Samuel G. (Samuel Goodwin) Barton.

Secular perturbations arising from the action of Saturn upon Mars : an application of the method of Arndt online

. (page 1 of 2)
Online LibrarySamuel G. (Samuel Goodwin) BartonSecular perturbations arising from the action of Saturn upon Mars : an application of the method of Arndt → online text (page 1 of 2)
Font size
QR-code for this ebook


LIBRARY

OF THE

UNIVERSITY OF CALIFORNIA.

RECEIVED BY EXCHANGE



Class



SECULAR PERTURBATIONS



ARISING FROM THE



ACTION OF SATURN UPON MARS



AN APPLICATION OF THE METHOD OF ARNDT



A THESIS

PRESENTED TO THE FACULTY OF PHILOSOPHY OF THE
UNIVERSITY OF PENNSYLVANIA

BY
SAMUEL GOODWIN BARTON

IN PARTIAL FULFILMENT OF THE EEQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY




PHILADELPHIA

1906



SECULAR PERTURBATIONS



ARISING FROM THE



ACTION OF SATURN UPON MARS



AN APPLICATION OF THE METHOD OF ARNDT




A THESIS

PRESENTED TO THE FACULTY OF PHILOSOPHY OF THE
UNIVERSITY OF PENNSYLVANIA

BY
SAMUEL GOODWIN BARTON

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY



PHILADELPHIA
1906



PRESS OF

THE NEW ERA PRINTING COMPANY
LANCASTER, PA.




INTRODUCTION.

The paper upon which this thesis is based is entitled " Re-
cherches sur le calcul des forces perturbatrices dans la theorie
des perturbations seculaires" by Dr. Louis Arndt. The paper
is published as a bulletin of the " Societe des Sciences Naturelles
de Neuchatel," an extract from tome XXIV, 1896.

Aside from the application of the author, as a check upon his
formulae, mentioned in the paper, so far as is known, no applica-
tion of this method has been made. The purpose of the author of
this thesis is to apply this method of computation and to compare
it with that of G. W. Hill and thus see whether Arndt's method
has the advantages claimed for it by its author.



I.
EXPRESSIONS FOR THE SECULAR PERTURBATIONS.

Let R, U, Z (each multiplied by the factor containing the masses,
i. e., m I(L -\- m) be the components of the perturbing force;
R acting in the plane of the orbit of the perturbing body and in
the direction of its radius vector, r, positive if r is increased ; U in
the plane of the orbit perpendicular to the radius vector, positive
in the direction of movement ; Z perpendicular to the plane of the
orbit, positive northward. Let fl be the longitude of node, i the
inclination of the orbit to a fixed ecliptic, a the semi-major axis,
e the excentricity, /A the mean daily motion, a> the angular distance
of perihelion from the ascending node, v 9 E, M, the true, eccentric
and mean anomalies and u = CD -j- v.

In accordance with the usual method of varying parameters we
have the following equations for the variations of the elements.

3



165260



4



map sec<t> _ .

sin (8fl) = j^ - - f sin u Z, (S) = -, r cos



[sin B + (cos v + cos



(So)) = (8a>,) COS i(Sfl),



(I)



* + 2 sin' (&0 + 2 sin

The secular part of these expressions will comprise those terms
which are independent of the positions of the bodies in their orbits,
that is of their mean anomalies M and M ' . Each of the above
variations is seen to be a function of 7?, 7, and Z which in turn
are functions of M and M' . They may therefore be expressed in
a Fourier's series, i. e., in a series of sines and cosines of multiples
of M and M ' . Now the terms of this expansion which are inde-
pendent of M and M' give the secular variation. By the known
properties of the series the only constant term with respect to M
is given by the integral



and the part of this which is constant with respect to M' will be



Thus the secular part of any variation (Se) for example is

TO '" V C S * i r r [sin vR + (cos v + cos E) U^lMdM '.

1 -|- m 47T JQ JQ

Since the limits of integration are constant, sin v, cos v and
cos E, may be treated as constants in the integration with respect
to M' i and this integral may be written

m'a*p cos <f> 1 ** . \ f-

- ^ fsm v ~~ RdM

1 + m 27rJ L 27rJ

-f (cos v -f cos E) ~ I**" UdM']dM.

" rjr Jo



Treating the other variations in the same manner we see that
the integration with respect to M' requires the three integrals



1 /

fcf



The integration with respect to Jlf can be obtained only by a
direct quadrature. Since this can be obtained more accurately in
terms of E we transform M and M' in our expressions into E
and E '. Then since

M=E-e sin E, dM= (1-6 cos E)dE=-dE,
dM' = (l-e r co%E')dE'.

Our desired integrals which which we shall now designate
7? , Z7 , Z respectively are

~\rR(\ - c' cos .E")dtf', ^ Par 7(1 - e' cos
(2)

-^ f ^ r 2 ^! - e' cos j0')cLtf'

where the factors a and r have been taken with the integrals.



II.
PRELIMINARY CONSTANTS.

Let the orbit of the perturbing body m be referred to that of
the perturbed body m. Let the distances of the perihelia from
the intersection of the orbits be II and II' and their mutual incli-
nation J. Let K and K' be the distances of the intersection
from the nodes. Solving the triangle thus formed we obtain
IT, K' and J in terms of i, i' and (ft' ft) whence we get II and
IT from the relations

n = TT ft - K, IT = if ft' - K'.

Let L' and B' be the longitude and latitude of m referred to
the orbit of m and the intersection. Let 77, % be rectangular
axes, origin at the sun, 77 and f lying in the plane of the orbit of



6

w, f passing through the body m. Let I be the longitude of m.
The coordinates of m are then given by the equations

I H -f v, f = / cos # cos (L r - Q,

T/ = r' cos .7?' sin (' Z), " = / sin B.

Now from the right triangle formed by the intersection, m\ and
the foot of the perpendicular from m to the orbit of m we have

cos L' cos E' cos (II' -f v), sin U cos .Z?' = sin (II' + v') cos J]

sin 5' = sin (IT + v) sin 7.
To simplify the formulae, write

A sin A' = sin II' cos <7, vl c = Ad cos (J/ + II + v),

4, = Ba! cos <' sin (^' + II + v),
J. cos A = cos II', B e = JLa' sin (J/ + H + v) r



(3)



B t = J?a' cos <#>' cos (J^ + ft + v),
sin B' = sin II', (7 C = a' sin II' sin /",

C7 t = a' cos $' cos II' sin 7,



Substituting these expressions in ' for example : by expansion
f ' = / cos II' cos (II -f- v) cos v + r sin II' sin (II -f v) cos 7cos v

r sin IT cos (II -f v) sin v -f r cos II' sin (II + v) cos </sin vV
but

r cos v = a'(cos ^" e'), / sin -y' = a' cos $' sin ^",
whence

- = ^.X 008 ^^ ~ e ') + ^ sin ^' and similarl y>

;' = C (cos ^' - e') + C. sin j^',



7



III.

EXPRESSIONS FOR JR Q , 7 , Z .
With axes as described it is not difficult to see that



A 3 ' ~A 3 ' ~A 3 '

where A is the distance between the two bodies. We see that
A 2 = (f ' rf + 77' -f J" which from (4) and (3) becomes

A 2 = A - 2# cos (e - E") + C Q coslE"
in which we have placed

' 2 -f r 2 + 2eWL c = ^ , eV 2 + r^4 c = ^ cos e,

V / ^ 7-) . ^2 /2 x>

rA s = ^ sin e, a e = (7 .

Substituting the values of 7?, 7, Z in (2) we obtain

" ^ c (cos ,E" - e') -f ^ sin ^ - r
== ~



(6)



J9 c (cos E' e) -f B t sin E'

(1 e' cos E'
C 2n C (cos ." e') 4. Q sin _"

'-i^J, r ~ -SF

(1 - e' cos E'}dE'.

These expressions cannot be directly integrated because of the
complexity of A in terms of E'. A transformation due to GAUSS
( Werke III, p. 333) makes integration possible by changing the
variable E' to a new variable T.



8



-r = -



aa" +



IV.

GAUSS'S TRANSFORMATION.
Let

E' = a + a' sin T + a" cos T,

E' = + /3'sin T+ j3" cos T,
= 7 + 7' sin J 7 + 7" cos J 7 .
Now let these new auxiliaries be subjected to the conditions

r - 77' = o,

77" = 0,
"' - 7'7" = 0,

2 rZ f/2 -t Q t ,y " Q" A

a a a = 1, ap ap a p = U,

? 2 ft' /3" = 1, ay a'y a"y" = 0,

7 2 y' Z y" Z = 1, fty ft'y' ft"y" = 0.

We now make these auxiliaries such that A 2 may take the form

(a) ^ 2 A 2 = G - G l sin 2 T+ G 2 cos 2 T.

Assuming this possible we shall see if real values of the
coefficients can be found such that ^V 2 A 2 may take the form

E' NCOS e

i^'- JVsin+ CJiNunEJ.

We now solve (7) for sin T and cos T in terms of E' and
observing that 1 = aN cos E' $N sin E' + yN and substi-
tuting these values in (a), then equating coefficients with (b) and
writing the resulting equations in three groups we have

I a G a a G l a -j- a" G 2 a" = C
SG-a-pGt-a' + P'Gz'a'^Q
yG'a y'G^a' + y'Gz-a'^ B Q cos e

The last two columns are the values of similar equations, the a's
being replaced by /3's and 7's respectively.



(9)





sin e



,0086

, sin e

A



9
Whence



GOL == C Q OL -f yB Q cos e,
G/3 = 7# sin e,

Gy = a^? cos e J9 sin e -f yA Q ,
GjLi etc., and 6r 2 a", etc.,

give the same expressions with a, /3, 7, replaced by a', y8', 7' and
a " " 7 " The coefficients of the a's, /3's and y's are of the same
form in each set of equations. The condition that these equations
may be consistent is expressed by equating the determinant of the
coefficients of a, /3, 7 to zero. This gives when expanded

#[(# + Q(# - A) + Bl cos 2 e] + Bl(G + <7 ) sin 2 e = 0.

Thus if N 2 A 2 is to be of the desired form (a), G must satisfy
this equation, and since the other two groups give the same equa-
tion, except for accents, G l and G 2 must also satisfy this equation.

Making the variable X and placing

(10) P^A.-C a P t -Bl-A t C. -P 3 =<7 ^sin'e

the cubic may be written

( C) X 3 - P^ 2 + P 2 X- P 3 =

GGi and G 2 are then the roots of this equation. The roots can
be shown to be real, two positive and one negative. We let them
be in order of descending magnitude G, G l and G 2 .

Multiply numerator and denominator of (6) by N z and the de-
nominator will be of the desired form (a). As shown by Gauss
NdE' dT and the limits for T are the same as those for E '.
Because of the limits terms containing sin T, cos T or sin Tcos T
vanish and our transformation gives

rm V _ R TT z __ i fV r+ r. sin* r+r,

^'^'^-2^1 M (G-G lS tf r+<?
In which we place

( T =/ 7 2 + bay + A/87 - da/3 - la\

(12) J r, =yy 2



=/ 7 " 2 + lay" + Wl" - da"/3" - la" 2 .



10






For S .
/ = Ae' r


For UQ.
~^X


For Z .
-Gf


= AJl + e 2 ) + re'


^ c (l + ^ /2 )


C c (l + e' 2 )


A = A


5.


c-.


d = ^x


Be


Gf


Z = Ae'


Be'


Gf


M ar


ar


r*



It is only in the integration of (11) that Arndt's method differs
essentially from that of Hill. Hill's method consists of integration
by means of elliptic integrals. The modulus of the elliptic in-
tegrals is &=*(G l + G 2 )/(G+ G 2 ). The computation of this
modulus necessitates a solution of the cubic ( (7) by approximations
for the roots 6r, 6^, 6r 2 . While the solution is not difficult it
becomes objectionable because of the number of times it must be
made. Arndt's method avoids the solution of this cubic by mak-
ing the integration depend upon the integrals of Weierstrass.



V.

AKNDT'S MODIFICATION.

LetTF= r, . dr _ = f ? ^.

Jo y 6^ G! sin T -\- G 2 cos ^Z 7 Jo ^ 5

then

dW fi dT



N*
TdT



Our expression for V then becomes



11

Now let sin 2 T= t and (G 1 + #,)/(# + #,) = ^ (modulus), then
= when r= 0, = 1 when T= Tr/2 and



and

z>

7 \jf

where



X 1 5TF r l dw r l

Qdt, SG=-P\ Qtdt, j = P Q(l-t)dt
U ^TI Jo v ^z Jo



4(G



and



In order to use Weierstrass's function for the integration we
make






where s is a new variable and m, 77i 1? m 2 constants which will be
disposed of. By differentiation we obtain

tfdt dt

^'



(k 2 l)dt
whence



, l2 -
When t = 0, 3 = (^ x when ^ = 1, s = (r 2 . Now writing

m = !/((? - G,) and (ff + G 2 )(G - ^)(^ + ^ 2 ) = (7



and i x 4(s G)(s G^s + G 2 ) = V S^ we have

dW G, + G n r~ G * ds
^G

dW



aw_ G+G 2 .r-* ds

3G= '20 J Gl ^



12

In order that S may be of the Weierstrassian form the second
term of the cubic ( (7) must be wanting, that is, each root must be
diminished by Pj/3. Hence

Vsi



where e^ e 2 , e 3 are the roots of the transformed equation. We
then have

8W G l+ G 2 r*~_^ ds



, + g, r

20 1



dG ' '2C ^ v/^

5TF G+ G 2 pa e a
'X VI

5TT Q-Gr'-^j.



SO,- ^ V~8

r

/*



a^ 2 ~ 2(7

and

_ L+_2(e lft > + ^)



7T






r ds
.. 1/



We must now obtain the coefficients of o> and 77 in terms of the
elements. For this purpose let

A = (& l + G 2 )T + (G
@ = (0, + GJGT + (
then, since e, = G - ^P,, e 2 - G, - JP 15 e s = - <? 2 -



A and must now be expressed in terms of the coefficients rather
than in terms of the roots of ( (7). Let



and from the symmetric functions of the roots

^ GG 2 - G,G 2 = P 2 ,



13

Substituting the values of F, T 1<t T 2 given in (12) we see that
the equation for A l reduces to A l f I which by (13) for
the three cases gives respectively r, 0, 0.

By substituting from (12) and reducing with (9) we find

B l =fA + bB cos e + hB Q sin - I <7 ,
whence from (13)



f = - A Q (A e e'+ r) + B cos {eV
(15)

B* A 9 C.e'+ -# cos e(7 c (l + e 2 ) + B Q O 8 sin e - <7 c <7/.



For 5^ it is preferable to substitute the primitive values, whence
we obtain



(16) J?f = - Jra' cos 2 ^ sin 2 ,/sin 2(11 + t>) - ^'^c-

For reducing C l we multiply the equations Got 2 0^+ G 2 a" 2 =z C Q
and G + #! - ^ 2 = P l by a' 2 -f a" 2 - a 2 = 1 and
GG 1 GG 2 0^2= P 2 respectively and add the products.
We thus have



a'G.G, - a 2 GG 2 -f a'GG, = P 2 -f P l C + a / (7 2 + a^ - 2 (7 2 .

The last three terms can be replaced by known quantities. Mul-
tiplying

Go. = C a + B Q cos 67 by Ga ,

^ cos 7 Gjot,



and adding the products we obtain



~ ^ 2 2 = Cl - ^ 2 cos 2 e.



Dividing by G&G = P 3 we get

a 2 a <t" 2 .ggin a 6



Similarly we obtain



14



dff a"/3" 1 ^ ,

-rr -f -7T- = ^ -#o sm e cos e,



7 a 7 ay
3y Py . /3" 7 "



cos e



sin.



-Cosine



The three terms in C^ occur in the form



V b + ^h a/3 d -I
b h G* ~ G ~ G

The other two quantities are given by the same equation with ap-
propriate accents. Substituting in the equation we find

- P s Ci = B Q sin e(cLS cos e - IB Q sin - hC Q ).
Substituting from (13) for each case we find






Ar'e'ABa' 2 cos <' cos (.4' - H),



To obtain A we consider the identity (v. HalpJien FonGtions
Uiptiques ch. VIII).



3



- r i\ r 2




111




0i -0 2





111
(? ^ ""^" 2


=


111




/^ /"* /^

tr Cr r Cr 2





Multiply by the equation GG 1 G 2 P B using a factor in each
column of the second determinant. Subtract the last column
from each of the first two and upon expanding we find it equal to
+ C. In the first determinant multiply the first column by



15



_ ( Q^ _j_ y and multiply the last two columns by ( G + 6r 2 ) and
(G l G) respectively and add them to the first column. We
thus find for this determinant thus modified :



-A



1


A




r,

ff,

i


r 2

-i
i


G 1+ G 2



Our identity when expanded then becomes



- B^P, - 9P 3 ) -
Now let
(18) P 1 P t -9P,- P , P?-3P 2

We then obtain for the three cases



J - 3P a ).



(19) H



If we write the transformed cubic in the usual manner
4 S 3 _ g^s g 3 = 0, we have the following equations for the roots.



e 2 + e 3 = 0,

4( ,-,-



P 2



If g z and g 3 are the invariants of the cubic whose roots are 6r,
the discriminant is g\ 27#* and C7 2 , i. e., the product of the
squared differences of the roots will be

(21) C".



(22)

g is the absolute invariant.



16



Again for



G



we consider


an identity




G 2 T 2




1


1


1


-G.
1


-


1

G 2 +6


1
^


1
G,


G


~ 0,


G,



.-A, A
P, 3



-p.-

Multiply this equation by GG t G 2 = P 3 , using a factor in each
column of the second determinant, the negative sign being used in
the last column. Next subtract the last column from each of the
first two ; the second determinant when thus treated and expanded
will be found equal to 2\( G l + 6r 2 ). In the first determinant
multiply the columns respectively by ( G^ + 6r 2 ), ( G -f 6^), and
(G l G). Subtract the last two columns from the first. The
determinant thus treated and expanded will become j(G l + G 2 ).
Our identity then becomes 2\O = A(P 1 P 2 9P 3 ) + C(P^
Whence we have for the three components



(23)



and for the components we have finally from (14)
2 FA*/ P,



(24)



2



P



The elliptic integrals w and 77 are computed by the method of
H. BRUNS.* Writing for the moment ^o> a = e a we know that

{K 4- u) - e a }(s - e a ) = (e a - e^(e a - e y ).



*"Ueber die Perioden der elliptischen Integrate," Math. Ann., Bd. 27.



17

Then
/ =



( s -



Let



. s =



5

ds = 7-7



, e e 2 g 3

e i H ir ,



When

5 = e a , s' =

ds



5 = e a , s = <? 3 , s = c s , s = e 2 ,



* ds r* a'da'

^ + ->7S"JL^



If now ^ 2 and ^ 3 are given functions of f , we obtain, by differ-
entiating the equations for co and 77,

d< p (?; + &),, ^_ f'(y; + y 3 ) ,.

df-J., 2^i rfs ' d|~ -J e , '
where

/ d $2 A ' d ff*

^-? and ^ 3 = ^.
But we have the identities



,- 6^-63) (.-,)



18

and introducing the integral I



sds



whence

(25) ~=-Pr,-Q a> , |

where



t
'



or expressed in terms of the symmetric functions of the roots

- - 86^; +



If now we substitute for &> and 97 in (25) respectively



10



the coefficients of the differential equations can be expressed by g
and its derivatives only and we obtain



n

-i) 1

and now for g'd% we write dg.

Eliminating H and H respectively from these equations we get

. dl 55



19



These equations are of the form of the differential equation
whose solution is the hypergeometric series, i. e.,

= x(x- 1) + [>(a+ fl + l)-7] + apy



where a = $, ft = T 6 2 , 7 = f , and a = T \, = ^ 7 = J. One
of the 24 solutions of this equation is



y



O) =



To determine the constants (7, (7' we place
= 1, 0r 3 = 1, then # 2 = 3, and e^l, e 2 = e 3 = - }, t? = e i~ e * = Q,



and since



1 C d6

CO = - =^:^ - ^

l/ 6i e 3 Jo VI /fc 2 si



TT



sin 2 <^ 1/6



l e 3 1/24



since when g = 1, (</ !)/# = 0, and F 1



.-. ^ ==^d I t> = || . , V^ ==^J I "t - = 4d i ,

1/6 iX!2 1/24 i!/1728

whence

(26) -j^



where JP, and ^ are the two hypergeometric series

For computation we first solve the triangle described in part II,
for the preliminary constants, then apply the formulae in the fol-
lowing order 3, 5, 10, 18, 20, 22, 21, 15, 16, 17, 19, 23, 26, 24.
The values of F m and F^ required in 26 may be taken from the



20

table given by Arndt or they may be computed directly. The
argument of this table is x = (g l)/gr.
We now compute the following expressions :

HU Z Q sin u, H { Z Q cos u, H e 7? sin v -f 7 (cos v + cos E),



1 + sin



r

1 + 5(1^



We now take the mean value of each quantity, which we repre-
sent by (fffi), etc., and obtain the variation of the elements from
the equations : m >

= 1 +ra'

sin (Sfl) = B sec <(^T ), (Si) = B sec
(8e) = B cos <t>(ff e ), e (^ w i) = -S cos

cos



(SJQ + 2 sin 2 ^^^) 4. 2 sin 2 1 (an).

X and TT are the mean longitude and the longitude of perihelion
measured from a fixed epoch.



VI.

COMPUTATION.
Saturn upon Mars.

The elements taken from Dr. G. W. Hill's " New Theory of
of Jupiter and Saturn " are

Mars p. 192. Saturn pp. 19,558.

7r=33317'51".74 TT'= 90 6'4r.37

i= 1 51 2.24 i'= 2 2940.19

H = 48 23 54.59 ft' = 112 20 49 .05

e = 0.09326803 e = 0.05606025

P = 689050".784 p' = 43996"21506

log a = 0.1828971 log a = 0.9794956

m 1/3093500 m = 1/3501.6
Epoch 1850.0 G.M.T.



21



The preliminary constants are found to be :



n = 17617'59".42
IT = 293 4 38 .78

K= 108 35 57 .73
1C = 44 41 13 .54

J= 2 2152 .11



A = 0.99927955
# = 0.9998659
^' = 6654'17".84
B' = 66 56 24 .55



The values of the various quantities given by the formulae are
given in the table below. The residual in Innes's test equation
was found to be 0.000,000,000,2. 2 t and 2 2 are the sums of
the values for the odd and even points of division



E


log r


V


log A.


log^ c


log A.


logJ? g





0.1403760


o o' o!6o


0.6331692*


0.9298510


0.9295583*


0.63221 33 n


30


0.1463201


32 47 24.62


9.9980540


0.9768010


0.97634 55 n


0.0001597


60


0.1621568


64 44 46.64


0.7680496


0.8760042


0.8753687*


0.7679641


90


0.1828971


95 21 5.91


0.9480287


0.5421705


0.5410633 n


0.9477062


120


0.2026919


124 31 47.16


0.9752134


0.1081758 n


0.1097082


0.9747498


150


0.2166314


152 34 23.40


0.8883581


0.7460753*


0.7460180


0.8877384


180


0.2216237


180 0.00


0.6331692


0.9298510n


0.9295583


0.6322133


210


0.2166314


207 25 36.60


9.0215124 n


0.9791564 n


0.9787259


9.0446347*


240


0.2026919


235 28 12.84


0.6602801 n


0.9223582 n


0.9217848


0.6603403*


270


0.1828971


264 38 54.09


0.9069056 n


0.7051686 n


0.7043164


0.9066458 n


300


0.1621568


295 15 13.36


0.9790254 n


9.4089485*


9.3984933


0.9786051 n


330


0.1463201


327 12 35.38


0.9148835 n


0.6835510


0.6835778 n


0.9142995 n


~sT


1.0916971


900 0.00










2,


1.0916972


1080 0.00











E


10g5


log^


p


\


g


logC"



30


1.0710308
1.1693425


1.9648845
1.9689849


10682.6599
18227.1670


8117.2267
8041.3776


1.0027367
1.0056470


8.3349592
8.6360598


60


1.2416284


1.9733790


26314.7192


7960.4005


1.0092812


8.8370927


90
120


1.2868582
1.3068094


1.9769311
1.9787352


32935.6960
36424.3060


7895.1366

7862.0877


1.0129818
1.0155661


8.9705083
9.0427814


150


1.3022786


1.9783135


35792.7330


7869.9176


1.0157721


9.0497000


180


1.2725893


1.9757393


31049.5750


7917.3434


1.0131888


8.9809495


210
240


1.2158096
1.1302863


1.9716558
1.9671494


23362.5420
14847.5260


7992.6957
8076.0143


1.0087905
1.0043632


8.8189886
8.5302017


270
300


1.0231950
0.9419157


1.9634684
1.9616475


7941.6038
4595.4802


8144.1096
8177.6560


1.0013537
1.0002558


8.0341618
7.3163722


330


0.9692790


1.9621814


5654.5662


8167.4836


1.0008465


7.8342202


1




11.8215349
11.8215351


123914.27
123914.31


48110.729
48110.721


6.0453918
6.0453916





22



E


lOgPstf/


log-P.Ci*


B*


B u


Bp


A*C





1.9169034


0.5318556


1.71845


.89671773


0.3721019


1273741.5


30


1.9755785


0.5328775


3.01775


-1.13441580


2.5062115


2172708.2


60


1.9062750


0.2070736


-4.62374


.98414281


4.7095692


3211393.9


90


1.6134503


8.6889828 n


6.08172


.44676768


5.6474343


4168957.1


120


1.2216849 n


9.5472262


6.93032


0.29124699


5.0684299


4791819.1


150


1.8858736,,


0.4342181


6.88475


0.95971490


3.1277779


4857869.0


180


2.0793986 n


0.6943511


5.96030


1.34015180


0.3456203


4287900.0


210


2.1185815 n


0.6960869


4.47493


1.35536890


2.5324875


3228683.9


240


2.0337615 n


0.4411836


2.90359


1.07265140


4.7354355


2023242.3


270


1.7767034


9.5893663


1.66622


0.62356451


5.6730943


1059477.9


300


0.4293996


8.6103629 W


1.00508


0.10695994


5.0942969


597208.9


330


1.6828108


0.2039025


1.01624


.42731476


3.1540537


697610.1


2,






23.14148


0.9301496


0.0782150


16185305.7








23.14161


0.9301500


0.0782128


16185306.2



E


30
60
90
120
150
180


A"C-


A Z C


*$


& u
log - c -


^


1,000 R


1331148.8
1499656.7
1257141.7
633680.8
- 251362.3
-1175895.1
1859502.8


51269.99
100538.88
149578.11
185230.51
-190157.71
154729.51
- 89067.22


8.0562601
8.1210086
8.1877509
8.2373935
8.2639876
8.2663024
8.2448728


7.5893686
7.5702475
7.4530372
7.1237659
6.6740144 M
7.3428058
7.5509401


5.9407458 n
6.8638621 n
7.095934U
7.1721544 n
7.1355044*
6.9590236 n
6.3943817n


1.7097689
1.7745204
1.9262896
2.1375744
2.3569481
2.5168682
2.5606371


210
240


2068748.7
1729849.5


20232.80
25701.71


8.2024945
8.1483792


7.6367383 n
7.6523844 n


6.6338873
6.9773713


2.4720554
2.2843136


270


969088.3


38725.85


8.1153154


7.6286630 n


7.0861514


2.0608640


300
330

I


43468.5
784500.8


24077.58

8288.07


8.2046672
8.0486528


6.7557748 n
7.5910007


7.1006984
7.7301623


1.8653206
1.7413404


1295892.6
1295893.9


430293.74
430293.93








12.70328
12.70322



E


1000 U


1000 Z


1000 # n


1000 H t


1000 H u


1000 He


-500 H Ml





+.0099129364


+.01219913


.01178899


+.00313663


1.7097689


+0.01982587


1.5503021


30


+.0070243377


.10023809


+.06747523


.07412671


1.4844357


+0.97300477


1.6311880


60


+.0054967025


-.20386986


+.03664317


.20054972


0.8120554


+1.74728269


1.8364587


90


+.0055926247


-.27213016


.09419268


.25530876


+0.2105525


+2.12773530


2.1375744


120


+.0032050993


.27013335


.20519340


1

Online LibrarySamuel G. (Samuel Goodwin) BartonSecular perturbations arising from the action of Saturn upon Mars : an application of the method of Arndt → online text (page 1 of 2)