Ernest W. (Ernest William) Brown.

The inequalities in the motion of the moon due to the direct action of the planets. An essay which obtained the Adams prize in the University of Cambridge for the year 1907 online

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LIBRARY



OF THE



UNIVERSITY OF CALIFORNIA.



Class



THE INEQUALITIES

IN THE

MOTION OF THE MOON

DUE TO THE

DIRECT ACTION OF THE PLANETS



CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
C. F. CLAY, MANAGER.

: FETTER LANE, E.G.
: 50, WELLINGTON STREET.




3Letp>i|j: F. A. BROCKHAUS.

Jftefo Sorfe: G. P. PUTNAM'S SONS.

Bombag anti Calcutta: MACMILLAN AND CO., LTD.



[All Rights reserved.}



THE INEQUALITIES

IN THE

MOTION OF THE MOON

DUE TO THE

DIRECT ACTION OF THE PLANETS



AN ESSAY WHICH OBTAINED THE ADAMS PRIZE
IN THE UNIVERSITY OF CAMBRIDGE FOR THE YEAR 1907



BY



ERNEST W. BROWN Sc.D. F.R.S.

(i

PROFESSOR OF MATHEMATICS IN YALE UNIVERSITY CONNECTICUT

SOMETIME FELLOW OF CHRIST'S COLLEGE CAMBRIDGE AND PROFESSOR OF

MATHEMATICS IN HAVERFORD COLLEGE PENNSYLVANIA




CAMBRIDGE

at the University Press
1908



Astron.










Cambrilrgt :

PRINTED BY JOHN CLAY, M.A.
AT THE UNIVERSITY PRESS.



TO

GEORGE HOWARD DARWIN

AT WHOSE SUGGESTION THE STUDY OF THE MOON'S MOTIONS

WAS UNDEETAKEN BY THE AUTHOR
AND WHOSE ADVICE AND SYMPATHY HAVE BEEN FREELY GIVEN

DURING THE PAST TWENTY YEARS
THIS ESSAY IS GRATEFULLY DEDICATED



a 3



173963



CONTENTS.



PAGE

INTRODUCTION . ix



GENERAL SCHEME OF NOTATION



SECTION I. THE EQUATIONS OF VARIATIONS 3

Numerical form of the equations 6

The variation of the moon's true longitude . 11

Abbreviation of the formulae 12

Final form of the abbreviated equations 14

SECTION II. TRANSFORMATION OF THE DISTURBING FUNCTION .... 16

SECTION III. DEVELOPMENT OF THE DISTURBING FUNCTION 26

Leverrier's expansion of I/A . . . . . . . . . . 26

Calculation of the functions $ p . 28

Solution of a difficulty 34

Calculation of the coefficients Mi and of their derivatives . . . . 35

SECTION IV. A SIEVE FOR THE REJECTION OF INSENSIBLE COEFFICIENTS . . 37

Construction of the sieve 41

Orders with respect to e, k, associated with the lunar arguments . . 45

Method for finding the periods to be examined ...... 45

SECTION V. AUXILIARY NUMERICAL TABLES 49

Adopted values for the constants of the solar system 49

Tables for A p , p , ..., for each planet 50

The values of M t and of their derivatives 61

An example of the computation of a primary coefficient .... 65

The terms retained by the sieve for computation ...... 67

SECTION VI. THE INEQUALITIES IN THE COORDINATES 70

Terms in the moon's longitude 70

Terms in the moon's latitude 87

Terms in the moon's parallax 88

Terms in the mean motions of the perigee and of the node ... 88

ADDENDUM. FINAL VALUES FOR THE TERMS IN LONGITUDE .... 89

ERRATA 93



<,!-* -^-'-tt YJ55

OF THE "

{ UNIVERSITY )

OF



INTRODUCTION.



THIS Essay aims at a complete calculation of the effects produced by the
action of a planet on the motion of the moon under the following
limitations and conditions:

(1) The problem of the motion of the moon under the action of the
sun (supposed to move round the centre of mass of the earth and moon in
a fixed elliptic orbit) and the earth, is considered to have been completely
solved.

(2) All the bodies are supposed to attract in the same manner as
particles of masses equal to their actual masses and situated at the centres
of mass.

(3) All the planets are supposed to move in fixed elliptic orbits,
i.e., the effect of the action of a planet transmitted either through the earth
or through another planet is neglected. '

(4) Perturbations of the first order with respect to the ratio of the
mass of a planet to that of the sun are alone calculated.

(5) The exception to the above limitations occurs in the periods of
revolution of the apse and node of the moon's orbit. These periods are not
exactly those arising from (1) but they are the observed periods or, what
amounts to the same thing owing to the close agreement between the
observed and calculated periods, the periods after all known causes have been
included. The point is only of importance in terms of very long period.

(6) All coefficients greater than 0"'01 in longitude, latitude and
parallax have been obtained. Many are also given which are less than
0"'01 whenever they have been accurately calculated. There are, in addition,
classes of terms of short period which run in series and which in the
aggregate will add up at certain times to much more than 0"'01 : these have
been found to be 0"'002.



X INTRODUCTION

(7) The maximum period considered is 3500 years, but as the sieve
in Section iv. retained a few terms of longer period, these were also included
in the general scheme.

The methods here adopted have been constructed mainly to overcome
the difficulties which have in the past prevented an accurate computation
of the long period terms. They were, however, found to be equally useful
for finding the terms which have periods of a year or less. These difficulties
include the development of the parts of the disturbing function which
depend on the coordinates of the earth and planet; the accurate calculation
of the derivatives of the moon's coordinates with respect to n, the moon's
mean motion ; the uncertainty arising from the possible omission of terms
of long period ; and the frequent appearance of small coefficients as the
difference of two large numbers.

In Section I., the equations of variations for the lunar elements have
been recomputed with the use of a semi- canonical system of elements. The
equations for the ordinary system of elements were first given by G. W. Hill
and independently, though at a later date, by S. Newcomb ; they were
recalculated in Hill's form by R. Radau. In the present system, errors
arising from the slow convergence of Delaunay's series have been avoided ;
in fact his literal expressions have only been used in small terms where
derivatives with respect to n were required but where the maximum possible
error could make no difference in the final results.

In Section II. Hill's method of dividing the disturbing function into
a sum of products in which the first factor of each product is independent
of the lunar coordinates and the other of the planet's coordinates, is exhibited
as part of a general theorem.

By referring these coordinates to the true place of the sun's radius vector,
I have obtained the first factors directly from the expansion of the inverse
first power of the distance between the planet and the earth (I/A) and of its
derivatives with respect to certain of the elements of the earth and planet.
Only one expansion is therefore required for all these factors, namely, that
of I/A, and this has been given by Leverrier in a literal form in powers of the
eccentricities and mutual inclination. The expansion also contains the
coefficients in the expansion of 1/A (the value of a'/ A when the eccentricities
and inclination are zero) and its derivatives with respect to the planet's mean
distance: the formulae for finding the coefficients and their derivatives are
here put into forms which admit of rapid and simple computation.

The factors containing the moon's coordinates, together with their deriva-
tives with respect to all the lunar elements except n, are found from the
results of my lunar theory; a special method which I gave five years ago and
which does not require the use of literal series in powers of ri/n has been



INTRODUCTION XI

used to find the derivatives with respect to n. These methods for finding
the planetary and lunar factors are set forth in Section in.

The only one of the difficulties previously mentioned which has not been
considered up to this point is the danger of omitting long-period terms with
sensible coefficients. In Section IV. formulae are constructed" which permit
one to find rapidly an upper limit to the magnitude of any coefficient. By
means of them, all terms having coefficients greater than 0"'01 and periods
less than 3500 years have been sifted out; there are about 100 of such terms,
excluding the terms of short period for which no sieve was required.

Sections V., VI. consist of numerical results. It may be noted that the
values of A p , B p , ... are also required for finding the perturbations of the
earth by the planets and of the planets by the earth, while those of Mi and
of their derivatives (as well as the equations of variations contained in
Section I.) are available for the computation of lunar perturbations other than
those due to the direct action of the planets.

No new inequalities sufficiently great to account for the observed dis-
crepancy between theory and observation appear from the direct action of
the planets, as shown in the tables of Section VI. Radau's well-known list
of terms in longitude has required considerable extension as far as the short
period terms are concerned, and a few new long period inequalities with
small coefficients have been computed. The more extensive developments
of this essay have shown that some of his coefficients require alteration, but
there is a general agreement for all those portions which he has taken into
account.

Only a few slight verbal changes and corrections of errors in copying have
been made to the first five sections, with two exceptions mentioned below,
since the award of the examiners. But I have gone over all the computations
for finding the short period terms and the larger long period terms during
the year that has elapsed and have made the following corrections to the
results in Section vi. :

Argument Former coefficient Corrected coefficient
1+3T-WV+33 - 0"-35 + 0"-35

-1-WT + 187-151 -15"-22 -14"-55

Z+29T-26F+ 112 + 0"-117 + 0"-108

2D-J + 21T-20F-87 + 0"-111 + 0"-126

2l-2T> + GM-5T+2ir-l + 0"-040 - 0"-038

together with their accompanying short period secondaries.

The signs of those coefficients containing h on p. 86 have been changed.

The annual mean motion of the perigee has been altered from 2"'66*
to 2"-69.

* I gave this value in a paper referred to on page 3 below.



xii INTRODUCTION

The values of M 2 and of its derivatives with respect to n, e, k under the
argument on page 61 have required a factor 2. This error necessitated
slight changes in some of the coefficients whose primaries were independent
of the lunar angles ; the largest correction was one of 0"'019 in the coeffi-
cient of sin (T - F).

A wrong sign in the computation of the equations of variations (see
Errata at the end of the volume) gave rise to a few almost insensible
changes in certain coefficients.

The additions are :

Argument Coefficient

Of-27 T +63 -0"-012

2l-2V + 4(M-T)-l +0"-017

21- 20 + 8^-67*+ 63 -0"-019

-l -0"'031



No other change or additional coefficient has been greater than 0"'010.

The Addendum containing the results obtained by adding together terms
of the same argument in Section VI. is also new.

During the summer of last year Professor Newcomb's new work* on
this subject was published. His methods differ so completely from those
given here that no comparison is made easily except in a few of the final
results where the indirect action is insensible or is separated from the direct
action. For the large inequality due to Venus he obtains a coefficient of
14"'83 while mine is 14"'55 ; a portion of the difference is probably due to
certain terms of the second order relative to the ratios of the masses of Venus
and of the earth to the sun which Professor Newcomb has included. On the
other hand, he states that the possible errors arising in his method may be of
the order of this difference, while such errors are excluded from my result.
His results and mine for the annual mean motions of the perigee and node
agree within 0"'01, which is the limit of accuracy to which I have obtained
these quantities.

* "Investigation of Inequalities in the Motion of the Moon produced by the Action of the
Planets." Carnegie Institution, Publication 72, Washington, D.C., June, 1907.



E. W. B.



NEW HAVEN, CONN., U.S.A.
1908 March 27.



OF THE

UNIVERSITY

OF



GENERAL NOTATION.



THE axes of x, y are taken in the ecliptic of 1850'0 and the centre of mass
of the earth and the moon is supposed to describe a fixed ellipse around the sun
in this plane. As it is more convenient to use the motion of the centre of
the earth than of this centre of mass, a slight well-known change, noted
below, is necessary in the disturbing function.

The axis of x is parallel to the line joining the earth and the sun.

In the scheme of notation which follows, two sets of constants are given
for the mean distance, eccentricity and sine of half the inclination of the
moon's orbit. The first set is that which I have used in the expressions for
the rectangular coordinates of the moon ; the second set is that of Delaunay
in the final form which he gives to the expressions for the longitude, latitude
and parallax. The longitudes of a planet, of its perigee and of its node
are as usual reckoned along the ecliptic to its node and then along the
orbit.



True long.
Mean long.
Mean anom.
Mean long, of node

Mean motion

Mean distance
Eccentricity
Sine half inclin.
Coors., origin earth
sun



Moon

V
1 = l + g + h



n

a, a

e, e

k, y

x, y, z, r



Earth
V
T

l' T-as'




Planet

V"

P
l" = P w"

h"

flP

at

a"

e"

y"

, rj, , A
x", y", z", r"



dw 1
n = mean motion of the moon = -j-= ,

& 2 = its perigee = ^ ,

b *= node ^'



BR.



2 GENERAL NOTATION

d, c 2 , c 3 are the canonical constants complementary to w lt w 2 , w 3 after the
problem of the moon's motion as disturbed by the sun, supposed to move in
a fixed elliptic orbit, has been solved.

R = the disturbing function of this problem, arising from the direct
attraction of a planet.

The symbols for the mean longitudes of the planets are : Mercury, Q ;
Venus*, F; Mars, M\ Jupiter, /; Saturn, S.

* No confusion will be caused by the use of the same symbol for the true longitude of the
moon and the mean longitude of Venus. The notation of Eadau has been adopted with a few
changes.



SECTION I.

THE EQUATIONS OF VARIATIONS.

LET w^ w 2 , w 3 represent the mean longitudes of the moon, of its perigee
and of its node, and suppose the problem of the moon as disturbed by the
sun, has been solved. Then it is well known that if a disturbing function R
be added to the force function of the moon's motion, the change in the latter
due to R can be obtained by solving the equations

dd dR dwi dR .



where 6 lf b 2 , b 3 are the mean motions of the moon, of its perigee and of its
node, and c l} c 2 , c 3 are the canonical constants corresponding to w ly w 2 , w s ]
the Ci are functions of the arbitrary constants n, e, k of the moon's motion,
and they also contain the constants n', e', depending on the sun's motion.
The substitution of the new values of d, Wi, thus found, in the expressions
for the coordinates will give the disturbed position of the moon at any
time.

The constant part of R only gives constant additions to the bt, i.e., to
the mean motions* : this part will be neglected, since it has no effect on
the new terms to be found. Hence & t = n.

Change to the semi-canonical system n, c 2 , c 3 , retaining the Wi un-
changed. Putting

~~7 ^ ^ Pi

dn
and remembering that

,.. , dci _ db 2 dCi _ db 3 db 2 db 3
dc 2 dn ' dc 3 dn ' dc 3 dc 2 '

* I have found these changes in an earlier paper : Trans. Amer. Math. Soc. , Vol. v. pp. 279
284. A fresh computation just made gives 2" - 69, -1"'42, for the mean motions of the perigee
and node respectively. The former is 0"'03 more than the value given in the paper.

12



4 THE EQUATIONS OF VARIATIONS [SECT.

we obtain equations (1) in the semi-canonical form

dn ldR, dbz dcdb 3 dcA dw l _ 1 dR .

' dt ~ a 2 3 dn



_ _

dw l dn ' dt dn ' dt ' dt ~ a 2 /3 dn

dc z dR dw, dR

di=^ ' w = -^

dc s dR dw 3 dR , fdw 1 , \ db



in these equations b 2 , 6 3) c a are supposed to be expressed in terms of n, c 2 , c 3
and R in terms of n, c 2 , c g , w lt w 2 , w 3 .
Consider any periodic term of R :

R n*tfA cos (qt + q) = n' 2 a?A cos (i l w l + t' 2 w 2 + i s w 3 + q"t + q'"},
where a is the linear constant of Hill's variational orbit and of my lunar
theory and A is a numerical coefficient (that is, its dimensions with respect
to time, space and mass are zero) ; q"t + q" is a combination of the solar and
planetary arguments. Then since qt + q' is independent of the Ci and A of
the Wi, the first three of equations (2) become

dn n' 2 a 2 . dq . , , ,,



It will now be supposed that R contains a small factor whose square may
be neglected. The coefficients in the right-hand members of the last set will

H

then be constants and we can integrate. Put m= , and let 8n, 8c 2 , Sc 3

iff

denote the increments of n, c 2> c 3 , due to R. Then



Bn ra a 2 dq n' . ,.

- = - . -^ - A cos (qt + q ),
n ft a? dn q



(3)



8c 2



= Lm . A cos (qt + q'),

r*& r* v -* * *



; 3 _ . a 2 n
tf 3 ' a 2 q

Again, if we put



A -
A ~



dA

-^
cfc 2



l] THE SEMI-CANONICAL FORM

the other three of equations (2) become
idw, n' 2 a 2 .



(3a)



dt ~nft ' a*~

dw 2 _n 2 a 2 /. AI dh
dt n a 2 \ ft dn

dw 3 n' 2 a 2 / . A! db 3 \ ,. ,

-T7- = -5 (A, + -g 1 . -,! ! cos (gtf + gr') -(- 6 3 .
dt n a 2 V P W



Let 8b l} Sb. 2 , S6 3 , Si^, Bw. 2 , Bw 3 denote increments due to M. Then

n' a 2 dq n'
$b l =*Sn = - 5 . -j 1 .

2



cos



/V ^l /i /^ fif*

{.4/lv \A/\S% ^*^- / 3

, a 2 n ' A ( 1 ^2 d^ 2 d6 2 2 d6 2 . ,

= n - . A\ TT . i . -= H a = . i 2 + a -7 . t 3 cos



= . - - . T- . -r . 2 -

a? q \ ft dn dn dc 2 dc 3

by equations (la).

Putting q 2 = - no? , <? 3 = -na 2 -7^- ,

ClCg CtCa

and using the last of equations (1 a), we find

, a 2 n A /I rf6 2 ^7 , \ ' N

86, = - n' . -Al-s . -j- . -~ + qz cos (qt + q ) ;
a 2 ^ \ an dn * J

... ., , a 2 w' /I 6^63 dq t \ ,,

similarly, d6 3 = - w . - ^ -5- . -7- . ^ + ^ 3 cos (qt + q ).
a 2 q V/3 dw <*ft /

The undisturbed values of b 1} b 2 , b s are of course equal to those of
T^ > ~T^> ~T^ > res P ec tively. To obtain Bw 1} 8w 2 , 8w a (the increments of



Wi, w 2 , w s ), it is necessary to replace Wi by w t + 8wi, and 6 t - by bi+Sbi in
(3 a), to substitute for Bbi the values just obtained, and then to integrate.
These operations give



(4)



1 a 2 / n' A dq n' 2 A
- A



^ l -

ft a 2 V q dn q 2



a 2 f w / . A ! d6 3 \ ?i 2 . /o s 1 d6 3 d<y \ ( . , /x

\m-(A 3 +~. l - 3 -) A p ^ + 5 -j- 1 ^ ^ H sin (qt + q ).

a 2 1 q \ )S an/ q 2 \n ft dn an/ 1



The equations (3), (4) constitute the solution of the problem.
The form in which a periodic term of R arises (see Section II.) is

R = \ ^L- ri*a?A cos (qt + q').
4 m



6 THE EQUATIONS OF VARIATIONS [SECT.

It is therefore necessary to multiply the right-hand members of (3), (4)
by m"/4>m'.

Next, let

s'= no. of seconds in the daily mean motion of the sun = 3548 //< 19,
. s= M argument qt + q'.

Then

Also, put for brevity



q s



(5)



=Z - 206265,
4 m a 2 /3





4 ra a 2






The coefficients of the right-hand members being thus expressed in seconds
of arc, equations (3), (4) become



(6)



Sn ,,,dqA ,

- = -/ -^ cos (at + q),
n dn s



.
dn



> - A ft + i . sin (q t + q') ;

2



.

n dn dn



s \ an '] s*

where, to recall certain definitions,

_ 1 dci

p = 5 . -j , C

a 2 dn

i&9.

g 2 = - na 2 -i q



Numerical form of the equations of variations.

It remains to be seen how these quantities may be put into numerical

a 2

form. In /, /' the factor is immediately obtained from Hill's results* for

a

m"
the variational orbit ; m is a well known quantity ; T is known as soon as

* Amer. Jour. Math., Vol. i. p. 249.



l] NUMERICAL FORM OF THE EQUATIONS OF VARIATIONS 7

the particular planet is chosen ; thus /, /' remain the same for a given
planet and ///' for all planets. The coefficients A, A lt A z , A 3 will be
found later on, while s is known as soon as the particular term of R has been
chosen.

There remain for calculation

.,_. dc-L db* dbs dbs db^_db 3 dba
dn ' dn' dn' dc 2 ' dc 3 dc z ' dc a '

which, depending only on the orbit of the moon as attracted by the sun and
earth, are the same for every perturbation of the moon's motion, and there-
fore apply not only to the present investigation but also to all investigations
where a disturbing function R is added to the moon's force function.

Some idea of the degree of accuracy required is desirable. The largest
known inequality is that with the argument 1 + I6T 18F, which has a
coefficient of about 15". For this ^ = 1, i 2 = - 1, i 3 = 0. The principal part
is given by

_/ 4 .._/ 4 (!_*!}__/ 4(1+ -01486),

J s 2 dn J s 2 V dn) J s 2 v

There is no other coefficient which is so great as 2". Since the degree of
accuracy aimed at is 0"'01, it will be sufficient to use four place logarithms
and four significant figures for the functions (7) so that the final results
will be accurate to at least three significant figures. But certain of the
functions are only needed to one or two significant figures, as will appear
immediately.

The functions c ly C 2 , C 3 are the same as Delaunay's L, G L, H G after
the final transformations and the changes to his final system of arbitraries,
n, e, 7 have been made. As my results will be used for the calculation of
the moon functions, it will be more convenient to transfer A^, A 3 to my
constants e, k.

Let -f* denote the derivative of a function Q with respect to n when it is

CtTl



expressed in terms of n, c 2 , c 3 , and jr when it is expressed in terms of

n, e 2 , 7 2 . Then the following equations serve for the transformation of the
derivatives of Q from one set to the other*:

dQ = /dQ\ _ dQ /dc 2 \ _ dQ

dn \dn ) dc 2 ' \dn ) dc s \dn

-
de z dc s ~de 2

dQ rdQ dQ

df '

The functions considered here involve e, y only in the even powers.



8



THE EQUATIONS OF VARIATIONS



[SECT.



the same equations will serve for the transformation from the set n, c 2 , c 3 to
the set n, e 2 , k 2 if we replace e 2 , <f by e 2 , k 2 , respectively.

It is first to be noticed that

C 2 = na?e 2 | J + power series in m, e 2 , y 2 , e' 2 , ( ) L
I \/ J

c 3 = naV {- 2 + } ,

G! = no? { 1 + } ,

6 2 = Tim? { f + } ,

6 3 = nra 2 {- 1 + } ;

the power series in each case vanishing with m. Hence when for Q in
equations (8) is put c 1} b 2 or 6 3) the principal part of the first term in each
equation will have a portion independent of e 2 , 7*, while the other terms will

have one of these quantities as a factor. Since e -fa, 7 = ^ approximately,

(J!G cLc
two significant figures will be fully sufficient for the values of ~ , -^ . It is

to be noted that the second and third of equations (8) do not depend on
derivatives with respect to n, and therefore that we may change to the variables
e, 7 or to e, k, from c 2 , c 3 without reference to the first equation.

From Newcomb's transformation* of Delaunay's values for L, G, H,
we have



Terms in


C 2


1 (dc 2 \
a 2 e*\dn)


Terms in


C3


1 /dc,\
0*1* \dn)


na 2 e 2


Jta 2 7 2


m


-50015


+ -1667


m


-1-99704


+ -6657


m 2


+ 01686


- -0393


m 2


- -00322


+ -0075


m 3
m 4


+ -00644
+ -00170


-0215
- -0074


m 3
m 4


+ -00049
+ -00005


-0016
- -0002


rem.


+ -00032


- -0020


rem.


- -00022


+ -0010


Sum


- -47483


+ -0965


Sum


-1-99994


+ 6724



The remainders in the first and third columns are obtained from the
values of c 2 , c s (calculated by processes independent of the nature of con-
vergence along powers of m) given by myself f ; those in the second and
fourth columns are estimated from them. In any case the results for

are correc * w ^hin two per cent, and this is all the accuracy



necessary for our purpose.



In a similar manner the values of (

\



^r 2 ), (-rM
dn) \dnj



can be obtained with



* "Action of the Planets on the Moon," Amer. Eph. Papers, Vol. v. Pt 3, pp. 201, 202.
t "Theory of the Motion of the Moon," Trans. R. A. S., Vol. LVII. pp. 64, 65.



I]



NUMERICAL FORM OF THE EQUATIONS OF VARIATIONS



more than needful accuracy from the series in powers of m given by
Delaunay*, Hillf, and Adams with my numerical values . The deri-
vatives with respect to e 2 , 7 2 are obtained immediately from the last named
reference. The derivatives with respect to e 2 , k 2 can be immediately derived

6 0^ -, ^

by inserting the values of , / which I have also given : the change in the

6 1C

derivatives with respect to n is insensible.

I find the following results to four places in the logarithms, to each of
which 10 has been added :



=)= + [7-1313] a 2 , p = + [67577] na 2 , g 2 = - [10-3039] m 2 ,


1 3 4 5 6 7

Online LibraryErnest W. (Ernest William) BrownThe inequalities in the motion of the moon due to the direct action of the planets. An essay which obtained the Adams prize in the University of Cambridge for the year 1907 → online text (page 1 of 7)