= — adu.sm.u; <?y =a.y/i —e2.du.cosM ;
'• ■' substituting all these values, we get,

[7872c ] ^^y — y^^ = a^'\/l—e^.du.{{cos.u — e).cos.M-{-sin.^w| = a^.\/l—e^.du. \ 1 — c.cos.w| .

Substituting [7872a, 6, c, 7871] in dl=de [7850, 7853], gives [7872] ; and in like
[7872i] manner, by substituting [7872a, 6, c 7869] in dh^ed-a [7849,7853], we get [7873].

dxa
[7873]

[7874]

da dn ds de dvs dy d6

IX. i. §3.] FORMULAS FOR COMPUTING T" » T » T » 3^ » 3^» X » T" • 373
■* ^ du du du du du du du '^

If we neglect the square of z, we shall have,*

consequently,

dR = — m'.adu,sm,u, \— — ^ "Z >

^ C «/ /"i/ y\ -i [7877]

+w'.a.v/L=:?.c^w.cos.w. ? ^ — ^^ ^^ S .

Hence we deduce, as in [78766],

da = 2m'.a^du.sin.u. < -^ — ' "Z^ [

( i/ (J y\

— 2m'.a^i/i— e2.<?w.cos.w . < -^ — ^^-r-^

[7878]

Then we have, as in [1181], by putting m- = 1 [7874],

dn = ^anAR ,• [7879]

consequently,t

/n</^ = ISIt +3f(ndt.fadR) ; [7880]

N being a constant quantity. Therefore we shall have, by the preceding
formulas f the variations of the excentricity and of the perihelion of the orbit; [7881]
also those of the greater axis, and of the mean motion of the comet.

[7876a]

* (3766) The differential of R [7802] relative to the characteristic d, neglecting z^
zdz, as in [7846'], gives the expression [7876] ; whence we get [7876'], by merely
changing the arrangement of the terms. Substituting in [7876'] the values of dx, dy
[78726], we obtain [7877]. Developing the first member of [7875] it becomes i-'°'^J
— a~^da = 2.djR ; multiplying this by — a^, and substituting the value of dR [7877]
we get the expression of da [7878].

t (3767) The integral of [7879] is n = JV-\-3fanAR; the constant quantity N
being added to complete the integral. Multiplying this by dt, and again integrating, we
get fndt = JVt-\-3fdt.fan.dR. Now a, n differ from their mean values by quantities of [78806]
the order of the disturbing forces, and JR is of the same order ; so that by neglecting [7880c]
quantities of the order of the square of these forces, we may bring n from under the sign j^aaoji
of integration, and put the preceding expression [78806] under the form [7880]. The
differential of this expression being divided by dt, gives n = JV-\-Snfa.dR ; and by [7880^']
neglecting terms of the order of the square of the disturbing force, it may be put under the r^ggQ, i
form n = JV.\l-{-3fa.dR}, which is used in [7901].

VOL. IV. 94

374 PERTURBATIONS OF A COMET. [M6c. C61.

To obtain the variation of the epoch s, we shall observe that, if the ellipsis
be invariable, the equation [7854] will give, by taking its differential,

[7882] ndt == c?w.(l— e.cos.w).

This equation holds good in the variable ellipsis [1167", &c.] ; therefore
we have,*

[7883] dz — dTS=zdu.(\ — e.cos.w) — dfe.sin.M;

supposing u to vary in this last equation only by reason of the variations of
the elements e and « [7885], whilst in the former case [7882] it varies
only with the variation of the time t. The equation [7856] gives, by
supposing £, xrf, to be the only variable quantities,!

[7883']

[7883a]
[78836]

* (3768) It appears from [7856] that m is a function of «, c, zrf ; and in the invariable
ellipsis V, t, are considered as the variable quantities ; but in the variable ellipsis we must
also consider e, w, as variable; so that u will vary with e, «, as in [7883']. Now taking
the differential of [7854], considering all as variable, and then subtracting from the result
the expression [7882], we get [7883], in which du varies only with e, zs, as in
[7883', 7885].

t (3769) The equation [7884] is the double of the differential of [7856], considering
u, e, TO, as variable. Adding 1 to the square of [7856], and substituting

[r884a] l+t^„^.^.^,-„) = —L__, i+,a„s.^J„=_i_ [34'" In..].

we get,

1 1 , 2e «,-

[78845] cosH-{v-^) = ^^^^ + jiTe -ta^S-'^"'

Substituting this in [7884], then multiplying by cos.^^u, and reducing by means of the

expressions [34', 31, 1] Int., namely,
[7884c] cos.2|M.tang.|w = cos.|M.sin.^M = ^.sin.w ; (cos.^M.tang.|M)2 = (sin.^M)2=|.(l-cos.M) ;

we obtain,

J ( 1 I «•(! — cos.u) ") /\A-t I «?c.sin.M

[7884<n -rf«. \ 1+ -iZT- \ = <««■ \/ i±f + (i_,).^i^ -

[7884e] The first member of this equation is easily reduced to the form — d-m. \ '■ — '— [ ;

substituting this, and then multiplying by \ y^_^i , we obtain the value of du [7885].

Finally, substituting this expression of du in [7883], we get,

, , rfw.fl — e.cos.M)2 rfe.sin.w.(l — c.cos.m) , .
[7884/] d^^d^ = — ^^7== 53^^ ^ — f?e.sm.w ;

which is easily reduced to the form [7886] ; and as d-a is known firom [7873], we shall
get dz from [7886].

, , <Z«.ri— e.cos.w)^ de.sm.u.(2 — e^ — e.cos.w)

ds — rfta = — - — . ^ — ^ - ^ ;

V/l— e2 1— e^ '

da dn ds de dm dv d6

IX.i.§3.] FORMULAS FOR COMPUTING :r» 7" » 3" ' T » 7" » /» T • 375

^ ■• du du du du du du du

^ du . y i + e , ^de.tang.^u y^ ^c^j^j^.^

cos.^.{v—vs) — cos.^u'V r:^e "^ (l-.e).v/l=^* !■ ^

Substituting for cos.^i.(u — «) its value, given by the same equation, we

shall have,

, dm. (I — e.cos.7i) de.s'm.u
du — —?^=-^ :. TT • [7885]

^Hence vre deduce, '^^ ^^^ woVl ..^ > .

as

[7886]

this equation determines ds — d^, consequently also the value of ds, [7886']

If we integrate, by quadratures, the differentials of e, «, a, n, s,
[7872, 7873, 7878, 7879, 7886], we shall have, for any time whatever, all
the elements of the comet'' s motion in its orbit ; whence we may obtain the ^'^^'^^"^
position of the comet in its orbit by means of the equations [7854 — 7857],
It will then only remain to determine the situation of the plane of this orbit,
relative to the ecliptic. For this purpose we shall resume the equations [1 173] ;

do = dt. J 3,.(f ) -a;.(f) I ; - [7887]

dd = dt. J z.(f.) _a;.(f ) I ; p888]

If vre put,
(p= the inclination of the comet's orbit to the plane of x, y [1173'] ; [7890]

d = the longitude of the ascending node of the comet's orbit upon the plane [789(ri
of a:, y [1173'];

we shall have, as in [1174, 1175], by using m- = 1 [7874],

tang.<p = ^^ J ■ ; [7891]

tang. 6 = -^ ; j-yggjy^

a.{l—e^) = c'+c'^+c"^. [7892]

When we take, for the fixed plane of xy, the primitive orbit of the comet,

d and c", as well as z, will be of the order of the disturbing forces ; '• ^

therefore by neglecting the square of these forces, and substituting for R its

376 T ' ^v r PERTURBATIONS OF A COMET. [Mec. Gel.

value, we shall have,*
^''''^ g=-m'.a.{cos...-e}..'.(l-i);

C7896] c = v/«.(l— ee).

17,'^. Now we have, as in [7882, 7857, 7874],

[7897] ndt = du.(l — e.cos.u) ;

[7897'] ^"^ "" ^ '

therefore by substitution we get,t

[7894a]

*(3770) The functions (^\ (^Y (~\ are of the order m' [7802]; hence

d(fj dd' [7888, 7889] are of the order m' ; and their integrals d, d', must also be of the
order id, as is evident from the equation [7891] ; and taking into view that, as the primitive
[78946] Qj-ijit of the comet is assumed for the plane of xy^ the angle 9 [7890 or 7891] must be of
the order m', as well as the ordinate z. Hence it is evident that if we neglect terms of the
order m'^, the equations [7888, 7889] will become,

dd fdR\ dd' /dR\

[7894d] Now from [7802] we have (^) = ^— ^^^1^^ using / [7870] ; and by neglecting

[7894c] the terra m'z, which is of the second order, it becomes ( — j = m'.«'.f-^ — -^j ;

substituting this in [7894c], we get,
[7894/] - ==_^'.^^.( - __); _=_^^^___ - j5

then substituting the values of x, y [7862, 7862'] in [7894/], we get [7894, 7895].
[7894g] Lggtiy^ ^g ^2^ ^/2^ a^Q Qf tije q^^q^ Qf the square of the disturbing force [7894a], they may

be neglected in [7892] ; and then taking its square root, we get c [7896].

[7897a]
[78976]
[7897c]

t (3771) From the equation [7897'] we get w=a ^ ; substituting this in [7897], and
then multiplying by a^, we obtain dt:=c^.du.{\ — e.cos.w); dividing this by c [7896],
we obtam — = \ — - . Multiplymg successively the equations [7894, 7895] by

this last expression of — , we get the equations [7898,7899] respectively. Integrating

these equations, we obtain the values of d, d' \ and by substitution in [7891, 7891'J, we
get the inchnation 9, and the longitude of the node ^.

dd m'.a^.du

(l-^e.cos.u).(cos.u—e),z'.Qr, — -^ ;

[7898]
[7898']

[7899]

c \/T^^

— = — m'.a^,du,(\ — e.cos.w).sin.w.2:'/-;3 — ^3 )•

integrating these two equations, we can determine, for any instant of time
whatever, the inclination of the orbit to the fixed plane, and the position
of its nodes,

4. The most important point in the theory of the perturbations of the Retum

motion of a comet, is the difference in the times of two successive returns to the p^jheuon.

perihelion. We shall now show hoio this can be determined, by taking for [7899']

an example the comet of 1682, which was at its perihelion in 1759, and gy^^^i^

putting, T, iV.

T = the interval between the times of passing the perihelion in 1682 [7900]
and 1759.
Then we may determine N by putting,

NT = 2* = the circumference of a circle whose radius is unity [7903a] ; [7900']
and we have, as in [7880e],

n==N.{l-}-3afdR]. ' [790i]

If we commence the integral fdR at the time of passing the perihelion in [790r]

1682, which we shall take for the epoch or origin of the time t, we may epoch of
suppose,

«= i\r.{l-|-69 + 3a/o'di2}; [^^1^0%'!"'] [7902]

6q being an arbitrary constant quantity* Now we have, as in [7858'], [7902']

F=/o'ntZ^ + s— «; [7903]

V being the cometh mean anomaly. Hence we obtain, [7903']

V=Nt.(\-\-^q) +3a.fo'.(Ndt.fo'dR) + b—zs + 6s—6zs ; [7904]

6 s, 5w, being the variations of s and to, from the time of passing the perihelion [7905]
in 1682; s, w, correspond to that epoch when e — « = ; since by hypothesis [7906]

* (3772) By means of this constant quantity 8q [7902'], we are enabled to satisfy the
assumed equation Ji^T= 2* [7900'], as will be seen in [7908, &£c.]. Substituting the [''903a]
value of n [7902] in V[190S], it becomes as in [7904] ; the terms SsSzs being added [79036]
on account of the variation of s — -cf, since the commencement of the epoch. If we
suppose that when t=:T, n becomes equal to N', as in [7910], the expression [7902] [7903c]
will become,

N' = N.{l+Sq-\-Sa.f^AR\ ; [7903d]

which is used in [7917/].

VOL. IV. 95

[7909]

378 PERTURBATIONS OF A COMET. [Mec. C61.

we then have V= [7901', 7903']. Moreover we have supposed, in
[7907] |.^g^^^ 7900'], that t being equal to T, F= 2*, and iVr=2^; therefore
we shall have,*

[7908] = NT.&q-}- 3a.fJ'.(Ndt.fJ'dR) + 6s — 5w ;

the variations 6s, (5w, as w^ell as the double integral, being taken between
the limits ^ = 0, and t = T. This equation gives the value of the constant
quantity 6q; therefore we shall have, at any time whatever, the value of n,
[7902]. This value will give that of the semi-major axis of the orbit, by

Online LibraryPierre Simon LaplaceMécanique céleste (Volume 4) → online text (page 50 of 114)
 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 | Links exchange | RSS feed  Online library ebooksread.com © 2007-2013