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Pierre Simon Laplace.

. (page 55 of 114)

first of tliese forces x^. â€” , by the second x^. â€” , we get, as in [8028], â€” â€” for the [8027/]

2**> T ^ lilt â€¢f^

ratio of the disturbing force of the sun, to the absolute force of the planet upon the comet.

* (3810) The quantity ^ "~ ^2 [8033], is easily reduced to the form,

^^-^^ ^ i ^'-r) {r'-^r) ^ ^8033a]

420 PERTURBATIONS OF A COMET, [Mec. Cel.

the limit of the sphere of activity of the planet upon the comet, this last

[8033'] . . m' . *

quantity must be very small in comparison with , .^ . We may satisfy

these two conditions by supposing ^ to be a mean proportional between

[8034] â€” and 2. - â€” ^ : which gives, for the radius r' â€” r of the sphere of activity
of the planet,

8

[8035] / â€” r = r.\/^m'2.

The error will be so much the less as the planet's mass is decreased.* The
radius of this sphere of activity may in fact be increased without any sensible
error. For if we resume the equation [8021 ],t

ddx, . (m-\-m').x, ^ ^ x x'

[8036] = -rf^ + p +73-;^;

we shall see that the term â€” j-^ adds to the value of Xi only the double

[8035']

and when / is nearly equal to r, or nearly equal to 2, it becomes ' â–  or â€” -5 â€” ,

[80336] "* r r ro

as in [8033]. The conditions required in [8032,8033'] are, that â– :;^^ rjZI^^' ^Â°^

â€ž,/ S.fr' r\

^â€” â– ' Both these conditions are satisfied by supposing, as in [8034], that

r8033c 1 -T- : 7-7-â€”.^ 'â–  â–  r-r^â€”7, 'â€¢ -^i!â€” â€” ^^ â€¢ because if the first of these terms is much greater than

Louootj ^2 (/ r)^ (jJ rf r^

the second, the third will be much greater than the fourth. From this proportion we get
^'rt's'"' (/ â€” ry = r^.lm'^; whence we easily deduce [8035]. We may remark, that this method
method. ^j. df^t^f-jfiijiifig (fiQ perturbations of the motions of a comet, when approaching very near to

a planet, was first proposed by D^Alembert, in his Opuscules, Tome vi, page 304.

* (381 1 ) The disturbing force of the sun , [8033], bemg divided by the absolute
[8035a] r3 -^

^' 2 ff' ^\3

force of the planet ,- â€” - [8032], gives . / for the ratio of this disturbing force to
the planet's action; but from [8035] we have 2.{r'â€”r)^ = 2r^.m''.{^y ==ii^.m'^ nearly;

1

hence the preceding ratio becomes |m'*, which is decreased by decreasing the mass m',
as in [8035'].

t (3812) The term of [8021] havmg the divisor r3, is ^ii^ = ^ [8020];
substituting this in [8021], it becomes as in [8036].

IX.ii. <Â§.ll.] APPROACHING VERY NEAR TO A PLANET. 421

integral* ffdt.l-^^ â€” â€”\ Now this double integral is very small, when it [8037]

X X

is limited to a small value of t ; for the function -75 ^ is very small, x'

and r' differing but very little from x and r respectively. Therefore we
mai/i in the calculation of the perturbations of a comet lohich approaches very
near to a planet, suppose the planet to have a sphere of activity, in ivhich the
relative motion of the comet is affected only by the planeth attraction ; and
that beyond this point the absolute motion of the comet about the sun is
performed in exactly the same manner as if the sun alone acted upon it.

Formulas

1 1 . We shall now develop this hypothesis, and determine the new elements wheTthe

comet IS

the relative orbit of the comet about the planet, while within the sphere of
that attraction. We have, in [572], the six following equations ;t

of the comefs orbit at the time it passes from the sphere of the planeVs ^phej" o^f

attraction. For this purpose we shall commence with the investigation of the'piane*t.

[8039]

[8040]

[8041]
[8042]
[8043]

V / - ..- y~ J ^3 , ^3 ^3 , ^ L^^^v^j [8037a]

may be reduced to the form = ~+Tâ€” T'; T' being much smaller than T, and [80376]

both being considered as functions of the very small portion of time t, and of constant
quantities. IMultiplying the preceding equation by dt"^, we may put it under the following
integral form, neglecting the constant quantities introduced by these integrations ;
Â«i = â€”ffT.di^-\-ffT'.dt'^ ; so that the part of x^ depending on T', is proportional to
the very small quantity ffT'.dt^, being the same as that in [8037] ; remarking, however,
that the introduction of the small disturbing quantity T', into the equation [80376],
affects also the terms composing the value of T, by terms of the same small order, and [8037e ]
by this means produces other similar variations in the value of x^.

x,.dy,â€”y,.dx,^ ^, x,.dzâ€”z,
^'~~ dt ' ^1- dt

.dx^ .
1

^ // yrdz,â€”z,.dy, ^
""^ - dt

, m'y, {dx,^+dz,^) X,
^' f ^ dt-

Jx,.dy,
dt-

z,.dz,.dy^
dt- '

. m'x, (dy,^-j-dz,^) y,
f ' dt-

.dy,.dx^
dt-

z^.dz^.dx^ ^
dt- '

m' 2m' {dx, - \-dy^ - \-dz,-)
a," f dt^

* ^3813^ If we nut for hrfivltv (â„¢+"Â»>i _ 7

. ^

â€” â€” = T' thp Pnnnflnn FftH.'^fiT

[8037c]
[8037rf]

t (3814) We have, in [78496], f' = ^h, f==iil', and in the present notation
li=zm-{-m' [530'''], or simply (x==m'; because the mass of the comet is small in

VOL* IV. 106

[8040o]

422 PERTURBATIONS OF A COMET, [Mec. Cel.

[8043'] Ci, f/, c/', hi, lij rti, being arbitrary constant quantities.

[8044] & = the longitude of the ascending node of the relative orbit of the

comet, counted from the axis of Xi [585'"] ;
[8045] <p = the inclination of the relative orbit of the comet upon the plane of

Then we shall have, as in [591],

"2

tangJ = -^ ; tang.? =

^ij c/) c/', being given, by means of the equations [8040], in functions of
[8047] J^ jy ^2!

the values of x^, Vi, Zy, ~ , ^, â€” ^, at the time of the entrance of the

at at at

comet into the sphere of activity of the planet ; and as these values are
supposed to be known, we shall have also the values of 6 and ?, at that time.
If we put 1 for the longitude of the projection of the perihelion [593'],
we shall have, as in [594', &c.],*

[8049] tang.i=y.

The semi-major axis a^ is given by means of [8043]. Then we have, as
in [714, &c.].t

m'.aâ€ž(l-^e,^) = 2m'/- ^' - -^^f ;

[8048]

[8050]
[805 J]
[8052]

[8053]

which gives the excentricity Ci. Thus we shall have all the elements of the
relative orbit.

We shall now^ refer the co-ordinates x^ and yi to the line of nodes. If
we suppose Xi, yl, z{, to be these new co-ordinates, we shall have,t

[80406] comparison with that of the planet ; so that we may put /' = m'h, f= nil ; or, for
brevity, /'= h^ , f= Zj. Substituting these values in the equations [572 lines 1, 3, 2, 5],
we get those in [8040,8041,8042,8043] respectively. The accents being placed below
X, y, z, c, cf, c", a, e, in order to conform to the present notation ; also changing r into
/, and making some slight reductions. In the same manner the equations [591] change
into [8046].

[8040c ]
[8040rf]

[8049a] * (3815) Substituting the values f = K, f=l, [8040c], in [594'], we get [8049].

t (3816) We have, in [714], {^a.(lâ€”e^)=2iirâ€” -â€” . Substituting iJ.^m',

[8051a] ' L jÂ» V y r â€ž ^^3

r=f [804tOa,d], and placing the accents below a, e, as in [8040c], we get [8051].

f (3817) The expressions of a?/, y^' [8054, 8055], are found in the same manner as
[8054a] jj^^gg ^^ ^,^ y j-^gg-j^ ^^^ multiplying [8054] by cosJ, and [8055] by â€” sin.^, then

IX.ii.^11.] APPROACHING VERY NEAR TO A PLANET. 423

[8056]
[8057]

[8060']
[8061]

a;/ = a^i.cos.^+z/i.sin.^; [8054]

yl = â€” ari.sin.()+2/i-cos.5 ; [8055]

zl = z^.

We shall now refer the co-ordinates a:/, y{, to the plane of the relative

orbit ; putting a:/', yC, for these new co-ordinates ; and we shall have,*

a:/ = x" ; [8058]

I// = y/'.cos.? ; ^ [8059]

z^ = yi'.s'm.cp, [8060]

Lastly, we shall refer the co-ordinates a;/', y" to the major axis, and

shall put "^ for the longitude of the perihelion counted from the line of nodes.

We shall have, by putting a;/", y/", for these new co-ordinates ; the axis

of x"' being the line drawn from the centre of the planet to its perihelion ; f

Xi" == a:/'.cos.is-f i^/'.sin.-ra ; [8062]

yr = â€” x/'.sin.Â« -I- y/'.cos.w. [8063]

These different equations give, t

taking the sum of the products, we get x^ [80546]. In like manner, by multiplying [8054]
by sin.d, and [8055] by cos.^, then taking the sum of the products, we get y^ [8054c] ;
these values of x^, y^ , will be of use hereafter ;

ojj = x^.cosJ â€” y/.sm.6 ; y, = aj/.sin.d-f-y/.cos.^. [80546 ]

* (3818) Here the axis of a;/' or x{ is not changed; but the ordinate y/' is taken in
the relative orbit, perpendicular to the line of nodes, and is then projected upon the plane [8059a]
of a?/, y/, into the ordinate y/ = y/'.cos.<p, as in [8059] ; and the distance z^ of its
extremity, from the plane of a?/, y/, is evidently represented by Zi=y".sin.cp, as [80596]
in [8060].

t (3819) The transformation, here treated of, consists in changing the rectangular
axes x", y", into the rectangular axes a?/", y/", in the same plane ; so that the angle ^

formed by the axes x", x"\ or by the axes y/', y/", may be represented by w. In
this case the reduction is made in the same manner as in [8054,8055], by changing & into [80626]
zj(j and adding two accents to the symbols x^ , y^ , a?/, y/ ; whence we obtain [8062, 8063].

X (3820) IVIultiplying [8062, 8063] by cos.?, and substituting the values [8058, 8059],

we get,

a?/".cos.(p=a?/.cos.w.cos.(p-f-yi'.sin.'ci ; [8065a]

y/".cos.<p == â€” a:/.sin.Â«.cos.(p-f-y/.cos.-5J. ["80656 ]

Substituting the values of a;/, y/ [8054, 8055], in [8065a, b], we get [8064, 8065]

respectively.

424 PERTURBATIONS OF A COMET, [Mec. Cel.

[8064]
[8065]
[8066]

Xi",cos.cp = Xi-l â€” siti.:3.sinJ+cos.t3.cos.d.cos.(p}

+2/i. { sin.w.cos.^-j-cos.ro.sin.^.cos.itp \ ;

y/".cos.(p = Xi.\ â€” cos.OT.sin.^ â€” sin.Trf.cos.^.cos.tp}

4-3/1. { cos.TO.cos J â€” sin.ts.sin.^.cos.cp} .

Therefore we shall have the values of x^'", y/", corresponding to the time

of the entrance of the comet into the sphere of the planet's activity. We

shall also have, by taking the differentials of these equations, the values of

dec '" dv '"

- i- , ^v- J relative to that point.
dt ^ dt ^ ^

The preceding equations give,*

x^ = Xi"'.{ cos.w.cos.5 â€” sin.TO.sin.5.cos.(p|

-\-yi"'{ â€” sin.:s.cos.5 â€” cos.Â«.sinJ.cos.cp} ;
yi = Xi".{ cos.TO.sin.^+sin.w.cos.^.cos.fp} ) (*S^)
+ ?//". { â€” sin.TO.sin.^-}-cos.'3j.cos.^.cos.(p} ;
[8069] z^ = a*i'".sin.OT.sin.9-l-2//"Â«cos.w.sin.9.

If we denote by x^, y^, x/", &c. the values of Xi, yi, x/", &c. at the
time of the entrance of the comet into the sphere of activity of the planet,
and by the same letters, with two lines at the top, their values when quitting
it, we shall evidently have,t

[8067]

Co-ordi-
nates.

[8068]

[8070]

[8067a]

* (3821) JMultiplying [8062] by cos.-z^r, and [8063] by â€” sin.trf, then adding the
products, we get, by making sonae small reductions, the value of a?/' [80676], or that of
a;/ [8058]. In like manner, by multiplying [8062] by sin.^, and [8063] by cos.-sr, then
adding the products, we get the value of y/' [8067c] ; multiplying this by cos.(p, and
substituting the result in [8059], we get 3// [8067cZ] ;

[80676] a?/' = a;/".cos.-zrfâ€” y/".sin.Â«=a?/; [8058]

[8067c ] y" = a?/".sin.w+y/".cos.TO ;

[8067c;] y/ =a?/".sin.zj.cos.(p-|-y/".cos.zJ.cos.(p.

IMultiplying the value of a:/ [8067^] by cosJ, and that of y,' [8061d] by â€” sin.^, then
taking the sum of the products, we get the value of x^ [80546], as in [8067] . In like
manner, multiplying a?/ [80676] by sin.^, and y/ [8067(/J by cos.^, then taking the
sum of the products, we get y^ [80546], as in [8068]. Finally multiplying [8067c] by

[8067/] gjj^^^^ ^^^ substituting the product in z^ [8060], we get [8069].

[80676]

[8071a]

t (3822) We shall suppose, in the annexed figure 103, that C is the centre of gravity

5

of the planet; DAA' a circle described about the centre C, with the radius r.\/km,'^

[8035], and representing the limit of the sphere of activity of the planet. Within this

[8071&] limit the comet is supposed, as in [8038], to move in an undisturbed orbit ABPA'B', about

lX.ii.<^ll.]

APPROACHING VERY NEAR TO A PLANET.

425

X^

dx^'

dx,"

yl"

=

-yl"'.

dvi"
dt

=

dyr

dt

dt dt '

By means of these equations, we can find, in the first place, the values of

[8071]
[8072]

the planet; and if P be the least distance of the comet from the planet, we shall have
the line drawn through CP for the axis of x"' [8061] ; and the line CY, which is
perpendicular to it, for the axis of y/". The comet enters the sphere of activity at A,
and quits it at A'. For both these points the value of x"' is CE', which is represented

by x"' for the point A, and x^" for the point A') hence we have xl"-:=.xl"y as in

[8071]. Moreover for the first of these points we have AE= â€” y/"; and for the second

point A'E=Jl"', and as AE = A'E, we get ^=â€” ^, as in [8071]. We shall
now suppose that the comet, upon entering the sphere of
activity, can describe the arc AB in the time dt ; and upon -^3-^"^
quitting the sphere of activity can describe the equal arc A'B',
in the same time dt. Then the co-ordinates of the point B

[8071c]

are CF=x,"'-\-dx;" ; BF = ^y;"-\-dy;" ; and those of -Â»

the point S' are CF' = x,'"â€”dx,"' ; B'F' = y,"' + dy,'".

Subtracting respectively from these four expressions the values

^. given in [8071(7, &c.], namely CE=^, AE = â€”'^,

CEz=x"\ A'E = y"', we get the increments of these co-ordinates respectively ; and
ifwe draw the lines BH, A'H', parallel to CP, we shall evidently have, for these
increments, the following expressions ;

EF=HB=1^', AH=d^'; Â£F' = ^'1Â£'==-^'; H'B' = ^,
But by construction we evidently have the triangles AHB, A'H'B', similar and equal;

L^ [807W]

[8071c]

[8071/]
[807]g-]
[8071A]

[8071i ]
[807U]

[807K ]

hence we have HB = A'H\ AH=H'B'; or in symbols [807 IZ], dx^" = â€”dx^"; [8071m]

dy"' =idy"''j dividing these by dt, we get the equations [8072] . Now if we have
â– ^ij Vi) ^1) ^^^ ^heir differentials, at the time the comet enters into the sphere of activity
of the planet, we may thence obtain x^'", y"', and their differentials, by means of

[8064, 8065]; and then <', yj^', and their differentials, from [8071, 8072],
Substituting these last expressions in the formulas [8067 â€” 8069], and in their differentials,
we obtain x^, y^, z^, and their differentials, at the time the comet quits the sphere of
activity of the planet [8074] . Thence we obtain the elements of the elliptical motion
about the sun, after quitting the sphere of activity of the planet, as in [8074', &c.].

VOL. IV. 107

[8071n]
[807101

[8071;>]

426 PERTURBATIONS OF A COMET, [Mec. Gel.

[8074]

[8074'J

[8076]

xl\ yl\ -^, -^, in terms of the values of x,, y^, z^, -^, -J^, -i-',

at the time of entrance of the comet into the sphere of activity of the
planet ; and we may thence deduce, by means of the equations [8067-8069],

[8073] and of their differentials, the values of x^^ y^, z^ -t^, -p, â– â€” , upon

quitting that sphere, in functions of their values at the time of the entrance

into it. Adding to these values those of x', \L z', -â€” , â€”^ -7-,

corresponding to the time w^hen the comet quits the sphere of activity, we

doo fiu dz

shall have the corresponding values of a:, ^, 2:, â€” , -^ , â€” [8020, &c.] ;

dZ CLZ (tl

consequently, by means of the formulas [572 â€” 597'], we can obtain the new
elements of the comet's orbit. To obtain the values of a/, 2/', z', and their
differentials, upon quitting the sphere of activity, we must find the time
required to traverse that sphere, which may be easily done by the formulas
[8075] of the elliptical motion, explained in the third chapter of the second
book [606, &c.]. ~

12. When the variations 5x, 6y, 8z, are very small, as is the case relative
to the motion of the comet of 1170, disturbed by the earth, it will be much
more simple to calculate the alterations of the elements of the orbit, by the
formulas of the preceding chapter. We shall consider the most important of
these variations, namely, thai of the comefs mean motion. We have, by
what has been said,*

[8077] rfn =3Â«n.d/i= 3/n .an. 3m .an. iâ€” -rz .

Y O TO

* (3823) Substituting in dn [7879] the value of R [7802], we get [8077], using/
[7870]. If we retain only the term containing f^, and substitute the differentials of
[8078], namely dx=^dt, dy = ^dt, dz = ydt, we shall get,

\(^x'-x).o.Jf-{y'-y).!B-\-{z'-z).yl

[8077a]

[80776] dn= â€” Sm'.andt.

Now by using the values [8078, 8079, 7870], and the abridged symbols [8081â€”8085],

we obtain successively,
[8077c] [:e~x).^-\iy'-y).^-^z'-z).y=\{A'-A).^-^{B'-B).^-{{C'-C).y\-\-\[^'^^^
[8077rfJ =F+IR;

[8077e] /2 = (x'-a:)2+(y-2/)H-(2'-zP={{^'-.^) + (a'-a).f5H-i(^ - B)+{^'-^)i}M-[(C''-C^
[8077/] = JW -f 2M + Lt\

Substituting [8077c?,/] in dn [8077 J], we get [8080] ; and by integrating, [8086].

IX.ii.<^12.] APPROACHIJNG VERY NEAR TO A PLANET. 427

In the interval of time during which the earth's action is sensible, we may
consider the motions of the planet and comet as being rectilinear. Therefore
if we put,

x=A + ^t; y=B + ^t; z = C + r^- [8078]

x' = A'-{-^'t; y' = B'+^'t; 2f=C'+ri; [8079]

we shall have, by noticing only the terms divided by /^, which is the only

one that can become sensible on account of the smallness of f,

, Sm'.an.(F-\-Ht).dt

the following symbols being used in this equation ;

F = (A'â€”A).^-\-(B'â€”B).^+(C'â€”C).y ; [808i]

H = (a'â€” a).a+(^'â€” /3).|3+(/â€”7).7 ; [8082]

M = (A'â€”Af-\-(B'â€”By-\-(C'â€”Cy ; [8083]

N = (^'_J).(a'_a)+(5'-.B).(f3'-|3)+(C"~C).(/-7) ; [8084]

L = (a'-a)Â«+(|3'-^)2+(/â€” ,)^. [8085]

Hence we shall have, by integrating [8080],

^"=-^'Â»'" /(^+S+lV)i - 'Â«Â»^^'

The integral must be taken during the whole time in which the action of

the planet upon the comet is sensible. Before and after this time the

distance \/M-\- 2Nt -\-Lt% of the comet from the planet, is considerable ; [8087]

and then the elements of the preceding integral become insensible, so that

it may be taken from t == â€” oo to i = oo, which gives,* [8087']

* (3824) We may easily prove the correctness of the following integral [80886], by
taking the differential of its second member ; then reducing all the terms to the common
denominator {LM â€” N^).{M-{-2Nt-\-Lf)^ ; since by this means the coefficient of t^, in [8O880]
the numerator, will vanish ; and the other terms of the numerator will become
{LMâ€”N^).{F-{-Ht).dt.

f dt.{F-{-Ht) {FJV-HM)â€”{HM^FL).t .

^M+^M-^Lt^)l = iLM-J^).iMi.2m+L^f +constant. [80886]

Multiplying [80886] by â€” Sm'.a7i, we get the expression of dn [8086] ; and if we

FJVâ€”TIM

substitute ^/{M+^Nt-i-Lt^)^/ [8077/]; also for brevity P = ^^l-^, [8088c]

^ HJ\r-FL

y = vir. â€” 7^ ; we shall get,

6n = Sm'.an. ^â€”j-\-q.j-{. constant. I . [8088rf]

The symbol / represents the distance of the comet from the planet [8025], which is [8088rf']

JiJ\

428 PERTURBATIONS OF A COMET, [Mec. Gel.

6m'.anAHNâ€”FL\

If we put f for the shortest distance of the planet from the comet, %oe

[8089] r Â»/ / ^ i /

shall have*
[8090] /'2.L = LMâ€”N^ ;

therefore,

[8091] ,^ = _4_____i,

[8093] we may here observe that \/L represents the relative velocity of the comet.\

always positive, while t varies from â€” oo to -j-co ; and at each of these limits the

P

quantity â€” â€” vanishes from the integral [SOSS^Z] ; so that it becomes,

[8088e] 6n=:3m'.an. \ Q-y + constant > .

IMow when / is very large, the terms depending on M, Nt [8077/], become very small
FROSSn ^^ comparison with Lt^ ; so that we shall have approximatively f=\/L^, or

/== =f t.\/L [8077/] ; the upper sign being used when i is negative ; the lower sign
[80885-] y^l^Q^ f j-g positive [8088cZ'J 5 so that when t = â€” cx) , we shall have / infinite, and

[8083^] â€” = . Substituting this in ^n [8088e], and taking the constant quantity so as to

/ /^

make the integral vanish at this limit < = â€” 00, we shall have constant = Q. -â€” ;
[8088i] whence 5w = 3m'.an.Q. 5 â€” + â€” ? [8088e]. At the other limit <=:-}- 00, we have

/= t.\/L [8088/], or â€” â€¢ = â€” ; substituting this in 5n [8088^], we finally obtain
5/1 = 6w'.an. Q. â€” ; which is the same as in [8088], using Q [8088c].

*.(3825) The least value of / is found by putting the differential of p [8077/]

JV*
[8090a] equal to nothing, which gives = 2?fdt-\-2Lt.dt ; whence t = â€” â€” . Substituting this

in /^ [8077/], and then multiplying by L, we get [8090]. Lastly, by substituting this
value of LMâ€”JY^ in the denominator of [8088], we get [8091].

f (3826) The velocity of the comet, reduced to a direction parallel to the axis of x,
[8092a] is â€” = a [8078]; that of the planet is -t-=^' [8079]; their difference, a'â€” a, is the

[80926 ] relative velocity of the bodies reduced to the same direction. In like manner the relative
velocity, in a direction parallel to the axis of y, is ^' â€” ^ ; and in a direction parallel to
the axis of z, is 7' â€” y. The whole relative velocity is equal to the square root of the

[80d2d] sum of the squares of these three expressions [39'J, or /(a' -a)2-j-(|3'â€” (3)2 + {yâ€”yf, which
is represented by \/L in [8085].

X ii.Â§ 13.] APPROACHING VERY NEAR TO A PLANET. 429

13. We shall apply these methods to the motion of the first comet of 1770, Remarka-

- able comet

which 2vas disturbed by the action of the earth and Jupiter. Astronomers Â»Â» i"o.
had made many abortive attempts to reduce its observed motion to the laws
of the parabolic theory. At length Lexell discovered that it described an
ellipsis, in which the duration of the revolution was only 5| years ; and he
found that this orbit satisfied all the observations of the comet. So
remarkable a result could not be admitted but after the most incontestible
proof; and to obtain it, the subject was proposed, as a prize question, by the
National Institute, in order to determine the theory of the comet by a new
discussion of the observations, and an examination of the positions of the
stars with which it had been compared. This has been done by Burckhardt
with the greatest care, in his paper which gained the prize ; and his
researches have produced nearly the same result as those of Lexell ; so that
we cannot now have any doubt relative to this point. A comet whose
revolution is so short, ought frequently to re-appear ; but it was not seen
before 1770, and has never been seen since. To explain this phenomenon
Lexell remaiked, that in 1767 and 1779 this comet approached very near to
Jupiter, whose great attraction could change the perihelion distance of the
comet so as to render it visible in 1770, instead of being invisible as it was
before, and afterwards render it invisible in 1779. But before we can
admit of this explanation we must prove that the same elements of the orbit,
which satisfy the first condition, will also agree equally well with the second ;
or, at least, that it is only necessary to make some very slight alterations,
and such as can be comprised within the limits of those which the attraction
of the planets may have produced, during the interval between 1767 and
1779. Burckhardt, at my request, has willingly undertaken to apply the
preceding formulas to the computation of Jupiter's action upon the comet of
1767 ; he supposed the elements of its orbit, at the moment of quitting the
sphere of activity of Jupiter, to be as follows. [Mem, Acad. 1806, page 20].

Time of passing the perihelion 1770, August 14'"'^0348 ; the day ) .

being supposed to begin at midnight. 5 r809f>i

Place of the ascending node upon the ecliptic in 1770, . . 146^,5327 ; 2 Eiemento

Inclination of the orbit, P,7377 : 3 orwtV

the comet,

Place of the perihelion in 1770, 395Â°, 8525 ; 4 ?i%r'-

Ratio of the excentricity to the semi-major axis, .... 0,785604 ; 5 ptiyVof

â€¢' .; ' 'I Jupiter m

Duration of the sideral revolution, 2050^''^%0d5. 6 "^â€¢

VOL. IV. 108

430 PERTURBATIONS OF A COMET, [Mec. Gel.

r8096'l ^Â® ^^^" ^^^^ "PÂ°" ^^Â® ^^^ ^^ Miy, 1767, at mid-day, for the time when
the comet quitted the sphere of Jupiter's attraction. Using these data, then

[8097] taking for the axis of x the radius vector of Jupiter at that epoch ; the
sun's mean distance from the earth for the unity of distance, and one day for
the element of the time dt ; he has found, for the moment of the comet's
quitting that sphere of activity [8096'], the foUovt'ing numerical values;*
x,== 0,086953; y^ = â€”0,2144740 ; z^ = â€”0,027 1 989 ; 1

dxi = â€”0,001286 ; dy, = 0,0036553 ; dz, = â€”0,00004212. 2