Xx
get [7549].
t (3679) The first terms of the equations [7533 — 7536] are given in [7541a], and the
second term in [7549] ; substituting these in [7533—7536], we get [7550—7553] [7550a]
respectively. Substituting the values of a, a', a", a'" [6797 — 6800], we get the values of
?' f^^ i'^ f [7554—7557].
312 MOTIONS OF THE SATELLITES OF JUPITFR. [Mec. Cel.
Substituting the values of a, a\ a" a'" [6797 — 6800], we obtain,
L7554] q = 111780";
[7555] q' = 69846" ;
[7556] q" = 43344";
[7557] q'" = 24524" ;
which give the following values for the half duration of the eclipses ;*
[7558] L Satellite, 4945", 87 ;
[7559] II. Satellite, 6205",93;
[7560] III. Satellite, 7801",30 ;
[7561] IV. Satellite, 10271",64.
The observed semi-durations are,
[7562] I. Satellite, 4713"; [7524]
[7563] II. Satellite, 5976"; [7485]
[7564] III. Satellite, 7419"; [7429]
[7565] ^ IV. Satellite, 9890". [7372]
They are all of them less than the calculated semi-durations, which ought
to be the case on account of the disks of the satellites ; for although these
disks are small, they are of a sensible magnitude, when viewed from Jupiter's
centre. A satellite does not therefore disappear at the moment of the
[7566] entrance of the centre of its disk into Jupiter's shadow, and the semi-duration
of the eclipse is diminished by the time which elapses between the entrance
of the centre, and the disappearance of the satellite. It may also be
decreased by the refraction of the sun's light, in passing through Jupiter's
atmosphere ; but it is increased by the penumbra. These various causes
are not however sufficient to account for the difference between the observed
and calculated semi-durations. This will appear by considering the eclipses
|.-_ Q • (3680) The synodical motion of the first satellite is n — M seconds [6025n] in one
Julian year, or 36525000" ; hence the time of describing a synodical arc of one second is
36525000"
— _ . IMultiplying this by the arc q [7554], we get the time [7558J. In like
[75586 ]
[7558c]
, • ^-,^.r^-, • J 1. 36525000" , , . ,-,^^^1 , 36525000" „
manner, the time [7559J is represented by — , — — — . q ; that m [75oOJ by — -^ — —- . q ;
36525000"
lastly, that in [7561] by — - ,,_ .g'". Substituting the values of n — M, &c. [6025n],
we get the values [7558 — ^7561], as given by the author very nearly.
VIII. xvi. § 34.] DURATION OF THE ECLIPSES OF THE SATELLITES. 313
of the first satellite, in which the effects of the penumbra, and of the refracted
light in Jupiter's atmosphere, are but of small moment. To obtain the
width of the disk, viewed from Jupiter's centre, we shall suppose the density [7567]
of the satellite to be the same as that of the planet ; then taking, for unity,
the semi-diameter of Jupiter, we shall find that the apparent semi-diameter
of the satellite, viewed from Jupiter's centre, will be equal to* ^—^
[7568]
Substituting the values of a, m [6797,7162], we shall obtain 2890",93, for
this semi-diameter. This angle being multiplied by the time of the synodical
revolution of the satellite, and the quotient divided by 400°, gives 127",913 r^gggi
for the diminution of the semi-duration of the eclipse, depending upon the
magnitude of its disk. Subtracting this quantity from 4945",870 [7558], [7570]
we get 4817",957 for the calculated semi-duration. This semi-duration is
greater than the observed time [7562], and yet there is reason to believe
that the satellite disappears before it is wholly immersed in the shadow. [7571]
[7568a]
* (3681) Jupiter's mass being 1, and that of the satellite m, their radii will be as
3
1 : ^m, supposing their densities to be equal ; so that Jupiter's radius being 1 [6786],
3
that of the satellite will be \/m. Dividing this by the distance of the satellite from Jupiter
a, we obtain very nearly the sine of the angle under which that radius or half-diameter
3 [75686]
appears when viewed from Jupiter's centre = — , as in [7568]. Substituting the values
of a, m [6797,7162], we find that it corresponds to sin.2890",93, as in [7568] ; observing
that, in the original work, this semi-diameter is erroneously marked 2809",93 j and the [7568c]
whole diameter is given in [7574] equal to 5619"'j86, instead of 5781"',86. The time of
describing the arc 2890"j93, by the synodical motion, may be obtained by the method
36525000"
pointed out in [7569], or by multiplying it by the factor ^-r— [75536], which gives [7568e]
[7568rf]
[7568/]
127",913 for the diminution of the semi-duration of the eclipse, as in [7569] ; and by
doubling it we get 255",826 for the time requisite for the whole disk to enter into ihe
shadow, as in [7574]. Proceeding in the same manner with the other satellites, we have,
/m' \/m!' s/in!" [7568g-]
as in [75686], for their diameters, the expressions 2. — p , 2. -^ , 2. -— ; and by
using the values of a', a", d" [6798—6800], and those of m', m", id" [7163—7165], [7568A]
they become as in the first column of the table in [7575 — 7577]. Multiplying these
expressions, as in [7568e], by 36525000", and then dividing the products by w' — M,
n" — M, ti" — M. [6025n] respectively, we get very nearly the times in the second column [7568&]
of the table [7575—7577].
VOL. IV. 79
314 MOTIONS OF THE SATELLITES OF JUPITER. [Mec. C61.
Hence it appears that we must decrease, by at least y^th part, the assumed
f''^^^^ diameter of Jupiter 120",3704 [6786], so as to reduce it to 118".
If we compute, in the same manner, the disks of the satellites viewed from
[7573] Jupiter's centre, and the times they require to enter perpendicularly into the
shadow, we shall obtain the following results.
Diska of the Satellites, viewed from Jupiter's centre, Times required for the whole disk
at their mean distances from the planet. to enter into the shadow.
[7574] I. Satellite, 5781",86, 255", 826 ;
[7575] . II. Satellite, 4007",30 356",055 ;
[7.576] III. Satellite, 3923",44 702",914;
[7577] IV. Satellite, 1749",04 732",567.
Hence it is easy to determine the times when the satellites, or their shadows,
[7578] enter into or quit the disk of Jupiter. Comparing these times with those
which are observed, we shall obtain the densities of the satellites of Jupiter,
when their masses shall be well known. The observations of these eclipses
of Jupiter by his satellites, may throw much light upon their theory. We
can almost always observe the beginning and end of them ; and these
observations can be made with the satellites and with their shadows, so that
they are really equivalent to four observations ; whilst, in general, we cannot
[7579] observe more than the beginning or the end of an eclipse of a satellite.
This kind of observations, which has been very much neglected, deserves
therefore the utmost attention of astronomers.
VIII.xvii.^35.] DIFFERENTIAL EQUATIONS. 315
CHAPTER XVII.
ON THE SATELLITES OF SATUBN.
[7580]
35. The theory of the satellites of Saturn is very imperfect, since we
have not a sufficient number of observations to determine their elements.
The impossibility of observing their eclipses, and the difficulty of measuring
their elongations from Saturn, have prevented the determination, with much
accuracy, of any of the elements, except the times of revolution and the
mean distances ; and even with respect to these distances there is a degree
of uncertainty, which leaves some doubt upon the value which results for
the mass of this planet. Now as we do not know the values of the
excentricities of the orbits of these satellites, it is impossible to give the [7580']
theory of their perturbations. But there is one phenomenon which deserves
the attention of mathematicians and astronomers, namely, the constant position [7581]
of the orbits of the satellites in the plane of the ring ; excepting, however, the
plane of the orbit of the outer satellite, ivhich varies sensibly from the plane of
the ring. This phenomenon is analogous to that which we have explained,
in the last chapter of the fifth book [3689, &c.], relative to the permanency
of Saturn^s rings in the same plane. We have already observed, in
[3689, 3692, &c.], that both these phenomena depend on the same cause,
namely, the oblateness of Saturn; whose action keeps the rings and the
satellites in the plane of its equator. We shall now explain the reason why
the orbit of the outer satellite varies from that plane in a very sensible
manner.
We shall resume the equation [6295], t
^ dds , r2 C fdR\ ds fdR\ }
If we neglect the ellipticity of the orbit, we shall have r = a, and h^ = a, [7584]
[7581']
[7582]
316 MOTIONS OF THE SATELLITES OF SATURN. [Mec. Cel.
[7584'] [6299a]. Moreover, if we take the primitive orbit of the satellite for the fixed
plane, s will be of the order of the disturbing forces ; therefore by neglecting
the square of these forces, ive may reject the products of s and — by these
forces. Hence the equation [7583] becomes,*
We shall suppose that this equation is for the outer satellite of Saturn, and
we shall proceed to ascertain the corresponding value of R. In the first
Part of ^ ^ ^ .
R place we shall have, by noticing only the sun's action,t
[7587] -'«' D~^ 2IP 2D^
We shall use the following symbols :
[7587'j The axis of x is the line drawn from the centre of Saturn to the ascending
node of the primitive orbit of the satellite upon the orbit of Saturn ;
[7588] X is the inclination of the primitive orbit of the satellite to the orbit
of Saturn;
[7588'] X', y, are the rectangular co-ordinates of the sun's place, referred to the
centre of Saturn, and to the relative orbit of the sun about Saturn ;
the axis of X being the same as that of x ;
[7588"] U is the angle included between the sun's radius vector D, and the axis x
or X.
* (3682) In this case R and (~) [6030,6039, 6042, &c.], are each of the order of
[7586a] ^^^^ ^^
the disturbing masses ; hence (~t~) '~r } is of the order of the square of these masses;
and by rejecting it from [7533], we obtain [7586].
t (3683) The expression of R, corresponding to the action of the sun S upon the
[7587o] outer satellite m of Saturn, as it is given in [7587], is of exactly the same form as that in
[6042e], relative to the sun's action upon Jupiter's satellite m ; the symbols being the same
[75876] ^^ ^^ [6021 — 6033], merely changing what relates to Jupiter and its satellite into the
corresponding quantities relative to Saturn and its satellite respectively. Thus X, Y, Z
r7587cl [^^^^1' i"^ the present article, represent the rectangular co-ordinates of the sun, referred to
the centre of gravity of Saturn as their origin, supposing this centre to be at rest, and the
sun to describe a relative orbit about Saturn. In like manner, x, y, z [6022] represent
[7587rf] t^® rectangular co-ordinates of the outer satellite m, referred to the centre of gravity of
Saturn as their origin, &c.
VI]I.xvii.<§»35.]
Then we have,*
X = a.cos.v
TERMS OF R
Y=Y'.cos.X;
Z = — F.sin.x.
X'=D.cos.U;
T=D.sm.U;
y=a.s\n.v; z=as;
317
[7589]
[7590]
[7591]
[7592]
[7593]
[7594]
JFi^. 96.
s'
-^â– â– :;:^ Sun
^y::^^
Y
/
y
i
* (3684) We shall suppose in ihe annexed figure, that C represents the place of the
centre of the planet Saturn, which is considered as the origin of the co-ordinates. The
plane of the figure is the primitive
orbit of the outer satellite, vvhicli is
taken for the fixed plane of ory,
[7584'] ; the axis of x being the
line CJV, drawn from the centre of
Saturn to the ascending node of the
primitive orbit above the sun's
relative orbit about Saturn, being ^ Lineojnodes, JV or axis of x.
the same as the descending node Saijirn
'of the sun^s orbit below the fixed plane of ocy [7587']. 5 is the place of the sun, whose
rectangular co-ordinates are represented, as in [7537cJ, by
CN= X, NS' = r, .S"*S'= —Z ;
the negative sign being prefixed to Z, because the line NS is supposed to fall below the
plane of xy [7589c?]. We have also, as in [7588'J,
CjY=X', JVS=Y', angle SiV,S" = X.
Now in the rectangular triangle J^S'S, we have NS' = NS.cos.SNS' =^ Y'.cos.'k;
SS' = NS.s\n.SjYS'=Y'.s\v..K. Substituting these in the values of Y, — Z [7589f],
we get [75£0, 7591] respectively. The equation [75S9J is obtained by putting the two
expressions of CA' [7589e, g] equal to each other. Again we have CS = D
[6027,75875], angle SCN=U [75S8"J ; hence in the rectangular triangle CJVS,
we have,
CN= CS.cos.SCN= B.cos.U; JVS=CS.sm.SCJV= D.s\n.U;
and by substituting the values of CN, NS [7589o-], we get [7592,7593] respectively.
If we substitute the values of X\ Y' [7592, 7593] in [7589—7591], we shall get,
X= jD.cos.C7;
Y = D.sin.C/.cos.X;
Z = — D.sin. t/.sin.X.
Finally, the values of x, y, z [7594], are easily deduced from [6034 — 6036], by rejecting
terms of the order s^ [7585], and substiuuing r=aj r'=a' [7584 or 6091] j neglecting
the excentriciiies, which are unknown [7580'].
VOL. IV. 80
[7589a]
[75896]
[7589c]
[7589(fl
[7589e]
[7569/]
[7589^]
[7589A]
[7589t]
[7589ft]
[7589Z ]
[7589m]
[7589n]
[7589o]
[7589J9]
318 MOTIONS OF THE SATELLITES OF SATURN. [Mec. Gel.
[7595]
therefore^ hy retaining in a.f-.-J no other terms than those depending on
sin.z; or cos.ij, ivhich are the only ones producing the secular motions of
the orbit [6298],* we shall have,
ry.Q^ /'dR\ SS.a^ . ^
\as / Si^"*
To determine the part of f'-f-r-)? which depends on the oblateness of
Part of Saturn, we shall observe that this part of R is found, in [6050, 6052],
^ ^^. to be,t
depending '
[7597] R=(p-h)'(y'-i)'
on the
oblateness
of tjatuiu.
[7596c]
* (3685) If we substitute the values of cc, y, z, X, Y, Z [7594, 7589/n— o] in the
first member of [7596/?], it becomes as in its second member ; and its partial differential
relative to s being divided by ds, gives [75966].
[7596a] {a?X+^r4-2:ZP=a2j)2 {cos.t^.cos.t)-|-sin.l7.cos.X.sin.«— s.sin.lZ.sin.xP;
[75966 ] jj — '"•^ ' — L = — 2a^. D^.sin. C^.sin.X . Jcos. i[7.cos.v4-sin. C7.cos.X . sin.w — s.sin. t^.sin.X? .
as *
Now if we put r=a [7534] in the second term of [7587], and then substitute this value
of R in the first member of [7596], we shall find that the first and second terms of [7587]
produce nothing, and the third term becomes like the first member of [75966], multiplied
O c* ^
by — ^-j^ 'j therefore the second member of [75966], being multiplied by the same factor,
gives,
[7596rf] a.( — ] = -^V~-sin.t7.sin.X. Jcos.CT'.cos.u+sin.C/.cos.X.sin.i; — s.sin.L^.sin.xL
\ds J D^ *
In noticing the terms depending on the secular equations, it will only he necessary to retain^
[7596e] ds in [6298], the terms depending on sin.u, cos.u, s. We may neglect the term
depending on s [7596c?], because it is of the order of the disturbing forces [7585], and is
multiplied by the factor - -^ , of the same order. We may also neglect the first term of
[7596/] ^
the second member of [7596^/], containing the factor sin.C7.cos.C/.cos.v = |.sin.2?7.cos.»,
producing angles of the form «^2C7, which differ from sin.r, cos.v [7596^]. The
[7596g-] remaining term of [7596(/J is the second, ' - .sin.X.cos.X.sin.v.sin.^C/; and by substituting
sin.^(7=^ — ^.cos.2C7, and neglecting the term containing 2 ?7, it becomes as in [7596].
t(3686) Substituting 72 = —(5 F [6052] in [6050], we get [7597]; M being the
[75D7a] mass of Saturn [6028,75876]; J?=the radius of the equator of Saturn [6045];
v = the sine of the decHnation of the outer satellite above Saturn's equator [6045', 75876].
VIII. xvil. ^ 35.]
TERMS OF R.
319
M = the mass of Saturn, which is taken for unity, or M = 1 ;
B = ihe mean radius of Saturn's mass, which is taken for unity, or B = \;
7 = the inclination of the primitive orbit of the outer satellite to the
plane of the ring ;
. Y = the distance of the descending node of the primitive orbit of the outer
satellite, relative to the plane of the ring, from the ascending node
of the same primitive orbit upon the orbit of Saturn ; the Jirst of
these nodes being supposed more advanced, according to the order
of the signs, than the second ;
V — Y= the distance of the outer satellite from the descending node of its
orbit upon the ring.
Then we easily find, by neglecting the square of 5,*
v^ = sin.^7.sin.^(«j — ^) — 2s.s\n.'y.cos.y.sin.(y — ^).
[7598]
[7598']
Symbols.
[7599]
[7600]
[7600']
[7601]
[7601a]
* (3687) In the annexed figure 97, C
represents the centre of Saturn, about which y^
is drawn the spherical surface NIHBGPP'. %,
The plane of the sun's relative orbit about
Saturn intersects this surface in the arc JVIN^ ;
the plane of the primitive orbit of the outer
satellite intersects this surface in the arc
JSHB ; and the plane of the rings intersects
it in the arc IHES ; P is the pole of the
orbit JVUB ; and P' the pole of the plane
of the rings IHE. Then from the above
notation we have, by supposing G to be the place of the outer satellite,
JVB = v; JVH=^; HB = v—^ = an^\e BPH, or GPH-,
PP' = y = 2.u^\eEHB', BG = s; PG = lOO°—s;
angle GPP'= GPH-f-100° = v-T+100°; P'G=100°-dec.; cos.FG=v[6045'].
Then in the spherical triangle PGP', we have, from [63] Int., the formula [760 1 A] ;
which, by using the symbols [7601e — g'\, become as in [7601t],
cos.P'G = cos.PG.cos.PP'-fsin.P(?.sin.PP'.cos.GPP' ;
V = sin. 5. COS. 7 — cos. s. sin.7.sin.(« — ^).
Neglecting terms of the order s^, as in [7589;?], we may put sin.s = 5, cos.«=l
[43,44] Int.; substituting these in [760U*], then squaring the result, and neglecting s^,
we get [7601].
[7601c ]
[7601/]
[7601g]
[760U]
[7601t ]
320
MOTIONS OF THE SATELLITES OF SATURN.
[Mec.Cel.
[7602]
[7603]
[7604]
[7605]
Hence we get,*
/dR\ 2.(p— |<p) . . , N
fl.f — j = —^ — - .sm.7.cos.7.sin.(t; — *).
It noio remains to consider the action of the rings and the six inner satellites.
If we consider an inferior satellite, whose radius is r' and mass w', supposing
its orbit to be in the plane of the ring, or in that of the equator of Saturn,
we shall have, as in [6030 line 1], relative to this satellite,
m'. {xx'-\-y]f -f- zsf )
R =
m
If we now take, for the axis of x, the intersection of the plane of the
primitive orbit with that of Saturn's equator or ring, we shall have,t
[7602a] *(3688) Putting r = a [7589p], ]VI = 1, B=l [7598,7598'], in [7597], then
taking its differential relative to 5, which is found in v^ [7601], we get,
Substituting the value of v^ [7601], we obtain [7602].
[7606a]
[76066]
[7606c ]
[7606d]
[7606e]
[7606/]
[7606g]
[7606/i]
[7606t ]
t (3689) The change of the axis of x from the line CiV'[7587'], in fig. 96, page 317,
to the line CH, fig. 97, page 319 [7605], has no effect on the function B [7604] ; as
we have seen in [949^, &,c.], where it is proved that this function R is wholly independent
of the planes of x, y, z; but the values of x, y, z, will differ from those in [7594]. To
obtain the values corresponding to the present case, we shall draw the lines CH, CB,
CG, and upon CG shall take Cg=r. From g let fall upon CB the perpendicular
gb', and from b let fall upon CH the perpendicular bh; then we shall have x=Ch,
y=hb, z=bg. Now in the triangle Cgb, wehave bg=Cg.sin.GCB=r.s'm.GCB=rs
nearly, as in [7608]; also Cb = Cg.cos.GCB=r.cos.GCB, or Cb = r, neglecting
terms of the order s^ [44] Int. Then in the triangle Chb we have Ch = Cb.cos.BCHj
hb=Cb.sm.BCH, or in symbols a; = r.cos.(v-^), y = r.sin.(t;-*^), as in [7606,7607].
We may find the values of x', y', z' [7609—7611], corresponding to the satellite m',
in the same manner as we have found those of X, Y, — Z, for the sun [7589»i — o],
merely changing the sign of Z, because the satellite m' is supposed to be at the point 5,
on the plane of the rings, above the plane of the primitive orbit x, y. In this case we have
the arc HS = v [7612], the angle SHB = '/y and the radius vector of the satellite
equal to r'. Hence it is evident that we may derive the values of x, y', z' [7609-7611],
from those of X, Y, — Z [7539m — o], by changing JD, Z7, X, into ?', v\ y,
respectively.
VIII. xvii.§35.] TERMS OF R, AND ITS DIFFERENTIALS. 321
X = r.cos.(i; — y) ; [7606]
y = r.siii.(v — y) ; [7607]
z = r.s ; [7608]
af = r'.cos.v' ; [7609]
y' = r'.cos.y.s'm.v' ; [7610]
zf = r'.sin.7.sm.v' ; [7611]
v' being the angular distance of the satellite m' from the descending node
of the primitive orbit upon the plane of the ring. If we change r, r' into
//77?\
a, a' respectively, in O'f-^)? rejecting the terms which do not depend [7613]
upon sin. I?, cos.y, or those which are multiplied by 5 [7596e], we shall have,* [7614]
Action
/dR\ m'.a'^a'.sm.y.sm.v' °^^^^
^- ":r = ~" ..,,,20/ 7 ^ . o f ^~^ ^^ — 77^- t7615]
\ds y [w'-j-a-^ — 2aa .cos.(r — f).cos.» — 2aa.cos.7.sin.(u-^).s)n.v p satellites.
If we suppose a' to be small in comparison with «, as is the case with
respect to the inferior satellites and the different points of the ring, we shall [7616]
have, by neglecting terms of the orderf — ,
*(3690) If we use for brevity W=x3f-{-yi/+z;^, also r'^=x^-\-f-{-z^ [60Q3],
/2==a.'2_|_y'2_|_^'2 [6039^], we shall find that the expression of R [7604] will become
m'.W m!
R = ~- ■2-i-r^—2lvii ' Its differential relative to 5, which is found only in Wj [76156]
S'^^' H^j = 7r-Vl^;-^r2 4-r'2_2fr|i-WJ- Now if we substitute the values [7615c]
[7606 — 7611] in TV [761 5fl], it becomes as in [7615d] ; and its partial differential, relative
to s, gives [761 5e],
^ = r/.{cos.(t; — ^).cos.r'-|-co3.7.sia.(t> — ^*-).sin.i>'4-5'Sin.7.sin.i;'] ; [7615<^]
\ds)-'
— —J =rr. sin. 7. sin.« . [7615e]
Substituting [7615e] in [7615c], and neglecting the first term which does not contain v
[7596e], we get,
«.(—-) = — 7-7- — 77— -— 73.r;-'.sin.7.sin.u'. [7615/"]
\ds J |r2-f-r'2— 2^|5 '
Now putting, as in [7589p], r = a, r'=a\ and then substituting the value of W
[7615(/], we get [7615] ; observing that we may neglect s, which occurs in the terra W \- ^^
of the denominator, because it produces only terms of the same order as those which we
have usually neglected.
1(3691) If we put for brevity ?F' = cos.(v— >F).cos.u'-f-cos.7.sin.(v — *).sin.»', we ry/^,^ i
shall find that the expression [7615] becomes, by development,
VOL. IV. 81
322 MOTIONS OF THE SATELLITES OF SATURN. [Mec. Cel.
If we consider the rings as a collection of an infinite number of satellites,
we shall have, in virtue of their mutual action, and of that of the satellites
which are within the orbit of the outer satellite,
[7618]
[7619]
^( — j = — jD.sin.7.cos.7.sm.(iJ — ^) ;
[7620] B being a constant coefficient depending upon the mass and constitution of
the rings, and also upon the masses and mean distances of the satellites from
Jupiter. Now if we put,
[7621] JSr=^'; K' = ^-^^+iB;
[76166] a.(^-^) = - "^•"'"•^''"•^•r"-^\ \ 1- -^^^.W ] -i
■• ■" \dsj (a24-a'2)i ( aS + a'S ^
The first term of [7616c] which does not contain W, is independent of v, and
[7616(/] may therefore be neglected [7614]. We may also neglect the term containing W'^j
because the square of W [7616a], when reduced by [1,6, 31] Int. relative to sin.^v',
sin.v'.cos.v', cos.^u', will become of the form A^-{-B^.sm.2v'-\-C^.cos.2v' ; A^, B^, C^,
[7616e] being independent of v'. This expression of W'^ is multiplied by sin.w', in [7616c];
so that if we reduce the product, by means of [17, 18] Int., it will produce terms of the
[7616/] forms ^*"* .v', ^'"'.3y', &;c., and no term independent of «' ; therefore it cannot be of
either of the forms sin.y or cos.v, which are retained in [7614]. Terms depending on
W^f W*, &c. are neglected, as in [7616], because they are multiplied by — and the
a' . . .
higher powers of — . Hence it is evident, that the only term to be retained in [7616c] is
the second, depending on the first power of W, which gives,
[7616g-]
[7616M «{-) = - ^^^:^. . W .s.n.y.sm.» .
Multiplying W [7616a] by sin.y', and substituting, in the products, the expressions
[7616t] sin.v'.cos.u' = J.sin.2y', s'xn.^v' = ^ — i.cos.2y' [31, 1] Int., then retaining only the terms
which are independent of r', we get sin.'y'./F' = ^.cos./ sin.(u — ir). Substituting this in