shall find that the equation [3408] becomes as in [6335]. We have placed an accent l®^*'*!
below n,, which is not in the original work, to distinguish it from the usual signification of
n, or the longitude of Jupiter [6024/].

f (3314) The terms of [6337], which are multiplied by L, excepting that which
depends on (0), mutually destroy each other, as is evident by inspection. Neglecting these [6337a]
terms and changing the signs of the rest, it becomes as in [6308]. In like manner we find
that [6338, 6339, 6340] are equivalent respectively to [6309, 6310, 6311] ; observing also

VOL. IV. 19



[6340]



74



MOTIONS OF THE SATELLITES OF JUPITER.



[Mec. Cef.



[6341']

[6342]
[6342']

[6343]
[6344]
[6345]
[6346]

[6347]

[6348]

[6349]
[6350]



We must connect these equations with those given by the displacing of
Jupiter's orbit [6347 — 6350], which give the values of p and L\ depending
upon this cause ; observing that the action of the satellites has no sensible
influence upon this secular change in the orbit.

The values of p^ corresponding to the displacing of Jupiterh equator, are
much less than those which depend on the mutual action of Jupiter'' s satellites,
as we shall hereafter see ; therefore we may neglect p in comparison with
(0), (1), &c. ; (0,1), &c.* in this case, if we suppose,

L 1 =X. f]^ ^') > [For the satellite »}

L V = X'. (L Zy') \ [For the satellite m']

Zj /" = X". CL U^ J [For the satelUte m"\

L /'" = >!".(L Z/') ; [For the satellite «'"]

we shall obtain from [6337 — 6340], the four following equations ;t
= [ (0) +[7]+(0,l)+(0,2)+(0,3) I .X _(0,1).X'-(0,2).X"_(0,3).X"'-|T] ;

= [ (l)-f [T]+(l,0)+(l,2)+(l,3) I .x'-(l,0).X-(l,2).x"-(l,3).X - [T] ;

= ^ (2) +[T]+(2,0)+(2,l)+(2,3) ^ .X''-(2,0).X_(2,1).X'-(2,3).X -



= \ (3)+|T]+(3,0)+(3,l)+(3,2) I .x - (3,0).X-(3,l).X'-(3,2}.X"-|T].
We can determine, bj means of these equations, the values of x, x'^ x", x'".



that [6338, 6339, 6340] may be derived from [6337], by changing reciprocally the elements
[63376] of ^ into those of to', rn, ml". The equation [6334] is easily reduced to the form
[6341], by developing the terms included under the symbol 2, and altering a little the
arrangement of the terms. It gives the following value of pL, which will be of use
hereafter;

3.(2C-.^.B)



[6337c] p/>=



4i.C'



^{i»i2.(i-i')+m.n2.(i-Z)+TO^n'2.(i^-Z')+»i^n''2.(2,-Z'')+m^n'''2.(L-r)f.



* (3315) This is evident by comparing the coefficients of t, in the values of fl', V

r6342o] [6928, 6929], corresponding to the motion of the equator, and representing the values of p,

with the values of (0), (1), (2), (3), &£c. [6864—6868], depending on the disturbing

forces of the sun and satellites, or the ellipticity of Jupiter. The former being much smaller

[63426] than the latter, may be neglected, as in [6342'], in finding the equations depending on these

values of p.

t (3316) Putting ^ = 0, in the equations [6337—6340], then substituting the
[6347a] values [6343— 6346], and dividing by — (i— jL'), we get [6347-6350] respectively.



Vin. iv. ^ 10.] INEQUALITIES IN THE LATITUDES. 75

The latitude of the satellite m, above the orbit of Jupiter, is represented
by a series of terms of the form (/ — L').sin.(v-\-pt-^A) ; therefore we shall [6351]
have,*
x'.(/ — L').sin.(v-i-p(-\-A) =s latitude of the satellite m, above Jupiter's orbit. (6352]

If we include only the part of this expression which depends upon the
displacing of Jupiter's equator and orbit, we shall have, as we have seen in [6353)
[6343], L—l = X. (L—L) ; hence we deduce, f

l—L' = (1— x).(L— L'). [6354]

Therefore we shall have, for the part of the latitude of wi, above the orbit
of Jupiter, and including the terms depending on the displacing of Jupiter's
equator and orbit ; the following expression ;

if,(l—L').sin.(v-\-pt-\-A) = (1— x).2'.(Z:^L').sin.(t?+p/-f A). [6355]

If the satellite move in the plane of Jupiterh equator, its latitude above
Jupiter* s orbit will be, t

2 '. (L — L). sin.(v-\-pti-A). [6356]



[6354']



[6355^]



♦ (3317) The latitude of the satellite m above the fixed plane, in [6298x line 1], is
j!.1.5'm.{v-\-pt-\-A) ; and if it were moving in the plane of the orbit of Jupiter, it would be l"^'^]
l!.L'.s\n.{v-\-pt-{-A), as in [6298rline5]. Subtracting this expression from the preceding,
we get Jf.{l — L').s\n.{v-{-pt-{-A)j for the latitude of the satellite m, above Jupiter's orbit ; [63526]
as in [6352].



[6355a]



t (3318) The equation [6343] gives 7 = L — X.(L — L') ; subtracting L' from both
sides of the equation, we get [6354]. Multiplying this by sin.(t>-|-p^-|-^)» *i"d prefixing
the sign 2', we obtain for the latitude [6352] the expression [6355]. We may observe
that the equations [6344, 6345, 6346] may be deduced from [6343] by increasing the [63556]
accents on /, X, by one, two or three marks respectively ; and by performing the same
process on [6.354], which was deduced from [6343], we obtain similar expressions for the ^ '
second, third and fourth satellites. Thus, by changing / into I'", and X into X'^, the
equation [6343] becomes like [6346] ; and [6354] changes into V'-L' = (l-X'").(Z:r-I,') ; [6355rf]
which is used in [7343].

X (3319) If the satellite m be supposed to move in the plane of the equator of Jupiter,
we shall have, by neglecting terms of the order 6^ as in [6316"], t^ [6051] for its
latitude above the fixed plane; and this is represented by 1' . L. sin.{v -\- pt -}- a) [6356a]
[6296x line 6]. In like manner, if the satellite be supposed to move in the plane of the
<n'bit of Jupiter, its latitude, above the fixed plane, can be obtained by changing U into v, ^
in [6298r line 5], which gives lf.L\s\n.{v-\-pt-\-A), Subtracting this from the expression
[6356a], we get [6356], which represents very nearly the latitude of the satellite m, above *â–  *



76



MOTIONS OF THE SATELLITES OF JUPITER.



[M6c. Gel.



The part

[6357] (1— X). 2'. (L—L'). sin. (?;+^^+ a),

of the expression of the latitude of the satellite^ above Jupiter^ s orbit [6366], is
therefore the latitude ivhich it would have, on the supposition that it moves
upon an intermediate plane, passing between the planes of the equator and the
orbit of Jupiter, through the common intersection of those two planes ; the
inclination of this intermediate plane, to the plane of Jupiterh orbit, being to
the inclination of Jvpiteih equator to the same orbit, as 1 — x to 1. We
shall suppose, as in [3108, 3104, &c.], that

d' = the inclination of the equator of Jupiter to its variable orbit ;
— t'= the longitude of the descending node of the equator of Jupiter, on

its variable orbit ; this longitude being counted from the axis of x.
Thus the part of the latitude of the satellite m, above Jupiterh orbit,
including the effect of the changes in this orbit and in the equator [6354'], is*

[6362] (X— I).^'.sin.(?; + ^').



[6358]



[6359]

[6360]
[6361]



[6356rf] Jupiter's orbit, supposing this satellite to move in the plane of Jupiter's equator. The
latitude [6356] corresponds to the inclination &' [6360] ; and if this inclination be decreased

r6356e] in the ratio of 1 — X to 1, as in [6359], the corresponding latitude will be decreased in
nearly the same ratio, and will become as in [6357].



[6357a]
[63576]

[6357c]
[6357rf]

[6357e]

[6357/]
[6357g-]

[6357/i]
[6357t]



* (3320) If we repeat the calculation in [6323a—/], changing {\\e fixed plane into the
plane of Jupiter's orbit, it will change 5 [6312] into 5' [6360], and ^[6313] into â– *'
[6361] nearly, as in [3104 — 3105]. By this means the expression — d. sin. («-}-'*')
[6323c?], which represents the elevation of the satellite m above the fixed plane, upon the
supposition that the satellite moves in the plane of the equator of Jupiter, will change into
— ^'.sin.(r-f-V), representing the elevation of the satellite m above the variable orbit of
Jupiter, the motion of the satellite being, as before, in the plane of the equator. Putting
the expression [6357 J] equal to the same latitude, found in [6356], we get,

2'. (L— L').sin.(«-j-pf+A) = — ^'. sin.(i)4-^')-
Multiplying this by 1 — X, and substituting the result in [6355], we get,

I.\{l—L').sm.{v-\-pt-\-A) = (1— X).2'.(L— jL').sin.(v+p<+A) = (X— l)J'.sin.(«+^')-
The first member of this expression represents, as in [6354', 6355], the part of the latitude
of the satelUte m, above Jupiter's orbit, including the terms arising from the displacing of
Jupiter's equator and orbit ; therefore it must also be represented by the last member of the
expression [6357/], as in [6362]. This expression [6362] corresponds, in the lunar theory,
to that in [5351 or 5357]. But there is this very essential difference, that in the lunar
theory the coefficient — 6%487, [5357], is but a very small part of the inclination of the
earth's orbit to its equator, so that X — 1 must be very small ; but in the theory of this
satellite (X — I). 6' is nearly equal to — 6'; X being very small [6923].



VIU. iv. '^ 10.] INEQUAUTIES IN THE LATITUDES. 77

This result is analogous to that which toe have found in the lunar theory

[5352] ; observing that for the moon, 1 — x is very small ; but for the [6363]

miellites of Jupiter, 1 — x differs but little from unity [6357 1].

ffe shall note determine the values of <», ♦, d', *', which depend upon

the displacing of the equator and the orbit of Jupiter, We shall in the first [6364]

place observe, that we may satisfy, very nearly, the equations [6337-6341],

by putting,*

L'=0, or L— L' = L; 1

/ =(1— x).L, L—l =xL; 2

/' = (I— X' ).L, L— /' = X' L ; 3 t®^J

/" = (1— x").L, L—l" = X" L ; 4

/'" = (1— x"').L, L— /'" = X'" L ; 5

then the equation [6341] gives a value of p, which we shall denote by 'p, [6365']

and we shall have,

'p = j.- ^^^"^'^'^ ' • ! ^'+»w.«'. ^ -\-m'. n'^. x'+m". n"^, x"+ w'". n'"^, x'" } . [6366]



• (3321) Taking as in [60^q or 6398], the orbit of Jupiter in 1750 for the fixed
plane, we shall have the sun's latitude S' [6040] of the same order as the disturbing
forces, which act upon the planet's orbit ; and if we neglect for a moment this latitude, we
shall have S' = 0; whence L' =0 [6300 line 5], as in [6365 line 1]. Substituting [eaesa]
this in [6343 — 6346], and making a slight reduction, we get the assumed expressions of
/, /', I", /'", [6365 lines 2 — 5]. From these we easily deduce the values of L^L',
L—l,kc. [6365] ; and by substituting them in [6337-6340], then dividing by —L, they t^^^^^ft]
become nearly satisfied by using [6347-6350]. Substituting the same values in [6341], then
dividing by JL, we get p, or 'p [6366] ; corresponding to the precession of the equinoxes
of Jupiter's equator, arising from the action of the sun and of the satellites, upon the ellipsoidal
figure of the planet. By means of this value of 'jp, the argument v-{- ^pt-\- a ^ '
[6300 line 6], for finding the latitude a,, supposing the satellite to move in the plane of
the equator, is augmented by the quantity *pt in the time t ; and as this argument must
evidently represent, as in [533a], the distance of the satellite from the equinoctial point. [6365</]
The part *pt must express the precession of the equinoxes [6323^, &c.] ; as this contains no
variable part, like that neglected in [631 7o], it may be considered as the mean precession
arising from the action of the sun and satellites, on the ellipsoid of Jupiter ; but not [6365«]
including the efiect of the change in the plane of Jupiter's orbit, from the value of L'
depending on the disturbing forces of the planets ; so that *pt corresponds to //, in the [6365/*]
expression of the precession of the equinoxes of the earth's orbit [3100]. Tlie efiVct
of the change in Jupiter's orbit, from the action of the planets, is noticed in
[6372, Stc., or 6928, 6929], being similar to that which changes 4. [3100] into 4-' [3107]. ^^^^^

VOL. IV. 20



[6372]



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

[6367] The value of L remains arbitrary, and we shall denote it by — 'L.

Hence we have, by noticing only this value, the latitude of the satellite
[6368] above the fixed plane, equal to — ^iy.sin.(?J + '^^ + ^A), * ^A being the

arbitrary constant quantity corresponding to ^p. But this latitude is also
[6369] equal to — &,sm.(v-\ - ¥) [6322] ; hence we deduce,
[6370] &, sin. ^ == iL. sin. Qpt + 'a) ;

[6371] ^. COS.* = 'L. cos. (y +U) .

^pt denotes the mean precession of the equinoxes of Jupiter ^ [QSQbe'] ; but the

true precession is modified by the displacing of Jupiter's orbit, as we have
[6372'j seen, in [3111 — 3115], that the change of the ecliptic modifies the precession

of the equinoxes relative to the earth. To determine these modifications,

we shall observe that the equation [6341] gives, J



* (3322) This expression of s^ is deduced from its assumed form [6300 line 6],
[6368a] changing L into — ^L [6367], and p into '^p [6365'J, in order to conform to the

present notation. Putting the value of s^ [6368] equal to that in [6323rf], we get,
[63686] —Ls\n.{v-\-^) = —^L.sm.{v-\-^pt-\-^A).

Now by means of [21] Int., we have,

sin.(i;-f-'^) ^^^sin.u.cos.Y-j-cos.iJ.sin.* ;
sin. (!;-[- ^V^-\- ^â– ^) = sin. v. cos.(*_p^-|- ^a)-|- cos. v. sin. {^pt-\- 'a).
Substituting these developments in the equation [63686], and then putting separately the
coefficients of — cos.«, — sin.v, in both members, equal to each other, we get [6370,6371],



[6368c]
[6368d]



[63726]



t (3323) We have observed, in [6365/], that the mean precession of Jupiter's equinox
[6372a] ^pt, deduced from [6366], is similar to that of the earth It [3100] ; or, in other words,
that ^p, in Jupiter's theory [6366], is equivalent to I [3098], in the theory of the earth ;
and for the sake of illustration, it may not be amiss to compare these two expressions.

Now the first term of I [3098], dependmg on the sun s action, is -— - . ( J.cos.A ;

and by changing m into M [3059, 6101'] ; also n into i [6317/], to conform to
[6372c] the present notation ; then putting cos. A = 1, on account of the smallness of h [7319],

we get the first term of [6366], depending on M^. In like manner, the term of [3098],

containing X, which depends on the moon's action upon the ellipsoid of the earth, is
[6372rf] similar to those in [6366], containing X, X', X", X'", and depending on the action of the

satellites upon the ellipsoid of Jupiter.

X (3324) Multiplying [6366] by L—L\ and substituting the values [6343—6346],
we get,

[6373a] ^p.{L-U)= ^ • ^^^^^^^.\M^l^U)^m.n^.{L-l)^\n'2.{l^')+m'\n''24I^V^



VIII. \y. ^ 10.] INEQUALITIES IN THE LATITUDES. 79

(P—'P)' L + 'pL=^0; [6373]

p being, in this case, one oj the values of p, relative to the displacing of
Jupiter^ s orbit. This equation gives, [6374]

L=z — -J——. [6375]

Therefore, if tee notice only the values of p relative to the displacing of [6375']
Jupiter^s orbit, we shall have,

6, sin. T = Jp. 2'. •^'"•(^<+^) . ^ggygj

^ , . L'.cos.(pt4-A)

^.COS.i' == p. If, ^^— ^ — ^ . [6377]

Connecting these two values with those in [6370, 6371], we shall have,*

6, sm. ^ = 'L. sin. ('pt + ^a) + 'p. if, Lpu . ^^^^

^cos.^ = ^L, cos.(^pt + ^a) + ^p. If. '^°^'\Pj-rV ^^^^^

Hence we easily obtain,t



The second member of this equation is the same as the value of pL [6337c] ; hence we [637351
obtain ^p.{Jb—L')=pL; and by transposition we get [6373]; whence we deduce L 1-6373^1
[6375] ; and by substituting it in [6326, 6327], we obtain [6376, 6377]. The symbol
2' being supposed to include the angles pt-{-Ai depending on the displacement of [6373rf]
Jupiter's orbit [6324'].

♦ (3325) The sum of the terms [6370, 6376] gives the value of ^.sin.>F [6378] ; and
the sum of those in [6371, 6377], gives 6.cos.^ [6379]. Observing that the values C^^^a]
[6370, 6371] are founded on X#' = [6365a], corresponding to a Jixed orbit of Jupiter ; [63786]
and the terms [6376, 6377] depend on the changes of that orbit ; consequently these sums
represent the whole values of ^.sin.'r, d.cos. ^r.



[6380a]



t (3326) From [22] Int. we get sin.^.cos.»/»/— cos..^.sin.'p< = sin.(.5 — ^pt) ; hence
if we multiply [6378] by cos.'p/, and [6379] by — sin.'p^, then take the sum of the
products, we shall get, by using [6380a],

which is easily reduced to the integral form in [6380], as is evident by taking the integral,
indicated by the sign / Tiie advantage of this form is seen in [6386], where we are [6380c]
enabled to introduce the symbol 7. In like manner, by multiplying [6378] by sin. *pt, [GSSOd]
tUo [6379] by cos. •/><, and reducing by means of the formula

Bin. A, sin. 'pt-\- COB. A.cos.'pt=co8.{A — *pt) [24] Int., [6380*?]



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

[6380] 6, sin. (^— >0 = 'L. sin. ^a + 'p. I'./L'dt. cos. (pt—'pt+ a) ;

[6381] 6. COS. l^—'pt) = 'L.cos. 'a—Y I'JL'dt. sin. (pt—'pt+A) ,

Now we have, as in [6330, 6331],

[6382] 7. sin. 7 = — 2'. L'. sin.(;?^ + A) ;

[6383] 7. cos. 7= 2'. L'. cos. (^^4-^);

and from these we deduce,*

[6384] 7, sin. (7 + y) = —2'. L'. sin. (pt—'pt-{-A) ;

[6385] 7. C0S.(7 + y) = 2'. L'. COS. (pt—''pt+^) 5

therefore,

[6386] 5. sin. (y— y) = ^L. sin. ^A -f >./7<f^. cos.(7 + 'pt) ;

[6387] ^. COS. (*— y) = ^jL. cos. ^A + 'p.frdt. sin. (7 + Y) .

[6388] j?!/ 'i^eans of these two equations we may obtain the excess f ^ — ^pt of the
true precession of Jupiterh equinoxes above the mean precession ; and the
inclination 6 of the equator of this planet to the fixed plane.

The latitude of the' satellite w, supposing it to move in the equator

[6390] of Jupiter, is — ^.sin.(i;+^) [6322] ; and its latitude above the same plane,

[6391] supposing it to move upon the orbit of Jupiter, is 7. sin. (-y — 7) ; f



we get,

/ NT /. . / Ir'.COS.fpf— Ipf+A)

[6380c] ^' cos.(*— . »jpf) = ^L. COS. A + ^p. 2'. _^ ;

which is the same as [6381], as is evident by taking the integral of its last term, as in
[6380/"] [6380c]. This integration procures the same advantage as above, of introducing the
symbol 7 in [6387].



[6386a]



* (3327) Multiplying [6382, 6383] by cos.^;?^, sin.';?^, respectively; taking the

sum of the products and reducing, by [21, 22] Int. ; we get [6384]. Again, multiplying

[6382] by — sin.^jo^, and [6383] by cos.^p^; taking the sum of the products and

[63866] reducing by [23, 24] Int., we get [6385]. Substituting the values [6385, 6384] in

[6380,6381], they become respectively as in [6386, 6387].



[6388a]



t (3328) The true precession being represented by ^ [6313], and the mean
precession by "^pt [6372]; the excess of the former, above the latter, will be y — ^pt,
as in [6388].



X (3329) The inchnation of the sun's relative orbit about Jupiter, to the fixed plane,

•' is 7 [6313']; and the longitude of its ascent?!/?^ node is 7 [6314]. Hence v — 7 is the

angular distance of the satellite from that node ; and by proceeding as in [6328c], we obtain

[63916] 7.sin.(i' — 7), for the latitude of the satellite m above the fixed plane, supposing it to move



VIII. iv. ^ 10.] INEQUALITIES IN THE LATITUDES. 81

subtracting this last expression from the preceding one, we get the latitude [639r]
of the satellite above the orbit of Jupiter, supposing the satellite to move in
the plane of the equator ; but this last latitude is — a'.sin.(tj4-*') [6357 </] ; [6399]
therefore we have,

— d'.sin.(r-fi'') = — ^.sin.(t-f i') — r.sin.(v — n) ; [€393]

r being indeterminate. Ifwe put successively v = — '/?/ and tJ=100° — ^pt, [6394]
we obtain from the preceding equation,

6', sin. (^' — ^pt) = 6, sin. (* — ^pt) — 7. sin. (7 -f ^pt) ; [6395]

6'. COS.(i'' — ^pt) = ^. COS.(i' — ^pt) + 7. cos. (7 + y) . [6396]

These equations will make known the precession ^' [6361], and the
inclination 6' [6360], of the equator, referred to Jvpiterh orbit.

It is sufficient, for the uses of astronomy, to have the values of these ^\i^^.
quantities in a converging series for two or three centuries. We shall take, *"*•
for the fixed plane, the orbit of Jupiter in 1 750 ; and that epoch for the
origin of the time t. We shall also take, for the axis of x, the line of
the vernal equinox of Jupiter at the same epoch. Then reducing the
expression to series, and neglecting the second and higher powers of i, we [6399]
shall have,*

7. sin. -7 = at ; [6400]

7.C0S. n = bt ; [6401]

a and b being constant quantities, which are easily found, as in [6901, 6902], [6402]



[6397]



[6398^



[6391c]



in the plane of the sun's relative orbit, or in the plane of the orbit of Jupiter. Subtracting

thb from the expression in [6390], we get the latitude of the satellite m above Jupiter's

orbit, supposing it to move in the plane of Jupiter's equator, as in the second member of

[6393]. Putting this equal to the value of the same latitude, found in [6357rf], we get the [6391rf]

first member of [6393]. Substituting in [6393] « = 200^— >/)/, we get [6395]; and

by putting i>=.300^ — ^pt, we obtain [6396], by making a few reductions. The same [6391e]

results are obtained by using the values of v [6394].



[6400a]



* (3330) The developments in [6400, 6401] are similar to those in the earth's orbit
[4332], neglecting terms of the order <"; observing that the assumed values of p", q"
[4249] for the earth, are similar to the expressions 7. sin. 7, 7. cos. 7, in the present [64006]
theory; because 9", 6", [4249, 4082, 4083], or rather 9/, «/' [4238c] are changed
into 7, 7, [6313', 6314]. The analytical values of a, b, are given in [6901, 6902] ; ^^^^^
and their numerical values in [6906—6908, or 6928, 6929].

VOL. IV. 21



82



MOTIONS OF THE SATELLITES OF JUPITER.



[Mec. C^.



rfi402n ^y means of the expression of the motions of Jupiter's orbit, given in [4246,]
From the preceding equations we obtain,*



* (3331) Putting z = 1 , and a=»p^, in [60, 61] Int., we obtain the developments
of sin.(7-}-*pi), COS. {l -\-^pt), according to the powers of t. Substituting these
expressions in the terms under the sign /, in [6386, 6387], and then the values of
y. sin7, 7. cos. 7 [6400,6401], we find that these integrals are of the order t^, or of a
higher order, which are neglected in [6399]. Therefore we may reject the terms under
the sign /, in [6386, 6387], and we shall have,

^. sin.(^ — ^pt) = ^i^.sin.^A ; ^.cos.(^ — ^pt) =^L.cos.^a.
The sum of the squares of these two equations, gives 6^ = '^L^, or 6=^L, as m
[6404] ; substituting this in the first of the equations [6403c], and then dividing by ^L,
we obtain,

sin.(Y — 'jp^) = sin. *A ; whence ^ — ^pt = ^A.
Now we find, in [6398'], that the line drawn from the centre of Jupiter, in the direction of
the vernal equinox of Jupiter's orbit, at the epoch 1750, is taken for the axis of x ; and
in [6313] Y represents the retrograde motion of the node from this axis, after that epoch ;
so that when ^ = 0, we shall have ^ = 0. Substituting these values of t and ^,
in the first member of the second equation [6403e], we obtain = ^a, as in [6403];
and by using this value of the constant quantity *a, we find that the second equation
[6403e] becomes generally, for any value of t, y — ^pt = 0, or ^ = ^pt, as in [6404].

In finding the values of *', &', from the equations [6395, 6396], we must observe that
the equation y — yt = [6403A] gives 6. sin. {^f — ^pt) = ; 5.cos.( •? — ^pt) =d=^L
[6403d]. Moreover, by using the developments [6403a], in connexion with the equations
[6400, 6401], and neglecting terms of the order t^, we have,

y. sm.(l -\-^pt) = at ; y. cos. {l -\-^pt) = hi.
Substituting the expressions [6403r, fc] in the equations [6395, 6396], we obtain,
6'.sin.{ir'—^pt) == —at ; d '. cos. {^'—^pt) — ^L-\-bt.

Dividing the first of these equations by the second, we get tang.(^' — ^pt) = - . If

we neglect terms of the order t^, we may put the second member of this equation under

the form —rr- = tang.f — — -j nearly, [45] Int. Hence the equation [6403m] becomes,

/ / \ / — (tt\ . at

tang.(* — ^pt) = tang.f ^y- j ; whence ^ — ^pt== — — .

From the last of these equations we easily deduce the value of ^' [6405]. Taking the
cosine of both members of the same equation [6403o], and developing the second member,
according to the powers of t, by means of [44] Int., neglecting terms of the order f^, we
shall have cos.(*' — ^pO = !• Substituting this in the first member of the second of the