Pierre Simon Laplace.

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If we substitute, in this value of 5h [7954], A + — instead of

Sr = -^-+y^^+-^ _ ^ . ^ r.(^9_^y8^^2)^(^^_^3^y4.^^)| [79546]

[7956]
[7956']

= 7 • H ^+{x^+yy') } = i wV + - . {xjif-\-yy') j [7954c ]

dSx = w'. I ^ dx-{-doi/ \ ; dSy== w!. [ ^ dy-\-di/ } . [7954(q

Substituting the values of Sx, Sy^ Sr, dSx, dSy [7951, 7954c, d], in [7953], and
91 transposing the terras so as to get the value of 6h, we obtain, without any reduction,

6h

^ ^ I r rff2 5 ^ t r dt^ S ^ X^r^ r3 ^ rf<a ^

m! .xdx.{\dy-\-dy') m'.xdy.{^dx-\-dxf) m'.dxdy.{^x-{-xf)
rf<2 i«2 rfia •

This expression consists of fifteen terms, and if we compare it with [7954] we shall find

that the third to the tenth terms of [7954] inclusively correspond respectively to the terms

3, 4, 6, 7, 9, 11, 13, 15 of [7954e]. The first and fifth terms of [7954e] destroy each

rfx2 dx^ dx^

other. The second and eighth are i^^'y* TJ + f '"V*";^ =*^V 3;2» ^^ i"^ the first of

[7954]. The three remaining terms of [7954e], namely, the tenth, twelfth and fourteenth,

being added together, give — m'. , as in the second and only remaining terms of

[7954] ; therefore the value of 6h [7954] is equal to that in [7954e].
voL» IV. 98

[7954c]

[7954/]

[7954g]

390 PERTURBATIONS OF A COMET, [Mec. Cd.

[7957] y,^ - x/-^ [7952], we shall get,* ,

Part of

6h

6h=^m'/h + ^^

, (xt/ — x'y) , Ax! (xdy— ydx)

[7958] «..^.. , ^j «**. ^ .... ^^, ^^

Jr""^^ _ , dx (xdi/—y'dx+3fdy—ydx')

^r ' dt' dt

Hence it follows, that to obtain the variation of h, from a given point of the
orbit to another point, so far as it depends on R^ [7956], or on the part

•• ^ of R which is independent of R', it is only necessary to subtract the value
of the second member of the preceding equation [7958], in the first point, from
its value in the second point.

If we change, in the equation [7958], h into I, x into y, xf into y, and
the contrary, we shall obtain the variation of I arising from R^, or from
the part of R which is independent of R' ; hence we get,t

[7960]

^"'J** iT^^>(i \^\ I _'.. [^—VA , _, f'y' [xdy^ydx)

[7961]

depending ^ Jy (xdy' — 'i/dx -\-x'dy—ydx')

"""jR,. ' "* * li ' ~~ dt

dt dt

[7958a]

[79586]

* (3790) Substituting iq [7954] the value of y.~ — x.'^ [7957], also

X T^ xx-\—yv 2^

— = -J = — 13— ? neglecting -^ , as in [7846], it becomes,

m'.xdxdy' m' .xdyix! m'^'.dxdy
rfia d^ -^ »

which, by a different arrangement of the terms, becomes as in [7958]. This value of 6h
£7958c] jg deduced from the general expression of hh [7953], by substituting the values of 5a?, 8y,
6z [7951], which were computed upon the supposition that jR' = 0, or R=R^ [7837e]5
it must therefore necessarily follow, that if we substitute Rz=R^ m [7849], we shall, by
integration, get the same value of Sh as in [7958], as is observed in [7956'J. This is so
£7958d] evident that we have not thought it to be necessary to repeat the calculations, by the
«iethod pointed out in [7956', &c.].

f (3791) If we change a?, x', y, y\ into y, tf, «, x', respectively, we shall find that
17961a] the expression of dh [7849] changes into that of dl [7850], and h [7952] into I [7952c] ;
also ^x into 6y, and 5y into ^x [7951]. Hence it is evident that if we make the same
changes in the expression of 6h [7958], we shall get the value of <5Z, as in [7961] ; the
arrangement of the terms being varied a little so as to make the factors appear in the same
forms as in [7958].

IX.i.<§>7.J FAR FROM THE SUN, DEPENDING ON R^. 391

Subtracting the value of the second member of this equation in a given point ryngoi
of the orbit, from its value in another point, we shall have in that interval the
variation of I depending on R^ [7956], or on the part of R [7949], which
is independent of R'. The variations of h and I give those of e and ztf,
observing that we have,*

e6e = h^h-\-Ul', e^6iz = l6h—h6l. £7963]

We have, in [1171 line 5], f

1 2 (dx^-\-dy^)

- = — — ^S-^ ; f^^3

a r at*

6a 25r 2dx.dSx-\-'ildi/My

which gives,

Substituting the values of ir, ddx, diy, which are given in [7954c, d], ryq^tr/i
we obtain,

^« „ »»' , .^ / i^^'+yy') . o / (dx^+dy^) , ^ , (dxdx'-{-dydif) ^„„^^,

If we substitute in this equation for — -rz-^ its value [79641,

dt^ r a ^ ■" I'HQQQ-i

and multiply by a^, we get, Pan of

r r "i depending

on
1 /?

Hence we may find (J ri, by means of the equation n^ = — [7897'], which '*
gives, as in [6662], ^ ■'

* (3792) The variations of the expressions of A, / [7847], relative to the characteristic
5, give, in like manner as in [7851,7852],

6e = (5A.sin.'3j -|- 6l.cos.-ui ; e5ttf = 8h.cos.zi — 5/.sin.w. [7963a]

Multiplying these by e, and substituting the values of e.sin.'zrf, e.cos.o [7847], we
get [7963] .

t (3793) Putting fA = l [7874] in the last of the equations [1171], and neglecting
dz^j as in [79526], we get the expression [7964]. Its variation relative to the
characteristic 6, gives [7965], by changing its signs ; and we may observe that terms of

the order m'^ are neglected in this expression of — . Substituting in [7965] the

<* [79656]

differentials of the values of 8x, Sy [7954f?], also Sr = ^.m'r-\ -{xxf-^-yy') [7954c], we

get [7966]. Multiplying this by a^, and substituting f .mV. ^ — Zl_£_l=:|. _ fm'a [7965c]

[7966'], we get, by a slight reduction, the expression [7967].

392 PERTURBATIONS OF A COMET, [Mec. Cel.

8n Sa

n -* a

[7969]

Part of substituting [7967] in [7969], and multiplying by n, we get,

Sn

[7972]

[«'»] sn=^-^^^+ m'n-3m:an. ^-^^±^ -Sm'.an. ('^-''^ +Jy'^ .

depending j. ' ^ ^^2

on

-R/ Subtracting the values of 6a, 6n [7967,7970], at a given point of the orbit,
[7971] from their values at another point, we shall obtain the variations of a and n
in that interval, arising from R^, or the part of R which is independent
of 72'.

To obtain the variation of the mean anomaly, depending upon the part of
/?, which we have represented by R^ [7837], we shall observe that this
i? variation is equal to* f5n.dt-\-Ss — 6ts, We shall put 6n for the ivhole value
of 6n, at the point oj the orbit where ive begin to consider separately this part
of R ; that is, the value of 6n which results from the preceding
perturbations. We shall have, by supposing t to commence at this point,
[7974] f6n.dt-\-5s-^zi =6n.t+f6'n.dt-]-5s—S,s ; [7972(^]

^'" 6'n being the variation of n, counted from the point mentioned in [7973'],
and depending upon R^ [7837], or upon the part of R ivhich is independent
of R'. We have, as in [7976c],

6'n,dt-\-6t — 5w = / u'n.dt ^ , — ^^^ — ^ \.

«/ I y/l—t^ l—e" )

Integrating by parts the second member of this equation, we obtain,!

[7973']

* (3794) The mean anomaly is expressed by V=fndt-\-s — zs [7858']. Its variation
relative to the characteristic 5, is 5V=f8n.dt-\-Ss — 8tss. The variations being supposed
to commence at the time t=0, when 8n is equal to 6n [7973] ; and the general value
[7972c] of Sn is Sn= 8n -{- 5h [7973, 7975]; substituting this in fSn.dt, we get,
f6n.dt = 87i.t-\-f6'n.dt; hence the value of &V [79726] becomes as in [7974]; the
integrals being supposed to commence at the time when Sn is equal to Sn.

[7972a]
[79726 ]

[7972<1

[7976a]

t (3795) We shall put for brevity,

, ^ (1 — e.cos.«)2 sin.«.f2— c2 — e.cos.uJ

and by substituting these symbols in [7886], we get,
[79766 ] ds — dia = — d-a. <p(u) — de.'iru.

Adding S'n.dt to both members of this equation, then integrating and putting fds=Ss,
[7976c] fd-a^S-a, fde = Se, we get [7976], Now if we integrate [79766] by parts, as in

[1716a], we obtain,

IX.i.§7.] FAR FROM THE SUN, DEPENDING ON It,. 393

^. -, , . . , 5to'.(1 — c.cos.«)2 Se.Bin.u.[2 — c2 — c.cos.m)

Jdn.dt4-Se—S-sj = constant . - ^— ^ ^ -

•^ ' yl—t^ l_c2

[7977]

[7979]

i

/•f d'n \ , rk r «?«-8in.tt.(l-e.cos.«) , 5e.rf«.{l-c.cos.M) (c+2.cos.m) ") ^

+y i - ^"'(i-^-^°^-")+2^^^- — Tfc^ — + — ■ — Y^ — v^^

ndt being, as in [7882], equal to c?w.(l-e.cos.w). We have, as in [7851e,/],

^ = ; 5h = e6vs; 1 = 6; 6l = 5e, [7978]

We shall put m'nq for the value of the preceding expression of Sn, at the
new origin we have assigned to the time t [7973] ; we shall then have for
the value of 6'n [7975], the following expression,

^n = 6n — m'nq\ [7980]

and by substituting the preceding values of &h, 61, we shall find,* [798(r|

6s — S-a = constant — Svs.<p(u) — 8e.'Jr{u)-\-J'Sis.(~ — \du-\-f8e.(-^ — -j.du ; [7976d]

considering (p{u), "^(m), as functions of u only, without noticing the variableness of e,
because it introduces only terms of the order m'^, which are neglected ; moreover from

/*6'n p6'n [7976e]

[7882] we have J 6'n.dt=J — .ndt=J — .rfM.(l— e.cos.w) j adding this to [1916d],

we get,

f6'n.dt-\-5s — Svi = constant — 5ot.9(m) — 5c.t(m) 1

The first line of this expression is the same as the first Une of [7977] ; and the integral in
the second line of [7976/] is the same as that in [7977 line 2] ; observing that, in finding
the differential of the expression ^(u) [7976a], we get, by successive reductions,
d. i sin.u.(2-e2-€.cos.M) | = rfu. J cos.M.(2-c2-e.cos.u) -}- csin.s u\= du. | co3.u.{2-e'^.co9M ) -\- c.(1-cos2m) \ [7976M
= du,i2.co3.u-\-e.{l — 2.cos.2m)— eS.cos.u^ = du.{l — e.cos.u) .(e-f-2.cos.M). [7976i ]

[7976/]
[7976^:];

* (3796) Substituting fe= — , Se = Sl [7978], in the terms in the first line of the

[7981a]

[7981&]

second member of [7977], we get the terms of the second member of [7981 line 1]. Again
if we substitute the expression of 6'n = — m'nq-\-5n [7980], in the first term of
[7977 line 2], we obtain the expression [79816] ; and by using the value of ndt [7882],
in the first terra of its second member, it is reduced to the form [7981c] ;

/ — .du.{l — ccos.m) = —-fm'q.du.{l — e.co3.u)-\- f — ,du.(l — e.cos.w)

= — m'nq.t'j-i — . du. (1 — c.cos.m). [7981c ]

If we now suppose the integral expression in [7977 line 2] to be represented by

^m'nq.t-\-Vji we shall find, by the substitution of [7981c] in [7977 line 2],

TT /•? X, \ ( <5»t 1 c 2.8in.u , - (e4-2.cos.«) ) ,«„«, «

V, =y;/M.(l- e.cos.«). ^ - + e5«. ^^;^== -f k. ' \__^^ ^ ; [7981rf]

VOL. IV. 99

394

PERTURBATIONS OF A COMET,

[Mec. C61.

Variation
of the
mean
anomaly

[7981]

depending
on Rj.

f6'n.at4-6s — isj = constant ^^ — ^^ -—I i

— mnq.t + ' . - - • 2

[7981c]
[7981/]

[7981ffl

I7981A]
[7981i ]

[7981i]
[7981Z ]

[7981m]

[7981n]

[7981o]
[7981;»]

[7981 g]
[7981 r]

and the autlior observes in [7980'] that if we use the values of 8h, 81, 8n [7958, 7951, 7970],
this integral expression will become of the form in the last terra of [7981 line 2], making

V,= ^2 /. '3 ; and it now remains to be proved that the differentials of these two

expressions of F"^, are identical ; or by using the values e8zi==8h, 8e = Sl [7978], we

must prove that the first member of the following equation [7981^] can be reduced to the

same form as its second member, by substituting the values of 8h, 81, 8n [7958,7961, 7970] ;

, ,, X ( ^n , M 2.sin.« , ., (e+2.cos.w) ) wi' , , , . .

The two first terras of 8h, 81, 8n, depending on the unaccented letters x, y, being
substituted in the first member of [7981^], mutually destroy each other. For if we retain
only these two terms, and use also the values h=0, l=e [7978], we shall have from
[7970, 7958, 7961], by neglecting for brevity the factor m', which is common to all
the tenns.

5n 3a

r

81= e^

and if we substitute the values of r, x, y [7855, 7862, 7862^], they will become.

6n
n

8h =

1 — e.cos.u
^1 — c^j.sin.M

; 81

( — 2 — e.cos.u)

-c.cos.u
cos.ti — c

_ (1 — e2).C03.M

e-j- = -— ,

' 1 — e.cos.u 1 — e.cos.u

1 — c.cos.u
Substituting these in the first member of [7981 o-], we get, without any reduction,

du. I ( — 2 — e.cos.u) -|-2.sin.^M -f-cos.M.(e-|-2.cos.w) | ;
and by putting 2.sin.^M-|-2.cos.^M = 2, we find that the terms mutually destroy each
other ; so that we shall now have to notice only those terms of 8n, 8h, 81, which contain
the accented letters x, dx', y', dx/, which we shall successively compute, taking them in
the same order.

First. Noticing only the terms which contain explicitly the finite quantity x', neglecting
for a moment, for brevity, the factor mfx', common to all these terms, and which will be
re-substituted in [79812;], we shall get, from [7970, 7958, 7960], the following expressions ;
6n 2ax

n r3 '

Now we have, in [7862, 7862'], the following values of a?, y, whose differentials are as
in [7981r] ;

x= aA cos.M — e | ; y = a.^l — eS.sin.M ;

xy dx dy

.J y^ ,dy^

IX.i.'§>7.] FAR FROM THE SUN, DEPENDING ON R,, 395

If we subtract the value of the second member of this equation, at the new ryqo,,,
origin of t [7973'], from its value at another point of the orbit, we shall

and if we substitute 1— e.cos.M= — [7855], in the expression of nf?^ [7882], we shall [7983s]

an.dt dt « i • •

get, by using [7897'], rfM= -y- = ^. Substituting this value of du in [79817*], and [7981*']

dividing by dt, we obtain,

dx a* . dy ai.v/rZTTa

-== — -. sm.M ; - == V ^ . cos.w. [7981« ]

Substituting the values [7981g', <] in [7981j?], we get, by a very slight reduction, the first
expressions in [7981m, Uji^^]. The second forms are deduced from the first, respectively,

by substituting in the terms between the braces, in [7981u, v)]^ the value of — [79815] ;

— =—. (e— cos.m) ; [7981w]

^j, ^ «^-vA:^.sin.M^ ^ (cos.2Z-e)+ ~.cos.^ ] = aWl==fj^ , {2.cos.M~e-e.cos.^M} [7981r]

SI = —^ — ■'. } — sin.^w-j .cos.^M > = -^ . I — sin.^M+cos.^w — c.cos.^mJ. [7981t*]

Substituting the expressions [7981m, v, w] in the first member of [7981^], it becomes as in
[7981a:] ; putting sin.% = 1 — cos.^m, and arranging according to the powers of cos.m, we

get the first form of [7981y] ; and by successive reductions, using the value of — [7981s],

and that of </y [798 Ir], we finally obtain the last expression in [798 ly] ;

2

^i^(l_e.cos.«). -.|3.(c-cos-«)-j-2.sin.2M.(2.cos.M-c-€.co3.2M)4-(e-j-2.cos.M).(-sui.2w+cos.at4-c.co8.3M)J [7981ar]

= f?M.(l — e.cos.tt). — . {-cos.M.(l-2e.cos.M-f-e^-cos.^M)| =:-du.{l-e.cos.uf. — .cos.w

, / r\^ a^ , COS.M dy

= — du.[ — ) . — .COS.M = — du. = — o . /.-

dy^
e2

[7981t/]

Connecting this last expression with the factor mW, which was neglected in [7981 o], we

dy [79812]

finally get — T/f^^'^'^'t for the term in the first member of [7981^], depending on

x' ; being the same as that which is given by the author, as in the second member
of [798 Ig-].

Second. We shall now compute the terms depending on dx'. If we retain only these
terms, and neglect for brevity the factor m!dx', which is common to all of them, taking [7982a]
care to re-substitute it at the end of the calculation, in [7982g"], we shall get from
[7970, 7958, 7960],

^ ^aJx^ ,__ {xdy—ydx) ydx Sl=2—^ [79826]

n dfi ' dt^ "*" dt2 » dt^'

5^6 PERTURBATIONS OF A COMET, [Mec. Cel.

have the variation of the mean anomaly during that interval, arising from
R^ [7837], or from the part of R which is independent of R\

[7982c]

Now from the values of a?, y, dx, dy [TOSlgf, r, <], we obtain by successive reductions,
using [7981s],

xdy — ydx = a^,du.\/i — e2. {cos.'u.(cos.w — e) -j- sin.^M|

= a^.du.\/l — c2. 1 1 — e.cos.wj = ar.du.s/l — e^ ;
1 substituting this, and di^ = at^.du^ [7981s'j, together with the values of dx, dy [7981rJ,
we get, from [79826],

[7982rf] ^_3a,sin^ 5A=-. «VlEi! J I. + sin.^wl ; ^^ _ _ ^f^flflil:!!:!!!:!! .

Substituting these last expressions in the first member of [7981_g-], it becomes as in [7982e].
This is successively reduced to the form [7982/*] by putting, in the factor between the

T

braces, sin.^M-{-cos.^M= 1, and 1 — e.cos.u = — [7981^] ; using also y [7981 5] ;

[7982c] (1-e.cos.u). "'^^"'^ 'p^"^*! a +sin-^")-cos.M.(e-t-2.cos.M) ^=(l-e.cos.«). ' ^°'^ .j l-c.cos.w-2.- >

rmnon^i r-, \ a.siii.M C r^ T a.sin.M r sin.u y r-,^^, -,

Umi =(l-e.cos.»).^^.| - ^= - .-^.-=-— =-^5^^, [7981,].

Multiplying this last expression by the factor mfdx', which was neglected in [7982a], we

[7982g] get — 2 /, g > foJ^ the part of the first member of [7981^] depending on dx' ; being

U .r/ JL — 6'*

the same as that given by the author in the second member of [7981^].

Third. We shall now take into consideration the terms which contain y' explichly.
[7982^ Noticing only these terms, and omitting, as above, the common factor m'y', which is
re-substituted in [7982p], we shall get from [7970, 7958,7961],

[7982i] ^ = -_?«^; sh=. - ^ + —; 6Z=^_^.^;

^ ■" n r3 » r3 ~ df2 » r^ dt dt *

dx dv

substituting the values of ^) Vi ^i -ir [7 931 g-, <], they become by successive reductions,
using ^ [7981s],

[7982.] ^ = _WHe!.3in..;

[7982n

dh = — A -(cos.M-e)^-j — . sin.^MV = - . j -(cos.M-e)2-}-(l-e.cos.M).(l-cos.%) j

= ~j.ll — 2.cos.'*M-|-e.cos.w-f-e.cos.^M — e'| ;

F7Qao«.i A7 a^Vl-c^.sin.M C^ . \ 1 *" ? a^.i/T^.sm.u <r, « >

[798^»i] 61 = — iL — _ . j (cos.M-e) -j . cos.w > = — ii-J_| 1 2.cos.M-e-e.cos.*M[.

Substituting the values [7982Zc, ?, m] in the first member of [7981^], it becomes as in
[7982n] ; and the terms between the braces may be reduced to

— 14-2e.cos.if— e^.cos.^M = — (1 — e.cos.w)^ ;

IX. i. ^1.] FAR FROM THE SUN, DEPENDING ON R^. 397

To obtain the variations of the inclination of the orbit and of the node,

[7981'"]
depending upon the same part of R, we must observe that we have, m

[1171 linel],

, xdz — zdx „ ydz — zdy [7982]

~ dt ' ~ dt *

hence we get [7982o], using the value of — [79815], and that of dx [7981r]

J ,, V a2sin.M C —3.(1 — e'^)-\-2.(l — S.cos.^M+e.cos.M+e.cos.^M — e^) >

du.{\ — e.cos.w). 3-7=== .^ ^ >' ' V , ' ' / c

v.y \.—e^ ^-j-(e+2.cos.w).(2.cos.i^^ — e — e.cos.^u) ^

= J«.(l— e.cos.u). ^^^Y=r^^ ' {— (i— ^•cos.m)^} = —du.{l-^e.cos.uf. ^-syj^^p

- / ^ V a^.sin.M adu.sin.u dx

[7982nl

[7982o]

Connecting this last expression with the common factor m'l/ [7982A], we get 3 /- — - ; [7982jp]
as in the third term of the second member of [7981^].

Fourth. The terms of [7970,7958,7961] depending on di/, give, by neglecting in like ri^ggg •.
manner as above, the common factor m'di/, and re-substituting it in [7982i;],

n dt^ ' di^ ' dl^ ^ dti l ^j

Substituting the values [7981g', r, 7982c, (/], they become as in [7982r, s], by using

^ [79815] in [79825];

^n 3a.\/i _ eS.cos.M fr a.sin.M.{cos.tt-cJ

n =- r^.du ' ^^= ^du 5 [7982r]

^^== ^^^^^^' '[^+ cos.w.(cos.w-0 I = '^:^?.|l-2e.cos.w+cos.3u|. [7982.]

Hence the first member of [7981^], depending on these terms, becomes as in [7982^] ;
and by putting sin.^M=l — cos.^m, we get the first expression [7982m]; then by
successive reductions, using a?, r [7981g', s], we get the last form of [7982m] ;

, .. . a C — 3.(1 — e2).cos.M-)-2.sin.2M.(cos.M — e)> r-roao*!

rf«.(l-e.cos.»).^52nvr^-^+(e4.2.cos.«Kl-2lcos.«+cos.^«) \ ' '

= (l-e.cos.«).,-3;^^=. j(l-e.cos.«).(cos.«-e) j ==(l-c.cos.«)^. ^^^^-g, = ^~=^^. [798a«]

Connecting this last expression with the factor ruldy' [7982g'], it becomes 3 /i . 2 ' r7982»l

as in the second member of [7981^]. From what has been said it appears that the second
member of [7981^] is equivalent to the first member of the same equation. Hence it
follows that the equation [7981] is equivalent to that in [7977].

VOL. IV. 100

398 PERTURBATIONS OF A COMET, [Mec. Cel.

'Hence we get,*

[79831 ' x.d5z-\-Sx.dz — z.d8x—-6z.dx

[7984] 6c" =

[7985]

dt
y.dSz-\-Sy.dz — z.dSy — Sz.dy

dt
If we substitute the values of 6Xj 5y, 5z, given in [7825 — 7827], we
shall have,t

[7986] ,c' = |mV+m'.(^^fi^^?=^^=^);

dt

^'/ _ 2 ^V'O.^' {yd^+y'dz—zdiZ—z'dy)
[7987] 6c=^mc-^m. — .

dz
We shall now observe that z, —, d and c", either vanish, or are of the

[7988] order of the disturbing forces [7893] ; therefore, by neglecting the square of

these forces, we shall have,t
[7989] ^'==^'. (^^^z::?;^;

at
[7990] ^" = ^'.(y^^=f:^).

Variations "'''

Liination From these equations ive may deduce, by means of the formulas [7891, 7891'],^

depending

on R,. • ■ ■ '

[7983al * (3797) The differentials of the equations [7982], taken relative to the characteristic
8, give the expressions [7983, 7984].

t (3798) Substituting the values [7825, 7827] in the numerator of [7983], and
reducing by means of the value of cf [7982], it becomes,
[7986a] ^ m'. (xdz-{-xdz — zdx — zdx)-\-m'. {xd2^-{-x*dz — zdx' — z^dx)

= f m'. (xdz — zdx)-{-m'.(xd!^-\-afdz — zdx' — sfdx)
= §m'.c'dt-^m'.{xd2f-\-x'dz — zdx' — sfdx) ;
[79S66] substituting this in [7983] we get [7986]. In like manner we get from [7984], by using
[7826, 7827], the value of 8d' [7987] ; or it is more easily derived from [7986] by
[7986c] changing x into y and y into x; for by this means Sc' [7983] changes into 5c" [7984],
and 8c' [7986] into Sc" [1981].

dz
t (3799) The quantities z, — [7846'J, are of the order m' ; therefore c', c" [7982]

are of the same order ; and by neglecting terms of the order m'^, we may reject
["98851 '"'^' ^'^'' m'.x'dz, m'.zdx, m'.y'dz, m'.zdy', from [7986, 7987] ; by this means they
will become as in [7989, 7990] respectively.

^ (3800) The values of Sd, Sc" [7989, 7990], being substituted for cf, c", respectively,
[7990a] in [7891,7891'], give the corresponding values of 9, ^ ; referred to the plane of the orbit
at the time of the commencement of the integral.

IX.i.<^8.] FAR FROM THE SUN, DEPENDING ON R\ 399

the variations of the inclinations of the orbit and the node, arising from R^ n^m\
in the part of the orbit now under consideration.

8. We can obtain the variations of the elements of the orbit, relative to

the part R' of R, by the formulas in [7872—7920], by changing R

1 [7991]

into R' [7838], in the expressions of dh, dl, d. - , dc', dc",

[7849, 7850, 7875, 7888, 7889], and integrating them by means of
quadratures. In the upper portion of the orbit, R' being very small, the
values of these integrals vs^ill also be very small ; but in this portion, w^here
it is so advantageous to divide R into two parts, we may determine without
quadratures, by means of converging series, the variations of the elements of
the orbit corresponding to R'. For this purpose we shall resume the [7992]
expression of R' [7838] ; and by developing it in a series, we shall have,*

R' = -—3 1 m'. ^ — ^^^ ' ^—^ f m', L-^^-Lj !_J &c. [7993]

Now we have,t

* (3801) If we put for brevity for a moment, to ^^r/.cos.y-^r'^ = X!xf-\-yi/-]-zzf-^T^^ [7993a]
[7817c], we shall have, from [78155],

/2 = (x'—xf-\-{y'—yY'}-{s/—zf = r^—2w. [79936 ]

Substituting this in the second term of R [7802], we obtain [7993c] ; and by development
by the binomial theorem, it becomes as in [7993£Z]. If we substitute in the second term of

w
this development — , the second expression of w [7993a], we shall get, by making a

slight reduction, the expression of R [7993e] ;

m'.{x3^-\-yy'+z2f) , „ ,

^= ^ m.(r^— 2u;)-* ^7993^]

m'.{xx'-{-yy'-\-zz') w' C , . tc , „ tc2 tc3 ")

=:=_-___;n'.(a:aj'+yy'-l-zz').(^-— -j + -^—-'\¥ ^ +^;i- +^- \ ' {7993e]
Substituting this last expression of R in the value of R' [7838], we easily obtain,

and by re-substituting the second value of w [7993a], it becomes as in [7993].

t (3802) If we put, as in [7851e], -orrrO, the expression of a?, y [7861] will become
as in [7994] ; we may also put 2;=0, by neglecting the square of z, as in [7846']. The [7995a]

400 PERTURBATIONS OF A COMET, [Mec. Cel.

[7994} X = r.cos.ij ; y = r.sin.r ; z ^ ;

l-}-e.cos.(t? — -n)
we have also a/, y, /, in functions of sines and cosines of t/ and its
multiples. We must substitute R' lor i?, in the differential expressions of
the elements of the orbit, and then develop them, using the following values
of r^Jy, r'^dv\ which are easily deduced from [1057], by putting
fx=l [7874];

[7996] r'^dv = dU\/a.{\ — e^) ;

[7997] r'^dx) = dt.\/ d .{\—^^) ;

and by this means we shall find that the part of each of these differential
expressions, corresponding to R\ will be expressed by a series of terms of

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