Pierre Simon Laplace.

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from the last of the equations [6975] we get [6997], being the equation of the part of the
surface which is touched by the tangent plane.

rgggg , "^ (3520) Tho equation [6998] can be deduced from the four equations [6994 — 6997],
in the same manner as [6991] is from the four similar equations [6985 — 6988]. The same
may be obtained more simply by derivation from [6991] ; observing that if we change

[699861 '^' ^' '^' ^' ^' ^^^^ ^' — ^' -^'j ^'^ ^'^ ^ — "^' I'^spectively ; the equations
[6985-6988] will change into [6994 — 6997] respectively ; and by making the same
changes in [6991], which was deduced from [6985—6988], we get [6998].

VIII. viii. ^26] ON THE DURATION OF AN ECLIPSE. 213

Therefore bj putting,

R

c—D

~ = \, [lOOOi] .' [6999]

we shall have,*
consequently,

c

= \ ; [7000]

C=^. [7001]

Now if we put, [7002]

•^=m(l-X.)«-^5 [7003]

the equation [6991] will give,t

b2=/2_a^; [7004]

*(3521) Dividing the equation [6998] by [6991], we get (^-^J = f^Y =z\^ j;70ooa]
[6999] ; whose square root is = ± x^ ; whence c = - — r- . Using the u^per [70006]

C J -4— Aj

signs, we obtain the equations [7000, 7001] ; but we may also use the lower signs ; and it
is easy to prove that the former correspond to the equation of the surface of the real [7000c]
shadow ; and the latter to that of the penumbra [6981]. For it is evident that the vertex
of the cone of the penumbra falls between Jupiter and the sun ; so that its distance from
the sun must be less than D ; but the vertex of the cone of the shadow falls beyond
Jupiter, making its distance from the sun's centre g-rert^er than D. Now if we suppose, [7000e].
as in [6992c], that the axis of X is the line drawn from the centre of the sun to that of
Jupiter, it is plain that the vertex of the cone of the shadow, or that of the penumbra, will
be on this line; and we shall have, at the vertex Y=0, Z=0; whence X=c
[6988]. Substituting this in the value of c [70006], we get, for the values of X, l^^^^fi

D

[7000(f|

corresponding to these vertices, X= - — — ■. If we use the upper sign of ^^i, we get [7000g']

X^D, corresponding to the shadow [7000e] ; and the lower sign gives X <CD, [7000A]

corresponding to the penumbra \1000d\. We may observe that the symbol Xj [6999, &c.]

is given without any accent in the original work ; but we have placed the figure 1 below L'OOOt ]

the letter, to distinguish it from the symbols X, X', &z,c. [6343 — 6346]. The same change

t (3522) Substituting the value of c [7001] in [6991], and then dividing by B?,

m . .... [^004a]

^^ S^^ 'T^ ;i = l~l~^^~hh^* Transposing 1, and substituting, in its first member,

the value of P [7003], we get /2= a^-f b^, as in [7004]. Substituting the values of
b, c, [7004,7001], in a; — c = ay + b2; [6958], we get [7006] for the equation of the ^''^^^^^
tangent plane.
VOL. IV. 64

214 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cel.

[7005] and the equation of the tangent plane [6958] will become,

[7006] X = 2iy-irz.\/p^^ .

Equation

tlnge^nt Taking the partial differential of this equation relative to a, we get,*

plane.

[7007] 0=2/— ^^

hence we deduce,
[7008] a =

fy .

[7009] b = v/7^irr2=- /-T-r-r,,

consequently,

[7010]

D r ,

or.

P0"1 (_^_^)- = /^(j,^ + ^^);

Equation Vl Xi

of the ^ '■

[7012] wMch is the equation of the surface of a cone,-\ y and z being nothing at
surface its vertcx. We shall have at this point,

of the

r7007 1 * (3523) The equation [7006] is equivalent to that in [6958], being the equation of
the tangent plane ; and in the same manner as [6960] is deduced from [6953], by taking

[70076] the differential, considering a, b, c as variable, we may find [7007] from [7006] ;
observing that as b, c, have been eliminated, there will remain only the variable quantity
a. Then the differential of [7006], considering a only as variable, gives

[7007c] := da.y—da. [ /j^-^^ ? dividing this by <?a, we get [7007], or y=— pL=.

[7007(f] Squaring this and then deducing the value of a^, we get a [7008] . Substituting this in
r7007 ^004], we get b [7009]. Substituting the value of a [7008] in the equation of the
plane [7006], it becomes as in [7010] ; whose square gives [7011].

[7011a]
[70116]

t (3594) We shall suppose, in the annexed figure, that CAV is the axis, and V the

vertex of the right cone VDGEF. Through any point D of its surface draw the

plane DGEF, perpendicular to the axis CF, and

intersecting the surface in the circle DGEF. We

shall suppose CAV to be the axis of x, CA=x,

CV=.D', and VA=;=D'—x; also AB^y,
[7011c] BD = z; these two lines being drawn parallel to the

axes of y, z, respectively. Then in the rectangular

plane triangle ABD we have,
[701 Id] AD^ = AB^-{-Bn^ = y^-\-z\

VIII. viil. <§.26.] ON THE DURATION OF AN ECLIPSE. 215

X = = distance from the sun to the vertex of the cone ; [7013]

1 — x^

and this expresses the distance from the vertex of the cone to the sun's

centre. Subtracting D from it, we obtain the distance from this vertex

to the centre of Jupiter, which is equal to

- — ~ = distance from Jupiter to the vertex of the cone. [7014]

1 — Xj

Noio in noticing the ellipticity of Jupiter, we shall suppose that the equator

of this planet coincides with the plane of its orbit. The error arising from

this supposition would vanish, if Jupiter were spherical ; it must therefore

be of the order of the product of the ellipticity of Jupiter, by the inclination

of its equator ;* therefore it must be insensible. This being premised, we

shall have, as in [6991 ],t

C = J?.v/l4-a2-}-b2; [7017]

[7015]

[7015']
[7016]

[7011/]

Now by putting tan^.AVD = — , we have, in the rectangular plane triangle VAD, [7011e]

AD == VA.tan^.AVD = -r . {D' — x). Substituting this in the preceding expression of

AD^, and then multiplying by f^, we get (U — x)^ =/^«(^^+^^)j which represents
the equation of a conical surface, whose vertex is V, and agrees with that in [7011], by

putting !>' = - — — . This value of D' being the same as the value of x, or CV, [7011g]
1— Xi

corresponding to the vertex V, where y==0, 2; = ; as is evident by substituting these

values of y, z in [7011] ; whence we get the value of x as in [7013J.

[7015a]

* (3525) If Jupiter were spherical, the shadow would be of the same conical form,
whatever be the position of the axis of revolution of the planet. The greatest possible
difference, when the form of the planet is elliptical, is of the order of the ellipticity p. Now [70156]
it is very evident that, if we obtain the form of the shadow, corresponding to the elliptical
form of the planet, and to the supposition that its axis of revolution is perpendicular to the
plane of its orbit, and then vary the inclination of the axis by a small angle ; this elliptical
part of the shadow p, will vary by a part which may be considered as of the same order
as the angle, or as the sine or tangent of this angle ; so that the whole effect will be of the [7015(i]
order of the product of the ellipticity p by this small angle, as in [7015'j.

t (3526) The equations [6985 — 6988] contain only the co-ordinates of the sun's
surface, and are wholly independent of those of Jupiter, which are found in the equation [7017o]
11=0 [7020,7018]. Therefore the equation [6991], which is deduced from the
equations [6985 — 6988], must also hold good when Jupiter is elliptical ; and by taking its
square root, we get [7017].

216 MOTIONS OF THE SATELLITES OF JUPITER. [M6c. C61.

Ellipsoidal

j5pue?g and the equation of Jupiterh surface will be,*

surface.

[7018] (X'-Df +Y" + (l+p)^(Z'^-i^'=) = ;

[7019] R' being the semi-minor axis of Jupiter. If we put the first member of this

[7020] equation equal to /, we shall have, by what has been said in [701 8/- A],

[7021] Y' + 2i.(X'—D) = 0;

[7022] (\+py.Z'-{-h,(X'—D) = ;

[7023] X'—D = 2iY'+ hZ'+ c— D.
Hence we deduce,t

[7024]

[7018a]

* (3527) The general equation of an ellipsoid, referred to its centre as the origin of
the rectangular co-ordinates x, y, z, is x^ -{-my^ -\- nz^ = 7c'^ [1363]. Its semi-axes,

k k _

[70186] parallel to x, y, z, being Tc, -7=, y= , respectively [1363"]. If we put ^^ = 1,

[7018c] \/n={l-\-p), Jc ={!-{- p) . R' , these semi-axes become {l-\-p).R'j {l-\-p).R' and
R', respectively; R' being the semi-axis of revolution [7019], and (l-f-p) - R' the

L Olodj gqiidforial radius. Substituting the values of m, n, Jc [7018c], in the equation of the

r7018 1 ellipsoid [7018a], it becomes, by making a slight reduction, x'^-{-y^-{-{l-{-pY.{z'^-R'^) = 0;
now changing a?, y, z, into X' — D, Y', Z', to conform to the present notation
[6992a — c], it becomes as in [7018]. Putting this equal to m-', we get,

[7018/] M'' = (X'-D)2-|-r2+(l-{-p)2.(Z'2-i2'2) ;

[7018,] (^f,) = 2.(X'-D); (^,) = 2r; Q = 2Z'.(I-fp).

Substituting these in [6973, 6974], we get [7021, 7022] respectively ; the other equation
[7018A] [^7023] is the same as [6997], being the equation of the part of the surface which is
touched by the plane.

t (3528) We have r' = - a.(X'— D) [7021]; Z' = -h.^-^^ [7022];
t7024a] ' V y V ; L j» (,_f_p)2

[70246]
[7024c]

substituting these in [7018, 7023], we obtain the two following equations respectively;

(X'-D).{l + a'+j-j|^| = c-D.
Dividing the equation [7024c] by the square root of the equation [70246],
iraUd] (X'-X»).^^l+aS+(j^]=(l+p).iJ', we obtain,

VIII. viii. «§.26.] ON THE DURATION OF AN ECLIPSE. 217

Now putting,

il±^ == X, [7000/] ; [7025]

we shall have, by neglecting the square of p,* " [7027]

Multiplying this by {l+?).R, we get c— D = (l+p).ir.ly/^ l+a^+^-^^l , as [7024/]

in the first expression of c — D [7024]. Substituting in its first member the value of c [7024g]
[7017], we get the last member of [7024]. This equation corresponds to the shadow
where X'"^ D [7000A] ; that of the penumbra is easily derived fi-om it by changing the [7024A]
sign of the radicalin [1024:d] ; or, in other words, by changing X' — D mto D — X' ; rynoA-i
which requires that we should change c — D [7024e] into — c-j-D.

* (3529) The notation used in [7025, 7026], is similar to that in [6999, 7003]. Now
if we put for a moment, for brevity, s^ = 1 -j-a^-f-h^ and neglect the square and higher
powers of p, we shall have successively,

1+^''+ (nTf = 1 + a2+ b2— 2pb2 = s^—2ph^ ; [70286]

substituting these in the two last forms of [7024], we get,

(l-}-p).2iV«^-2pb2 = Rs-D; or {1 + p).rAs—^-^1 = Rs—D, [7028c]

Dividing this last equation by R, and substituting the value of X^ [7025], we get,

( pb2) D

pb2 D
(1 __x,).5 + X,. — = ;k . [7028d']

Squaring this last equation, and re-substituting in its first term the value of s^ [7028a],
then dividing by (1 — Xj,)^, we get,

2X 2)2

1+ a" + b^4- ^ ■ pb" = ^ (i_^_ ^ ; or, pme]

The square root of this last equation gives b. < 1 -f- - ^ > = \/f^—sfi ; dividing it by the
coefficient of b, we obtain [7028]. Substituting this value of b, in the expression of c,
[7017], and then making successive reductions, using 1+/^ = p.^., . ,o [7026], we get, [7028g]
VOL. IV. 55

218 MOTIONS OF THE SATELLITES OF JUPITER. [M^c. Gel.

[7028] b = (^1— Y^^ . v/r=72 ;

[7029] c = -— — \p. — . (/'— a^) ;

which gives, for the equation of the plane [7028Z],
[7030] ^ = ai, + (l - ^yz.v^r=?+ -^- ^.(/^-a').

Equation \ 1 \/ A — \ ^

piaSl!" Taking its differential relatively to a only, we get,

Eliminating a, by means of these equations, we shall obtain the equation
of the surface of the shadow. But we may simplify the computation, if we
suppose,

[r032] a = ^+qp.

fv
: I g being the value of a in the spherical hypothesis [7008], we shall

have,*

q

[7028/1]
[7028t ;

= fl.^^l+a^+(l-^^)(/»-a^)|=R.^{l+/=-^.(/'-a=)|

[7028^] =11::^^— '"'?•(/''— ^'')' 3" •

This last expression of c agrees with that in [7029]. Now substituting the values of b, c,
[7028, 7029], in the equation of the tangent plane [6958], it becomes as in [7030]. This
equation is similar to that in [7006], corresponding to the spherical form of Jupiter ; and by
[7028ffi] taking its differential relative to a, then dividing by t?a, we get [7031] ; in the same
manner as we have deduced [7007] from [7006J.

* (3530) Substituting the value of a [7032], in \/{^f^ — a^), and making successive
developments and reductions, rejecting terms of the order p^, we get [70336J.

[7033c ] Multiplying this by 1 — —^ , we obtain [7033] ; and by again multiplying by z, we

1 — Xj

VIII. viii. <§>26.] ON THE DURATION OF AN ECLIPSE. 219

Hence the equation of the tangent plane becomes.

Hence we deduce, S'^he'""

2- \ [7035]

from [7026] we have /= — \y/ _^ — 1 ; observing that the [7036]

gn — ; I
we shall have, very nearly.

radical ought to have the sign — ; because x is less than .* Thus

1— \

get [7033/]. Now multiplying [7032] by y, we get [7033e]. Neglecting the part of
a [7032] containing p, we get a value of a, whence we deduce P—2?= ;

multiplying this by ^^— , we get [7033^]. Now adding — — to the sum of the [7033<fl

first members of [7033e,/,^], we get the second member of [7030], or the value of oo ; [7033rf']
and the term [7033cr] being added to the similar sum of the second members of [7033e,/,^],
becomes as in [7034], by making some slight reductions.

A X,p\ /z2 X,/pz2

\^- TlJ^y-VP-^^ = ^^, -• qpy - (i-xo./y.+ z2 ^' [7033/].

Transposmg - — — in [7034], and squaring the result, we get [7035] : wiiich represents

^—^^ [7Q33/i]

the equation of the shadow of Jupiter, /ree from radicals, except in the last small term of

its second member.

* (3531) The extreme point of the real shadow being at the vertex of the cone, the
greatest value of x must correspond to that vertex ; and if Jupiter be spherical, this [7036a]

distance is - — - [7013] ; therefore, in the case of p = 0, the general value of a?, within

1— Aj

the limits of the shadow, must be less than - — — ; so that x — - — r- must be negative.

l-"— Aj 1 — Aj "

Now if we put p = in [7034], we shall get x — — — = /y/pipiS j therefore / must [70366]

220 MOTIONS OF THE SATELLITES OF JUPITER. [M&. Cel.

[7038] D being much greater than R [6982, 6982"] ; therefore we shall have,*
[7039] B!'.(l-\f f_D_ V_. , ^ „ < R . ) .

Jupiter.

which is the equation of the figure of the shadow of Jupiter. We find by the
same analysis, that the equation of the penumbra is, as in [yOSdg],

be negative, supposing the radical \/y'i^z^ to be positive. Moreover, it is evident from
[7036c] the definitions in [6982, &c., 7025], that — , jr and X,, are very small quantities;

and if we develop the expression of / [7036], according to the powers of — , we shall

[7036d] ijave /= — — + —777^ — + ^-c- ; which, by neglecting the small terms of the

it.(i — Xi) ■iJj

R

order — , becomes as in [7037] ; observing that the symbol \ denotes the ratio of the

[7036e] , , . .

equatorial diameter of Jupiter to that of the sun [7025, 7018a, 6982, &ic.], which is very

[7036/"] small ; so that the expression of / [7037] is of the same order as — , which is very

great in comparison with the neglected terms of the expression [7036cZJ.

[7039a]
[70396]

* (3532) Dividing the equation [7035] by /^ then substituting the value of / [7037],

we get [7039], which represents the equation of the surface of the real shadow of Jupiter.

The equation of the penumbra is obtained in a similar manner ; for the equations

[6983 — 6991] corresponding to the sun's surface, and those in [7017 — 7023] relative to

the surface of Jupiter, require no alteration. But the quantity c — D [7024] changes its

[7039c] signs in the first and third members of [7024], as in [7024A, i]. This produces a

change in the signs of the second members of the equations [7028c], which were deduced

from [7024], as well as in the resulting equation [7028</] ; and this change may evidently

be produced, by supposing the sign of X^ to be changed in [7028^, he.]. Therefore the

equations [7028 — 7035] may be made to correspond to the penumbra, by changing

the sign of X . Moreover, in [7036] we must change the sign of /, and put
[7039rf] J-

/= \ / — — - — - — 1, because x > — — - [7000^, &;c., 70366, he] ; so that for
y^ R^.{l-f-\f -l + Xi

[7039e] the penumbra we must have /=——— — instead of [7037]. This change in the sign

■"•(l+Xi)

of /, produces a change in the sign of the last term of [7035], which contains the divisor
^i/2-|-z2 ; so that in this term the radical may be supposed to change its sign, instead of f^.
Finally, by making these changes in the signs of X^, y/^ipia, in the equation of the
[7039g-] gi^ajQ^ [7039], we get that of the penumbra [7040].

VIII. viii. ■J 26.] ON THE DURATION' OF AN ECLIPSE. 221

We shall now consider a section of Jupiter^s shadow, by a plane
perpendicular to the axis, at the distance r from the centre of the planet.
We shall have in this case, x^ D + r [6982"] ; therefore,* [7041]

We have at first, by taking the square root of [7042], and neglecting terms
of the order p,

v/y"^+?= §-{D\-r.(l-h)} = R. I h-"-'^ \ .

Substituting this approximate value in the small term of [7042], which is
multiplied by p, we obtain, for the equation of the shadow, the following
expression ;

[7042]

[7043]

[7044]

DXx

i^+i)

This equation appertains to an ellipsis, whose ellipticity is f — ^^ —^ . [7045]

^ r.{l-\)

P9 ^,

* (3533) The first member of [7039] may be put under the form — . {D—x.(l—\)],

and then substituting x = D-\-r [7041], it becomes as in [7042]. Neglecting p in [7042a]
[7042], and extracting tlie square root, we get the approximate value of \/i/2 -\-z^ [7043].

Substituting this in the small term containing p in [7042], and putting also R == ■ - —

[7025], we get,

—^-—.lU\-r.^l \)l — y H-^-t-^_^^.pz.) r.{\-^.,) U; [70426]

I'D S

which is easily put under the form [7044] ; observing, that by reducing the last factor of
[70426] to a common denominator, we have,

1 t (3534) We have in [2088], P = a;^-}-X„2y9, for the equation of an ellipsis, whose [7045o]

VOL. IV. 56

222 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cel.

r

[7046] The quantity jr- is very small, even in its greatest value corresponding

to the fourth satellite ; hence we see that this ellipsis, or the figure of the

section of the shadow , is very similar to the generating ellipsis of Jupiter,

[7045A;]. We shall use the following symbols ;
[7047] a == the semi-major axis of the section of the shadow ;

[7047'] p' = the ellipticity of the section of the shadow ;

[7047"] R' = the polar semi-axis of Jupiter's mass [6982] ;

R'.(\-\-f) = the equatorial semi-diameter of Jupiter, [7018c] ; its ellipticity
being p;

then we shall have,

[7048]

rectangular co-ordinates are x, y, greatest semi-axis Tc, and least semi-axis — [2088a].

If we change x into y, and y into z, in order to conform to the present notation, this
[70456] equation will become T<^ = y^-\-\^'Z^ =iy^-^z^-\-[\^ — V).z^. Comparing this with the

equation [7044], we find that they become identical, by putting Tc^ equal to the first
'■ ■' member of [7044], and X^^^ — 1 equal to the coefficient of z^, in the last term of the

second member of [7044]. Hence we get,

V. DXi ^ Z>X,

The greatest semi-axis Ic is the same as a [7048^]. The ellipticity, which is represented

by the difference of the two semi-axes Tc, — , divided by the first of these quantities, is

equal to 1 ^^ ~\ — = \i — ^j nearly; and by using X^^ [7045s], it becomes as

[7045^] ^"

m [7045 or 7049]. Now we have \ = ijs nearly [7547]; substituting this in the

[7045/i] ellipticity of the shadow p' [7049], it becomes nearly p' = p. ( ^ ) ^ p. ( 1 -|- J 0. ^ ) ;

and if we take for r its greatest value, corresponding to the distance al" of the fourth

satellite, we shall have very nearly, from [6785 1, r = d" = D. sin. 1530",864 == xi^. D ;
[7045t] y

hence 10.— = ji2- nearly; consequently the eUipticity p' [7045A], becomes p'=|fp;

[7045*] which differs only jV part from the ellipticity p of Jupiter's surface [7048], as
in [7046] .

i

VIII. viii. <§.26] ON THE DURATION OF AN ECLIPSE. 223

. = (,+P).ie'.Ji_^i;-=^)J,.

[7048'J

P-(l+^) [7049]

p' = — = the ellipticity of the shadow ; Eiiipticuy

r ( 1 \ \ ^ "^ of the

we shall also have, for the equation of the section of Jupiter^ s shadow,* of the see-

and 2a will be the greatest width of this section. [70511

If we suppose \ to be negative, in the values of « and p', the equation
[7050] will become that of the section of the penumbra. f Now the greatest t'^^^^l
width of the penumbra, at the distance r from Jupiter's centre, being, equal
to the difference of the two values of «, relative to the shadow and the
penumbra, it will be given by the following expression [7053] ; R being [7052']
the semi-diameter of the sun [6982] ;

greatest penumbra = (l-fp)./?'.^ = ?^ [7052e]. [7053]

* (3535) Substituting in the first member of [7044] its abridged value a^ [7045^7],
and in the last term of its second member, the symbol p' instead of its value [7049], we
get a2 = i^2_|_^2_^2/.za, or a2-2/2 = ^2 (i_^2/) =^2.(i^^')2^ nearly, as in [7050]. [7050a]
Putting z = 0, we get y=a==tlie semi-major axis, as in [7051].

t (3536) Changing the signs of X,, R, in the equation of the shadow [7039], we get
that of the penumbra [7040]. The equation [7043] becomes,

V/^^^+r^ = /?. X, . ^ 1 + '-^^^^ I . [7052a]

This change in the signs of X^ , R, requires a similar change in the values in [7048', 7049] ;
andwe*shall put a^, ^/, for the vahies corresponding to a, /, respectively; then we [^^0526]
shall have, from [7048', 7049], observing that the value of R' [7025] is not altered,

«, = (I+,).i?.(l+^^j; p.=^-^__.

We shall also have, for the equation of the surface of the penumbra, the following
expression, which is similar to [7050] ;

Subtracting the value of a [7048'] from that of a^ [7052c], we get, by using

Xi

a,_a=(14-/.)./2'.-^ = '^; asin[7053]. [7052e]

[7052c 1

224 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. Cel.

Symbol*.

[7053']
[7054]
[7055]
[7056]

We shall now put,
Z = the height of a satellite, above the orbit of Jupiter, at the moment of

its conjunction ;
r = the distance of the satellite from the centre of Jupiter, at the time of

the eclipse ;
Vi = the angle described by the satellite, by its synodical n^otion, in the

orbit of the planet, from the time of its conjunction ;
X, The axis of x is the projection of the radius vector of the satellite, on

the orbit of Jupiter, at the instant of the conjunction; or, in other

words, it is the continuation of the radius vector of Jupiter's orbit,

at the same instant ;
then we shall have,*
[7057] f = (f — r^).sinAu,.

Hence the equation of the surface of the shadow becomes.

We shall neglect the quantities of the order 2^ and r^.sin.^t?i, which reduces
[7059] this equation to the following form ;

[7060] r^, sin.2^, = a2_(l_|-p')2..22.

Now we have,t

rfZ , ^ . 3 ddZ
- — hi.sm. .v.. — -

[7058]

Equatiooa
of the

[7061] z = Z + sm. Vy. — -\- |.sm A i^i • x^ + &c. ;

* (3537) The satellite being at the distance r from the centre of the planet, and at
the height z above the plane of a?, y, the projection of r upon this plane will be
represented by r^ = \/r-i — z^ ; and as this projected line r^ forms the angle v^ , with the
axis of X [7055,7056] ; and the angle 100^ — v^, with the axis of y; we shall evidently
have y = rj.sin.»j = ^r2 — z-^.sin.Uj ; whose square gives y^ [7057]. Substituting this
in the second of the equations [7050], we get [7058] ; and if we neglect the term
z^.sin.^Vj, on account of its smallness, it becomes as in [7060].

[70576]
[7057c]

t (3538) Considering z Bsa. function of v^ , and developing it according to the powers
[7061o] ^2 djz

of t>i, by Maclaurin's theorem [607a], we get z = Z-}-Vi- — \-iv^^.-— -\-hc.;

substituting Ui = sin.'»j-{"i*sin.^Vi -f- &c. [46 Int.], and neglecting terms of the order
[70616] . . . rfZ rfrfZ

mentioned in [7059], we get the expression of z [7061] ; observing that - — , — — are of

the same order as z, as is evident from the consideration that Z is of the order as [7065] ;

Online LibraryPierre Simon LaplaceMécanique céleste (Volume 4) → online text (page 36 of 114)
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