A.STRONOMY DEPT
INVESTIGATION OF INEQUALITIES Efc*
s^*/
MOTION OF THE MOON PRODUCED BY
THE ACTION OF THE PLANETS
SIMON NEWCOMB
ASSISTED BY
FRANK E. ROSS
WASHINGTON, D. C.:
PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON
JUNE, 1907
ACTION OF THE PLANETS ON THE MOON.
TABLE III. Concluded.
MUTUAL PERIODIC PERTURBATIONS OF VENUS AND THE EARTH.
The term of long period is omitted. The tabular unit is o".oi in tu and <5f', and 10 8 in Sp and <)//.
System 9.
System^ 10.
System ii.
i
du
8v'
dp
S t o>
du
dv'
dp
V
du
dv'
fy
8p>
- 508
+447
+ 196
+563
- 548
+407
+ 300
+495
- 205
+ 158
+ 341
+471
I 1 544
+3i8
+ 174
+583
- 616
+355
+ 295
+506
- 371
+ 169
+ 372
+454
2 482
+ 176 + 237
+552
660 j +285
+ 355
+469
540
+ 175
+ 468
+393
3
444
+ 36 + 357
+487
-698
+200
+ 465
+398
- 718
+ 179
+ 591
+306
4
446
90
+ 493
+394
747
+ 109
+ 579
+301
-897
+ 176
+ 700
+207
5 493
196 + 613
+285
- 811
+ 19
+ 673
+ 193
-1066
+ 165
+ 767
+ 103
6 576
-278 + 699
+162
-889
- 62
+ 727
+ 82
1208
+ 141
+ 783
+ o
7 -690
-338 i + 744
+ 26
980
-127
+ 732
32
1321
+ 109
+ 757
-104
8
- 828
370
+ 736
121
1073
171
+ 681
142
1409
+ 77
+ 662
-205
9
-981
371
+ 667
272
1164
194
+ 577
-254
-1470
+ 51
+ 5i6
-300
10 1138
335
+ 522
42O
1249
193
+ 4i6
364
-1497
+ 39
+ 342
-384
ii 1274
260
+ 298
-553
1290
-166
+ 203
-472
1484
+ 41
+ H4
-456
12 1360
147
3
661
1310
113
60
-569
-1423
+ 59
147
517
13 1372
i
351
740
1274
- 29
- 364
-648
-1314
+ 95
427
-566
M
-1288
+ 164
- 715
-783
1176
+ 83
-693
-699
1154
+ 147
7H
-603
IS
1106
+339
1055
-786
IOOO
+222
1015
-716
- 938
+220
-987
624
16
-835
+504
-1328
-753
741
+374
1301
-695
- 669
+311 1237
620
17
493
+654
1520
-682
- 415
+530
1511
-637
346
+420 1438
-583
18
- 106
+779
1609
-576
40
+671
1617
549
+ 19
+536 -1561
-Sio
19
+ 308
+876
-1582
440
+ 353 +791
1605
-431
+ 413
+651 1579
403
20 + 710
+938
-1438
-282
+ 726 +878
-1469
293 + 793
+749
1479
270
21 +IO72
+904
1180
-108
+ 1049
+933
1225
135 +1126
+821
1260
-116
22 +I3S9
+948
- 820
+ 68
+ 1323
+949
- 880
+ 33 +1387
+862
940
+ 40
23 | +IS32
+899
399
+242
+ 1501
+927
467
+207
+ 1547
+867
544
+202
24 \ +I580
+816
+ 54
+405
+ 1572
+863
13
+376 +1600
+835
IOI
+359
25 +1493
+711
+ 494
+551
+ 1525
+765
+ 446
+530 +1546
+769
+ 355
+509
26 +1278
+592
+ 885
+677
+ 1353
+632
+ 872
+659 ; +1379
+667
+ 79'
+644
27
+ 957
+466
+ I2IO
+778
+ 1070
+479
+ 1221
+759 + 1116
+533
+ "74
+757
28
+ 558
+336
+ 1436
+849
+ 70i
+3i6
+ 1471
+833 : + 766
+372
+ 1460
+841
29
+ 106
+205
+ 1545
+884
+ 271
+ 153
+ I&00
+874 < + 353
+ 192
+ 1627
+887
30
355
+ 73
+ 1534
+883
179
i
+ 1609
+886 - 84
+ 6
+ I6S9
+896
3!
791
59
+ 1404
+848
- 621
142
+ 1501
+867 - 5i5
-171
+ 1565
+870
32
1164
-183
+ 1169
+78o
1026
-270
+ 1278
+815 - 909
-330
+ 1357
+814
33
1439
297
+ 861
+690
-1358
-383
+ 96l
+733 -1234
-466
+ 1050
+735
34
-1594
-393
+ 497
+58i
1592
477
+ 574
+621 : 1476
573
+ 673
+632
35
1633
471
+ 134
+465
1704
550
+ 154
+491 1629 652
+ 250
+5"
36
1560
537
225
+345
-1684
-602
256
+351 1661 ! 707
- 183
+370
37
1402
594
- 544
+226
1549
630
620
+209 : 1579 733
- 596
+216
38
-1173
-645
- 818
+ "5
1317
639
923
+ 77 1384 730
959
+ 60
39
- 879
-694
-1058
+ ii
1017
-637
-1149
43 1098 703
1232
- 85
40
540
-740
1237
-84
674
-631
1301
-150 - 752 : -654
1403
219
41
- 159
-783
1347
169
303
627
244 375 595
1473
-324
42
+ 244
-813
1390
-245
+ 80
-626
324 1 + 3 532
1450
403
43
+ 651
-838
-1365
-315
+ 465
-626
1342
391 ; + 3f>4 477
1357
462
44
+ 1039
-849
-1275
-383
+ 835
623
1225
442 + 697 432
1208
504
45
+ 1399
-845
II2I
449
+1181
-617
1049
478 +1001 396
1009
531
46 +1708
-821
911
509
+1482
-607
- 832
502 ; +1264 368
- 778
542
.17 ' +1946
-768
642
-562
+ 1728
-591
- 580
517 : +1480 346
520
-536
48 +2089
-681
322
60 1
+ 1908
-566
303
526 +1640 328
248
515
49
+2122
SV
+ 27
621
+2014
-524
- 15
527 +1743 312
+ 17
-481
50
+2O26
393
+ 372
-6n
+2027
460
+ 276
514 +1784 i 298
+ 265
434
51
+ 1803
207
+ 678
570
+ 1939
-371
+ 548
479 i +1761 i 283
+ 487
-378
52
+ 1482
ii
+ 912
499
+ 1744
254
+ 779
418 , +1679 259
+ 673
314
53
+ 1093
+ 178
+ 1054
396
+ 1454
-116
+ 942
327 +1528 i 221
+ 813
-236
54
+ 679
+344
+ 1097
269
+ 1097
+ 27
+ 1017
213 +1313 169
+ 895
145
55
+ 279
+477
+ 1042
-117
+ 709
+ 166
+ 995
80 +1044 ioi
+ 900
3i
56
71
+569
+ 906
+ 49
+ 33i
+284
+ 884
+ 66 +74327
+ 825
+ IOO
57
344
+612
+ 7io
+217
+ 3
+37i
- 716
+209 ; + 451 i + 43
+ 686
+234
58
- 523
+603
+ 497
+370
254
+418
+ 536
+337
+ 189 +99
+ 525
+354
59
- 60 1
+548
+ 312
+489
434
+430
+ 385
+437 - 27 +137
+ 397
+437
COEFFICIENTS FOR DIRECT ACTION.
TABLE IVa.
PERTURBATIONS OF THE G-COORDINATE X OF VENUS.
The tabular unit is io- 8 .
53
Sys-
tem
I
2
3
4
5
6
7
8
9
IO
II
i
o
+ 47
+ 57
+ 92
+ 137
+ 181
+217
+218
+184
+142
+107
+ 71
+ 52
i
+ 37
+ 50
+ 81
+118
+ 159
+ 197
+212
+ 183
+140
+ 92
+ 52
+ 30
2
ii
+ ii
+ 41
+ 71
+ 109
+ 149
+ 176
+ 158
+ 114
+ 59
+ 8
23
3
92
59
24
+ 5
+ 39
+ 75
+ 112
fill
+ 72
+ 9
52
- 97
4
193
154
in
79
50
14
+ 26
+ 46
+ 14
-48
122
-183
5
305
-269
217
-183
-156
122
79
- 41
55
ill
-191
-268
6
412
396
-338
290
-267
-237
197
146
-136
179
26l
-348
7
506
519
465
405
-380
354
-318
-263
-230
251
-326
-421
8
-581
-628
-586
-487
-468
-437
-384
331
330
-386
482
9
-636
-715
-696
629
-583
-569
546
-502
-437
410
443
530
IO
-668
-763
-780
-718
-661
650
-636
-599
539
-488
494
-565
ii
674
777
-827
-783
-718
-699
-697
672
619
556
534
-582
12
-660
759
-828
814
745
714
-722
710
-669
606
-561
-580
13
-621
710
790
-803
745
699
-707
712
-682
-625
-569
-560
14
559
-639
-713
-749
-709
-652
652
-671
659
610
-552
523
15
479
541
614
-659
640
-580
564
-591
-596
-559
507
-466
16
-383
431
-495
-542
540
-487
452
-475
-496
476
432
-389
17
279
-308
364
410
-417
-374
-325
337
-368
366
331
-298
18
171
-183
-231
273
-283
-251
195
-187
221
236
-213
-187
19
54
61
-98
137
149
123
68
- 38
66
95
84
64
20
+ 60
+ 56
+ 25
13
26
__ T
+ 48
+ 93
+ 80
+ 46
+ 41
+ 58
21
+ 170
+160
+ 136
+ 100
+ 82
+106
+203
+209
+ 179
+ 155
+ 171
22
+269
+253
+229
+ 197
+ 174
+192
+236
+288
+312
+291
+262
+266
23
+350
+330
+305
+275
+249
+259
+299
+350
+384
+377
+348
+342
24
+406
+392
+363
+333 +307
+306
+341
+387
+430
+434
+412
+398
25
+441
+436
+407
+373 +347
+338
+367
+412
+451
+466
+456
+437
26
+461
+460 1 +435
+400
+374
+361
+378
+418
+457
+475
+476
+464
27
+468
+471
+451
+417
+388
+372
+38o
+418
+454
+478
+483
+478
28
+476
+475
+459
+426
+394
+381
+379
+411
+447
+472
+481
+484
29
+480
+475
+463
+432
+397
+381
+379
+405
+440
+465
+475
+485
30
+484
+479 +467
+438
+401
+380
+380
+401
+434
+459
+473
+479
31
+490
+484 +476
+445
+406
+380
+38o
+397
+431
+457
+470
+476
32
+491
+492 +483
+455 ' +414
+380
+379
+392
+426
+456
+469
+473
33
+485
+497 +488
+460 i +417 +379
+375
+387
+419
+451
+467
+466
34
+468
+492
+489
+463 +420 +377
+362
+375
+403
+437
+456
+456
35
+439
+468
+479
+458 \ +414 +366
+341
+354
+377
+4"
+432
+436
36
+397
+426
+448
+437
+392 +342
+3io
+314
+338
+368
+391
+399
37
+340
+360
+394
+396
+353
+301
+262
+260
+282
4-311
+331
+343
38
+263
+270
+309
+327
+293
+240
+ 196
+ 184
+208
+2 3 "6
+ 2 55
+264
39
+ 167
+ 166
+ 199
+230
+211
+155
+ 109
+ 90
+ 112
+141
+161
+166
40
+ 56
+ 47
+ 69
+ 107
+104
+ 5i
+ 2
20
4
+ 33
+ 55
+ 57
41
67
- 80
74
- 38
22 67
121
-145
127
90
59
58
42
-188
-206
215
188 j 161 192
-254
-283
-265
222
-180
172
43
301
-325
344
336
-306
319
-381
420
-407
357
304
282
44
-397
430
459
464
443
-438
494
-549
543
486
423
-385
45
473
510
545
-568
-557 -544
-589
-656
-665
-609
-534
-475
46
-528
-560
-605
634
-638 -626
-655
729
-760
712
628
555
47
-566
-578
-626
-665
681 i -678
694
764
-818
791
702
-617
48
-585
574
-611
-660
-685
-692
703
759
-833
-833
749
-657
49
-588
-547
-566
619
-653
-668
679
-723
-805
-838
-775
-675
50
572
-508
-506
549
-588
609
627
659
-737
-796
-766
-668
5i
539
457
431
-454
499
525
-545
-571
639
-716
-723
639
52
491
401 347
-350
391
423
447
-470
-525
-605
644
-589
53
431
339 268
248
276
3"
333
-357
401
-477
-535
-518
54
-362
274 193
152
-162
197
222
247
-281
-346
-414
428
55
-283
205 122
-69
-58
- 88
III
137
-167
219
-286
324
56
109
-134 - 58
+ 2
+ 31
+ 14
14
- 37
- 64
106
164
211
57
114
67 3
+ 60
+ 101
+ 102
+ 73
+ 48
+ 20
14
60
105
58
- 38
- 8 ; + 44
+ 104
+151 +167
+ 144
+ II5
+ 85
+ 54
+ 16
18
59
+ 19
+ 36 | + 78
+ 130
+179 +208
+ 194
+ 161
+ 128
+ 94
+ 59
+ 36
. . fc . >
INVESTIGATION OF INEQUALITIES IN THE
MOTION OF THE MOON PRODUCED
THE ACTION OF THE PLANETS
BY
SIMON NEWCOMB
ASSISTED BY
FRANK E. ROSS
WASHINGTON, D. C.:
PUBLISHED BY THE CARNEGIE INSTITUTION OF WASHINGTON
JUNE, 1907
Astron. Oept.
CARNEGIE INSTITUTION OF WASHINGTON
PUBLICATION No. 72
ASTRONOMY
CONTENTS.
INTRODUCTION I
PART I. DEVELOPMENT OF THE THEORY 3
CHAPTER I. Fundamental Differential Equations 5
1. Notation 5
2. Dimensions of Quantities in Terms of Time, Length, and Mass 6
3. Fundamental Differential Equations 6
4. Transformation to the Moving Ecliptic 6
5. Preliminary Form of the Potential Function 8
6. Reduction of the Terms of the Potential Function for the Indirect Action 9
7. Reduction of /?, the Potential of Direct Action 10
8. Completed Form of the Fundamental Equations n
CHAPTER II. Development and Integration of the Differential Equations for the Variation of the
Elements 13
9. Fundamental Variables 13
10. Canonical Form of the Differential Equations 14
n. Transformation of the Canonical Elements 14
12. Form of the Partial Derivatives 15
13. Numerical Values of the Fundamental Quantities 15
14. Formation of the Transformed Differential Equations 16
15. Elimination of the Time in Certain Cases 19
CHAPTER III. Definitive Form of the Differential Variations of the Elements 21
16. General View of the Problem 21
17. Reduction of the Equations for the Direct Action 21
18. Notation of the Planetary Factors 23
19. Notation of the Lunar Factors 23
20. Numerical Form of the Fundamental Coefficients 24
21. Fundamental Differential Equations for the Direct Action 26
22. Reduction of the Equations for o, e, and y to Numbers 26
23. Reduction of the Equations for /, IT, and 6 to Numbers 26
24. Development of the Indirect Action 28
25. Abbreviated Coefficients for the Indirect Action 30
26. Integration of the General Equations 31
27- Inequalities of /, T, and 9 32
28. Treatment of the Non-periodic Terms in /? 33
29. Adjustment of the Arbitrary Constants 35
30. Opposite Secular Effects of the Direct and Indirect Action of a Planet near the Sun 35
PART II. DEVELOPMENT OF THE PLANETARY COEFFICIENTS 37
CHAPTER IV. Coefficients for the Direct Action 39
31. Remarks on the Method of Development by Mechanical Quadratures 39
32. Action of Venus, Systems of Coordinates 41
33. Action of Venus, Fundamental Data for the -^-coefficients 42
34. Explanation of Tables of ^-coefficients for Venus 44
35. Mechanical Development in a Double Periodic Series., , , ,, 46
36O5S8 m
PREFACE.
THE immediate incentive to the present work was the hope of explaining by
gravitational theory the observed variations in the mean longitude of the Moon,
shown by more than two centuries of observation to exist, but not yet satisfactorily
accounted for. The author has published a number of papers and memoirs on this
subject during the last forty years, terminating with a summary of the case, which
appeared in the Monthly Notices of the Royal Astronomical Society for March,
1904. The deviations in question offer the greatest enigma yet encountered in
explaining the motions of the heavenly bodies, and the present paper may be
regarded as a contribution to the study of the problem thus offered.
While the work was in progress the completing chapter of Professor Brown's
Theory of the Moorfs Motion appeared. The actual work being based on De-
launay's theory, it seemed to be desirable to revise and correct it by Brown's
results. In doing this the imperfections of Delaunay's theory as a basis became so
evident, and the later theory proved to be so much better adapted to the purpose
of the investigation, that the completed work gradually became step by step prac-
tically based upon Brown's theory, except in those parts requiring derivatives
which could not be readily obtained except from Delaunay's literal expressions.
Acknowledgment is due to Professor Brown for courteous advice and assistance
which facilitated the use of his work for the purpose.
The theory of the action of the planets on the Moon being, in several points, the
most intricate with which the mathematical astronomer has to deal, it is important
that its development should be presented in a form to render as easy as possible the
detection of errors or imperfections. In the arrangement of the work this end has
been kept constantly in view. It is hoped that any investigator desiring to test the
processes will find few difficulties except those necessarily inherent in the nature
of the work.
To form a general conception of the arrangement it may be stated that the work
naturally divides itself into four parts. One of these treats of the theory of the
subject, including under this head not only the general equations, but the numerical
details on which all the computations are based. In this part the fundamental
quantities are reduced to products of two factors, one of which depends upon the
coordinates of the planet; the other upon the geocentric coordinates of the Moon.
The first factors, termed planetary, are numerically developed in Part II. This
development falls into two parts, one treating the direct action of the planet, the
other the indirect action through the Sun. In Part III is found the numerical
VII
VIII ACTION OF THE PLANETS ON THE MOON.
development of the factors depending upon the Moon alone, and of their partial
derivatives as to the lunar elements. In Part IV is presented the combinations of
these two factors and the final results of the work.
A more complete summary in detail is found in the table of contents. An effort
has been made to lessen the trouble of finding the definitions of the symbols used
by collecting in the introduction definitions or references to these symbols as to
the meaning of which doubt might be felt.
A word may be added as to the part taken by the author's assistant. At an
early stage in the work Dr. Ross made a practically independent computation of
the principal periodic inequalities, using the methods of Hill and Radau. In doing
this he discovered the error of the Jovian evection as computed by them, which
arose from the omission of what we may call the side-terms in the indirect action.
His result for the coefficient was i".i6, in exact agreement with that originally
found by Mr. Neville. In this early stage of the work the writer did not intend to
do much more than revise these computations, and make a thorough investigation
of the terms of long period. But he found the theory of the subject so interestingj
and the opportunity for recasting the methods so attractive, that he was led to
carry the work through, with Dr. Ross's assistance, on the basis of his own
developments.
The next step in logical order is the rediscussion of the moon's mean longitude
since 1650, as derived from occultations of stars, with a view of learning what
modifications will be produced by the use of the more rigorous data now available,
and the addition of thirty years to the period of available observations. This redis-
cussion will, the writer hopes, be his next contribution to the subject of the motion
of the Moon.
It remains to add that the work has been prosecuted under the auspices of the
Carnegie Institution of Washington, without the help of which it could not have
been undertaken.
SIMON NEWCOMB.
WASHINGTON, MAY, 1907.
ACTION OF THE PLANETS ON THE MOON.
INTRODUCTION.
MORE than thirty years ago the author proposed to treat the action of the
planets on the Moon by using the Lagrangian differential equations for the variation
of the elements by considering as simultaneously variable, not only what are com-
monly called the elements of the Moon, but those of the orbit of the centre of mass
of the Earth-Moon around the Sun also.* Twelve elements would thus come in,
and the coordinates both of the Moon and of the Sun would be expressed in terms
of the osculating values of all these elements.
Notwithstanding the favorable opinion of this method expressed at the time by-
Professor Cayley, and later, as to some of its processes, by Professor E. W. Brown,
the author found that, in applying it unmodified, which he did during the years
1872-77, very long and complex computations were required in its application.
The result was that the work, so far as it was carried, remained unpublished for
nearly twenty years. Hoping that the general developments of the work and some
of the details might be of use to subsequent investigators, the incomplete work
was finally published in 1895.
About the same time with the publication of this work appeared the very elabo-
rate one of Radau. f This work contains a seemingly exhaustive enumeration of
possible inequalities of long period, and the numerical computation of a great num-
ber of lunar inequalities due to the action of the planets which had not previously
been suspected.
On recommencing the work in 1904 it became very clear to the author that its
completion by his former method, unmodified, would be impracticable, and that
satisfactory results could best be reached by regarding the solar elements as con-
stants, or known variables from the beginning. In the present investigation, there-
fore, the method has been modified so that the final values of the coordinates of
the Moon, instead of being expressed as functions of the instantaneous elements
of the Earth's disturbed motion, are expressed as functions of the mean elements.
As thus modified it is substantially a continuation of that of Delaunay, as applied
* Liouv Hie, Journal des Mathematiques, 1871, March.
t Annales tie I' Observatoire de Paris, Me'mofres, vol. XXI.
2 INTRODUCTION.
first by Hill and then by Radau. In this method the coordinates of the Sun,
relative to the centre of gravity of the Earth and Moon, are regarded as known
functions of the time. Then, when the action of the Sun alone is considered,
the coordinates of the .Moon relative to the Earth are found by the method of
Deteiin2y^>c9ftiijBet^d.if necessary, as functions of six purely arbitrary constants.
This solution of the problem of three bodies is supposed to be complete in
advance. When the action of the planets is then taken into consideration, the
only elements whose variations are to be determined by the Lagrangian equations
are the six final elements of the Moon's motion. The variations in the coordinates
of the Sun, due to the same action, are derived with great ease, and enter into the
differential equations. In this way a system of six differential equations for the
determination of the changes in the lunar elements is all that is necessary.
In setting forth the subject it is deemed unnecessary to repeat the derivation of
the equations already found in astronomical literature. For this branch of the
subject, reference may be had to Hill's paper in the American Journal of Mathe-
matics, Vol. VI, and to Chapter XIII of the Treatise on the Lunar Theory by
E. W. Brown. It is deemed necessary only to explain fully, at each point, the
application of the method, and the meaning of the symbols introduced.
PART I.
DEVELOPMENT OF THE THEORY.
CHAPTER I.
FUNDAMENTAL DIFFERENTIAL EQUATIONS.
i. Notation. The following notation is mostly used in this work:
G, when designating a point, centre of mass of Earth and Moon; m', mass of the
Sun; ; 2 , mass of the Earth; / 3 , mass of the Moon; m 4 , mass of the Planet.
H = m 2 + m 3 fj.' m' + /*
x, y, z, r, geocentric coordinates and radius vector of the Moon, referred
to the moving ecliptic;
x', y', z', r', coordinates and radius vector of the Sun, referred to the point
G and the moving ecliptic;
, 17, , and p, the ratios of x, j, z, and r of the Moon to the mean dis-
tance of the latter: x = at;, etc. When unmarked the coordinates
are referred to a moving J^-axis directed toward the mean Sun;
#1, jj, Moon coordinates referred to the mean Moon as the Jf-axis;
A, distance of the Planet from G;
S, cosine of angle between rand r'\
S', cosine of angle between r and A ;
/>(,, potential function of mutual action of Earth and Moon;
11, potential function for action of Sun on Moon;
7?, potential function for action of Planet on Moon;
/, TT, 6, mean longitude, longitude of perigee, longitude of node of Moon;
TTj, #!, motions of IT and in unit of time (quantities of dimensions T~ l ) ;
N, motion of argument in unit of time;
n, ratio of motion of an argument to n, the mean motion of the Moon;
v, the integrating factor, generally =W/N;
a, e, g, defined in (43), 22;
K> C, D, planetary coefficients for the direct action, defined in 20;
p, y, K 4 , lunar coefficients, 20 Eq. (36) ;
6r,y, /, planetary coefficients for the indirect action, defined in 24;
G is also used for a combined lunar and planetary argument;
a, logarithm of a, the Moon's mean distance;
v, M, j, s, the mean longitudes of the respective planets, Venus, Mars,
Jupiter, and Saturn measured, in each case, from the Earth's peri-
helion: TT' for i8oo = 99.5.
5
6 ACTION OF THE PLANETS ON THE MOON.
The abbreviation " Action " has been used to designate the previous work of
the author on this subject "Theory of the Inequalities in the motion of the Moon
produced by the Action of the Planets"; forming Part III of Astronomical Papers
of the American Ephemeris, Vol. V.
2. Dimensions of quantities. In this subject it will be found helpful to the
reader and investigator to have, in the case of the principal equations, a statement
of their dimensions in terms of the fundamental units of Mass, Time, and Length.
In strictness an independent unit of mass is not necessary in gravitational astronomy,
because the most convenient unit is that mass which, on an equal mass at unit
distance, exerts a unit force of gravitation. But it is still sometimes convenient to
use this unit in the equations, although it is a derived one.
In the case of each system of equations which are regarded as fundamental will
be found the dimensions of the terms which form its members, the signification
being as follows:
T, Time ; Z, Length ; M, Mass.
The definition of the unit of mass just given leads to the relation
In this way it will be much easier than it would be without this help to appreciate
the degree of magnitude of small quantities. Considered by itself, no concrete
quantity can be regarded as small or great; it is so only when compared with other
quantities of the same kind, or, to speak more accurately, of the same dimensions
in fundamental quantities. The ratios of two fundamental quantities of the same
kind are pure numbers, and these may be large or small to any extent.
3. Fundamental differential equations.
Putting
x \t y z \i tne geocentric coordinates of the Moon referred to any system
of fixed axes,
P, the total potential
the differential equations to be integrated may be written
dP dP dP
- [ Dimensions = AfZ.-' = LT~*1 (j)
dx l dy l oz l
4. Transformation to the moving' ecliptic. In the preceding equations the
coordinates are referred to fixed axes. In astronomical practice the coordinates
of the heavenly bodies are referred to the moving ecliptic. The latter carries the
plane of the Moon's orbit with it in its motion. It therefore seems desirable to
refer the motion, in the first place, to the moving ecliptic.
FUNDAMENTAL DIFFERENTIAL EQUATIONS. 7
To do this let us put
x, y, z, coordinates referred to the moving ecliptic;
K, the speed of motion of the plane of the ecliptic;
II, the longitude of the ascending node of the moving on the fixed ecliptic,
or of the instantaneous axis of rotation of the ecliptic. At the
present time we have II = 173, nearly.
Then, regarding nt as infinitesimal, the expression for the moving coordinates in
terms of the fixed ones will be
x = .*, zjic sin II
y = y l + Z^K cos II
z = z l + x^tic sin II y^K cos 13
Putting for brevity
p = K sin II q = K COS II [Dim. of /, y, and= T~ l ],
these expressions become
(2)
Differentiating them twice as to the time, regarding p and q as constant, we have
Dfx = Df x ,-ptDfz l -2pD^
?y, + qtDfz, + 2qD t z, (3)
Dfz = Df
Regarding P, originally a function of x i9 y ly and z lt as becoming a function of*, y,
and z through the substitution (2) we have
dP dP dP dP dP dP dP dP dP dP
dx~ == '6 X + **& =~d~ qt te aF " dz ~^dx + qt S
Substituting these expressions for D\x^ D\y^ and DIZ^ in (3) and dropping terms
of the second order in pt and qt we find
dP
+ *qD fl (4)
dP
D ? z =-B-z +
Equations of this form were used by Hill for the same purpose.
* Annalfyf Maihematict, vol. I, 1890,
8 ACTION OF THE PLANETS ON THE MOON.
It follows that if we add to P the terms
A* = 2p(zD^ - *D fl ) + 2q(yD^ - zD t y,} (5)
so that the potential shall become
P + LR [Dim. = 3/z,-' = z," r-']
the fundamental differential equations in x, y, and z, will retain the form (i)
unchanged, and the coordinates referred to the moving ecliptic will be determined
by the general equations
dP dP BP
>'x = -f D?y = if- D t 2 z = - (6)
dx dy dz
In A/? the symbols x 1} y 1} and z i have the same meanings as x, y, and z, but they
are to be regarded as constant when AT? is differentiated as to the lunar elements.
5. Preliminary form of the potential function.
We put fl for the part of the potential P due to the action of the Sun. This
part is developed in a series proceeding according to the powers of r\r' in the
well-known form
where S, the cosine of the angle between the radii vectores of the Moon and Sun
from the point G, is determined by the equation
rr'S = xx' + yy' + zz'
When we assign to x', y', z', and r 1 their elliptic values, we have what may be
called the Delaunay part of the potential. We put
f! , the Delaunay part of fi.
Op, the increment of fl produced by the action of the planets on the Earth.
The part /? of P, due to the direct action of the planet in changing the coordi-
nates of the Earth relative to the Moon, may be formed from fl in (7) by replacing
m' ' , r' , x', y', and z'
by
m t , A, X, Y, and Z
where ;., is the mass of the planet, and A, X, K, and Z its distance and coordi-
nates relative to the point G. Putting R for this part we have for its principal term
FUNDAMENTAL DIFFERENTIAL EQUATIONS. 9
where S' is determined by the equation
r*S' = (*' + xjx + (y' + yjy + + z^z
*4> J an d ^4 being the heliocentric coordinates of the planet.
We have thus separated the potential of all the actions changing the coordinates
of the Moon relative to the Earth into the following five parts.
A. The part generated by the mutual action of the Earth and Moon, /> = /*/ r,
which taken alone would give rise to an undisturbed elliptic motion of the Moon
around the Earth.
B. The part fig generated by the action of the Sun, assuming the point G to move
in an elliptic orbit.
C. The part ft p , the increment of ft due to the action of the planets on the
point G.
D. The part /? due to the direct action of the planet. Developed in the same
way as the highest term of ft the principal term of this part is formed from SI by
replacing m', *', y', and z' by the mass and G-coordinates of the planet. The value
of its principal term is given in (7).
E. The part A/? arising from the reference of the coordinates to the moving
ecliptic.
The complete value of P thus becomes
p= p e + n + n p + j? + A/? (8)
and we are to consider this expression as replacing P in the equations (6).
6. Reduction of the terms of the potential function for the indirect action.
By substituting tor S in (7) its value, the first and principal term of ft becomes a
linear Junction of the six squares and products of the lunar coordinates *, y, and z,
which we may write
ft = 7> J + TJ + 7X + * T t xy + a 7>* + 2 T t yz (9)
Moreover, since we form the part ft p of the potential by assigning increments to
T, and the part R by making T & function of the elements of the planet, it follows
that both of these parts as well as ft are of this same form.
For the first and principal term of ft in which the higher powers of rjr' are
dropped we have
-^ x ' y>
-,-,-- -. . (10)
r ' 3 \2 r>* 2J 2 r '* r '*
(S -\ T - 3 m> y' z '
r ' s \2 r>> 2) 2 '~ 2 r >*' r >*
io ACTION OF THE PLANETS ON THE MOON.
The study of the second term, which it may be advisable to examine for sensible
results, is postponed, and ft is taken as equal to its principal part. The value of fl p
is then found by adding to the preceding values of T t their increments produced
by the action of the planets upon the coordinates #', y' , and z' of the Sun. If we
put
v', the longitude of the Sun
and take the moving ecliptic as the plane of reference, we may regard z', the
periodic perturbations of the latitude, as infinitesimal and write
x' = r' cos v' y' = r' sin v' z' = r' sin /S'
where /3' is the Sun's latitude, a minute purely periodic quantity.
Substituting these values in (io), the expressions for the coefficients T become
T, - ^,(t + f cos 2') T t = ~ (i - | cos 2*') T *=-~
(LOO)
3 m' . 3 ;' sin /3' cos z/ ~ 3 '' sin /3' sin r'
y, = -- s sin 2z' / =-
r
i = -- 5
3 2 r ' 3
If we assign to these quantities their elliptic values, (7) will become fl for which
the integration is assumed in advance. We have now to assign to v' and /' the
increments 8v' and r'8p', p' being the Naperian logarithm of r' . The resulting
increments of the coefficients are
87;= ^j {-2 sin 2w'8'-3 cos 2v'&p'-&p'} 87 >= ^73 { 2 sin 2^'Sz;' + 3 cos 2'8/'-V}
(ii)
5 r 3 = -3 V 8 T t = ^3 { 2 cos 2r'Sw' - 3 sin iv'Bp' }
2^* 4^*
The values of 8T 5 and 87" 6 will be the original values (9) of 7" 5 and T 6 as they
are due wholly to the action of the planet. With them the expression for fl p
derived from (9) becomes
fl f = 8 7> 2 + 8 T 2 y 2 + 8 7> + 28 7>j + 2 7>* + 2 7^* (12)
7. Reduction of R, the potential of direct action.
By substituting for S' in the principal term (7) of ./? its expression in terms of
the G-coordinates of the planet we shall have
where
-f Cz 2 -f 2/?*_y -f lExz + iFyz [Dim. = z,-'
FUNDAMENTAL DIFFERENTIAL EQUATIONS. n
the values of the coefficients being
(*' + * 4 ) 2 i i n (*' +
- ~ 3
[Dun. = Z, ] (14)
3 A 3
It should be noted that these coefficients require the factor f to make them directly
comparable with ZJ, T 2 , etc., in (10).
8. Complete form of the fundamental equations.
Comparing the expressions (12) to (14) we see that fl p and R are of the
same form, and that the principal terms of each are products of two factors, of
which one depends solely on the heliocentric coordinates of the Sun and planet,
and the other is a square or product of the coordinates of the Moon. Moreover, if
we put, for brevity,
/ = o, + R + A;? (is)
the fundamental differential equations may be written
where x, y, and z are coordinates referred to the moving ecliptic as the fundamental
plane.
We shall now consider these differential equations as solved for the case when
PI is dropped from the second members. The problem will then be that of the
solution when P t is included; and this problem will be attacked by the Lagrangian
method of variation of elements.
CHAPTER II.
DEVELOPMENT AND INTEGRATION OF THE DIFFERENTIAL EQUATIONS
FOR THE VARIATION OF THE ELEMENTS.
9. The problem being to integrate equations (16), we shall regard as known
quantities the coordinates x', y', z' of the Sun, which enter implicitly into the equa-
tions, as well as those of the planets relative to the Sun. The problem then is to
express the values of x, y, and z in terms of the fundamental constants implicitly
contained in the differential equations, and six other arbitrary constants which we
regard as elements of the Moon's motion.
The solution of the equations is separated into two parts by applying the La-
grangian method of the variation of elements. We have first the Delaunay solu-
tion, in which P t is dropped. This solution gives the orbit of the Moon around
the Earth under the influence of the Sun's and Earth's attraction alone. From it
we are to pass, by the method of variation of elements, to a solution when P is
taken account of.
We accept the results of Delaunay, as found in his work, as forming the basis of
the first solution, the results needing only certain modifications in the terms depend-
ing on the Sun's parallax, arising from the tact that he did not take into account
the mass of the Moon, and certain reductions, to reduce them to the required form.
This being done we have values of the Moon's coordinates satisfying the differential
equations in the case P=p/r-\-tl and expressed as functions of six arbitrary
constants
c, 7 4)' "o> ^o
and of the time /. The latter enters only through the quantities /, IT, and 0, named
and defined thus
Mean longitude : lt^+nt Long, of perigee : TTTr^+irJ Long, of node : 0=0 +fy (17)
where n, ir t , and 0, are functions of a, e, and y.
I use the quantities 7, IT, and 6 instead of Delaunay's /, g, and h, which are the
mean anomaly, the angle node to perigee, and the longitude of the node. The
expressions for the symbols used here in terms of those used by Delaunay are
therefore
/ =3 Delaunay's // + g + J TT = Delaunay's h + g 6 = Delaunay's h (18)
13
14 ACTION OF THE PLANETS ON THE MOON.
The fundamental idea of the Lagrangian method, which we propose to apply
to the present problem, is that the six arbitrary elements are to become such
functions of the time that the solution which satisfies (16) when P t = o shall still
satisfy it when the variable values of the elements are substituted for the constant
values in the expressions for the coordinates. The derivatives of the elements as to
the time may be formed by known processes, but the details of these processes are
unnecessary, because Delaunay gives their results in a form most convenient for
our purpose.
10. Canonical form of the differential equations.
We see from (5), (12), (13), and (15) thatP t is a function of given quantities and of
the Moon's coordinates. By substituting for the latter their expressions in terms of
the six arbitrary constants of the first integration, /\ becomes a function of a, e, y, /,
TT, and 0. The differential variations of the elements are then expressed in the most
condensed form by replacing a, e, and y by three other quantities c lf c 2 , and c 3 , func-
tions of a, e, y, so chosen that the differential equations to be solved shall be
(19)
The variable elements c a , c 2 , and c 3 are functions of Delaunay's Z, G, II.
CI = L C,= G-L CS = H-G (20)
[Dim. = L\M* =
ii. Transformation of the canonical elements.
The canonical elements c w c 2 , and c s can not be used explicitly in the processes
of solution. We have therefore to express them in terms of a, e, and y. The
values of Z, ?, and H are not given by Delaunay in terms of the final a, e, and y,
but of preliminary ones from which the required expressions are to be derived as
follows :
1. In Vol. II, pp. 235-236, Delaunay gives the expressions for Z, G, and // in
terms of the a, e, and y which resulted immediately from his processes of
integration.
2. On p. 800 he gives the transformation of these a, e, y, into the final values of
these quantities which appear in the expression for the Moon's coordinates, which
are those we are to use.
DEVELOPMENT OF THE DIFFERENTIAL EQUATIONS. 15
To find from these data the expressions for the derivatives of Z, G, H in terms
of the final a, e, y, I shall write a, e, g, n, for the quantities a, e, y, , as found on
pp. 235-236 of Delaunay, Vol. II, and shall also put
n'
m = -
n
The forms which we have to use are:
L, G, Jf=/(a, e, g, m) a, e, g =/(, e, 7, z) (21)
Noticing that m is a function of a and m of a, we shall then have
dL (dL dLdm\da dL de dL dg
^ I i I i i __o. / 2 2 ^
da \ da dm da J da de da dg da
with similar forms for G and H.
12. Form of the partial derivatives. Two points in the use of the partial
derivatives are these:
a. In taking the partial derivatives I use the logarithm of a and of a instead of
these quantities as the variables with respect to which derivatives are to be formed.
Homogeneity in the equations is thus secured, the variables being all pure numbers,
or quantities of dimensions o. We put
a log a whence a = e a
ft. The quantities n and n are defined as functions of a and of a respectively by
the equations
a'n 2 = oW = p
It follows that if we have an expression M developed in powers of m or m,
we shall have
-fc = a* (iM 9 + (i + f ) Mjn + (i + f ) Mpi* + ) (23)
13. Numerical v alues of the fundamental quantities.
Instead of effecting the preceding transformations analytically, to put the equa-
tions (21) into numbers, we use the numerical values of e, y, and m given by
Delaunay in his Vol. II, pp. 801-802, namely
e = .054 8993 7 = .044 8866 m = .074 8013 (24)
1 6 ACTION OF THE PLANETS ON THE MOON.
We then find from his expressions on p. 800
a = 0.996 4930 = [9.998 474] a
an = i.ooi 758072 = [o.ooo 763] an
a 2 n = 0.998 245 2 = [9.999 237] J
m = 0.994 7437/2 = [9.997 7n]? = 0.074 4082
e = 0.054 86 7 g = 0.044 993
We also find, from these numbers, the following values of the required partial
derivatives for the numerical transformation
^- = 0.986 6910 s~ = ~ -7 37 e o-ooo 404 -j = + 0.006 857 = o.ooo 308
da de dg
^ = -0.0013750 ^= + 0.99961 ^ = +0.000202
da de , dg
^- = +0.0013530 5- = o.ooi 22e= 0.000067 _j- = + 1.002 324
Then, from Delaunay, II, p. 236, we find
L = i .000 197 a 2 n G = 0.998 586a 2 n // = 0.994
dL dG dH
a = tS IS a a "" = <499 97 a " =
f^ T r\ f~* f) J-T
-^ = o.ooo o88a'n -^- = 0.052 4ioa 2 n = 0.052 185 a 2 n
dL dG dH
-v- = 0.0000073 n -- = 0.0000353 n ^- = 0.17947430
14. Formation of the transformed differential equations.
Let us now return to the equations (19), in which we have to replace c^ c 2 ,
and c 3 by a, e, and y. We have, for any c,
dc dc da dc de dc dg
di = da dt + de dt + dgdt
and
dc dc da dc de dc dg
f i [ O
da ~ da da de da dg da
In the case of c^ we have from (20)
da ~ da
DEVELOPMENT OF THE DIFFERENTIAL EQUATIONS. 17
so that the numerical expressions need not be repeated. For the derivatives of
c 2 and c 3 we find
a ~-* = o.ooo 46ia z n ~ = 0.052 322a 2 n ~ = o.ooo O28a 2 n
da. de eg
dc. dc, Be,
a.fT- = 0.002 ooi a 2 n -^ = + o.ooo 225 a 2 n ~ = 0.179 43a z n
By substitution in the form (22) we now find
dc dc. dc'
-r = 0.494 39 M ~= = o.ooo 777' -~ = o.ooo &jifrn
-r- 2 = o.ooo 435 2 ~B* ~ ~ -5 2 2 9 2 ~B~~ ~ - oo O2 S a * n ( 2 5)
., 3 = 0.002 033 2 ~ = o.ooo igicfn ^~ 3 = 0.179
We now have the data for transforming the equations (19), p. 14, so as to express
the differential variations of tf, , and y instead of c^ c 2 , and c 3 , and to express those
of / , TT O , and in terms of the partial derivatives of R as to , , and y. For this
purpose we need the nine partial derivatives of a, e, and y as to c lt c 2 , and c s . We
shall express these nine derivatives by means of the nine numerical factors
a < e 1* (* = 1:2:3)
defined by the equations
da &? ^7
a. = a z ^ c . = a 5 7. = aw 3-
dc t dc f dc t
The numerical values of these coefficients are most expeditiously found in the
following way. Multiplying the first three equations (19) in order by the respective
lactors
da da da
&; ft; ^
we have
da dP. da dP. da dP.
li_ \
with similar equations in D t e and D t y. From the same three equations we have
d Cl dc. dc. , dP
da de
dc, 7 dc, . dc, , dP
..^ D t a + 5- 3 De + ~ Z>,7 = -33 1
da ^ de ' dy " dO
It follows that if we solve these three equations for D t a, D t e, and D,y the nine
partial derivatives required will be the coefficients of the second members in the
1 8 ACTION OF THE PLANETS ON THE MOON.
solution. Replacing the coefficients of the unknowns by their numerical values
(25), we may reduce the solution to that of three numerical equations
0.494369^ . 000777 Y+ 00067 *Z = P
.000435^ . 05 2209 Y . 000025.?= Q
.002033^+ . 000191 Y .179538^= ff
The solution of these equations so as to express JT, Y, and Z as linear functions
of P, Q, and R gives the following values of the factors which we seek. Along
with these values is given for comparison the values found in Action of Planets,
Clffi
p. 196, where the numbers are the coefficients of - . The two determinations
JWj*J
are completely independent, in that the earlier one is derived from the analytic
expressions for the coordinates of the Moon, while these last have been obtained
from Delaunay's expressions of the canonical elements L G H in terms of a, e, y.
a, = -f 2.0228 Former value : -f 2.0225
a 2 = 0.0301 0.0293
3 = + 0.0075 + 0.0075
e l = 0.0168 0.0169
^=-19.1534 -ip^Si (26)
e 3 = + 0.0026 +0.0017
7, = 0.0229 0.0233
7 2 = O.O2OO O.O2I6
7 3 =-5-5700 -5-5704
The fundamental differential equations for the variations of the elements now
become
BP. dP. dP.
,_-+ v _ + o,^-
*-st+ t *-ti+ t it
dP. BP. dP l
-
BP l
(27)
= a -3 e. -,- 7. -^~
2 da * ce 2 cy
--
3 da 3
[Dim. = A/-']
DEVELOPMENT OF THE DIFFERENTIAL EQUATIONS. 19
In order that we may, so far as possible, handle only pure numbers, with speci-
fications of the units as concrete quantities, we shall substitute nt, the total motion
of the Moon in mean longitude, and therefore a pure number, as the independent
variable. The first numbers will then take the form a 2 n 2 D Hl a, etc.
Since
a'n 2 = -
a
the equations will now give
' <*>
[Dim. =o]
with five others formed in the same way from (27) which need not be written.
15. Elimination of t from the partial derivatives of I, TT, and 6.
An important remark at this point is that since P l is a function of /, TT, and 6,
the three quantities a, e, and y enter into P l not only explicitly but implicitly
through n, TT, and 0, so that the complete differential variations of these functions
are
dl dL dn dtr dir dtr. dO d6 n dd.
dt = df + " + 'dt *3t+*> + <-3i ar-rf + . + 'J
P l being a function of the six quantities
a, e, 7, / + nt, TT, + TT,*, + Qj
its complete derivatives as to a, e, and y are
_ , ,
da ~ da. ^ dl Ba """ chr da "*" d0 da
with similar expressions for dP^Se and dP^dy. Thus we have for any canonical
element c
dc \ dc
^to^a^dPidK
dl dc + dir dc " h 60 dc)
20 ACTION OF THE PLANETS ON THE MOON.
The complete derivatives of /, TT, and are therefore
. t (dn BP^dn BP, d-^ dP^d
\dt~' dl d Cl dw d Cl " d0 d Cl
dt ~ dt
dir dir ( cfrr dP dn dP STT BP 80
V d _P^" ^3^ a^e^\
" dl ~6c 3 S-n- dc 3 " d0 dcj
dt - dt ' v dt
It is a fundamental theorem of the development of the planetary coordinates in
periodic series that the terms of these equations containing t as a factor all vanish.*
The values of /, TT, and 6 are therefore
* A demonstration of this theorem in the most general case is found in the author's paper On the General Inte-
grals of Planetary Motion : Smithsonian Contributions to Knowledge, 1874.
CHAPTER III.
DEFINITIVE FORM OF THE DIFFERENTIAL VARIATIONS
OF THE ELEMENTS.
16. The differential equations (27) in the form (28) are the fundamental ones
of our problem, the integration of which is to be effected. This need be done only
to terms of the first order as to the disturbing function. This amounts to saying that
we regard the second members of the equation as known functions of the time, and
that the required integration is to be performed by simple quadrature.
We begin by studying the general form of the function f\. Besides A^?, this
function consists of two parts, one, 7?, arising from the direct action shown in 7,
and the other fl p arising from the indirect action. We have reduced both these
parts to the general form
Ax 2 + By* + Cz* + 2Dxy + lExz +
The coefficients, A, J5, etc., are functions of the heliocentric coordinates of two
points: the centre of gravity G of the Earth and Moon, and that of the planet.
They are, therefore, regarded as independent of the elements of the Moon's orbit.
The variables x 2 , y 2 , etc., being functions of the geocentric coordinates of the Moon,
are independent of the position of the planet, and contain, besides the six lunar ele-
ments proper, the major axis and eccentricity of the Earth's orbit around the Sun.
The arguments on which the coefficients A, B, etc., depend are g and g' . The
coordinates # 2 , y 2 , etc., depend on the four arguments /, IT, 0, and g' . It follows
that the terms of P l depend on the five arguments
Although the two actions, the direct and indirect, admit of being treated together
by combining the corresponding coefficients of # 2 , y 2 , etc., yet the coefficients are
so different in their form and origin that it will be better to treat them separately.
17. Reduction of the equations for the direct action.
We begin with the development of ^?, as given by (13) and (14). Since
x 1 , y 1 , etc., each = a pure number x a 2
A, B, C, etc., each = a pure number -=- a'
21
22 ACTION OF THE PLANETS ON THE MOON.
it follows that R may be developed in the form
R-fr^H (30)
H being a pure number.
When the fundamental equations are taken in the form (28), and /*, is replaced
by R expressed in terms of A, the second members will all take the common
constant numerical factor
37<
2 p a >*
This factor may be simplified by the fundamental relations
V as ft a' V 2 = m' + p
where p. and m' are the respective masses of Earth -j- Moon and of the Sun.
Owing to the minuteness of fi relative to m' (i 1330000 -(-) we may drop it from
the quotient, thus obtaining
m' <? n ft 2
i = "- ^= 7W
M a' 3 *
The factor thus reduces to the pure number
* ^ nf
2 me
The ratio m^\ m' is what is commonly taken as the numerical expression of the
mass of the planet. We shall write
M= ? ->' = 0.008 392 86 ~ 4
2 ' 2 r
The numerical values of M for the four planets whose action is to be determined
are as follows:
'
^ M
Venus 408 ooo o".oo4 242
Mars 3 093 500 o .000 560
Jupiter 1047.35 i .653
Saturn 35OO o .4947
We have next to consider H and its derivatives. As this quantity has been
above introduced we have
(31)
The terms in E and fare omitted here, owing to their minuteness.
DIFFERENTIAL VARIATIONS OF ELEMENTS. 23
We have now to deal with two sets of factors :
1. The planetary fac
2. The lunar factors
i. The planetary factors, a' 3 A, a' 3 !?, etc.
x 2 y" 1 z 1 2xy
~tf' ~a"~tf' ~~J r> CtC<
for which we use
f 2 , if, 2 , 2l~r), respectively
1 8. Notation of the Planetary Factors. The development of these requires
numerical processes which, owing to their length and their distinctive character,
are given in Part II. We shall therefore assume this development to be effected,
referring to Part II for the methods and numerical results. Considering the latter
in their general form, we remark that these coefficients being of dimensions Z~ 8 ,
if we compute their values, taking the Earth's mean distance as unity, the numbers
obtained for the several coefficients A^B, etc., will readily be the values of a' 3 A,
a' 3 B, etc. We shall therefore put
a'" A = 2 (A. cos JV 4 + A, sin JV t )
a' 3 S = 2 ( c cos 7V 4 + B. sin 7V 4 ) (33)
a'*C = 2 (C t cos 7V 4 + C. sin
where each argument is of the general form
/ 4 being the mean longitude of the planet, measured from a point which we shall
take as that corresponding to the earth's perihelion.
19. Notation of the lunar factors. We have shown in Action, Chapter II,
how, from Delaunay's results, the squares and products of the Moon's coordinates
may be developed in the general form
r (34)
2% t] = 2* 4 sin N 2%% == 2* s sin N 277? = 2* 6 cos N
Here the K are functions of , e, y, a', and e', and the arguments JV may be
expressed in the general form
N= il +I'TT + i"
These developments comprise all the quantities necessary to the formation of
and its derivatives.
24 ACTION OF THE PLANETS ON THE MOON.
20. Numerical form of the fundamental coefficients. The condition
A + B + C = o
enables us to reduce by one the number of terms in //, and at the same time to
simplify the computation. We have the identity
A? + B,f
Replacing A -\- B by C there results
A? + Br? +
Putting, for brevity,
K= \a'\A - B) C; = 0' 3 C p = .J = p + i,+(?
which will make K, C and D l pure numbers, we shall have
The planetaiy factors, A, C^ and D^ are taken as developed in a double trigo-
nometric series from the equations (33), by putting
We shall then have for //the double trigonometric series
//= 2 (K c cos 7V 4 + K t sin 7V 4 ) (*, - ,) cos 7V^
-^(\C C cos A r 4 + ^ C. sin /VJ (, + * 2 - 2 s ) cos .V (35)
+ 2 (Z> c cos 7V 4 + /?, sin 7V 4 )* 4 sin N
Introducing, for brevity,
-/* = M*! - * 2 ) ?= a J(l+*l)-"s (3 6 )
the terms of the lunar factors will be expressed by
(f 2 7/ 2 ) = 2/> cos 7V p 1 3? 2 = 2^ cos N ifr = K t sin N.
Every combination of a planetary argument N t with a lunar argument yV^ will give
rise to a set of terms in H of the form
H= h. cos (JV+ .A 7 ,) + A. sin (A T + /VJ + /// cos (TV- 7V 4 ) + /// sin (IV- N t ) (37)
where
^c = K.I ~ \ C,g - JZ? * 4 ^ = - KJ - J C f? + JZ?.* 4
(38)
//. = K.p-\ C,g + \D f , h> = -JT t p + i C.g
DIFFERENTIAL VARIATIONS OF ELEMENTS. 25
The partial derivatives of If as to a, e, and y are to be found from
Dh c = K c Dp - J C c Dq -
de
a%
dy
with three other sets formed by replacing p, q, and /c 4 in (38) by their partial deriva-
tives. These derivatives of A c , h n A c ', and hi being substituted in (37) give the
required partial derivatives of H. In forming the derivatives as to /, TT, and 6 we
note that these quantities enter only through the arguments jV, in which they have
the respective coefficients
Their formation is therefore a simple algebraic process after H is developed.
The elements e and y also enter R only through H. But a appears both in //",
which is a function of m, and in the factor a 2 / a' 3 . We therefore have from (30)
dR a 1 / dH
For consistency in form and notation we shall put
D'H= 2H+ (40)
It may be remarked that the formation of D' H may be effected by the general
operation indicated in (23), by supposing H developed in powers of m and putting
M=d>H
so that
/= 2
We then have
and
dH
The sum of this -\-^H gives D' H as above expressed. In forming this sum we
need not use the analytic development of 2//, which is necessary to form dH/da,
but may use the numerical development when it is more accurate.
The partial derivatives of R as to a, e, and y are
a 2 dR a 2 dH dR a 1 dH
3 ij __ T)t If
- 9 ffl A '< *-* J -*
__ _ _
1 - 9 ffl A '< *-* J -* ~-^ - iS Irl A ~~^ ~^ tffff, * ~^
Da 2 t a > i de 2 4 fl ' 3 de dy 4 a ' s dy
26 ACTION OF THE PLANETS ON THE MOON.
21. The fundamental equations in the form (28) for the direct action now
become
dH dH djf\
dH dH dH\
i ^ ~^i ~T~ ^~a i" ^a a ZT I
ol OTT uu f
dH\
-DJ, =
+e 2 d +y 2 (42)
22. We have next to show how the second members of these equations may be
most readily reduced to numbers. There being a certain number of lunar argu-
ments N and also a certain number of planetary arguments N 4 , it will conduce to
simplicity to carry forward the quantities depending on the argument of each class
as far as possible before making the combination.
Each lunar argument being of the general form
and each planetary one of the form
N t = k'g> + kl t
it follows that by putting G for the general value of the combined final argument,
NN
G=il+ i'-n + i"9 + (j k')g' k? t
the general iorm (37) of H may be written
osG + /i i sin G)
The derivatives of H as to 7, IT, and 6 are
-^j- = 2 ( t7i c sin G -f ih s cos G) -= = S( i'h e sin G + i'h t cos G)
-QQ = 2(- f'h t sin G + i"h t cos G)
Substituting these values in (41) and putting
a = /a, + t'a 2 + i"a^ e = ie l + i'e^ + i"e t g = f^ + t'j 2 + S"y 3 (43)
DIFFERENTIAL VARIATIONS OF ELEMENTS. 27
the equations (41) become
Dp. = M(a.h t cos G a/i c sin G)
D nt e = M(eh t cos G eA c sin G) (44)
Z? n( 7 = M(gh t cos G gh c sin G)
Every combination of a lunar argument TV with a planetary argument N 4 gives
rise in each derivative of an element to four terms, which we shall express in the
form
D M a = /&, . cos (IV + N.) + * . sin (JV+ 7V 4 ) + A, t .' cos (2V- N.) + *.. / sin (2V- Nj (45)
Replacing ^, and //,. in (44) by their values (38) we have for each combination
(46)
(47)
- \MC,gq
+ \MD & *<
(48)
23. We now reduce in a similar way the group (42). We have for each
argument,
D'H = D'h c cos G + D'h, sinG en e + r*6 = f COS C + sin 6^
CC oe 5g ^77 C7 Cy
Replacing h c and ^ g by their values (38) and substituting the resulting partial
derivatives in (42) we have results which we may write in the form
- Z>,,/ = //,, . cos (N+ 7V 4 ) + A,,, sin (N+ 7V 4 ) + //,,/ cos (IV- N t ) + h lt .' sin (IV- 2Vj
-Z> n( 7r = h, ie cos (N+ N,} + h, it sin (^V+ yV 4 ) + h, t .' cos (IV- N,} + A, t ,' sin (IV- JV t ) (49)
-1)JS 9 = //, e cos (7T+ A 1 ,) + //,, . sin (yY + JV t ) + h ti / cos (JV- vV 4 ) + A.. / sin (N- JV t )
28 ACTION OF THE PLANETS ON THE MOON.
where the values of the coefficients are found by the following computation. For
each lunar argument we form
(50)
Then for each pair of arguments
//,, . = MK C L' - \MC C L" - \MD.LI h lt / = + MK C L> - \MC C L"
h lt , = MK,L' - \MC,L" + \MD C L, h lt / = - ^ff;Z' + \MC t L"
h fi . = J/^P' - \MCJP" - \MD.P, h, t / = + J/7T/" - \MC C P"
h, t , = MK t P' - \MC,P" + \MD C P, A Wi .' = - MKf + \MC.P" + \MDJ\
h tt , = MK C R' - \MCR" - \MD t Ri h tt c ' = + MK C R' - \MC C R"
h lt , = MK,R> - \MC,R" + \MD C R, /<, ,' = - MK t R ' + \MC.R"
24. Development of the indirect action,
The fundamental equations for the indirect action are found from (28) by replac-
ing PI by the function fl p defined in (12). We first replace the coefficients SZ'by
the following:
v etc.
Taking, as we do throughout this work, the mean Sun as the origin of longi-
tudes, the true longitude, v', will be replaced by the Sun's equation of the centre
= E. We also put
r'
With these substitutions the equations (n) will be replaced by others which may
be written thus: Put
G = f 7-j- s sin zElv' + %r- 3 cos
/- f^rV (52)
/= | ri -3 cos 2 ESv'-$r- s sin
Then
A' = -G-J B' = G-J C' = 2j D' = I
E' = f^- 3 cos E sin /8' F' = \r~* sin E sin '
DIFFERENTIAL VARIATIONS OF ELEMENTS. 29
These substitutions lead to the replacement of expression (12) by
n,-=*' (54)
where
H' =A'? + B>'n*+C>?+2D'l;r,+ -.- (55)
This function //', a pure number in dimensions, will hereafter be used as a fun-
damental quantity instead of flp.
By replacing P, by this value of fl p in (27) the second members in the form (28)
take the common factor
Ha' 3
and the differential variations of the elements become
dH' dH' dH'
dH' BH
D nt e =
BH'\
-^ J ( 5 6)
dH' dH' dH'
(57)
We have next to develop the values (52) of G, J, and / in terms of the mean
anomaly '. This may be done by means of Cayley's tables in the Memoirs of
the Royal Astronomical Society, Vol. XXIX, or the development given by
Leverrier in Annales de V Obseivatoire de Paris, Vol. I. Dropping unnecessary
terms and powers of e' we have
r~* cos lE = i |e' 2 + (3^' tye'*) cosg' + Ij-e' 1 cos 2g'
r~ 3 sin 2E= ($e r - *e'*) sing-' + -^V'sin 2g'
r~* = i+ | e' 2 + ( 3 e' + Qe' 3 } cos g' + \e<* cos 2g (58)
r~* cos E = i + $e' cos g'
r~* sin E= "i
30 ACTION OF THE PLANETS ON THE MOON.
The expressions for G,j, and /thus become
G = {(6e' - - 9 /V 3 ) sing-' + -\V 2 sin 2g'}&v'
+ {9 __ 45. e / 2 + (]gt - -VgV 3 ) COS"' + lf^' 2 COS 2g'}Sp'
/=-(! + &'* +(!' + IX 3 ) cosg-' + -\V 2 cos 2-'}V (59)
/= {| - -- e ' 2 + (K - W e/s ) cos "' + e/2 cos 2-'}&>'
- {(9*' - -VgV) sing-' + ifV 2 sin ig'} &p'
In reducing these expressions to numbers I take, with Delaunay and Brown, the
value of e' for 1850
e' = .016 771
With this datum the expressions for G,J, etc., become
G = (0.10058 sing-' + 0.00359 sin 2g')8v'
+ (2.24842 + O.II3I3 COS-' + O.OO538 COS 2g')Sp r
J= (0.75032 + 0.03775 cos g 1 + 0.00095 cos 2g-')Sp' (60)
7= (1.49895 + 0.07542 cos^-' + 0.00359 cos 2g')Sv'
(0.15087 sin"' + 0.00538 sin 2g-')$p'
25. Abbreviated coefficients for the indirect action. Since
A' + B' + C> = o
we have, as in the direct action,
H> = \(A'- B')(? - rf) - \C> (
Replacing A', ', and C' by their values (53)
H> = - G? - J -
As the last two terms of H' are important only in some exceptional cases, we
postpone their development to Part IV.
With the notation of (36), we have for each lunar argument
H' = (- 2 Gp cos N 2jq cos N+ 7* 4 sin N) (61)
The planetary factors, G,J, and /are to be developed in a periodic series of the
same form as that for A, B, and C, so that, for each planetary argument N we
shall have
G = G c cos JV t + G, sin 7V 4 / = J t cos JV t + J. sin 7V 4 / = /. cos 7V 4 + /. sin 7V 4 (62)
DIFFERENTIAL VARIATIONS OF ELEMENTS. 31
With these values we shall have H' developed in a double series in which for
each pair of arguments j^Vand JV 4 , H' will have the four terms
H' = h t cos (N+ JV 4 ) + h c ' cos (JV- N t } + h t sin (N '+ 7V 4 ) + h,' sin (IV- vVJ (63)
where
.i V = ~G e p- J c q + K.<
h. = -G.p- J.q + J/A A/ = G,p + J t q + K* 4
Expressing the differential variations of the elements in the same form as before
we shall find
Ka/0 h J = \ G.zp
( 6 4)
with two other sets of equations found by replacing a and a by e and e for the set
in e, and by y and g for the set in y. Also,
h t> . = nf(- G e L' -J C L" - J/.A) h^' = m\- G C L' - J C L"
(65)
A lt . = m*(- G.L' -J.L" + KA) *,.' = m 2 (G,L' + J,L" + \I c L t )
with two other sets formed by replacing / and L by TT and P, for the set in IT, and
by 6 and R for the set in 0.
Comparing these with the corresponding coefficients (51) for the direct action
we see that the equations for the indirect action may be formed from those of the
direct action by replacing
K, \C and D by G, J, and /; and also Mby m?
It also follows that the two actions may be combined by replacing in the expressions
tor the coefficients h, given in (46), (47), (48) and (51),
MK by MK- m 2 G ; \MC by \MC + n?J ; MD by MD + ntl (66)
We shall make this combination to save labor in the formation of the products, but
shall give the separate parts of the coefficients, so that the parts of each term due
to the respective actions may be readily found.
26. Integration of the equations. The integration is effected by multiplying
each coefficient by the quotient of the mean motion of the Moon by the motion of
the argument itself, which factor is
n . , ,
" = in + t'lr, + i"0 l (J + k')n' kn 4 ^ 7 '
32 ACTION OF THE PLANETS ON THE MOON.
The reciprocal of this factor, which we may use as a divisor, is
a form most convenient for numerical computation.
We shall thus have for the perturbations of the elements corresponding to each
pair of lunar and planetary arguments
Sa = vh^ e sin (IV N^ ) - vh^ t cos
Se = vh, iC sin (N JV t ) - vh tt . cos (IV JV 4 ) (68)
S 7 = vh y<e sin (N JV 4 ) - vA y> . cos (JV JV 4 )
/ = - />&,_ sin (JV JV t ) + vh ti , cos (7V^ 7V 4 )
r = - I/A,,, sin (1V JV t ) + ^ ffi . cos (IV 7V 4 ) (69)
^ = - i/A, ie sin (N 7V 4 ) + f// 9 ,, cos
Practically we use the perturbation of n, the mean motion, instead of a. From
the relation of 12, {$, we have
Dfi = \nDp.
Thus the first equation (68) is replaced by
Sn = - \vnh^ e sin (IV 7VJ + f wiA.,. cos (^ 7V 4 ) (70)
27. We pass next to the inequalities of the actual mean longitude, /, and of the
perigee and node, IT and 6. Taking the equations (29) for these quantities
TT = TT O + jrf/ = 0,,
the complete expressions are
8/ = S/ + fSndt Btr = 57r + fSv^t 80 = 80 + fse^t (71)
The motions n, TT^ and ^ are functions of the elements a (or a), e, and y. n is
given by the relation a 3 2 = /x, while TTJ and ^ have been developed by Delaunay,
whose results are found in Comptes Rendus, Vol. LXXIV, 1872, I, and are repro-
duced in part in Action, p. 190.
*.-*++* >-%+%><+%*>
From (70) and (71) we thus have, in the variation of/, the terms
0J* = - ! < . sin (^v W + 1"*., . cos (^ ^*) (7 2)
DIFFERENTIAL VARIATIONS OF ELEMENTS. 33
arising from the variation of n. Integrating and including the value of 8/ we
shall have for the complete perturbation of the mean longitude
U = l t cos (N JV t ) + I, sin
where
l c = IM ai c + V h lt , = v (f< . + //,, .) /. = f itt., . - vh lt , = v (f< . - A,, .) (73)
From the Delaunay developments in powers of m are found
433 1
(74)
BIT. BIT dir.
- = - .01480 -+ = - .ooio 4 2 = - -433 1
B0. B0. 80 l
-^ = + .00377 j* -- .ooi2 9 2 &j = + -
Substituting these values and the values (68) and (70) we find that by putting
7T, = .02220/J a .00104/& e .00433^, e 7T 1|4 = .02220/1,, .OOIO^, .00433^,.
(75)
t , .00129^, + .
we shall have
STT, = v {ir lt e sin (vV^i TV 7 ;) ir^ . cos
l = I/M {,, c sn , - lt , cos
Then by integrating we have the terms
STT = - j/V, cos (TV 7 "* ^V 4 ) - i/V, . sin
(7 6 )
cos i 4 - 1/,, . sn
Adding the values (69) we have the complete periodic perturbations of IT and
expressed in the form
STT = TT C cos (JV .VJ + w, sin (TV^i 7VJ 8^ = o cos (^Vd= JVJ + 0, sin (JV JVJ
where
"c = "^-r, . - "X, e = K 7 ^, . - ^1, ) 7T. = - !///,_ - Z^TT,, , = - !</*, + OTT,, ,)
(77)
28. Treatment of the non-periodic terms in R.
In the preceding integration we have supposed all the arguments to be of the
form GQ + N/. We have now to consider the special case in which N vanishes.
In the case of the direct action this occurs when, in the pair of arguments which
form G,
i=i' = i' = k = o j k' = o
We shall then have in H a term, H= th n which we shall call h c simply. It will
be affected by a minute secular variation which we need not consider at present,
34 ACTION OF THE PLANETS ON THE MOON.
In the case of the indirect action we note that the coefficients of Sz/, Sp', and S/3',
as found in (59), are developed in the general form 2/, sin ig' or k t cos ig', in
which /& , being a function of the eccentricity of the Earth's orbit, is a function
of the time. The coefficients of all the terms arising from the indirect action are
therefore aftected by a secular variation.
The perturbations 8v ' and 8p' contain terms independent of the mean longitude
of the disturbing planet, which may be treated separately, namely:
(1) A constant term in 8p'.
(2) Terms of the form c ^ ig' in 8v' and S/>'.
(3) The secular variation of e' and of p' .
Omitting for the present the powers of t above the first, we shall have in 8t> ' and
8p' terms of the general form
(c + c'f) ig'
The product of these into (59) gives rise to terms of G, J, and /of the same form.
When we form the products of these terms by f 2 , rf, etc., we shall have in H' terms
of the form
h + h'nt +;
Substituting the derivatives of the non-periodic direct term in (41) and (42), and of
the indirect term in (56) and (57), omitting terms in /, and putting for brevity
P % = Mh c + nth
we find
- Djr. -
(78)
- . , 3 y 3 . A"
Adding in the terms multiplied by /, these three equations may be written
DJ, - - A, - h,Ht D n r, = - h> - h'nt DJ = - V - A^'nt (78')
The integration of (78) and (78') will give
?>a = V ; S/o = V - V - J V 2 ' 2
Se = V ; &r. = V. ~ >>'* ~ P'V (78")
By = 8 o7 ; S0 = B a d - h 'nt - J/;/V/<
8 designating, in each case, the arbitrary constant of integration.
DIFFERENTIAL VARIATIONS OF ELEMENTS. 35
The completed expressions ior /, rr, and are to be found by the equations
d * &*, dTr, . dir. , 60. . 80. . 60. .
Sn = i- &a STT. = ^-' 8a + -, .-' &e + -=- 1 87 80. = -^ Sa + -^ Se + -^ 87 (79)
da. da de dy da de tty
87 = 8/ + f $>ndt Sir = STT O + / Str^t $0 = S<? + f W^t (80)
In these equations the perturbations (78") are to be substituted. In doing this
the arbitrary constants 8/ , 8 7r , and 8 # , being merely constant corrections to /, TT,
and 0, may be dropped as unimportant to the theory. We shall then have from
(78") and (80)
' V + 1 V + *o7 - *
29. Adjustment of the arbitrary constants. Values are next to be assigned to
the arbitrary constants 8 a, S g, and 8 y. We shall do this so as to satisfy the condi-
tions that the coefficient of/ in S/, of sin g in the mean longitude, and of sin (16}
in the latitude, shall all remain unchanged. The first of these conditions gives
- f V = V or V = - l ( 82 )
We thus have
The determination of 8 e and 8 y must await the computation of the periodic
terms depending on the arguments^ and 10, which is found in Part IV. The
increments in the motions of TT and now become
(83)
30. Opposite secular effects of the direct and indirect action of a planet
near the Sun.
An important theorem of the planetary action on the Moon is that as the planet
is nearer the Sun, not only does each form of action become smaller, but the two
forms tend to cancel each other, so that when the mass of the planet can be con-
sidered as simply added to that of the Sun, the non-parallactic perturbations vanish.
36 ACTION OF THE PLANETS ON THE MOON.
To find the effect of the direct action in this case, let the values of x 4 , j 4 , and 2 4
in (14) be so small that they may be neglected in comparison with x', y', and z'.
Then A will merge into r' and we shall have
For the indirect action we remark that the only effect of the action of the planet
on the position of the Earth, after so adjusting the constants of integration that the
mean motion shall remain unaltered, is to increase the mean distance, so that
instead of
.3 .2 .
a' n' = m'
we shall have
This gives, for the perturbation of a'
.3 ,2 .
a' n' = m -f- / 4
and the eccentricity e' being unaltered
-
3 m>
The corresponding part of fl p is found from (7) by assigning the increment r'Bp'
to r'. We thus have
,--
This cancels the value of R found above.
PART II.
NUMERICAL DEVELOPMENT OF THE PLANETARY
COEFFICIENTS.
40 ACTION OF THE PLANETS ON THE MOON.
that we could not be sure of this point without actual computation. In the case
of the Hansenian inequality of long period due to the action of Venus it was shown
that the perturbations in question, considered individually, were nearly of the
same order of magnitude as the coefficients to be determined. This proceeded
from the fact that, even when we consider only the formulae of the elliptic motion,
the coefficients of the term in question are in the nature of minute residual differences
of large quantities. In view of the undoubted fact of some apparent inequalities
of long period in the motion of the Moon of which theory has yet given no expla-
nation, it seems necessary to exhaustively discuss every possible mode of action
which might affect the result.
The most effective and certain way which the author could devise to over-
come this difficulty was to employ the purely numerical development sometimes
called "mechanical quadratures," but, more exactly, that of induction of general
formulas from special values. It is true that the numerical computations required
by this method would be very voluminous, possibly more so than those by other
methods. But the use of the method has the great advantage that the computations
are made on a simple and uniform plan, which can be executed by routine com-
puters, and in which the complexity incident to the analytic treatment does not enter
at all. Another important advantage of this purely numerical method is that the
mutual periodic perturbations of Venus and the Earth can be taken account of from
the beginning. This will readily be seen by a statement of the method.
The values of the planetary coefficients A, S, etc., being functions of the geocen-
tric coordinates of Venus, can be computed for any assigned mean longitude of the
Earth and Venus. They are therefore to be computed for a certain number of equi-
distant values of the mean longitude of each planet. For each of these values
there will be a definite perturbation of the coordinates of each planet, which may
be computed and applied in advance. Thus the first computation gives at once
numerical values of the coefficients in which the effect of periodic perturbation is
included. From these are developed by well-known formulae the coefficients of the
sines and cosines of the multiples of the mean longitudes.
The perturbations of Mars are so small that it was assumed that undisturbed
values of the coefficients would suffice. But the same method was used owing to
its simplicity in theory.
In the case of Jupiter the analytic development would not have involved the
difficulty which I have pointed out. But it was so convenient to apply the numer-
ical method that it was adopted for this planet also.
The action of Saturn is so minute that a very simple development suffices. It
was therefore unnecessary to employ the numerical method in this case.
COEFFICIENTS FOR DIRECT ACTION.
4 1
A. ACTION OF VENUS.
32. We shall now show how the computations were arranged in the case of
Venus. Let us first suppose that the orbits of both planets are circular. Then
assume the Earth to be in zero of longitude. We assign in succession 60 equidistant
longitudes to Venus, 6 apart. For each of these positions we compute the values
of the four principal coefficients. Numerical induction from these special values
will then give the values of A, B, etc., in a series proceeding according to the cosines
of the multiples of the differences of the mean longitudes.
Now assign to the Earth a mean
longitude equal to any multiple of 6.
If we start with Venus at inferior con-
junction we shall have the same series
of values of the coefficients as before,
provided that we now take the line
joining the Sun and Earth as the axis of
X. Supposing all our coordinates re-
ferred to this axis we should then have
A, J3, etc., developed according to
cosines of multiples of the difference
of the mean longitudes.
It follows that in the actual case of
the two orbits having a small eccen-
tricity and inclination the other terms
which we require will be of the order of
magnitude of these quantities and will therefore be smaller than these principal
terms. It is therefore not necessary to divide the circle into so many parts in order
to obtain them.
The actual process was to take the direction of the solar perigee for 1800 as the
initial line, or axis of X. The way in which the coordinates were defined will then
be seen by the diagram. Here on the left, ir' marks the position of the Earth's
perihelion. The positive direction of X passes through the Sun and is therefore
directed toward the solar perigee. The Earth being in this ( fixed position, the
coordinates of Venus are computed for 60 equidistant values of the mean longitude
of Venus differing by increments of 6. The initial or zero value corresponds to
the mean inferior conjunction of Venus, marked o in the figure, which determines
all the other values; a few of the others are numbered in order.
For each of these mean longitudes, the actual coordinates of Venus, including
the effect of perturbations by the Earth, were computed. The position of the Earth
at TT', corresponding to the 6 positions of Venus, was then corrected in each of the
60 cases by the periodic perturbations due to each position of Venus. With
these coordinates 60 numerical values of the ^4 -coefficients are computed. I
Arrangement of Coordinate Axes, in Systems o, I, etc.,
for Venus.
42 ACTION OF THE PLANETS ON THE MOON.
designate this system of 60 values, corresponding, perturbations aside, to one
position of the Earth, by the number o; and I distinguish the values by 60 indices
o, i, 2, ... 59.
In the next system, called system i, the Earth has moved through 30 of mean
longitude, or mean anomaly, to the position E. The set of 60 heliocentric coordi-
nates of Venus to be used will be the same as before, except lor the perturbations,
which will now be those for JS lt or for ^-=30. But the position corresponding
to the inferior conjunction in this system will be that corresponding to the index 5
in system o. A new axis of ^ is now adopted, again passing through the mean
Sun, and therefore making an angle of 30 with the initial axis. The coordinates
of Venus are all transformed to this axis, and another set of 60 values of the
yl-coefficients are computed.
The remainder of the process consists in assigning to the mean longitude of the
Earth successive increments of 30 until it is brought around to the position E IV
in mean anomaly 330. In each case the axis of X is taken to pass through the
mean Sun.
From these 720 special values of the ^-coefficients the general values are sepa-
rately developed for each of the 12 systems. Then the general development for any
system is effected by a second quadrature. The final result will be the values of
A y By etc., referred to an axis always passing through the mean Sun.
Were we to adopt a fixed system of coordinate axes, it would now be necessary
to transform these values referred to the moving axis, to the adopted fixed system.
But the necessity of this transformation is avoided by referring all the coordinates,
those of the Moon as well as of the planet, to the mean Sun from the beginning.
This is fully as simple as, perhaps even simpler than, referring them to a fixed axis.
The ease of doing it is all the greater from the fact that, in the actual computation of
the lunar coordinates, they are first referred to the mean Moon. The transformation
from the mean Moon to the mean Sun is probably simpler than the transformation
to a fixed axis.
33. Development of the A-coefficients for Venus.
The computations relating to Venus are shown in tabular form in Tables I- VIII,
and will now be explained. To obtain the 12 undisturbed values of the Sun's
coordinates, we derive the equation of the centre and the logarithm of the radius
vector from the tables of the Sun found in Astronomical Papers, Vol. VI.
For the argument of mean anomaly of the Sun the initial value is
corresponding to g' = o. The increment for each 30 is
b.M =30.43830
resulting in the value 188.0000 for^-' = 180.
COEFFICIENTS FOR DIRECT ACTION. 43
With the 12 values of M thus found are taken the equation of the centre, , for
1800, and log r'. Then
x' = r 1 cos E y' = r' sin E
The resulting values of x' and y' are shown in Table I.
For Venus, we have
Initial mean longitude = 99 30' 7"
this being the longitude of the Earth's perihelion for 1800. . For the same epoch
we have
Longitude of perihelion of Venus = 128 45' i7".4
Initial mean anomaly of Venus = 330 44' 50". 2
To find the tabular argument corresponding to this mean anomaly we proceed thus:
Adding 5 increments of 6, we have
Mean anomaly of Venus for index 5 = o 44' 50". 25
For this mean anomaly the precepts of Tables of Venus, pp. 278-279, give
Tabular Arg. K ; K^ = 1. 11601
Increment of K ior 6 = 3.745014
We now add one period to A' and subtract 5 increments
K & = 1. 11601
P = 224.70084
225.81685
5 increments 18.72507
Initial K, 207.09178
which corresponds to the inferior conjunction of Venus in system o.
The resulting values of K are found in Table II. With the values of K thus
formed the equation of the centre and log r in the elliptic orbit of 1800 are taken
from the tables.
The data lor the rectangular coordinates are:
Node of Venus, 1800 H= 74 52' 48". 75
Perihelion of Earth -IT = 99 30 7 .6
Node referred to Perihelion #=335 22 41 .2
Inclination for 1800 7 = 3 2 3 33 -45
The values of the coordinates x, y, and z in the initial system are now computed
by the formulae
u = Eq. Cent, -f 24 37' i8".8o m sin M '= cos 7 sin m cos M= cos 6
m' sin M' = sin m' cos M' = cos 7 cos
leading to
Jl/=2 4 35'i".8 4 ^' = 2 4 3 9 '35". 9 i
log m = 9.9998680 log m' = 9.9993706
x = ntr cos (M + ) y = m'r sin (M 1 -fa) z = r sin 7 sin u
44 ACTION OF THE PLANETS ON THE MOON.
Designating the systems by suffixes, and putting c = cos 30, these coordinates
were transformed to the axes of the other 1 1 systems by the formulae
x i = cx <> T jjXo y\ = c y<> % x v
and then, in general,
x n == x *-t y* = j-6
34. Explanation of the tables. The periodic perturbations of the longitudes
of the Earth and Venus, and of the logarithms of their radii vectores, omitting
terms of long period, are now to be found.
TABLE III: Mutual periodic perturbations.
For the perturbations of Venus by the Earth, Su and 8/>, the arguments of the
double entry Tables VIII and XVII are:
Hor. Arg. g = K o d .65o = 206^44 + 3.745*
Vert. Arg. II for System o and /= o, 104.35
Increment of II for each system All = 20
" " " " " index A 2 II = 2.461
For the single entry Tables XI and XX we have
Arg. A = i.62203(---')
For the index i g 330. 75 + 6?'
For the/th system g' = 30 j
Hence, for / = o, j = o, Arg. A = 536.49
Increment for each unit of t, &A = -f 9.732
" " " " " j, AM = 48.661
With the values of the arguments thus formed the periodic perturbations of Venus
by the Earth are taken from the Tables VIII, XVII, XI, and XX.
For the corresponding perturbations of the Earth by Venus, we have
Hor. Arg. -=30.43837
Vert. Arg. II for/= o; t = o 165.375
Increment for each unit of /; JA_-= 3
" " " /; -24.383
Argument A has the same value as in the Tables of Venus.
COEFFICIENTS FOR DIRECT ACTION. 45
The perturbations of the longitude and log. radius vector of the Earth found with
these values of the arguments are given in the columns $v' and 8p.
TABLE IVa AND IV. The perturbations in Table III are transformed into
increments of the rectangular coordinates of Venus and the Earth.
Neglecting the cosine of the inclination we have for Venus when referred to the
initial system of axes
AA; O = y sin i"Su -f x&p Ay = x sin i"Bu + ySp
the tabular 8p being multiplied by the modulus of logarithms. For the other
systems the transformation is made by the formulas for the transformation of the
coordinates themselves. The results are given in full, in units of the 8th place of
decimals, in Table IV. Applying them to the undisturbed coordinates, we have
the coordinates of Venus for each position of the two bodies.
TABLE V. The values of the solar coordinates in Table I, of the Venus coor-
dinates in Table II, after being transformed to the axis of the system, and of the
increments in Table IV, are added so as to form the disturbed geocentric coordi-
nates of Venus in each system for each position of Venus.
TABLE VI. With the perturbations of latitude in the different systems the dis-
turbed geocentric coordinate Z was computed and tabulated.
With these geocentric coordinates are computed the 720 values of the four
coefficients A, Z?, C, and D defined in 7. Since
A + + C=o
the computation of C might have been dispensed with. It was, however, carried
through as an additional check on the accuracy of the work. The latter was, how-
ever, done in duplicate, the check being incomplete.
TABLE VII gives the values of the coefficients thus computed.
The coefficients E and Flead to appreciable inequalities only in the case of the
argument 0, and have been treated separately. Their special values were computed
for six systems and thirty indices only, and are found in Table VIII.
35. The process of developing the general value of each coefficient in a periodic
series is given by Briinnow in his S-ph'drischen Astronomic, Taking A as an
example we first develop the value for each system in the form
A k = ' (a k cos iL + b k sin t'L)
where k is the number of the system and L the difference of the mean longitudes
of Venus and the Earth,
L-v-f
We thus have 12 values of each of the coefficients a k and b k , one corresponding
to each value of g' . These values are then again developed in the form
' + b k<j s'mjg-') b h = 2 (a*,/ cos ig 1 + b ki
44 ACTION OF THE PLANETS ON THE MOON.
Designating the systems by suffixes, and putting c = cos 30, these coordinates
were transformed to the axes of the other 1 1 systems by the formulae
and then, in general,
34. Explanation of the tables. The periodic perturbations of the longitudes
of the Earth and Venus, and of the logarithms of their radii vectores, omitting
terms of long period, are now to be found.
TABLE III: Mutual periodic perturbations.
For the perturbations of Venus by the Earth, 8u and 8p, the arguments of the
double entry Tables VIII and XVII are:
Hor. Arg. g = K o d .65o = 2o6 rf .44 + 3-745*
Vert. Arg. II for System o and i= o, 104.35
Increment of II for each system All = 20
" " " " " index AJI = 2.461
For the single entry Tables XI and XX we have
For the index i g= 330. 75 + 6/'
For the/th system g' = 30 /
Hence, for / = o, j = o, Arg. A = 536.49
Increment for each unit of i, &.A = + 9.732
" " " " " _/, AM = 48.661
With the values of the arguments thus formed the periodic perturbations of Venus
by the Earth are taken from the Tables VIII, XVII, XI, and XX.
For the corresponding perturbations of the Earth by Venus, we have
Hor. Arg. = 30.43837
Vert. Arg. II for/ = o ; * = o 165.375
Increment for each unit of / ; JA < g'= 3
" " " " " /; 24.383
Argument A has the same value as in the Tables of Venus.
COEFFICIENTS FOR DIRECT ACTION. 45
The perturbations of the longitude and log. radius vector of the Earth found with
these values of the arguments are given in the columns v' and Sp'.
TABLE IVa AND IV. The perturbations in Table III are transformed into
increments of the rectangular coordinates of Venus and the Earth.
Neglecting the cosine of the inclination we have for Venus when referred to the
initial system of axes
A# = y sin i"Su + xSp Ay = x sin i"Su -f- ySp
the tabular 8p being multiplied by the modulus of logarithms. For the other
systems the transformation is made by the formulae for the transformation of the
coordinates themselves. The results are given in full, in units of the 8th place of
decimals, in Table IV. Applying them to the undisturbed coordinates, we have
the coordinates of Venus for each position of the two bodies.
TABLE V. The values of the solar coordinates in Table I, of the Venus coor-
dinates in Table II, after being transformed to the axis of the system, and of the
increments in Table IV, are added so as to form the disturbed geocentric coordi-
nates of Venus in each system for each position of Venus.
TABLE VI. With the perturbations of latitude in the different systems the dis-
turbed geocentric coordinate Z was computed and tabulated.
With these geocentric coordinates are computed the 720 values of the four
coefficients A, B, C, and D defined in 7. Since
A + B + C=o
the computation of C might have been dispensed with. It was, however, carried
through as an additional check on the accuracy of the work. The latter was, how-
ever, done in duplicate, the check being incomplete.
TABLE VII gives the values of the coefficients thus computed.
The coefficients E and F lead to appreciable inequalities only in the case of the
argument 6, and have been treated separately. Their special values were computed
for six systems and thirty indices only, and are found in Table VIII.
35. The process of developing the general value of each coefficient in a periodic
series is given by Briinnow in his Spharischen Astronomic. Taking A as an
example we first develop the value for each system in the form
A li = 'Z (a k cos iL + b h sin iL)
where k is the number of the system and L the difference of the mean longitudes
of Venus and the Earth,
Z = v--'
We thus have 12 values of each of the coefficients a k and b k , one corresponding
to each value of g'. These values are then again developed in the form
' b u = 2 (<**,/ cos ig' + b k '
46 ACTION OF THE PLANETS ON THE MOON.
These being substituted in the general expression given above for A k gives the
value of A itself in the form
A = 22 [> cos (iL +jg') + b sin (iL +jg')]
The development was effected in this way up to i = 8 only, this being the limit
for possible sensible terms other than the Hansenian term of long period depending
on the argument
2g' -g
36. The Hansenian Venus-term of long period. The computation of this
inequality requires the determination of the coefficients for i = 18, which we obtain
trom the general formulae thus. Putting, in any one system,
AH A lt A v A M
for the 60 values of A, and
A cos i8Z + A sin i8Z
for the pair of terms depending on the argument i8Z, the general formulae give
A + A l cos 108 + A 2 cos 216 + .
T > oA i = A l sin 108 + A t sin 216 +
the angles increasing by 108 in each term. The fifth angle will be 180 -j- 2ir,
so that the only numerically different values of the coefficients which enter into
the series besides i and o are
sin 18, cos 18, sin 36, and cos 36
For example, we have
A tl A l sin 18 A t cos 36 + A 3 cos 36 + A< sin 18 +
30^4, = A l cos 18 A t sin 36 A 3 sin 36 + AI cos 18 +
From the cyclic order of the coefficients the method of computing A c and A,
is as follows:
With the 60 values of any one coefficient, say A, in any one system,
A , A lt A v , A M
compute
' = A
=
A lt + A it
Next:
Next:
COEFFICIENTS FOR DIRECT ACTION.
J'-Aj-AJ A>'=A,'-A S >
47
We then have, in each system
2oA c =A " + A Cil s\ni8 + A
_ . . OQ A * .Q
with similar values for B, C, and D.
The numerical results of these processes for each system are shown in Table IX.
The next step is to develop each set of numerical values of any one pair of coeffi-
cients, say A c , and A, in the form
30^4 c = ,, + , cos g' + a 2 cos 2g' + ft l sin g' + ft t sin 2g'
3oA t = a g ' + a/ cos g' + a t f cos 2g' + /3/ sin g' + ft 2 ' sin 2g'
These are to be substituted in the general form
A A e cos i8L + A t sin i8L
Retaining only terms which may be wanted for our purpose, we shall have
30^4 = a cos i8L + <*' sin i8L
+ 1 (a, - /) cos (iSL + g') + \ (a/ + /3J sin (i8L + g')
+ i(a, - /8,') cos (i8Z + 2P-') + i(a,' + ft.) sin (i8Z, + 2^')
()
+ K 3 - ft') cos (i8Z + 3 -') + HO,' + ft) sin (iSL + Zg')
TABLE I.
SUN'S GEOCENTRIC COORDINATES IN THE MEAN ORBIT OF 1800,
REFERRED TO MEAN SUN AS DIRECTION OF Axis OF X.
System.
g'
x'
/
o
i
o
o
30
+0.983 2075
+0.985 3853
O.OOOOOOO
+0.016 8542
2
60
+0.991 3897
+0.029 1452
3
90
+0.0997183
+0.033 5823
4
1 20
+1.008 1877
+0.029 0233
5
ISO
+1.0144741
+0.016 7321
6
180
+1.0167929
O.OOOOOOO
7
2IO
+1.0144741
0.016 7321
8
24O
+1.0081877
0.029 0233
9
27O
+0.999 7183
0.033 5823
10
300
+0.991 3897
0.029 1452
ii
330
+0.985 3853
0.016 8542
ACTION OF THE PLANETS ON THE MOON.
TABLE II.
COMPUTATION OF RECTANGULAR COORDINATES OF VENUS IN THE ELLIPTIC ORBIT OF
l8OO, REFERRED TO SOLAR PERIGEE AS AXIS OF X.
i
Arg. K.
Eq. Cent.
log. r
log. X
\o g .y
log. Z
o
207.0918
/ n
23 15.02
9.856 7321
0.856 5920
+7.7269130
+8.242 1742
i
210.8368
18 47.49
9.856 5914
9.8542879 8.8490901
+8-33I 7492
2
214-5818
14 7-30
9.8564812
9.8470688 9.1638652
+8401 8640
3
218.3268
9 17.60
9.856 4028
9.834 7638
9.341 2292
+84579729
4
222.0718
4 21.63
9-8563570
9.8170717
-9.463 1513
+8.503 3027
5
1.1160
+ o 37.28
9.8563445
9-793 5309 9-554 3215
+8.539 9034
6
4.8610
+ 5 35.76
9-856 3653
9-763 4657 9-625 5304
+8.569 1404
7
8.6060
+ 10 30.48
9.8564193
9.725 9000 9.682 4522
+8.591 9486
8
12.3510
+ 15 18.10
9-856 5057
9.679 4042 9-728 4252
+8.608 9752
9
16x1961
+ 19 55-43
9.856 6236
9.621 8221 9.765 5624
+8.6206617
10
19.841 1
+24 19.34
9-8567717
9-549 7350 9-795 2686
+8.627 2934
ii
23.5861
+28 26.93
9.8569481
9457 3022 9.818 5057
+8.629 0280
12
27.3311
+32 15-39
9-857 1509
9-333 3365 9.835 9401
+8.625 9108
13
31-0761
+35 42.29
9-857 3778
9. 1 5 1 9096 9.848 0289
+8.6178781
14
34-8211
+38 45-29
9.857 6261
8.824 3320 9-855 0707
+8.6047511
IS
38.5661
+41 2244
9-8578931
+7-961 5592 -9-857 2356
+8.5862194
16
42.3112
+43 32.07
9-858 1756
+8.929 0946 9-854 5813
+8.561 8070
17
46.0562
+45 12.78
9.858 4706
+9.203 5288 9-847 0574
+8.530 8192
18
49.8012
+46 23.54
9-858 7747
+9.367 1085 9-834 5015
+8492 2492
19
53.5462
+47 3-66
9.859 0846
+9.482 0030 9.816 6224
+8.444 6187
20
57.2912
+47 12.78
9.859 3968
+9.5688886 9.7929712
+8.385 6881
21
61.0362
+46 50.89
9.859 7079
+9.637 21 13 9.762 8896
+8.3118851
22
64.7812
+45 58.29
9.860 0146
+9.692 0696
9-725 4250
+8.2170551
23
68.5263
+44 35-65
9.8603134
+9.7365149
9.679 1841
+8.0892497
24
72.2713
+42 43.92
9.8606011
+9.772 5022
9.6220691
+7.900 1224
25
76.0163
+40 24.44
9.8608748
+9.801 3428
-9-550 7649
+7.5469461
26
79.7613
+37 38.73
9.861 1314
+9.823 9406
9.459 6400
6.971 8246
27
83.5063
+34 28.65
9.861 3684
+9.840 9243
9.338 oooi
7-731 3881
28
87-2513
+30 56.31
9.861 5830
+9.852 7264
9.161 4423
7.9903231
29
90.9963
+27 4.01
9.861 7733
+9.859 6294
-8.8495051
8.148 1795
30
94.7414
+22 54.28
9.861 9370
+9.861 7933
+7-6393795
8.260 1046
31
98.4864
+ 18 29.84
9.862 0726
+9.859 2710
+8.8997388
8.345 1900
32
102.2314
+ 13 53-52
9.862 1786
+9.8520116
+9.1862719
8.412 3233
33
105.9764
+ 9 8.29
9.862 2540
+9.8398567
+9.354 2638
8.466 3544
34
109-7214
+ 4 17.19
9.862 2779
+9-822 5254
+9-471 5502
8.5102060
35
113.4664
o 36.64
9.862 3009
+9-799 5863
+9.5600170
8.545 7583
36
117.2114
5 30.07
9.862 2899
+9.770 4104
+9.629 5228
8.574 2757
37
120.9564
10 20.01
9.862 2381
+9.734 0928
+9.6853431
8.596 6294
38
124.7015
-15 3-29
9.862 1550
+9.6893188
+9-7306171
8.613 4244
39
128.4465
19 36.93
9.862 0415
+9.634 1256
+9-7673464
8.625 0747
40
132.1915
23 57-o8
9.8618988
+9-565 4464
+9-7968687
-8.631 8465
41
135.9365
-28 3-60
9.861 7283
+0.478 1499
+9.82O 1022
-8.6338851
42
139.6815
31 5i-i6
9-861 5318
+9.362 7126
+9.8376839
8.631 2325
43
143.4265
-35 18.17
9.8613113
+9.1982317
+9.85O O5OO
8.623 8210
44
I47.I7I5
-38 22.35
9.861 0693
+8.921 4921
+9.8574837
-8.6114784
45
150.9166
41 1.70
9.8608081
+7.912 5788
+9.860 1443
8.593 9051
46
154.6616
43 14.42
9.860 5307
8.8273540
+9-858o8l7
8.570 6520
47
158.4066
44 58.99
9-860 2398
9.1518035
+9.851 2413
8.541 0589
48
162.1516
46 14.23
9.8599388
-0-3323188
+9.8394597
8.504 1837
49
165.8966
46 59.22
9-859 6307
9-455 9020
+9.822 4495
8.458 6539
50
169.6416
47 13.42
9.8593190
9.548 1734
+9-7997714
8.402 4145
51
173-3866
46 56.56
9.8590071
9.620 2216
+9.770 7859
8.332 2360
52
I77.I3I7
46 8.76
9.8586983
9.6778438
+9-734 5737
8.242 6677
53
180.8767
44 50.45
9.8583962
9.724 4363
+9-689 7900
8.1234525
54
184.6217
-43 2.43
9.858 1040
9.762 1421
+9.634 4660
7-951 5304
55
188.3667
40 45.82
9.857 8250
-9-792 3808
+9.565 4503
7-653 3423
56
192.1117
-38 2.06
9.857 5624
9.816 1217
+9.477 5085
4.049 3707
57
195-8567
34 52.90
9.8573192
9.834 0339
+9.360 8616
+7.6516300
58
109.6017
31 20.43
9.8570981
9.846 5740
+9-I93 8645
+7-950 6732
59
203.3468
27 26.94
9.8569016
9.854 0381
+8.9098324
+8.1228613
COEFFICIENTS FOR DIRECT ACTION.
49
TABLE III.
MUTUAL PERIODIC PERTURBATIONS OF VENUS AND THE EARTH.
The term of long period is omitted. The tabular unit is cf'.oi in <! and dv f , and 10 8 in Sp and
i
System o.
System i.
System 2.
8u
8v' dp
dp'
Su
dv'
dp
dp'
du
dv'
dp
8 P '
o
+ 239
169 + 347
+464
+ 534
-390
+ 32i
+472
+ 543
402
+ 229
+537
I
+ 65
151 + 394
+440
+ 432
423
+ 398
+424
+ 545
507
+ 331
+467
2
146
116 + 515
+365
+ 267
423
+ 540
+334
+ 479
573
+ 5H
+349
3
406
- 65 + 675
+254
+ 23
-379
+ 7i6
+214
+ 315
590
+ 724
+203
4
- 705
+ 4 +814
+ 126
- 295
303
+ 880
+ 78
+ 58
-558
+ 910
+ 42
5
1018
+ 82 +897
+ o
- 662
194
+ 988
- 62
-276
-477
+ 1045
-"7
6
1304
+ 158 + 898
Ill
1052
62
+1023
194
659
357
+ 1093
261
7
1541
+219 + 823
206
1420
+ 80
+ 941
-308
-1058
204
+ 1044
-388
8
1702
+264 , + 687
284 ! 1728
+221
+ 779
-397
1435
27
+ 895 -49
9
-1796
+289 + SOI
352
-1941
+348
+ 551
459
1760
+ 161
+ 658
-562
10
1820
+301 ; + 281
414
2049
+449
+ 273
495
-1991
+348
+ 349
602
n
1777
-j-307 + 26
-468
2047
+522
21
-5H
2103
+517
+ 5
612
12
1674
+313 234
510
1950
+566
-316
-518
2089
+657
- 343
594
13
-1508
+323 5o6
-538
-1767
+589
- 596
-515
-1956
+757
661
557
14
-!286
+343 767
549
-1517
+599
-851
504
1721
+818
- 935
506
15
1013
+370 1002
544
i2ii
+605
-1066
-485
1417
+847
1146
447
16
- 695
+403 1205
524
- 857
+607
1236
454
1057
+850
1298
-387
17
350
+446
I36l
-486
- 475
+610
-1348
-406
- 666
+836
-1386
322
18
+ 19
+498
1462
433
77
+611
1399
346
- 260
+812
1407
255
19
+ 407
+559
I486
353
+ 320
+613
-1382
-269
+ H9
+78i
1357
181
20
+ 787
+622
1417
-247
+ 70i
+614
1302
-178
+ 535
+741
1236
99
21
+1139
+681 1244
-113
+1047
+616
1148
72
+ 883
+695
1053
6
22
+1431
+723 966
+ 40
+I35I
+617
921
+ So
+ 1182
+643
825
+ 96
23
+1629
+740 597
+205
+1586
+611
6l9
+ 189
+ 1417
+586
541
+209
24
+1709
+726 180
+367
+1725
+590
248
+338
+ 1578
+524
226
+328
25
+1660
+682 + 263
+5i6
+1748
+550
+ 164
+489
+ 1652
+458
+ 121
+453
26
+1489
+606 + 694
+646
+1636
+484
+ 586
+630
+ 1623
+3 &
+ 488 ; +578
27
+ 1215
+504 +1074
+752
+ 1394
+395
+ 973
+750
+ 1478
+288
+ 846
+695
28
+ 854
+37<5 +1381
+836
+ 1045
+285
+ 1287
+844
+ 1212
+179
+ 1162
+799
29
+ 438
+226 +1583
+890
+ 616
+160
+ 1503
+GOO
+ 8 4 2
+ 56
+ 1399
+875
30
9
+ 58 +1667
+912
+ 148
+ 24
+ 1608
+922
+ 397
- 77
+ 1532
+918
3i
451
122 + 1616
+898
321
122
+ 1596 +909
79
214
+ 1544
+923
32
- 859
301 +1439
+843 - 764
272
+ 1463
+863
549
-342
+ 1437
+887
33
1197
468 +1142
+755 i 145
426
+ 1210
+784
- 971
467
+ 1213
+815
34
1440
610 + 765
+637
1429
-571
+ 860
+673
-1309
-581
+ 896
+709
35
1577
-7i8 + 336
+502
-1598
7OO
+ 436
+536
1540
-684
+ Soi
+576
36
1605
793 103
+355
-1636
-80 3
14
+377
1643
-777
+ 66 +420
37
1529
833 524
+204
1557
870
- 453
+206
1610
-849
- 378 +247
38
1357
-837
911
+ 51
1360
-898
- 856
+ 33
-1443
-893
797
+ 07
39
lotjS
-811
1221
102
-1085
-889
-1183
132
1163
-907
1148
"5
40
- 770
-755
1440
248
747
-844
1422
284
- 800
-883
1404
-288
41
- 393
-675
-1548
-380
370
-770
-1557
418
- 389
824
-I55i
443
42
- 8
-575
1540
-485
+ 19
-669
1579
532
+ 31
-733
-1578
573
43
+ 360
467
1430
559
+ 395
547
1491 620
+ 426
620
1503
-673
44
+ 681
359
1243
599
+ 730
410
1303 678
+ 780
-487
-1328
741
45
+ 950
264
1003
-608
+ IOO2
-269
1035 ! 702
+ 1068
340
1067
-775
46
+1160
-187
739
593
+ 1192
135
- 731 -688
+ 1273
-188
- 750
772
47
+ 1318
130
463
-561
+ 1302
20
415 641
+ 1383
- 38
403
732
48
+1424
92
186
-Si8
+ 1347
+ 68
- 106 -566
+ 1399
+ 99
59
656
-19
+1485
-69
+ 72
-465
+ 1334
+ 127
+ M8
-476
+ 1335
+212
+ 240
552
50
+J497
59
+ 304
-398
+ 1284
+ 155
+ 367
374
+ 1212
+291
+ 472
422
Si
+1467
61
+ 450
319
+ 1206
+ 160
+ 537
-268
+ I066
+331
+ 627
277
52
+ 1401
- 74
+ 641
232
-j-IIII
+ 148
+ 655
-163
+ 904
+336
+ 709
129
53
+1306
95
+ 739
-136
+ 1007
-(-116
+ 719
- 55
+ 757
+308
+ 739
+ 14
54
+1191
122
+ 783
-38
+ 903
+ 65
+ 731
+ 52
+ 634
+254
+ 701
+ 144
55
+1050
147
+ 777
+ 63
+ 799
+ i
+ 688
+161
+ 535
+ 182
+ 627
+264
56
+ 889
-I6 5
+ 717
+ 168
+ 728
- 76
+ 603
+267
+ 467
+ 92
+ 504
+371
57
+ 723
-175
+ 6n
+272
+ 670
-162
+ 494
+362
+ 443
- 17
+ 390
+46l
58
+ 556
-179
+ 488
+367
+ 630
251
+ 388
+435
+ 462
-142
+ 276
+527
59
+ 398
-177
+ 387 +436
+ 591
-329
+ 320
+474
+ SGI
-274
+ 212
+556
5
ACTION OF THE PLANETS ON THE MOON.
TABLE III. Continued.
MUTUAL PERIODIC PERTURBATIONS OF VENUS AND THE EARTH.
The term of long period is omitted. The tabular unit is o".oi in 6u and 6v', and 10 8 in 6p and
i
System 3.
System 4.
System 5.
3u
8v'
ip
iff
du
dv'
9p
V
du
dv'
dp
V
o
+ 391
295
+ 106
+639
+ 249
194
20
+749
+ 110
- 87
121
+839
i
+ 452
431
+ 220
+555
+ 326
337
+ 89
+669
+ 223
- 253
- 31
+777
2
+ 456
- 543
+ 424
+419
+ 349
457
+ 293
+533
+ 273
391
+ 170
+649
3
+ 375
619
+ 667
+251
+ 299
- 548
+ 547
+357
+ 245
493
+ 424
+477
4
+ 197
647
+ 893
+ 66
+ 170
604
+ 802
+ 156
+ 128
- 556
+ 665
-1-277
5
- 68
623
+ 1059
-118
- 38
- 621
+ 1008
- 53
- 65
579
+ 903
+ &
6
393
551
+ "32
287
310
- 596
+ 1132
251
3i4
- 568
+ 1053
144
7
744
435
+ 1106
436
614
529
+ H45
430
- 595
- 524
+ iiii
344
8
-1097
- 283
+ 977
557
924
421
+ 1042
-576
- 882
- 447
+ 1062
523
9
1424
103
+ 700
-646
1208
278
+ 838
-688
1148
339
+ 902
-672
10
1690
+ 97
+ 463
-701
1441
- 108
+ 547
-761
1362
202
+ 637
-781
it
-1876
+ 305
+ 109
720
1604
+ 79
+ 200
799
-1499
44
+ 293
-845
12
-1958
+ 507
- 271
-702
-1688
+ 276
- 180
799
-1545
+ 126
95
-863
13
1919
+ 692
647
-654
-1681
+ 476
- 563
-765
1501
+ 303
- 489
-844
14
1761
+ 843
977
-578
-1582
+ 667
921
-695
-1372
+ 478
- 859
-788
IS
-1505
+ 952
1226
-487
-1393
+ 839
1219
594
-1174
+ 648
1171
-701
16
i 173
+ 1014
-1378
-388
II2O
+ 979
-1433
-471
- 914
+ 805
1412
-588
17
- 800
+ 1035
-1458
-287
- 787
+ 1078
1541
-332
610
+ 944
1554
449
18
- 412
+ 1019
-1448
-188
429
+ 1127
1539
194
284
+ 1051
-1588
291
19
23
+ 080
-1366
- 95
- 6p
+ 1129
-1431
- 62
+ 45
+ II2I
1503
-125
20
+ 344
+ 922
1219
5
+ 260
+ 1089
1245
+ 56
+ 347
+ "43
1313
+ 37
21
+ 679
+ 853
1014
+ 84
+ 547
+ 1021
999
+ 162
+ 595
+ II2O
1036
+ 188
22
+ 967
+ 774
- 758
+ 172
+ 788
+ 931
713
+254
+ 778
+ 1056
706
+313
23
+ "95
+ 682
- 466
+262
+ 975
+ 827
403
+340
+ 893
+ 959
356
+416
24
+ 1349
+ 582
I5i
+353
+ IIO2
+ 712
- 81
+4'9
+ 942
+ 841
- 7
+495
25
+ 1428
4 477
+ 171
+446
+ 1161
+ 587
+ 241
+492
+ 936
+ 710
+ 322
+557
26
+ 1427
+ 364
+ 489
+542
+1148
+ 449
+ 546
+558
+ 875
+ 569
+ 623
+609
27
+ 1339
+ 248
+ 786
+635
+1067
+ 307
+ 817
+617
+ 765
+ 422
+ 879
+650
28
+ 1164
+ 126
+ 1059
+726
+ 920
+ 163
+ 1043
+674
+ 608
+ 258
+ 1086
+684
29
+ 003
4
+ 1273
+804
+ 719
+ 13
+ 1208
+726
+ 412
+ 107
+ 1216
+707
30
+ 562
143
+ 1416
+860
+ 473
- 138
+ 1313
+770
+ 194
- 56
+1278
+720
31
+ 160
- 288
+ 1454
+886
+ 186
290
+ 1341
+803
- 36
219
+1263
+726
32
- 273
429
+ 1379
+876
- 137
442
+ 1282
+814
- 269
379
+ "77
+720
33
-692
- 562
+ "83
+829
- 472
591
+ 1117
+796
491
- 536
+ 1003
+707
34
1052
-675
+ 887
+743
- 781
730
+ 854
+742
- 692
-685
+ 773
+676
35
-1325
- 764
+ 514
+625
-1038
- 848
+ 500
+652 - 857
824
+ 461
+624
36
-1484
-832
+ 96
+479
1206
- 937
+ 94
+526 - 966
- 948
+ 93
+540
37
-1523
- 878
334
+3"
1269
991
- 329
+376 1002
-1047
- 309
+425
38
-1427
902
751
+ 129
1221
1007
- 740
+207 - 945
1106
- 718
+281
39
1212
906
III2
59
I06S
991
-1095
+ 26 - 797
1124
1082
+ "4
40
- 888
-884
1392
244
809
946
1376
158 564
1095
1373
- 65
41
-482
-837
1560
417
477
- 880
-1558
339 267
1026
1562
242
42
44
- 759
1603
-567
- 89
793
1622
507 + 61
920
1635
-4"
43
+ 392
- 655
1524
-689
+ 322
- 685
-1563
-654
+ 408
- 79i
-1586
564
44
+ 778
530
-1338
-777
+ 715
- 558
-1385
-769
+ 743
- 643
1428
-696
45
+ 1095
- 388
1067
-828
+ 1054
418
1106
846 +1043
- 484
"59
-800
46
+ 1319
240
749
-841
+ 1304
- 268
762
-880
+ 1279
319
819
-869
47
+ 1445
- 89
392
-815
+ 1453
114
- 384
-872
+ 1429
- 151
427
-896
&
+ 1474
+ 57
33
-751
+ J495
+ 33
- 5
824 +1481
+ 12
22
-876
49
+ 1405
+ 194
+ 292
-651
+ 1438
+ i/o
+ 338
-740 +1430
+ 104
+ 349
-813
5<>
+ 1256
+ 3"
+ 56l
-516
+ 1297
+ 292
+ 618
620 ! +1288
+ 296
+ 661
-706
51
+ 1048
+ 398
+ 741
354
+ 1085
+ 392
+ 829
-467 +1078
+ 404
+ 886
-565
y-
+ 820
+ 446
+ 825
177
+ 836
+ 468
+ 930
286 + 829
+ 484
-j-IOII
-395
53
+ 609
+ 452
+ 819
+ 4
+ 575
+ 5ii
+ 932
- 87 + 564
+ 534
+ 1031
-205
54
+ 432
+ 414
+ 734
+ 177
+ 342
+ Sii
+ 839
+116 + 313
+ 551
+ 955
- 3
55
+ 308
+ 342
+ 600
+330
+ 159
+ 468
+ 672
+313 + 97
+ 534
+ 790
+206
56
+ 237
+ 245
+ 451
+463
+ 5i
+ 385
+ 465
+490 - 55
+ 479
+ 556
+4"
57
+ 219
+ 127
+ 292
+569
+ 26
+ 264
+ 246
+629 125
+ 384
+ 297
+597
58
+ 247
- 6
+ 161
+640
+ 66
+ "9
+ 72
+726 105
+ 249
+ 66
+742
59
+ 312
148
+ 89
+665
+ 151
- 38
- 26
+768 - 15
+ 87
- 89
+827
COEFFICIENTS FOR DIRECT ACTION.
TABLE III. Continued.
MUTUAL PERIODIC PERTURBATIONS OF VENUS AND THE EARTH.
The term of long period is omitted. The tabular unit is o".oi in fu and 6v', and 10 8 in Sp and Sp'.
i
System 6.
System 7.
System 8.
du
8v'
*t
iff
9*
3v'
if
dp'
du
3v'
if
ip*
57
+ 45
130
+838
225
+ 170
33
+759
422
+ 3i9
+ 73
+655
i
+ 73
129
94
+826
no
+ 5
39
+773
- 316
+ 156
+ 59
+682
2
+ 166
292
+ 58
+74i
12
- 155 + 56
+726
- 226
3
+ 134
+654
3
+ 186
426
+ 296
+596
+ 38
301 + 231
+629
177
148
+ 272
+58l
4
+ 113
- 519
+ 558
+406
+ 21
422 + 446
+487
179
- 273
+ 434
+471
5
- 55
-565
+ 794
+195
82
509 +663
+312
- 238
- 375
+ 596
+334
6
294
- 565
+ 96i
- 23
- 266
552 + 840
+ "5 353 449
+ 733
+ 178
7
573
- 526
+ 1039
232
5"
547 + 943
94 522
490
+ 827
+ 5
8
- 860
- 453
+ 1017
421
790
493 I + 950
-298
735
490
+ 845
175
9
1128
351
+ 892
-588
-1066
400
+ 855
-483
973
442
+ 776
349
10
-1349
228
+ 668
-723
1302
275
+ 651
-639
I2OI
- 348
+ 605
-518
ii
1496
- 88
+ 36i
-823
1470
131
+ 368
-758
-1378
217
+ 343
-659
12
1549
+ 64
9
-879
1549
+ 23
+ 23
-836
1480
- 62
+ 13
-762
13
1404
+ 223
405
-893
1524
+ 182
352
-876
1481
+ 107
- 350
-828
14
1335
+ 38i
- 790
-860
1390
+ 339
- 732
-875
-1378
+ 277
7U
-850
IS
1089
+ 533
-1125
-788
1150
+ 489
1076
-834
1178
+ 440
1048
-831
16
- 775
+ 671
-1382
-682
-815
+ 624
-1355
-757
-875
+ 588
1327
774
17
427
+ 796
-1541
-547
419
+ 742
1534
640
- 500
+ 7i8
1521
-680
18
73
+ 902
1592
-396
+ 2
+ 833
1595
496
75
+ 821
-1606
555
19
+ 266
+ 087
-1528
-231
+ 409
+ 90i
-1538
-328
+ 370
+ 895
-1567
406
20
+ 563
+ 1046
-1357
59
+ 759
+ 944
1364
151
+ 782
+ 934
1402
-238
21
+ 799
+ 1073
-1083
+ 114
+ 1030
+ 966
1098
+ 29
+ "24
+ 941
1126
- 58
22
+ 957
+ 1062
- 731
+277
+ 1205
+ 967
- 754
+203
+ 1362
+ 920
- 766
+ 123
2.3
+ 1023
+ 1009
349
+425
+ 1274
+ 943
359
+366
+ 1479
+ 876
- 356
+298
24
+ 1001
+ 919
+ 47
+544
+ 1232
+ 890
+ 4O
+512
+ 1467
+ 812
+ 70
+459
25
+ 899
+ 801
+ 412
+632
+ 1087
+ 809
+ 461
+634
+ 1333
+ 735
+ 486
+599
26
+ 737
+ 663
+ 726
+687
+ 842
+ 696
+ 816
+726
+ 1083
+ 641
+ 861
+713
27
+ 540
+ 5'7
+ 962
+7l6
+ 541
+ 565
+ 1000
+784
+ 742
+ 531
+ "65
+795
28
+ 3l8
+ 35
+ "42
+729
+ 207
+ 419
+ 1266
+809
+ 340
+ 406
+1368
+846
29
+ CO
+ 213
+ 1251
+728
121
+ 270
+J339
+802
- 91
+ 268
+1457
+861
30
139
+ 57
+ 1282
+721
422
+ 121
+1321
+772
- 498
+ 126
+1428
+843
3i
- 359
IO2
+ 1236
+705
- 677
23
+1230
+727
- 849
n
+ 1294
+796
32
- 562
262
+ 1118
+679
-884
164
+1071
+673
1126
140
+ 1084
+726
33
73i
422
+ 926
+645
1037
- 306
+ 855
+615
1312
260
+ 813
+642
34
-852
574
+ 675
+599
1128
445
+ 589
+553
1406
- 370
+ Sio
+548
35
923
- 718
+ 374
+545
"53
579
+ 284
+487
1418
474
+ 187
+453
36
- 935
-852
+ 39
+480
1108
709
40
+414
1346
577
- 136
+357
37
- 886
972
315
+399 991
- 828
365
+334
I2O2
- 676
449
+265
38
- 773
1071
-683
+300 803
933
- 688
+246
-982
- 767
743
+ 177
39
- 589
i 143 ; 1024
+ 178 - 558
1024
-983
+ 150
697
-850
997
+ 88
40
335
1177 1317
+ 33 - 262
1093
1240
+ 45
362
920
1203
3
41
- 28
1165 j 1526
130 + 79
"39
-1436
72
+ 12
977
1354
-96
42
+ 313
noo 1622
297 + 450
"47
1550
201
+ 407
1017
-1438
-192
43
+ 650
989
1599
-458
+ 825
III2
1556
337
+ 80S
1036
-1454
-291
44
+ 959
- 837
1456
-600
+ "72
1024
1451
-473
+"90
1024
-1381
393
45
+ 1215
658 1207
-717 +1462
-887
-1233
-599
+1535
974
1213
-491
46
+ 1400
- 466 - 83 1
80 1 +1660
709
930
703
+1805
- 878
- 961
-581
47
+ 1505
269 496
850 +1753
502
564
773
+ 1974
- 737
629
-655
48
+ 1524
- 77 - 87
-863 +1738
- 285
- 167
807
+2021
- 555
- 248
-70S
49
+ 1454
+ 106 + 307
-837
+1629
- 67
+ 222
803
+ 1941
342
+ 135
725
50
+ 1298
+ 270
+ 651
-769
+1433
+ 137
+ 576
-758
+ 1741
125
+ 488
-706
Si
+ 1070
+ 407
+ 909
-653
+ 1168
+ 319
+ 866
-675
+ 1450
+ 9i
+ 785
650
52
+ 802
+ 5i3
+ 1062
500
+ 861
+ 471
+ 1063
-556
+ 1100
+ 287
+ 998
-555
53
+ 52i
+ 582
+ IIOI
-316
+ 534
+ 586
+"5i
403
+ 720
+ 454
+ 1118
425
54
+ 256
+ 610
+ 1045
117
+ 225
+ 654
+ "29
225
+ 346
+ 58o
+"38
-271
55
+ 34
+ 599
+ 890
+ 93
43
+ 677
+ 988
- 27
+ 7
+ 662
+ 1051
97
56
- 135
+ 553 + 671
+299
239
+ 651
+ 773
+ 180
- 266
+ 693
+ 875
+ 90
57
224
+ 472 + 414
+49i
348
+ 579
+ 520
+379
446
+ 670
+ 639
+273
58
237
+ 356 + 163
+656
371
+ 467
+ 272
+552
- 522
+ 592
+ 392
+440
59
- 174
+ 212 : 33
+776
322
+ 328
+ 76
+683
504
+ 471
+ 188
+572
ACTION OF THE PLANETS ON THE MOON.
TABLE III. Concluded.
MUTUAL PERIODIC PERTURBATIONS OF VENUS AND THE EARTH.
The term of long period is omitted. The tabular unit is o".oi in tu and dv', and to 8 in <$/> and
System 9.
Systen/io.
System II.
i
du
dv'
dp
9ff
du
Sv'
*P
9p>
du
ilp'
df>
dp'
- 598
+447
+ 196
+563
- 548
+407
+ 300
+495
205
+ 158
+ 341
+471
i
544
+3i8
+ 174
+583
- 616
+355
+ 295
+506
- 371
+ 169
+ 372
+454
2
482
+176 ; + 237
+552
660
+285
+ 355
+469
- 540
+ 175
+ 468
+393
3
444
+ 36 + 357
+487
-698
+200
+ 465
+398
- 718
+ 179
+ 591
+306
4
446
90
+ 493
+394
747
+ 109
+ 579
+301
-897
+ 176
+ 700
+207
5
493
-196
+ 613
+285
- 811
+ 19
+ 673
+ 193
-1066
+ 165
+ 767
+ 103
6
576
-278
+ 699
+ 162
-889
- 62
+ 727
+ 82
1208
+ 141
+ 783
+ o
7
- 690
-338
+ 744
+ 26
980
127
+ 732
32
1321
+ 109
+ 757
104
8
- 828
370
+ 736
121
1073
-171
+ 681
142
1409
+ 77
+ 662
205
9
-981
-371
+ 667
272
1164
194
+ 577
-254
-1470
+ 51
+ 5i6
300
10
-1138
335
+ 522
42O
1249
193
+ 416
364
1497
+ 39
+ 342
-384
II
1274
260
+ 298
-553
1290
-166
+ 203
-472
-1484
+ 41
+ "4
456
12
1360
-147
__ ny
-661
1310
113
60
-569
-1423
+ 59
147
-517
13
1372
i
351
740
1274
- 29
- 304
648
1314
+ 95
427
-566
M
-1288
+ 164
715
-783
1176
+ 83
693
699
"54
+ 147
- 7"
603
IS
1106
+339
1055
-786
1000
+222
1015
-716
- 938
+220
-987
624
16
- 835
+504
-1328
753
- 741
+374
1301
-695
- 669
+3H
-1237
620
17
493
+654
1520
-682
415
+530
15"
-637
346
+420 -1438
-583
18
- 106
+779
1609
576
40
+671
1617
-549
+ 19
+536 -1561
510
19
+ 308
+876
-1582
440
+ 353
+791
1605
-431
+ 413
+651
1579
403
20
+ 7io
+938
-1438
-282
+ 726
+878
-1469
293 + 793
+749
1479
270
21
+ 1072
1180
-108
+ 1049
+933
1225
135 +1126
+821
1260
-116
22
+ 1359
+948
820
+ 68
+ 1323
+949
- 880
+ 33
+ 1387
+862
940
+ 40
23
+ 1532
+899
399
+242
+ 1501
+927
467
+207
+ 1547
+867
- 544
+ 202
24
+ 1580
+816
+ 54
+405
+ 1572
+863
13
+376
+ 1600
+835
IOI
+359
25
+ 1493
+7U
+ 494
+551
+ 1525
+765
+ 446
+530
+1546
+769
+ 355
+509
26
+ 1278
+592
+ 885
+677
+ 1353
+632
+ 872
+659 +1379
+667
+ 79i
+644
27
+ 957
+466
+ 1210
+778
+ 1070
+479
+ 1221
+759 +1116
+533
+ U74
+757
28
+ 558
+336
+ 1436
+849
+ 701
+3i6
+ M7I
+833 + 766 > +372
+ 1460
+841
29
+ 106
+205
+ 1545
+884
+ 271
+ 153
+ 1600
+874 + 353
+ 192
+ 1627
+887
30
355
+ 73
+ 1534
+883
- 179
I
+ 1609
+886 - 84
+ 6
+ 1659
+896
31
- 79i
- 59
+ 1404
+848
- 621
142
+ 1501
+867 - 515
-171
+ 1565
+870
32
1164
-183
+ 1169
+780
1026
-270
+ I2 7 8
+815
- 909
-330
+ 1357
+814
33
1439
297
+ 861
+690
-1358
-383
+ 96l
+733
1234
-466
+ 1050
+735
34
1594
393
+ 497
+58l
1592
-477
+ 574
+621
1476
-573
+ 673
+632
35
1633
-471
+ 134
+465
1704
-550
+ 154
+491
1629
652
+ 250
+Sn
36
1560
-537
- 225
+345
-1684
602
+351
-1661
-707
- 183
+370
37
1402
-594
+226
1549
630
620
+209
1579
-733
- 596
+216
38
-1173
-045
818
+ H5
-1317
639
923
+ 77 -1384
-730
959
+ 60
39
- 879
694
-1058
+ n
1017
-637
1149
43
1098
703
1232
- So
40
- 540
-740
-1237
-84
674
-631
1301
-ISO
- 752
654
1403
219
4i
- 159
-783
-1347
169
303
-627
-244
- 37S
595
1473
-324
42
+ 244
-813
1390
-245
+ 80
-626
-324
+ 3
532
1450
-403
43
+ 651
-838
1365
315
+ 465
-626
1342
391 , + 364 477
1357
462
44
+ 1039
-849
-1275
-383
+ 835
623
1225
442 + 697 432
1208
504
45
+ 1399
-845
II2I
449
+ 1181
-617
1049
478 +1001 306
1009
531
46
+ 1708
821
911
509
+ 1482
-607
- 832
502 +1264 368
- 778
-542
4?
+ 1946
-768
642
562
+ 1728
-591
- 580
517 +1480 ; 346
- 520
-536
48
+2089
-681
- 322
60 1
+ 1908
-566
- 303
526 +1640 328
248
515
49
+2122
-S<S
+ 27
621
+2014
-524
IS
527 +1743 312
+ 17
-481
50
+2026
39J
+ 372
-611
+2027
460
+ 276
-514
+ 1784 -298
+ 265
434
Si
+ 1803
-207
+ 678
-570
+ 1939
371
+ 548
479
+ 1761 i 283
+ 487
-378
52
+ 1482
II
+ 912
4C9
+ 1744
-254 i + 779
418 +1679 259
+ 673
314
53
+ 1093
+ 178
+ 1054
-396
+ 1454
116 i + 942
327 +1528 221
+ 813
236
54
+ 679
+344
+ 1097
269
+ 1097
+ 27
+ 1017
213 +1313 l69
+ 895
145
55
+ 279
+477
+ 1042
-117
+ 709
+ 166
+ 995
8O : +IO44 j IOI
+ 900
31
56
71
+569
+ 006
+ 49
+ 331
+284
+ 884
+ 66 + 743 27
+ 825
+ IOO
57
344
+612
+ 7io
+217
+ 3
+371
- 716
+209 + 451 + 43
+ 686
+234
58
- 523
+603
+ 497
+370
254
+418
+ 536
+337
+ 189+99
+ 525
+354
59
- 601
+548
+ 3t2
+489
434
+430
+ 38S
+437
- 27
+ 137
+ 397
+437
COEFFICIENTS FOR DIRECT ACTION.
53
TABLE IV.
PERTURBATIONS OF THE G-COORDINATE X OF VENUS.
The tabular unit is io- 8 .
Sys-
tern
i
o
I
2
3
4
5
6
7
8
9
10
II
+ 47
+ 57
+ 92
+ 137
+ 181
+217
+218
+ 184
+142
+107
+ 71
+ 52
i
+ 37
+ 50
+ 81
+ 118
+ 159
+197
+212
+ 183
+140
+ 92
+ 52
+ 30
2
ii
+ ii
+ 41
+ 7i
+ 109
+149
+ 176
+ 158
+H4
+ 59
+ 8
23
3
92
59
24
+ 5
+ 39
+ 75
+ 112
+ iii
+ 72
+ 9
52
97
4
193
154
in
79
50
14
+ 26
+ 46
+ 14
-48
122
-183
5
305
-269
217
-183
-156
122
79
- 41
55
in
-191
-268
6
412
-396
-338
290
-267
237
197
146
-136
-179
-26l
-348
7
506
-519
465
405
-380
-354
-318
-263
230
-251
-326
421
8
-581
-628
-586
519
-487
-468
437
-384
331
330
-386
482
9
-636
715
-696
629
-583
-569
-546
-502
-437
410
443
530
IO
-668
763
-780
-718
-661
650
-636
-599
539
-488
494
-565
ii
674
777
-827
-783
-718
-699
697
672
619
-556
-534
-582
12
-660
759
-828
814
745
714
-722
710
-669
606
-561
-580
13
621
710
790
-803
-745
699
-707
-712
-682
-625
569
-560
14
-559
-639
713
749
709
-652
652
-671
659
610
552
-523
IS
479
-541
614
-659
640
-580
564
-591
-596
559
507
-466
16
-383
431
495
-542
-540
-487
452
-475
496
476
432
-389
17
279
-308
364
410
417
-374
-325
337
-368
-366
331
-298
18
171
-183
-231
-273
-283
-251
-195
-187
221
-236
-213
-187
19
- 54
- 61
-98
137
149
123
- 68
-38
66
- 95
- 84
- 64
20
+ 60
+ 56
+ 25
13
- 26
i
+ 48
+ 93
+ 80
+ 46
+ 41
+ 58
21
+ 170
+160
+ 136
+ IOO
+ 82
+ 106
+151
+203
+209
+179
+155
+ 171
22
+269
+253
+229
+197
+ 174
+192
+236
+288
+312
+291
+262
+266
23
+350
+330
+305
+275
+249
+259
+299
+350
+384
+377
+348
+342
24
+406
+392
+363
+333
+307
+306
+341
+387
+430
+434
+412
+398
25
+441
+436
+407
+373
+347
+338
+367
+412
+451
+466
+456
+437
26
+461
+460
+435
+400
+374
+36i
+378
+418
+457
+475
+476
+464
27
+468
+47i
+451
+417
+388
+372
+380
+418
+454
+478
+483
+478
28
+476
+475
+459
+426
+394
+381
+379
+411
+447
+472
+481
+484
29
+480
+475
+463
+432
+397
+381
+379
+405
+440
+465
+475
+485
30
+484
+479
+467
+438
+401
+38o
+380
+401
+434
+459
+473
+479
31
+400
+484
+476
+445
+406
+380
+38o
+397
+431
+457
+470
+476
32
+491
+492
+483
+455
+414
+380
+379
+392
+426
+456
+469
+473
33
+485
+497
+488
+460
+417
+379
+375
+387
+419
+451
+467
+466
34
+468
+492
+489
+463
+420
+377
+362
+375
+403
+437
+456
+456
35
+439
+468
+479
+458
+414
+366
+341
+354
+377
+411
+432
+436
36
+397
+426
+448
+437
+392
+342
+3io
+314
+338
+368
+391
+399
37
+340
+360
+394
+396
+353
+301
+262
+260
+282
+3ii
+331
+343
38
+263
+270
+309
+327
+293
+240
+ 196
+ 184
+208
+236
+255
+264
39
+ 167
+ 166
+ 199
+230
+211
+ 155
+ 109
+ 90
+ 112
+ 141
+ 161
+ 166
40
+ 56
+ 47
+ 69
+ 107
+ 104
+ Si
+ 2
20
4
+ 33
+ 55
+ 57
41
- 67
- 80
74
- 38
22
-67
121
-145
-127
90
59
- 58
42
-188
206
215
188
-161
192
254
-283
265
222
-180
-172
43
301
325
344
-336
-306
319
-381
420
407
-357
-304
-282
44
-397
430
459
464
443
-438
494
549
-543
-486
-423
-385
45
473
510
545
-568
557
544
-589
-656
-665
609
534
-475
46
-528
-56o
-605
634
-638
-626
-655
729
-760
-712
-628
-555
47
-566
-578
-626
-665
-681
-678
694
-764
-818
-791
702
-617
48
-585
574
-611
-660
-685
692
703
759
-833
-833
749
-657
49
-588
547
-566
619
-653
-668
-679
723
-805
-838
-775
-675
50
572
-508
-505
-549
-588
609
-627
-659
-737
-796
-766
-668
Si
539
457
-431
454
-499
-525
-545
571
639
-716
-723
639
52
491
401
347
-350
-391
423
-447
470
-525
-605
-644
-589
53
431
339
-268
248
276
3n
333
357
401
-477
-535
-518
54
-362
-274
193
-152
162
197
222
247
-281
-346
414
428
55
-283
-205
122
-69
- 58
-88
III
137
-167
219
-286
-324
56
-109
134
- 58
+ 2
+ 31
+ 14
14
37
-64
-106
164
211
57
114
- 67
+ 60
+ IOI
+ 102
+ 73
+ 48
+ 20
- 14
- 60
-105
58
- 38
- 8
+ 44
+ 104
+151
+ 167
+ 144
+ H5
+ 85
+ 54
+ 16
- 18
59
+ 19
+ 36
+ 78
+130
+179
+208
+ 194
+ 161
+ 128
+ 94
+ 59
+ 36
54
ACTION OF THE PLANETS ON THE MOON.
TABLE IV*.
PERTURBATIONS OF THE G-COORDINATE Y OF VENUS.
The tabular unit is 10 8 .
Sys-
tem
i
I
2
3
4
5
6
7
8
9
10
II
o
164
370
379
274
-176
- 78
+ 42
+ 160
+300
+423
+385
+ 146
i
100
357
436
-365
-275
-199
- 87
+ 39
+181
+337
+377
+202
2
22
310
454
430
-350
-290
203
- 77
+ 68
+239
+348
+251
3
+ 70
224
424
-458
394
-345
-287
175 30
+ 145
+304 +294
4
+172
ill
350
-438
405
-360
330
246
-108
+ 65
+252 , +324
5
+273
+ 22
-236
373
-378
341
-328
-281
-162
+ 3
+200 +339
6
+355
+ 164
97
-272
-318
-294
-289
277 -190
- 38
+ 154 +334
7
+410
+298
+ 53
148
231
228
225
239
-193
- 65
+ ii5 +313
8
+435
+406
+202
II
124
146
144
171
-167
- 72
+ 89 +286
9
+436
+ 4 8l
+340
+ 131
- 6
- 54
- 59
- 87 -114
62
+ 73 +258
10
+415
+523
+452
+267
+ 114
+ 46
+ 32
+ 7 - 38
- 31
+ 70
+232
ii
+389
+531
+533
+390
+230
+ 146
+ 120
+ 98 +49 +19
+ 78
+214
12
+36l
+519
+583
+490
+339
+244
+205
+ 185 i +139 + 86
+ 101
+205
13
+339
+495
+602
+571
+442
+338
+200
+267
+227 +165
+ 142
+207
14
+331
+472
+600
+626
+533
+429 , +372
+348
+315 +254
+201
+229
IS
+339
+457
+590
+660
+610
+515 +454
+427
+309 +348
+279
+ 268
16
+363
+457
+579
+673
+673
+597 i +536
+508
+484 +440
+371
+329
17
+406
+472
+575
+679
+721
+673
+6l5
+588
+568 +534
+473
+409
18
+469
+502
+582
+678
+749
+738
+600
+667
+650 +624
+579 , +505
19
+550
+548
+602
+680
+76o
+787
+757
+735
+729
+710
+675 +610
20
+639
+606
+629
+685
+759
+812
+809
+792
+793
+786
+76i
+710
21
+729
+668
+660
+692
+747
+814
+837
+832
+837
+845
+829
+795
22
+804
+729
+691
+698
+729
+788
+835
+847
+858
+877
+875
+855
23
+848
+779
+715
+606
+704
+743
+799
+830
+846
+875
+888
+880
24
+854
+801
+725
+681
+671
+681 +729
+778
+801
+833
+861
+864
25
+812
+786
+717
+653
+621
+606 +630
+683
+7i7
+752
+792
+809
25
+723
+722
+676
+602
+552
+SI7 +510
+551
+594
+632
+676
+7io
27
+594
+609
+593
+53i
+465
+416 +383
+397
+441
+478
+522
+567
28
+428
+455
+467
+428
+360
+297 +2.48
+228
+262
+301
+34i
+390
29
+237
+269
+301
+295
+239
+174 +H3
+ 66
+ 70
+ 106
+ 138
+ 187
30
+ 26
+ 67
+ 107
+ 132
+ 100
+ 40 22
po
-116
92
- 65
- 27
31
-187
142
099
54
- Si
- 98 1 -155
-227 -280
-283
-261
-235
32
-388
340
300
250
216
-237 ! -284
346 416
449
440
424
33
-565
523
-484
439
-389
-375 I -403
451 i 520
-580
-589
-582
34
702
-673
-637
-602
547
-505
-509
-538
594
-667
-702
703
35
-796
-785
-754
-729
-685
625
-601
609
644
-711
-770
-787
36
-846
-852
-835
814
-789
-728
-679
-664
-675
-723
-788
829
37
-857
-875
-872
-860
-850
-807
743
704
691
712
774
-830
38
-832
-857
-869
-867
869
-850
791
729
-695
-688
729
794
39
-779
810
-833
840
850
-858
-820
749
-690
659
-669
729
40
704
-741
-769
-787
802
-835
-827
-761
-685
631
-608
647
41
-613
659
-690
-713
-736
-778
-810
-767
-682
607
-554
-561
42
-518
-568
-601
629
-655
704 766
-763
-680
-589
512
-483
43
428
-475
513
-539
-568
618 702
743
-686
-582
-484
421
44
-350
-383
426
-452
-478
528 ; 622
705
-689
-586
-468
379
45
-289
299
341
-369
-393
435 ' 530
-645
-68 1
-597
-469
353
46
249
-226
-260
291
310
344 ! 432
569
-659
612
-480
345
47
227
-168
-183
214
229
254 329
472
610
-618
499
352
48
220
-126
in
-139
-152
166 ; 223
360
532
-606
521
-363
49
227
103
47
- 64
- 77
- 79
-116
-235
425
-566
-535
-387
50
240
- 94
3
+ 9
2
+ 8
10
104
-295
-488
-530
410
51
-2 5 8
- 96
+ 25
+ 75
+ 74
+ 88
+ 97
+ 30
148
377
497
424
52
-2 7 8
-105
+ 40
+ 124
+ 143
+ 163
+ 192
+ 160
+ 8
235
427
427
S3
297
-124
+ 35
+ 151
+200
+226
+270
+276
+ 161
- 77
324
407
54
314
-150
+ 13
+ I5I
+234
+272
+327
+368
+299
+ 84
194
-359
55
319
184
21
+ 124
+237
+298
+358
+428
+409
+235
- 49
-285
56
311
226
68
+ 74
+205
+291
+360
+449
+483
+361
+ 92
192
57
2pO
-273
-132
+ 8 +136
+247
+329
+427
+509
+4-18
+215
94
58
257
-319
212
__ **
+ 42
+ 164
+264
+365
+481
+487
+305
+ I
59
-215
-355
-298
-173
- 66
+ 50
+ 165
+274
+407
+477
+36l
+ 81
COEFFICIENTS FOR DIRECT ACTION.
55
TABLE V.
RECTANGULAR G-COORDINATES X AND Y OF VENUS, REFERRED, IN EACH
SYSTEM, TO AN Axis OF X PASSING THROUGH THE MEAN SUN.
i
System o.
System i.
System 2.
X
r
X
r
X
r
+0.264 4388
+0.005 3159
+0.267 8642
+0.0172810
+0.273 5900
+0.024 1368
i
+0.2682411
0.070 6564
+0.271 9461
0.058 8025
+0.277 8772
0.0518927
2
+0.280 0227
-0.1458383
+0.2840031
0.1340369
+0.290 1029
0.1270162
3
+0.299 6585
0.219 3893
+0.303 8978
0.207 5810
+0.310 1265
0.200 3988
4
+0.326 9346
0.290 4863
+0.331 4038
0.278 6154
+0.337 7133
0.271 2285
5
+0.361 5487
0.358 3344
+0.366 2082
0.346 3496
+0.372 5452
0.338 7248
6
+0.403 1159
0.422 1763
+0.4079159
0.4100328
+0.414 2234
O.4O2 1483
7
+0.451 1712
0.481 2993
+0.456 0557
0.468 9619
+0.462 2754
0.460 8077
8
+0.505 1754
0.535 0445
+0.5100841
0.522 4898
+0.5161594
0.5I4067I
9
+0.564 5219
0.5828139
+0.569 3924
0.570 0313
+0.575 2710
0.56l 3528
10
+0.628 5438
0.624 0792
+0.633 3160
0.61 1 0692
+0.6389518
O.6O2 l6O2
ii
+0.696 5229
-0.658 3852
+0.701 1382
0.645 1617
+0.7064955
0.636 0564
12
+0.767 6965
0.685 3576
+0.772 1023
0.671 9440
+0.777 1566
-0.662 6856
13
+0.841 2692
0.704 7061
+0.845 4198
0.691 1353
+0.850 1572
0.68 1 7730
14
+0.9164199
0.716 2270
+0.920 2784
0.702 5389
+0.924 6978
0.693 1251
15
+0.9923125
0.7198055
+0.995 8539
0.706 0456
+0.9999648
0.696 6325
16
+ 1.068 1057
0.7154169
+ 1.0713157
0.701 6335
+ 1.075 1404
0.692 2716
17
+ 1.1429619
-0.703 1247
+ 1.1458394
-0.6893674
+ 1.1494106
0.680 1025
18
+ 1.2160577
-0.683 0801
+ 1.2186136
0.669 3977
+ 1.2219735
0.6602691
19
+ 1.2865933
0.655 5200
+ 1.2888493
-0.641 9577
+ 1.2920488
0.632 9963
20
+ I-353799I
0.620 7640
+ 1.3557887
0.607 3599
+ 1.3588839
-0.5985883
21
+ 1.4169464
0.579 2086
+ 1.4187118
0.565 0943
+ 1.421 7633
0.557 4240
22
+ 1.4753529
0.531 3238
+ 1.4769448
0.5183206
+ 14800138
0.5099523
23
+ 1.5283911
0.477 6470
+ 1.5298660
04648651
+ I.5330I33
04566884
24
+ 1-5754942
0.418 7748
+ 1.5769123
0.4062148
+ 1.580 1954
0.398 2084
25
+ 1.6161628
-0.355 3577
+ 1.6175843
0.343 0090
+ 1.6210558
0.335 1418
26
+ 1.6499691
0.2880919
+ 1.6514516
0.275 9346
+ 1.655 1564
0.268 1673
27
+ I.6765593
0.2177116
+ 1.678 1568
0.205 7162
+ 1.682 1302
0.1980049
28
+ 1.6956592
0.1449820
+ 1.6974188
0.133 1099
+ 1.7016857
0.1254065
29
+ 1.7070734
0.070 6903
+ 1.7090351
0.058 8960
+ 1.7136087
0.051 1511
30
+ 1.7106804
+0.0043615
+ 1.7128842
+0.016 1297
+ 1.7177659
+0.023 9652
3i
+ 1.7064775
+0.079 3664
+ 1.7089249
+0.091 1628
+ 1.714 1050
+0.099 1360
32
+ 1.6944891
+O.I535I90
+ 1.6971990
+0.165 3988
+1.702 6556
+0.1735529
33
+ 1.6748587
+0.226 0244
+ 1.6778288
+0.238 0408
+ 1.6835308
+0.2464124
34
+ 1.6478008
+0.296 1060
+ 1.6510177
+0.308 3085
+ 1.6569253
+0.3169261
35
+ 1.6136080
+0.3630127
+ 1.6170473
+0-375 4452
+ 1.623 1144
+0.384 3269
36
+ 1.5726476
+0.426 0265
+ 1.5762761
+0.4387253
+ 1.5824506
+0.4478781
37
+ 1-5253583
+0.484 4694
+ 1.529 1350
+0497 4622
+ 1.5353622
+0.506 8825
38
+ 1.4722450
+0.537 7122
+ 1.4761240
+0.5510155
+ 1.4823484
+0.560 6879
39
+ 1.4138753
+0.585 1789
+ 1.4178080
+0.598 7973
+ 1.4239759
+0.608 6959
40
+ 1.3508731
+0.626 3539
+ 1.3548096
+0.640 2802
+ 1.3608720
+0.650 3687
41
+ 1.2839122
+0.660 7876
+ 1.2878040
+O.675 OO2O
+ 1.2937184
+0.685 2350
42
+ 1.2137108
+0.6880995
+ 1.2175124
+0.702 5721
+ 1.2232454
+0.7128968
43
+ 1.1410227
+0.707 9845
+ 1.1446940
+0.722 6753
+ 1.1502220
+0.7330331
44
+ 1.0666304
+0.7202157
+ 1.070 1388
+0.735 0768
+ 1.0754484
+0.745 4066
45
+0.991 3369
+0.724 6478
+0.994 6591
+0.739 6239
+0.999 7488
+0.749 8650
46
+0.9I5957I
+0.721 2182
+0.919 0806
+0.7362498
+0.923 9587
+0.746 3434
47
+0.841 3093
+0.709 9494
+0.844 2332
+0.724 9743
+0.848 9201
+0.7348666
48
+0.7682082
+0.690 9488
+0.770 0409
+0.705 9042
+0.775 4674
+0.7155486
49
+0.6974541
+0.664 4077
+0.7000152
+0.679 2337
+0.7044196
+0.688 5925
SO
+0.629 8261
+0.630 6012
+0.632 2439
+0.645 2428
+0.636 5703
+0.654 2871
Si
+0.5660715
+0.5898843
+0.568 3830
+0.604 2938
+0.572 6800
+0.6130068
52
+0.5068987
+0.542 6896
+0.509 1472
+0.5568286
+0.5134653
+0.565 2062
53
+0.452 9685
+0.489 5225
+0.455 2029
+0.503 3636
+0.459 5903
+0.5114133
54
+0.404 8861
+0.4309573
+0.407 1576
+0.444 4847
+0.4116602
+0.452 2270
55
+0.363 1948
+0.3676314
+0.365 5549
+0.380 8404
+0.3702130
+0.388 3082
56
+0.328 3680
+0.300 2365
+0.330 8664
+0.313 1351
+0.335 7125
+0.320 3708
57
+0.300 8042
+0.2295127
+0.303 4856
+O.242 I2IO
+0.308 5436
+0.249 I75i
58
+0.280 8207
+0.1562403
+0.283 7236
+O.I68 5895
+0.289 0073
+0.1755176
59
+0.268 6504
+0.08 1 2302
+0.271 8049
+0.093 3621
+0.277 3 1 70 +o. loo 2240
ACTION OF THE PLANETS ON THE MOON.
TABLE V .Continued.
RECTANGULAR G-COORDINATES X AND Y OF VENUS, REFERRED, IN EACH
SYSTEM, TO AN Axis OF X PASSING THROUGH THE MEAN SUN.
I
System 3.
System 4.
System 5.
X
r
X
T
X
J'
o
+0.279 8926
+0.024 4020
+0.285 2603
+0.0184830
+0.2886591
+0.0080878
I
+0.284 2769
0.051 3907
+0.289 6702
0.0569777 +0.2930191
0.067 0880
2
+0.296 5601
0.1262430
+0.301 9298
0.131 4966 +0.305 1838
0.141 3542
3
+0.316 5918
0.1993308
+0.321 8895
0.204 2627
+0.325 0091
0.213 9093
4
+0.344 1354
0.269 8527
+0.3493160
0.274 4873
+0.352 2710
0.283 9728
5
+0.378 8721
0.337 0406
+0.383 8974
0.341 4123
+0.3866655
0.350 7937
6
+0.420 4078
0.400 1668
+0425 2457
0404 3190
+0.427 8152 0413 6573
7
+0.468 2736
0458 5510
+0472 9020
0.462 5340
+0.475 2721 0.471 8919
8
+0.521 9346
0.5115674
+0.5263418
0.5154368
+0.528 5222 0.524 8756
9
+0.5807952
-0.5586507
+0.5849803
0.5624651
+0.5869911 -0.5720427
10
+0.644 2076
0.5993022
+0.648 1798
0.603 1207
+0.650 0504
0.612 8887
ii
+0.7114758
0.633 0942
+0.715 2549
0.636 9740
+0.7170241
0.646 9765
12
+0.781 8659
0.659 6737
+0.785 4819
0.663 6672
+0.787 1933
0.673 9389
13
+0.854 6132
-0.678 7647
+0.858 1036
0.682 9185
+0.859 8058
0.693 4842
14
+0.928 9294
0.690 1730
+0.932 3397
-0.694 5257
+0.934 0825
-0.705 3977
IS
+1.0040113
0.693 7852
+1.0073925
-0.698 3667
+1.0092254
0.709 5459
16
+1.0790492
0.6895714
+ 1.0824565
0.694 4006
+ 1.0844261
0.705 8759
17
+1-1532351
0.677 5823
+ 1.1567246
0.682 6691
+1.1588744
0.6944182
18
+1.2257719
-0.657 9526
+ 1.2293983
0.663 2956 + 1.23 1 7646
0.675 2864
19
+1.2958808
0.630 8962
+ 1.2996944
0.636 4839 +1.302 3064
0.648 6759
20
+1.3628093
0.596 7058
+ 1.3668546
0.6025160 +1-3697311
0.6148635
21
+14258394
0.555 7489
+ 1.4301522
0.561 7502
+14333011
0.574 2026
22
+1.4842931
0.508 4047
+14889006
0.5146175
+1.4923178
O.527 1222
23
+1.5375412
0455 3593
+1.5424596
0.461 6180
+1.5461296
0474 1214
24
+1.5850084
0.397 0007
+ 1.5002425
0403 3157
+ 1.594 1387
0.415 7661
25
+1.6261801
0.334 0124
+ 1.631 7225
0.340 3342
+ 1.6358083
0.352 6827
26
+1.6606072
0.267 0689
+1.6664388
0.273 3482
+ 1.6706690
-0.285 5523
27
+1.6879113
0.1968867
+ 1.6940012
0.203 0779
+ 1.6983230
0.215 1035
28
+1.707 7882
0.1242202
+ 1.7140957
0.1302819
+ 1.7184514
0.142 1049
29
+ 1.7200122
0.049 8508
+ 1.7264883
0.055 7493
+ 1.7308162
0.0673551
30
+1.7244388
+0.0254188
+ 1.7310274
+0.0197085 +1.7352660
+0.008 3225
31
+1.7210059
+0.100 7745
+ 1.7276465
+0.095 2669
+ I-73I 7363
+0.084 0928
32
+1.7097359
+0.175 3989
+1.7163660
+0.1700960
+ 1.7202518
+0.1591154
33
+1.6907351
+0.2484792
+ 1.6972920
+0.243 3714 +1.700 9265
+0.232 5558
34
+1.6641950
+0.3192167
+ 1.6706195
+0.314 2840
+ 1.6739638
+0.303 5951
35
+1.6303893
+0.3868336
+1.6366271
+0.382 0454
+ 1.6396529
+0.371 4384
36
+1.5896721
+0.450 5830
+ 1.5956761
+0.445 9010
+ 1.5983674
+0.435 3259
37
+1.5424753
+0.509 7560
+ 1.5482079
+0.505 1363
+ 1.5505612
+0.4945411
38
+1.4893032
+0.5636915
+ 14947388
+0.5590849
+ 1.4967627
+0.548 4197
39
+14307300
+0.6117835
+ 14358543
+0.607 1386
+ 1-437 5701
+0.596 3565
40
+1.3673923
+0.653 4880
+ 1.3722027
+0.648 7530
+1.373 6443
+0.6378131
41
+1.2999821
+0.688 3306
+ 1.3044890
+0.683 4552
+ 1.3056998
+0.672 3257
42
+1.2292412
+0.7159113
+ 1.2334657
+0.710 8508
+ 1.2344980
+0.699 5075
43
+1.1559507
+0.7359II4
+ I.I599243
+0.730 6274
+ 1.1608358
+0.7190571
44
+1.0809236
+0.748 0961
+ 1.0846868
+0.742 5595
+ 1.0855389
+0-730 7598
45
+1.0049938
+0.752 3188
+ 1.0085958
+0.7465108
+ 1.0094492
+0.7344915
46
+0.929 0085
+0.748 5233
+0.932 5029
+0.742 4365
+0.933 4172
+0.7302183
47
+0.853 8157
+0.736 7446
+0.857 2595
+0.730 3837
+0.8582903 +0.7179976
48
+0.780 2560
+0.717 1082
+0.783 7064
+0.7104897
+0.784 9033 +0.697 9763
49
+0.709 1529
+0.6898295
+0.712 6639
+0.682 9817
+0.7140686 +0.6703895
50
+0.641 3017
+0.6552115
+0.644 9229
+0.648 1733
+0.646 5668 +0.635 5557
5i
+0.5774611
+0.6136402
+0.581 2344
+0.606 4605
+0.583 1378 +0.593 8734
52
+0.5183431
+0.565 5804
+0.522 3027
+0.5583153
+0.5244736
+0.545 8162
53
+0464 6055
+0.5115714
+0.468 7755
+0.504 2817
+0.471 2105
+0.491 9264
54
+0.4168456
+04522194
+0.421 2379
+0.444 9681
+0.423 9224
+0.432 8084
55
+0.375 5907
+0.388 1916
+0.380 2062
+0.381 0400
+0.3831147
+0.369 1218
56
+0.341 2944
+0.320 2069
+0.346 1218
+0.3132132
+0.349 2206 +0.301 5727
57
+0.3143306
+0.249 0281
+0.3T93477
+0.242 2440
+0.3225952 +0.2309071
58
+0.204 9887
+0.175 4508
+0.300 1638
+0.1689220
+0.303 5124 +0.157 9020
59
+0.283 4712
+0.1002967
+0.288 7653
+0.094 0597
+0.2921646 +0.0833577
COEFFICIENTS FOR DIRECT ACTION.
57
TABLE V '.Continued.
RECTANGULAR G-COORDINATES X AND Y OF VENUS, REFERRED, IN EACH
SYSTEM, TO AN Axis OF X PASSING THROUGH THE MEAN SUN.
i
System 6.
System 7.
System 8.
X
r
X
r
X
r
o
+0.2893812
0.004 3547
+0.287 0415
0.0159849
+0.281 8724
0.023 8026
i
+0.293 5931
-0.0793938
+0.291 OOI2
0.091 0510
+0.285 5340
0.099 0059
2
+0.305 5780
-0.153 578i
+0.302 7254
0.1653184
+0.2969815
0.1734542
3 +0.32S 2014
0.226 1096
+0.322 0914
0.2379885
+0.316 1026
0.246 3419
4 +0.352 2490
0.296 2092
+0.348 8955
0.308 2783
+0.342 7024
-0.3168787
5 +0.3864284
0.363 1251
+0.382 8548
-0.375 4297
+0.376 5054
0.384 2966
6 +0.427 3728
0.426 1400
+04236113
0438 7161
+0.417 1580
-0.4.178587
7 +0.474 6443
0.484 5776
+0470 7341
04974515
+0.464 2316
0.5068671
8 +0.527 738o
-0.537 8098
+0.523 7240
0.5509962
+0.5172268
0.560 6696
9 +0.586 0872
0.585 2627
+O.5820I78
0.598 7649
+0.575 5777
-0.6086687
10 +0.649 0693
0.6264213
+0.6440946
0.640 2315
+0.6386584
-0.650 3275
n +0.7160118
-0.6608369
+O.7II 9802
0.674 9360
+0.705 7897
0.685 1772
12 +0.786 1986
0.688 1308
+0.782 2554
0.702 4883
+0.776 2436
0.7128211
13 +0.8588769
0.707 9983
+0.855 0617
0.722 5740
+0.849 2528
0.732 9398
14
+0.9332651
0.720 2135
+0.929 6105
0.734 9582
+0.924 0172
0.745 2958
15
+1.0085598
0.724 6313
+1.0050902
0.7394890
+0.099 7145
-0.7497373
16
+1.0839453
0.721 1895
+ 1.0806753
0.736 0095
+1.075 5o85
0.746 1991
17
+1.1586020
0.709 9106
+ i. 155 5347
0.724 8102
+I.I505579
0.734 7062
18
+1.231 7142
0.6909018
+ 1.2288424
0.705 7280
+1.2240268
-0.715 3728
19
+1.3024807
0.664 3547
+ 1.2997857
0.679 0484
+1.2950946
0.688 4024
20
+1.3701219
0.630 5443
+ 1-3675740
0.645 0509
+1.3629645
0.654 0862
21
+1.4338001
0.589 8264
+ 1.4314510
0.604 0981
+14268752
0.612 7087
22
+1.4930762
0.542 6339
+ 1.4907009
0.5566323
+ 1.4861086
0.564 9945
23
+1.5470187
0.4894723
+ 1.5446576
0.503 1709
+1.5399987
0.5112033
24
+1.5951122
0.4309158
+ 1.5927131
0.444 2908
+1.5879409
0.452 0237
25
+1.6368140
0.367 6003
+ 1.6343252
0.3806684
+1.6293973
0.388 1 167
26
+1.6716503
0.3002166
+ 1.6690214
0.312 9805
+1.6639048
0.320 1963
27
+ 1.6992228
0.229 5034
+ 1.6964089
0.241 9865
+1.691 0789
0.249 0223
28
+1.7192138
0.1562412
+ 1.716 1761
0.1684765
+ 1.7106192
-0.1753907
29
+1.731 3808
0.081 2404
+ 1.7280986
0.093 2689
+ 1.7223122
o.ioo 1249
30
+1.7356043
0.005 3345
+ 1.7320410
0.0172049
+ 1.7260340
0.024 0644
31
+1.7318010
+0.070 6309 +1.727 958o
+0.0588662
+1.721 7514
+0.0519430
32
+1.7200145
+0.1458077 +1.7158966
+0.1340934
+1.7095212
+0.1270511
33
+ 1.7003702
+0.2 19 3560 +i .695 0944
+0.207 6356
+1.6894904
+0.2004263
34
+1.673 0827
+0.290 4526
+ 1.6684777
+0.278 6726
+1.6618933
+0.271 2560
35
+1.6384553
+0.358 3016
+ 1.6336507
+0.3464130
+1.6270482
+0.3387587
36
+1.5968743
+0.422 1439 +I-59I 9353
+0.410 1049
+1.5853540
+0.402 1930
37
+1.5488048
+0.4812659 +I.S437778
+0469 0434
+1.5372837
4-0.4608658
38
+1.4947865
+0.535 0089 + 1 489 7309
+0.522 5796
+14833802
+0.5141397
39
+1.4354258
+0.582 7755 +i .430 4045
+0.570 1266
+ 1.4242480
+0.561 4397
40
+1.3713900
+0.6240380 +1.3664651
+0.611 1675
+1.3605472
+0.602 2588
41
+1.303 3980
+0.6583431
+ 1.2986290
+0.645 2002
+1.2929865
+0.636 1634
42
+1.2322125
+0.685 3171
+ 1.2276529
+0.672 0417
+1.2223115
+0.662 7978
43
+1.1586310
+0.7046608 +1.1543266
+0.691 2326
+ 1.1493005
+0.681 8865
44
+1.0834752
+0.716 1979
+ 1.0794622
+0.702 6377
+ 1.0747540
+0.693 2381
45
+1.0075811
+0.7197864
+ 1.0038858
+0.706 1489
+0.9094847
+0.696 7453
46
+0.931 7009
+0.7154100
+0.9284277
+0.701 7444
+0.9243115
+0.692 3855
47
+0.8569412
+0.703 1324
+0.853 9128
+0.689 4895
+0.850 0486
+0.680 2209
48
+0.7838553
+0.683 1047
+0.781 1516
+0.669 5340
+0.7774975
+0.660 3960
49
+0.713 3338
+0.655 5634
+0.7109317
+0.642 1 1 1 1
+0.7074383
+0.633 1359
50
+0.646 1446
+0.620 8269
+0.644 0104
+0.607 5322
+0.640 6223
+0.558 7436
Si
+0.5830165
+0-579 2912
+0.581 1065
+0.566 1862
+0.577 7638
+0.5575971
52
+0.524 6297
+0.5314234
+0.522 8929
+0.5185316
+0.5195340
+0.510 1441
53
+0.471 6 no
+0.4777588
+04699907
+0.465 0927
+0466 5545
+04568979
54
+0.424 5246
+0.418 8929
+0.422 9616
+0.4064538
+0.4193902
+0.3084327
55
+0.383 8706
+0.355 4747
+0.382 3050!
+0.343 2525
+0.378 5456
+0.335 3763
56
+0.350 0760
+O.288 2O02
+0.3484501!
+0.276 1738
+0.3444581
+0.2684051
57
+0.323 4952
+0.2178039
+0.321 7545
+0.205 9419
+0.3174943
+0.1982368
58
+0.3044032
+O.I450SI2
+0.302 4906
+0.133 3140
+0.2079461
+0.1256232
59
+0.292 9944
+0.070 7305
+0.2908879,
+0.0590724
+0.286 0278
+0.051 3438
ACTION OF THE PLANETS ON THE MOON.
TABLE V .Concluded.
RECTANGULAR G-COORDINATES X AND Y OF VENUS, REFERRED, IN EACH
SYSTEM, TO AN Axis OF X PASSING THROUGH THE MEAN SUN.
i
System 9.
System 10.
System n.
X
r
X
r
X
r
o
i
2
3
4
+0.275 0523
+0.2784844
+0.2897521
+0.308 7484
+0.335 2831
-0.025 3633
o.ioo 7462
0.1754000
0.248 5086
0.3192704
+0.268 5973
+0.271 9768
+0.283 2537
+0.302 3220
+0.328 0878
0.019 7819
0.095 3562
0.1702047
0.243 5018
0.3144354
+0.264 6365
+0.268 1640
+0.279 6432
+0.298 9610
+0.325 9149
0.0084260
0.084 2045
0.1592361
0.232 6860
-0.303 7353
5
6
8
9
+0.369 0820
+0.409 7903
+0.4569758
+0.510 1331
+0.568 6886
0.386 9062
0.450 6682
0.509 8485
-0.563 7854
0.6118737
+0.362 9727
+0403 9145
+0.451 3723
+0.504 8294
+0.563 7000
0.3822158
04460864
0.5053317
0.559 2848
0.607 3382
+0.3602162
+0401 4913
+0.4492861
+0.503 0724
+0.5622517
0.371 5891
0.435 4874
0.494 7126
-0.548 5982
0.596 5386
10
ii
12
13
14
+0.632 0062
+0.6993951
+0.7701160
+0.8433898
+0.918 4056
-0.653 5698
0.688 4000
0.7159656
0.735 9488
0.748 1159
+0.6273358
+0.695 0329
+0.766 0396
+0.839 5657
+0.914 7912
0.648 0481
0.683 6429
0.711 0281
0.730 7919
0.742 7091
+0.626 1636
+0.694 0946
+0.765 2841
+0.838 9356
+0.914 2243
0.637 9955
0.672 5042
0.6996795
0.719 2203
0.7309118
3
17
18
19
+0-994 3301
+ 1.0703171
+ I.I455I78
+1.2190910
+1.2902123
0.752 3209
0.748 5084
0.736 7126
0.717 0597
-0.689 7649
+0.990 8753
+ 1.0669676
+ 1.1422168
+ 1.2157813
+ 1.2868399
0.7466441
0.742 5523
0.730 4812
0.7105689
0.683 0438
+0.990 3091
+1.0663406
+1.1414717
+1.2148681
+1.2857175
-0.734 6303
0.730 3419
0.718 1042
0.698 0645
0.670 4585
20
21
22
23
24
+ 1.3580846
+ 1.4210480
+ 1.481 0876
+ 1.5348440
+ 1.5826192
0.655 1320
0.613 5482
0.565 4803
0.511 4688
O.452 I2IO
+ 1-3545999
+ 1.4183087
+ 1.4772619
+ 1.5308092
+ 1.5783646
0.6482193
0.6064921
0.558 3354
0.504 2948
0.444 9805
+1.3532374
+14166861
+1.4753700
+1.5286519
+ 1-5759570
0.635 6060
0.593 9072
0.545 8365
0.491 9379
0432 8169
25
26
27
28
29
+ 1.6238856
+ 1.658 1899
+ 1.685 1598
+ 1.7045055
+ 1.7160249
0.388 1040
0.320 1363
0.2489795
0.175 4284
0.1003034
+ 1.6194111
+ 1.6535064
+ 1.6802882
+ 1.6994769
+ 1.7108776
0.381 0500
0.3132470
0.242 3001
0.1690056
0.094 1744
+ 1.6167795
+ 1.6506865
+ 1.6773221
+ 1.6964120
+ 1.7077640
0.369 1332
-0.301 5947
0.230 9478
0.1570687
0.083 456i
30
3i
32
33
34
+ 1.7196036
+ 1.7152172
+ 1.7029292
+ 1.6828904
+ 1.6553370
0.024 4386
+0.051 3259
+0.126 1551
+O.I992270
+0.269 7422
+ 1.7143826
+ 1.7099702
+ 1.6977055
+ 1.6777386
+ 1.6503021
0.0186200
+0.056 8022
+0.131 2957
+0.204 0425
+0.274 2547
+ 1.7112698
+ 1.7060075
+ 1.6947377
+ 1.6740043
+ 1.6476325
0.008 2204
+0.066 9225
+0.141 1607
+0.213 6945
+0.283 7444
35
36
37
38
39
+ 1.6205873
+ 1.5790366
+ I-53I 1536
+ 14774737
+ 1.4185926
+0.3369322
+0.400 0673
+0.4584650
+0.5II4975
+0.558 5979
+ 1.6157077
+ 1-5743442
+ 1.5266706
+ 14732125
+ I.4I45550
+0.341 1756
+0.404 0865
+04623116
+0.5152296
+0.562 2757
+ 1.6132252
+ 1.5720603
+ 1.5245861
+ 14713167
+ 14128279
+0.3505588
+0.413 4229
+0.471 6640
+0.524 6595
+0.571 8423
40
41
42
43
44
+ 1.355 1605
+ 1.2878735
+ 1.2174671
+ 1.1447074
+ 1.0703837
+0.599 2658
+0.633 0725
+0.659 6638
+0.678 7636
+0.690 1770
+ I.35I337I
+ 1.2842449
+ 1.2140031
+ 1.141 3690
+ 1.067 1246
+0.602 9494
+0.636 8197
+0.663 5280
+0.682 7924
+0.694 4103
+ 1.3497496
+ 1.2827595
+ 1.2125774
+ I.I399554
+ 1.0656731
+0.612 7065
+0.646 8129
+0.673 7929
+0.693 3538
+0.705 2806
4|
46
47
48
49
+0.995 2985
+0.920 2620
+0.8460814
+0.773 5541
+0.703 4583
+0.693 7915
+0.689 5775
+0.6775884
+0.657 9598
+0.630 9076
+0.992 0676
+0.9170042
+0.842 7410
+0.7700760
+0.699 7507
+0.6982589
+0.6042980
+0.682 5604
+0.663 1965
+0.636 3845
+0.090 5284
+0.915 3290
+0.8408858
+0.768 0039
+0.6974731
+0.709 4400
+0.705 7790
+0.604 3282
+0.675 2018
+0.648 5938
50
51
52
53
54
+0.636 5464
+0.573 5356
+0.515 1027
+0.461 8752
+0.4144269
+0.506 7255
+0-555 7804
+0.5085110
+0.455 4212
+0.397 0772
+0.632 6437
+0.5693612
+0.5106299
+o. 157 0893
+0409 3243
+0.6024170
+0.561 6533
+0.5145258
+0461 5341
+0.403 2415
+0.6300613
+0.566 5049
+0.507 5018
+0.453 7038
+0405 7084
+0.614 7816
+0.574II95
+0.527 0362
+0474 0329
+04156762
55
56
57
58
59
+0-373 2719
+0.3388588
+0.311 5656
+0.291 6064
+0.2794770
+0.334 10 12 +0.367 86 1 1
+0.267 1652 +0.333 1597
+0.196 9846 +0.305 6091
+0.124 3117 +0.285 5228
+0.049 9280 +0.273 1348
+0.340 2695
+0.273 2007
+0.203 0240
+0.1302265
+0.055 6874
+0.364 0524
+0.329 2053
+0.301 5630
+O.28l W/\7.
+0.209 0848
+0.352 5927
+0.285 4627
+0.215 0136
+0.1420126
+0.067 2585
COEFFICIENTS FOR DIRECT ACTION.
59
TABLE VI.
G-COORDINATE Z OF VENUS.
Sys-
tem
i
o
I
2
3
4
5
+.0174641
+.034 6635
+.0423808
+.038 5648
+.0243031
+.003 5232
i
+ .021 4657
+.037 0784
+.042 5597
+.0364563
+.0205041
.0009384
2
+.025 2276
+.0390785
+.042 2557
+.033 9453
+.0164810
.005 3894
3
+.028 7075
+.0406417
+.041 4818
+.0310603
+.OI2 2785
.009 7824
4
+.03I866S
+.041 7509
+.040 2471
+.027 8339
+ .0079423
.014 0692
5
+ .0346689
+.0433939
+.0385664
+.024 3022
+.003 5202
.018 2044
6
+ .0370834
+.042 5642
+.036 4589
+.020 5043
.OOO 94OO
.022 1436
7
+ .0390828
+.042 2603
+.033 9486
+.0164824
-.005 3898
.025 8445
8
+.040 6450
+.041 4861
+.031 0641
+.0122809
.009 7816
.029 2675
9
+ .041 7527
+.040 2507
+.0278378
+.0079454
.014 0672
.032 3763
10
+.042 3943
+.038 5650
+.024 3057
+.003 5235
.018 2018
035 1374
ii
+.042 5630
+.0364602
+.0205072
.000 9368
.022 1406
.037 5214
12
+.0422579
+.033 9485
+.0164842
.005 3871
.025 8416
039 5028
13
+.041 4828
+.031 0627
+.012 2814
.009 7798
.0292651
.041 0601
U
+.040 2473
+.0278355
+.007 9446
.014 0665
.032 3746
.042 1764
15
+.038 5660
+.024 3029
+.003 5218
.Ol8 2O22
.035 1366
.042 8392
16
+.036 4580
+.020 5046
.000 9389
.022 1419
037 5215
.043 0410
17
+.033 9476
+.016 4822
.005 3893
.025 8434
-.039 5037
.042 7792
18
+.0310631
+.012 2804
.009 7815
.029 2671
.041 0620
.042 0556
19
+.027 8370
+.007 9450
.0140675
.032 3762
.042 1783
-.040 877S
20
+.0243054
+.003 5233
.0182021
.035 1377
.042 841 1
.039 2568
21
+.020 5072
.000 9367
.022 1407
.037 5218
.043 0424
.037 2103
22
+.0164847
.005 3866
.025 8415
039 5032
.042 7800
034 7593
23
+ .OI2 2824
.009 7789
.029 2648
.041 0604
.042 0556
.031 9296
24
+ .0079461
.0140654
0323739
.042 1765
.040 8769
.0287515
25
+.003 5237
.018 2007
035 1357
.042 8391
-.039 2556
.025 2500
26
.000 9370
.022 1401
.037 5204
.043 0406 .037 2089
.021 4899
27
.005 3875
.025 8415
.039 5023
.042 7786 .034 7578
.0174848
28
.009 7799
.029 2652
.041 0600
.042 0548
.031 9285
.OI3 2876
29
.0140663
.032 3746
.042 1766
.040 8766
.028 7508
.0089440
30
.0182014
.035 1364
.042 8395
.0392557
.025 2588
.0015014
31
.022 1404
.037 5209
.0430411
.037 2091
.021 4900
.0000091
32
.025 8416
.039 5026
.042 7789
-.0347581
.0174851
+.004 4833
33
.0292651
.041 0601
.042 0550
.031 9287
.0132878
+.0089260
34
.032 3744
.042 1765
.040 8765
.028 7508
.008 9438
+.013 2696
35
035 1363
.042 8394
039 2555
-.025 2585
.004 5009
+.0174653
36
.0375212
.0430411
.037 2089
.021 4895
.0000084
+.0214663
37
039 5032
.042 7792
.0347581
.0174846
+.004 4842
+.025 2276
38
.0410611
.042 0556
.0319288
.013 2874
+.0089269
+.028 7069
39
.042 1777
-.0408775
-.0287513
.0080438
+.013 2702
+.0318651
40
.042 8407
.0392567
.025 2504
.004 5015
+.0174654
+.0346668
41
.043 0424
.037 2103
.0214908
.000 0095
+.021 4658
+.0370806
42
.042 7801
034 7594
.0174861
+.004 4827
+.O25 2265
+.0390797
43
.042 0559
.031 9300
.013 2889
+.0089251
+.028 7052
+.0406420
44
.0408771
.0287519
.0089451
+.013 2684
+.031 8631
+.041 7501
45
039 2555
.025 2594
.004 5022
+.0174641
+.034 6648
+.042 3922
46
.037 2085
.021 4900
.0000095
+.021 4650
+.0370791
+.042 5618
47
.034 7572
.0174846
+.004 4835
+.025 2265
+.0390788
+.042 2575
48
.031 9278
.0132869
+.0089269
+.028 7061
+.0406418
+.041 48.34
49
.028 7504
.0084931
+.013 2707
+.031 8649
+.041 7508
+.0402487
50
.025 2588
.004 5007
+.017 4664
+.034 6672
+.042 3939
+.038 5680
Si
.021 4907
.0000088
+.02 1 4671
+.037 0818
+.042 5641
+.036 4606
52
-.0174867
+.004 4829
+.025 2277
+.0390815
+.042 2603
+033 9503
53
.013 2004
+.008 9247
+.028 7061
+.040 6439
+.0414862
+.031 0659
54
.008 9473
+.013 2672
+.031 8635
+.041 75i8
+.0402511
+.0278396
55
.004 5052
+.0174621
+.0346645
+.042 3935
+.038 5697
+.024 3076
56
.0000130
+ .021 4624
+.0370779
+.042 5624
+.036 461 1
+.020 5092
57
+.004 4797
+.025 2232
+.0390768
+.042 2573
+-33 9495
+.0164863
58
+.008 9228
+.028 7024
+.0406300
+.041 4822
+.0310636
+.012 2834
59
+.013 2669
+.031 8610
+.041 7474
+.0402465
+.0278361
+.007 9465
6o
ACTION OF THE PLANETS ON THE MOON.
TABLE VI. Concluded.
G-COORDINATE Z OF VENUS.
Sys-
tem
i
6
7
8
9
10
II
.0182000
.035 1342
.042 8369 .039 2532
.025 2572
.004 5010
i
.022 1401
.037 5194
.043 0386 .037 2062
.021 4874
.000 0076
2
.025 8423
.039 5018
-.042 7767 -034 7550
.0174818
+.0044858
3
.029 2668
.041 0601
.042 0532 .031 9256
.013 2840
+.0089294
4
.032 3768
.042 1773
.040 8754 .028 748 1
.008 9400
+.013 2734
5
035 1389 .042 8410
039 2552 .025 2564
.004 4976
+.0174692
6
.037 5236 .043 0430
.037 2094 .021 4882
.000 0058
+.021 4697
7
.039 5053
.042 7812
0347591 .0174841
+.004 4860
+.025 2302
8
.041 0626
.0420571
.031 9302 .013 2877
+.0089278
+.028 7085
9
.042 1786
.040 8785
.028 7525 .008 9446 +.013 2704
+.031 8658
10
.0428409
.039 2571
.0252602 .0045023 +.0174650
+.0346667
ii
.0430419
.0372100
.0214909 .000 oioo +.0214652
+ .0370801
12
.042 7792
.034 7584
.017 4855 +.004 4828
+.025 2260
+ .0390788
13
.042 0547
.031 9285
.013 2877 +.008 9260
+.028 7051
+.O4O 64IO
14
-.0408759
.028 7504
.0089434
+.0132608
+.031 8637
+.041 7493
15
.0392548
.025 2577
.004 5004
+.0174658
+.034 6659
+.042 3920
16
.0372083
.021 4886
.0000078 +.0214667
+.037 0804
+.042 5621
17
-034 7576
.0174838
+ .004 4849 + .025 2279
+.039 0801
+.042 2582
18
.031 9285
.013 2868
+.008 9276 +.028 7070
+.040 6429
+.041 4843
19
.0287511
-.0089433
+.013 2707 +.031 8652
+.041 7515
+.0402494
20
.025 2591
.004 5010
+.0174659 +.0346669
+.0423938 ! +.0385683
21
.021 4905
.0000089
+.021 4663 +0.37 0806
+.042 5634
+.0364601
22
.0174858
+.004 4834
+.025 2269 +.039 0798
+.042 2588
+.033 9492
23
.013 2884
+.0089260
+.028 7058 +.040 6420
+.041 4841
+.031 0639
24
.0089445
+.013 2693
+.031 8639
+.041 7503
+.040 2485
+.0278371
25
.004 5016
+.017 4648
+.034 6657 +.042 3926
+.038 5671
+.024 3046
26
.0000090
+.021 4657
+.037 0798 +.042 5622
+.036 4589
+.020 5061
27
+.004 4836
+.025 226>
+.0390792 +.0422578
+.033 9480
+.0164835
28
+.0089263
+.028 7059
+.0406418 +.0414834
+.031 0630
+.OI2 28l2
29
+.013 2697
+.031 8640
+.0417503 +.0402483
+.0278365
+.0079453
30
+.0174651
+ .0346658 +.0423926 , +.0385671
+.024 3044
+.003 5232
31
+.02 1 4656
+ .0370798
+.042 5622 +.036 4589
+.020 5061
.000 9372
32
+.025 2265
+.0390700
+.042 2577 +.033 948i
+.0164835
.005 3874
33
+.028 7055
+.O4O 6414
+.041 4832 +.031 0629
+.OI2 2813
-.009 7798
34
+.031 8637
+.041 7499
+.040 2479 +.027 8362
+.0079452
.014 0663
35
+.034 6657
+.042 3924
+.038 5666
+.0243039
+.003 5227
.0182016
36
+.037 0803
+.042 5624
+.0364587
+.020 5053
.000 9380
.022 1410
37
+.039 0800
+.0422582
+.033 948i
+.OT6 4829
-.005 3885
.025 8425
38
+.040 6429
+.041 4843
+.031 0634
+.012 28lO
.0097809
.029 2663
39
+.041 75i6
+.040 2496
+.027 8373
+.007 9454
.014 0673
.032 3757
40
+.042 3941
+.038 5686
+.024 3059
+.003 5236
.0182018
-.035 1375
41
+.042 5637
+.0364607
+.020 5076
.000 9364
.022 1405
.037 5218
42
+.042 2590
+.033 9500
+.0164854
.005 3861
.0258411
.039 5031
43
+.041 4843
+.031 0647
+.012 2833
.009 7782
.029 2643
.041 0603
44
+.040 2488
+.0278377
+.007 9472
.014 0645
.032 3732
.042 1761
45
+.0385672
+.0243051
+.003 5245
.018 1997
.035 1349
.042 8386
46
+.0364590
+.020 5062
.000 9367
.022 1394
0375197
.043 0401
47
+.033 948o
+.0164835
-.005 3875
.025 8414
.039 5020
.042 7782
48
+.031 0631
+.0122812
.009 7805
.029 2657
.041 0604
.042 0548
49
+.0278367
+.007 9452
.0140671
.032 3758
.042 1777
.040 8770
50
+.0243049
+.0035231
.018 2023
-.035 1381
.042 8413
.0392569
51
+JO20 5O7I
.000 9371
.022 1413
.037 5229
.043 0437
.0372112
52
+.0l6 4852
.005 3869
.025 8422
.039 5045
0.42 7820
.034 7610
53
+.012 2837
.009 7786
.029 2654
.041 0619
.042 0581
.031 9323
54
+.007 9482
.0140642
.032 3741
.042 1778
.0408794
.028 7548
55
+.003 5264
.018 1987
.035 1353
.042 8401
.039 2580
.025 2624
56
.000 9340
.022 1376
.0375193
.0430411
.037 2108
.021 4931
57
.005 3844
-.025 8388
.039 5006
.042 7784
.034 7591
-.0174875
58
.009 7771
.0292624
.041 0578
.042 0538
.031 9290
.OI3 2804
59
.014 0642
.032 3720 .042 1740
.040 8748
.028 7503
.0089447
COEFFICIENTS FOR DIRECT ACTION.
6l
TABLE VII.
VALUES OF A, B, C, AND D FOR THE ACTION OF VENUS ON THE MOON.
System o.
Sjstem i.
1
A
B
C
D
i
A
B
C
D
+35.541 47
17.87647
-17.66499 + 1.07427
o
+32-593 73
16.607 73
-15.98599
+ 3-18745
i
+27.684 74
12.48321
15.20155 11.36932
i
+27.372 10
-13.08833
14.28378
9-17780
2
+ 14.10943
- 3-81823
10.291 18 12.81204
2
+ 14.89321
- 4-857 J7
10.03602
11.99256
3
+ 6.07222
+ 0.26235
6.33456 9.16752
3
+ 6.704 15
0.37485
6.32930
- 9.06478
4
+ 2.63936
+ 1-25133
- 3-89071 - 5.85772
4
+ 3-00334
+ 0.935 15
3.93850
- 5-93021
5 + 1.27090
+ 1.20398
2.47487 3-74692
5
+ 147429
+ 1.04697
2.52125
3.830 19
6
+ 0.70797
+ 0.93776
1.64574 2.48604
6
+ 0.82746
+ 0.85385
1.68130
2.54954
7
+ 0.45996
+ 0.68285
I.I428I ! 1.72274
7
+ 0.53500
+ 0.63364
1.16865
1.76704
8 + 0.341 13
+ 0.48405
0.82517 I.2433I
8
+ 0.391 31
+ 045253
-0.84383
1.27360
p
+ 0.27866
+ 0.33795
0.61661
0.92936
9
+ 0.31404
+ 0.316 17
0.63022
0.95006
10
+ 0.24248
+ 0.23228
- 0.47476 \ 0.71540
10
+ 0.26852
+ 0.21633
0.484 84
0.72959
ii + 0.21941
+ 0.15584
0.37525 0.56421
ii
+ 0.23923
+ 0.14361
0.382 83
0.57395
12
+ 0.20337
+ o.ioo 16
0.30352
0.45390
12
+ 0.21885
+ 0.09046
0.30932
046055
13
+ 0.191 38
+ 0.05922
0.250 58 0.371 12
13
+ 0.20373
+ 0.051 35
0.255 07
0.375 58
14
+ 0.181 go
+ 0.02881
0.21071 0.30744
14
+ 0.191 91
+ 0.02231
0.21422
0.31032
15
+ 0.174 12
+ 0.00601
0.180 13 0.25736
15
+ 0.18233
+ 0.00058
0.18250
0.259 10
16
+ 0.16756
o.on 23
0.15632 0.217 19
16
+ 0.17435
0.01582
0.158 53
0.21809
17
+ 0.161 93
0.02438
0.13755 0.18439
17
+ 0.16757
0.02827
0.13930
0.18466
18
+ 0.15705 0.03447
0.122 s8 0.157 18
18
+ 0.161 75
0.03778
0.123 98
0.15697
19
+ 0.15280
0.04226 - 0.11054 0.13423
19
+ 0.15672
0.04508
O.HI 65
0.13368
20
+ 0.14909
0.04831 ; 0.10078 0.11461
20
+ 0.152 35
0.05068
o.ioi 66
0.11379
21
+ 0.14586
0.05301 ! 0.09286 0.09760
21
+ 0.14854
0.05501
0.093 54
0.09658
22
+ 0.14307
0.05666 i 0.08640 0.08265
22
+ 0.14524
0.05833
0.08692
0.081 48
23
+ 0.14067
0.05951 : 0.081 17 0.06933
23
+ 0.14239
0.06086
0.081 53
0.06804
24
+ 0.13864
0.061 71 0.076 93 0.057 30
24
+ 0.13995
0.06278
0.077 17
0.05594
25
+ 0.13693
0.06338 0.07356 0.04629
25
+ 0.13789
0.064 20
0.07368
0.04487
26
+ 0.13558
0.06464 0.07094 0.03606
26
+ 0.13616
O.O6522
0.07096
0.03461
27
+ 0.13453
- 0.065 55
0.06898 0.02643
27
+ 0.13478
0.06589
0.06850
0.02497
28
+ 0.13377
0.066 15
0.06761 0.01722
28
+ 0.133 71
0.06626
0.06744
0.01578
29
+ 0.13330
0.06649
0.06682 0.00827
29
+ 0.13294
0.06638
0.06655
0.00688
30
+ 0.133 12
0.06657
0.06655
+ 0.00051
30
+ 0.13245
0.06626
0.06620
+ 0.001 87
31
+ 0.13322
0.06641
0.06681 + 0.00930
31
+ 0.13225
0.06589
0.06636
+ 0.01060
32
+ 0.13361
0.06600
0.06760 + 0.01823
32
+ 0.13234
0.055 28
0.06706
+ 0.01944
33
+ 0.13428
0.065 32
0.06896 + 0.02744
33
+ 0.132 70
0.06438
0.06831
+ 0.02854
34
+ 0.13525
0.06433
0.07091 + 0.03706
34
+ 0.133 34
O.O63 19
0.070 16
+ 0.03803
35
+ 0.13651
0.06299
0.07353
+ 0.04728
35
+ 0.13427
O.O6 1 63
0.07264
+ 0.04808
36
+ 0.138 10
0.061 21
0.07688 : + 0.05827
36
+ 0.13549 ! 0.05963
0.07586
+ 0.05887
37
+ 0.14001
0.05892
o.cSi 09 + 0.07027
37
+ 0.13702
0.057 10
0.07991
+ 0.07063
38
+ 0.14227
0.05597
0.086 30 + 0.083 55
38
+ 0.13886
0.05392
- 0.08495
+ 0.08361
39 + 0.14490
0.052 19
0.09272
+ 0.09844
39
+ 0.14103
0.04989
0.091 15
+ 0.098 14
40 + 0.14795
0.047 35
0.10059
+ 0.11535
40
+ 0.14356
0.04479
0.09876
+ 0.11461
41 + 0.151 43
0.041 17 0.11028
+ 0.13484
41
+ 0.14646
0.03834
0.108 13
+ 0.133 55
42 + 0.15543
O.O33 21 O.I22 23
+ 0.15760
42
+ 0.14978
0.030 10
0.11968
+ 0.15562
43 + 0.16000
O.02294 O.I37O6
+ 0.184 57
43
+ 0.153 57
0.01953
0.13403
+ 0.181 71
44 + 0.165 26
0.00960
0.15566
+ 0.21700 44
+ 0.15789
0.00590
0.15199
+ 0.21301
4S + O.I7I35
+ 0.00785
0.17920
+ 0.25665
45
+ 0.16286
+ O.OII 84
0.17471
+ 0.251 18
46 + 0.17854
+ 0.03087 0.20941
+ 0.30597
46
+ 0.16867
+ 0.035 14
0.20381
+ 0.29854
47 + 0.18724
+ 0.061 51 0.24875
+ 0.368 55
47
+ 0.17561
+ 0.06601
0.241 63
+ 0.35846
48
+ 0.198 20
+ 0.10269 0.30091
+ 0.44999
48
+ 0.18428
+ 0.10735
0.291 64
+ 0.43590
49
+ 0.21280
+ 0.15866 0.37145
+ 0.55751
49
+ 0.195 77
+ 0.16331
0.35907
+ 0.53846
50
+ 0.23370
+ 0.235 44 0469 14
+ 0.70484
50
+ 0.21225
+ 0.23985
0.452 1 1
+ 0.67805
51
+ 0.26643
+ 0.341 67 0.608 10
+ 0.91263
Si
+ 0.23832
+ 0.34549
- 0.583 82
+ 0.87409
52
+ 0.32291
+ 0.48903 0.81195
+ 1.21644 i 52
+ 0.28404
+ 0.491 88
- 0.77591
+ 1-15930
53
+ 0.43039
+ 0.691 21
i. 121 59
+ 1.67867
53
+ 0.37252
+ 0.69304
1.06556
+ 1-59084
54
+ 0.65464
+ 0.95589
1.61052
+ 241220
54
+ 0.55982
+ 0.95881
1.51864
+ 2.271 42
55
+ 1.16322
+ L25I I?
2.414.40
+ 3.621 88
55
+ 0.98862
+ 1.26617
- 2.254 78
+ 3.38575
56
+ 2.39922
+ 1.385 II
- 3.784 32
+ 5-65379
56
+ 2.03482
h 145600
349084
+ 5-25163
57
+ 5.5H93
+ 0.63808
6.15001
+ 8.90000 57
+ 4.67267
+ 0.91643
- 5-58909
+ 8.243 78
58
+ 12.93286
2.92359
10.00926
+ 12.77721 58
+ 11.00705
2.04409
8.06205
+11.98839
59
+26.185 67
11.259 17
14.92649
+ 12.46123 59
+22.761 17
- 9-475 57
13.28560
+12.554 18
62
ACTION OF THE PLANETS ON THE MOON.
TABLE VII. Continued.
VALUES OF A, B, C, AND D FOR THE ACTION OF VENUS ON THE MOON.
System 2.
System 3.
f
A
B
C
D
I
A
B
C
D
o
+29.624 12
15.17831
1444582
+ 3.98350
O
+28.095 38
14.29079
13.80459
+ 3-72368
I
+26.178 17
12.85822
13-31998
- 7-55334
I
+25.IO8 12
12.24065
12.86746
6.97990
2
+15.05897
5-37464
- 9.68431
11.06821
2
+ 14.804 22
- 5-33468
9.46955
1047031
3
+ 7.08250
0.82744
6.25507
8.775 53
3
+ 7.I427I
0.97149
6.I7I 21
8.464 1 1
4
+ 3.272 10
+ 0.673 17
3.94529
- 5-88003
4
+ 3.36804
+ 0.54538
- 3-9I34I
5.74732
5
+ 1.63783
+ 0.90532
2.543 15
3.84260
5
+ 1.70907
+ 0.82100 2.53007
- 3-78667
6
+ 0.92681
+ 0.77463
I-70I45
2.571 56
6
+ 0.97332
+ 0.721 94
1.69525
2.546 14
7
+ 0.59776
+ 0.58640
1.18416
1.78590
7
+ 0.627 53
+ 0.553 13
1.180 66
1.77284
8
+ 043283
+ 0.42237
0.855 20
1.28747
8
+ 0452 10
+ 0.40075
- 0.85285
1.27974
9
+ 0.34273
+ 0.295 72
0.63846
0.95970
9
+ 0.35545
+ O.28l 22
0.63665
0.95445
10
+ 0.28909
+ 0.201 73
040082
0.73606
10
+ 0.29760
+ O.I9I 71
048933
0.732 10
ii
+ 0.25443
+ 0.13276
0.38720
- 0.578 14
ii
+ 0.26020
+ 0.12569
- 0.38589
0.57491
12
+ 0.23036
+ 0.082 14
0.31252
0.463 12
12
+ 0.23428
+ 0.07706 0.3II33
0.4^36
13
+ 0.21260
+ 0.04482
0.25741
0.37700
13
+ O.2I5 22
+ 0.041 IO : 0.25634
0.37458
14
+ 0.19884
+ 0.01709
0.21594
0.31093
14
+ 0.20055
+ 0.01437
0.21494
0.30876
J|
+ 0.18778
0.00364
0.184 16
O.259 12
15
+ 0.18883
O.OO5 63
0.18321
0.257 16
16
+ 0.17866
0.01925
0.15942
0.21769
16
+ O.I792I
O.02O68 0.15852
0.21591
17
+ 0.17099
0.031 07
0.13991
0.18397
17
+ O.I7I 14
O.O32O8 O.I3906 I 0.18235
18
+ 0.16444
0.04006
0.12437 0.15606
18
+ 0.16429
0.04074 0.12355
0.15459
iQ
+ 0.15880
0.04693
0.11187
O.I3262
19
+ O.I584I
0.04733
O.I 1 1 08
0.131 28
20
+ 0.15392
0.052 17
o.ioi 74
O.II264
20
+ 0.15333
0.05236
0.10098
0.11143
21
+ 0.14968
0.056 17 i 0.093 50 1 0.095 37
21
+ 0.14894
0.056 1 8
0.092 76
0.09427
22
+ 0.14600
0.05921
0.08679 0.08024
22
+ 0.145 14
0.05907
0.08606
0.07925
23
+ 0.14282
0.06 1 50
0.081 32
0.06680
23
+ O.I4I 84
0.06 1 24
0.08060
0.06593
24
+ 0.14009
0.06320
0.07689
0.05470
24
+ O.I390I
0.062 82
0.076 17
0.05304
25
+ 0.13775
0.06441
0.07334
O.O4366
25
+ 0.13659
0.06395
0.07263
O.O.J300
26
+ 0.13579
0.06524
0.07055
- 0.03345
26
+ 0.13454
0.06470
0.06985
0.03289
27
+ 0.134 17
- 0.065 74
0.06844
0.02386
27
+ 0.13286
0.065 13
0.06773
0.02341
28
+ 0.13288
0.06595
0.06692
0.01473
28
+ 0.131 50
0.065 29
0.06622
0.01439
29
+ 0.131 89
0.06592
0.06598
0.00591
29
+ 0.13046
O.O65 21
0.06526
000568
30
+ 0.131 21
0.06564
0.06556
+ 0.00275
30
+ 0.12972
0.06489
0.06483
0.00287
31
+ 0.13081
0.065 13
0.06567
+ 0.01 1 37
31
+ 0.12927
0.06434
0.064 92
+ o.oi i 38
32
+ 0.13068
0.06438
0.06631
+ 0.02009
32
+ 0.129 10
0.06357
0.065 54
+ 0.01998
33
+ 0.13084
0.06336
0.06749
+ 0.02905
33
+ 0.12921
0.06253
0.06669
+ 0.02880
34
+ 0.131 28
0.06203
0.06925
+ 0.03838
34
+ 0.12960
0.061 19
0.06842
+ 0.03800
35
+ 0.13199
0.06035
0.071 65
+ 0.04825
35
+ 0.13028
0.05950
0.070 77
+ 0.04771
36
+ 0.13299
0.05823
0.07476
+ 0.05883
36
+ 0.131 23
0.05740
0.07384
+ 0.058 14
37
+ 0.13428
0.05558
0.07869
+ 0.07035
37
+ 0.13248
0.05478
0.077 7i
+ o.o '19 47
38
+ 0.13586
0.05228
- 0.08359
+ 0.08304
38
+ 0.13404
0.051 51
0.08252
+ 0.081 97
39
+ 0.13775
0.048 15
0.08962
+ 0.09723
39
+ 0.13590
0.04744
0.08846
+ 0.09594
40
+ 0.13998
- 0.04295
0.097 02
+ 0.11330
40
+ 0.138 10
0.04235
0.095 76
+ 0.11177
41
+ 0.14255
0.03642
0.106 13
+ 0.131 75
4i
+ O.T4066
0.03593
0.10473
+ 0.12993
42
+ 0.14550
0.028 12
- 0.11737
+ 0.15323
42
+ O.I4360
0.02781
0.11580
+ 0.15108
43
+ 0.14885
0.01755
0.13131
+ 0.17857
43
+ 0.14658
0.01746
0.12953
+ 0.17604
44
+ 0.15269
0.00395
0.14875
+ 0.20895
44
+ 0.15085
0.004 19
0.14668
+ 0.20595
45
+ 0.15709
+ 0.013 68
0.17077
+ 0.24592
45
+ 0.15533
+ 0.01300
- 0.16833
+ 0.24236
46
+ 0.16222
+ 0.03671
0.19894
+ 0.291 73
46
+ 0.16059
+ 0.03542
0.10601
+ 0.28747
47
+ 0.16834
+ 0.067 12
0.23547
+ 0.34957
47
+ O.I66O2
+ 0.06406
0.231 89
+ 0.34443
48
+ 0.17598
+ 0.10768
0.28365
+ 0.424 17
48
+ 0.17489
+ 0.10427
0.279 15
+ 0.41786
49
+ 0.186 10
+ 0.16234
0.34843
+ 0.52271
49
+ 0.18554
+ 0.15709
0.34264
+ 0.51483
50
+ 0.20072
+ 0.23676
- 0.43747
+ 0.65644
50
+ 0.201 00
+ 0.22874
0.42976
+ 0.64633
Si
+ 0.22403
+ 0.33893
0.56296
+ 0.84360
51
+ 0.22568
+ 0.32664
0.55232
+ 0.830 17
52
+ 0.265 29
+ 0.47966
0.74495
+ 1.11472
52
+ O.269 1 1
+ 0.46054
0.72966
+ 1.00602
53
+ 0.34571
+ 0.671 72
1.01742
+ 1.52277
53
+ 0-35301
+ 0.641 40
0.99442
+ 1.49508
54
+ 0.51642
+ 0.92341
143982
+ 2.161 97
54
+ 0.52902
+ 0.87436
1-40338
+ 2.11762
55
+ 0.90636
+ 1.21194
2.11829
+ 3-20054
55
+ 0.02556
+ 1-13169
2.05725
+ 3-12267
56
+ 1.85077
+ 1.39039
3.241 16
+ 4-91924
56
+ 1.87094
+ 1.262 21
3-133 16
+ 4-76908
57
+ 4.20451
+ 0.91034
5-11484
+ 7-64885
57
+ 4-183 37
+ 0.73285
4-91623
+ 7-34183
58
+ 979293
- 1-71451
- 8.07843
+ 11.07242
58
+ 9.55247
1.83400
- 7-71848
+ 1047949
59
+20.207 75
8.331 97
-11.87575
+ 11.86403
59
+ 19-331 99
- 8.023 85
11.308 13
+ 11.06309
COEFFICIENTS FOR DIRECT ACTION.
TABLE VII. Continued.
VALUES OB- A, B, C, AND D FOR THE ACTION OF VENUS ON THE MOON.
System 4.
System 5.
i
A
B
C
D
f
A
B
C
D
+27-753 75
13.94094
13.81281
+ 2.71293
+27.63990
13.80676
-13.833 14
+ 1.162 19
i
+24.10807
11.43147
12.67662
7-27193
i
+22.709 15
10.43790
12.27123
8.00809
2
+ M.OgO 02
4-81177
9.22826
10.18594
2
+12.86967
4.11827
8.75142
10.01752
3
4- 6.82696
- 0.83885
- 5.988 13
8.14397
3
+ 6.17336
0.53043
5.64295
- 7.784 12
4
+ 3.24571
+ 0.55049
3.79620
5.53629
4
+ 2.93411
+ 0.64851
- 3.58261
5.261 65
5
+ I.6593I
+ 0.79841
- 2.45770
3-661 70
5
+ 1.50482
+ 0.82485
2.32966
- 3.48648
6
+ 0.94972
+ 0.700 16
1.64989
2471 69
6
+ 0.86547
+ 0.70639
1.57185
2.36290
7
+ 0.6l3gO
+ 0.53734
1.15126
1.72669
7
+ 0.56289
+ 0.53922
1. 102 12
I.658o8
8
+ 0.442 70
+ 0.39037
0.83307
1.24977
8
+ 0.40883
+ 0.392 14
0.80098
1.205 16
9
+ 0.348 10
+ 0.27476
0.62286
- 0.934 13
9
+ 0.32390
+ 0.27725
0.601 14
0.90424
10
+ 0.291 44
+ 0.18792
0.47937
0.71780
10
+ 0.273 1 6
+ 0.191 03
0.464 1 8
0.69723
ii
+ 0.25483
+ 0.12366
- 0.37848
0.56454
ii
+ 0.24042
+ O.I27 12
0.36755
0.55008
12
+ 0.22948
+ 0.076 19
0.30568
045265
12
+ 0.21778
+ 0.07980
0.29759
0.44233
13
-j- 0.21089
+ 0.041 01
0.25191
- 0.368 74
13
+ 0.201 16
4- 0.04461
0.245 78
0.361 30
14
+ 0.19660
+ 0.014 79
0.211 39
0.30429
14
+ 0.18836
4- 0.01828
0.20664
0.29891
IS
+ O.lSS 20
0.00489
0.18031
0.25370
15
+ 0.178 13
o.oo i 56
0.17656
0.24982
16
+ 0.175 85
0.01974
O.I56 12
0.21322
16
+ 0.16973
0.01661
0.153 10
0.21047
17
+ O.I68O3
0.031 01
0.13703
0.18025
17
+ 0.16267
0.028 1 1
0.13457
0.17836
18
+ 0.161 41
0.03959
O.I2I 80
0.15297
18
+ 0.15668
0.03692
0.11977
0.151 73
19
+ 0.15571
0.046 15
0.10956
0.13005
19
+ 0.151 53
0.04369
0.10785
0.12933
20
+ 0.15082
0.051 17
0.09964
0.11051
20
+ 0.147 10
0.04892
0.098 19
O.IIO20
21
+ 0.14657
0.05501
O.O9I 56 O.O93 62
21
+ 0.14328
0.05295
0.09031
0.09364
22
+ 0.14290
0.05792
0.08498 0.07883
22
+ 0.13997
0.05607
0.08390
O.O79 12
23
+ 0.13975
O.O6O 12
0.079 62 0.065 70
23
+ 0.137 13
0.05847
0.07868
0.06621
24
+ 0.13703
0.06 1 75
0.07528 0.05388
24
+ 0.13473
0.06028
0.07444
0.05457
25
+ 0.13473
0.06293
0.071 80 0.043 10
25
+ 0.13270
0.06 1 64
0.071 06
0.04394
26
+ 0.13280
0.06374
0.069 07 0.033 13
26
+ 0.131 03
0.062 62
0.06842
0.03409
27
+ O.I3I 22
0.06423
0.06700 ! 0.02377
27
+ 0.12970
0.06328
0.06641
0.02484
28
+ 0.12997
0.06445
0.065 51 0.01486
28
+ 0.12869
0.06369
0.06500
0.01602
29
+ O.I29O4
0.06444
0.064 59 0.006 25
29
+ 0.12798
0.06385
0.064 13
0.00748
30
+ 0.12840
0.06421
0.064 19 ! + O.O02 19
30
+ 0.12758
0.06379
o.o53 79
+ 0.00092
3i
+ O.I28o6
0.063 75
0.06430 + 0.01061
31
+ 0.12747
0.06351
0.06396
+ 0.00930
32
+ 0.12799
0.06306
o. 6494 . 4- 0.01912
32
+ 0.12764
0.06300
0.06465
+ 0.01779
33
+ 0.12822
O.O62 12
0.066 ii 4- 0.027 87
33
+ 0.128 12
0.06225
0.06587
+ 0.02652
34
+ 0.12875
0.06089
0.06784 4- 0.03698
34
4- 0.12890
O.O6 1 22
0.06768
+ 0.03565
35
+ 0.12955
0.05933
0.07022 + 0.04663
35
+ 0.12998
0.05987
0.070 1 1
+ 0.04533
36
+ 0.13066
0.05736
0.07329 4- 0.05699
36
+ 0.131 39
O.058 12
0.07326
+ 0.055 75
37
+ 0.13207
0.05490
0.077 18 + 0.068 27
37
+ 0.133 13
0.05589
0.07723
+ 0.067 ii
38
+ 0.13381
0.05 1 82
0.082 oo 4- 0.080 73
38
+ 0.135 23
0.05305
0.082 17
+ 0.07968
39
+ 0.13590
0.04794
0.08795 4- 0.09466
39
+ 0.13772
0.04945
0.08825
+ 0.09378
40
+ 0.13834
0.04308
0.09527 + o.i 10 47
40
+ 0.14062
0.04489
0.09573
+ 0.10981
41
+ O.I4I 20
0.03693
0.10426 + 0.12864
4i
+ 0.14400
0.03907
0.10493
+ 0.12828
42
+ 0.14449
0.029 13
0.11536 + 0.14982
42
+ 0.14793
0.03163
0.11628
4- 0.14986
43
+ 0.14829
0.019 17
0.129 14 4- 0.17486
43
+ 0.15247
0.02208
0.13040
+ 0.17544
44
+ O.I527I
0.00634
0.14636 4- 0.20492
44
+ 0.15779
0.00972
0.14808
+ 0.20620
45
+ O.IS785
+ 0.01028
0.168 13
+ 0.241 57
45
+ 0.16407
+ 0.00638
0.17044
+ 0.24382
46
+ 0.16399
4- 0.032 oo
0.19597 + 0.28704
46
+ 0.171 62
+ 0.02748
0.199 ii
+ 0.29063
47
+ O.I?! 48
4- 0.06063
0.23209 + 0.34455
47
+ 0.18098
+ 0.05539
0.23637
+ 0.34998
48
+ O.lSl 02
+ 0.09873
0.27974 + 041883
48
+ 0.19304
+ 0.092 6 1
0.28564
+ 042686
49
+ O.I939I
+ 0.14990
0.34380 + 0.51708
49
+ 0.20945
+ 0.14264
0.35207
+ 0.52885
So
+ O.2I265
+ 0.219 19
- 0.431 84 + 0.650 55
50
+ 0.23330
+ 0.21035
0.44366
+ 0.66781
SI
4- 0.24236
4- 0.31354
0.55590 + 0.83740
51
+ 0.27084
+ 0.30233
0.573 16
+ 0.86291
52
+ 0.29393
+ 0.441 78
0.73571 + 1.10789
52
+ 0.33507
+ 042652
0.761 61
+ 1.146 ii
53
+ 0.391 71
+ 0.61299
1.00472 + 1.51406
53
+ 0.45469
+ 0.59001
1.04470
+ 1-572 14
54
+ 0.59284
+ 0.82829
I.42I 12 : + 2.I47O2
54
+ 0.69605
+ 0.78884
1.48488
+ 2.23629
55
+ 1.03724
+ 1.051 ii
2.08836 + 3.16503
55
+ I.2I939
+ 0.97368
2.19308
+ 3.301 12
56
+ 2.07620
4- 1.11203
3.18823 + 4.81735
56
+ 241926
+ 0.94390
3-363 18
+ 5.01076
57
+ 4.561 ii
+ 0.45028
- 5.0II40 + 7-34432
57
+ 5-21973
+ 0.077 57
5.29730
+ 7-54759
58
+10.15489
2.290 72
7.864 16 + 10.250 29
58
+ 11.281 01
3.00564
- 8.275 35
+ 10.19086
59
+19.85615
- 8.398 13
11.45803 4-10.29566
59
+21.04269
9-19381
11.84889
+ 9-39027
64 ACTION OF THE PLANETS ON THE MOON.
TABLE VII. Continued.
VALUES OF A, B, C, AND D FOR THE ACTION OF VENUS ON THE MOON.
System 6.
System 7.
i
A
B
C
D
i
A
B
C
D
+27.16805
13.66015
-13.50787
0.614 53
o
+26.70986
13-59511
13.11476
2.251 50
i
+20.933 58
9.36452
11.56906
8.839 67
i
+19.44882
846686
10.981 95
9.68240
2
+11.41364
3.29223
8.121 39
9.88868
2
+10.20337
2.547 15
7.65624
9.92208
3
+ 5.381 20
0.147 12
5-23409
744099
3
+ 4.702 14
+ 0.24741
4-94955
7.24930
4
+ 2.541 06
+ 0.80284
- 3.34388
4.99086
4
+ 2.18926
+ 0.993 76
3.18300
4-81724
5
+ I-30I93
+ 0.88985
2.191 79
3-31041
S
+ 1.11288
+ 0.98787
2.10076
3-191 27
6
+ 0.751 59
+ 0.738 57
M90I5
2.25264 ' 6
+ 0.64227
+ 0.79477
1.43702
2.17589
7
+ 0.49304
+ 0.55884
1.05187
1.58825 7
+ 042478
+ 0.59479
1.01958
1-53905
8
+ 0.36249
+ 040635
0.76885
- I.I5995 8
+ 0.31691
+ 0.43149
0.74840
1.12806
9
+ 0.291 15
+ 0.28868
0.57982
0.87426 9
+ 0.25897
+ 0.30739
- 0.56637
0.85330
10
+ 0.24883
+ 0.20073
0.449 54
0.67695 10
+ 0.225 10
+ 0.21527
0.44039
0.66303
ii
+ 0.221 64
+ 0.13556
0.357 19
0.53617 II
+ 0.20354
+ 0.14721
0.35075 ; 0.52689
12
+ 0.20285
+ 0.08721
0.29006
043271 12
+ 0.18867
+ 0.09673
0.28540
042657
13
+ O.I89O2
+ 0.051 15
0.240 17
0.35465 13
+ 0.17768
+ 0.05904
0.23672
0.35068
14
+ 0.17833
+ 0.02406
0.202 37
0.24935 14
+ 0.169 10
+ 0.03066
- 0.19977
0.291 91
IS
+ O.I0972
+ 0.003 54
0.17325
0.24679 15
+ 0.162 13
+ O.O09 12
0.171 25
0.24543
16
+ 0.16259
O.OI2 IO
0.15049
0.20855 16
+ 0.15630
0.00738
0.14893 i 0.20799
17
+ 0.15658
O.O24 12
0.13248
0.17728 17
+ 0.151 34
O.02O IO
0.131 25
0.17729
18
+ O.I5I46
0.033 39
0.11807
0.151 29
18
+ 0.14708
O.O2998
0.11709
0.151 73
19
+ 0.14705
0.04059
0.10646
0.12937
19
+ 0.14338
O.0377I O.IO568
0.130 i^
20
+ 0.14324
0.04621
0.09703
0.11061
20
+ 0.140 19
O.04378 0.09642
O.I 1 1 60
21
+ 0.13995
0.05060
0.08936
0.09435
21
+ 0.13745
0.04856 0.08887
0.09551
22
+ 0.137 13
0.05404
0.083 10
0.08005
22
+ 0.13509
O.05236 O.08273
0.081 34
23
+ 0.13472
0.05672
0.07800
0.06731
23
+ 0.133 II
- 0.055 37 - 0.077 73
0.06868
24
+ 0.13270
0.05881
0.07388
0.05581
24
+ O.I3I47
0.05775
0.073 71
0.05724
25
+ 0.131 04
0.06043
0.07060
0.04528
25
+ 0.130 15
0.05964
0.07051
0.04674
26
+ 0.12970
0.061 66
0.06804
0.03551
26
+ 0.129 14
0.061 13
0.06803 0.03698
27
+ 0.12869
0.06257
O.066 12
0.02631
27
+ 0.12844
0.062 27
o.o65 19 0.027 77
28
+ 0.12799
0.063 20
O.O6478
0.01752
28
+ 0.12804
0.063 12
0.06493 0.01895
29
+ 0.127 58
0.06359
O.O64OO
0.00899
29
+ 0.12793
0.063 72
0.06421 0.01037
30
+ 0.12747
0.06375
0.063 73
0.00059
30
+ 0.128 II
0.06408
0.06402 0.00191
31
+ 0.12766
0.06368
0.06397
+ 0.00782
31
+ 0.12853
0.06423
0.06436 + 0.00658
32
+ 0.128 14
0.06340
0.06475
+ 0.01635
32
+ 0.12937
0.064 14
0.06523 + 0.01522
33
+ 0.12893
0.06287
0.06606
+ 0.025 1 6
33
+ 0.13046
0.06382
0.066 66 , + 0.024 IS
34
+ 0.13003
0.06207
0.06796
+ 0.03439
34
+ 0.13188
0.06320
0.06867 + 0.03352
35
+ 0.13146
0.06095
0.07052
+ 0.044 19
35
+ 0.13365
0.06228
0.071 36 + 0.043 SO
36
+ 0.13325
0.05943
0.07380
+ 0.05476
36
+ 0.135 77
0.06097
0.07481 + 0.05429
37
+ 0.13538
0.05746
0.07793
+ 0.06633
37
+ 0.13831
0.059 19
0.079 12 + 0.066 ii
38
+ 0.13794
0.05488
0.08305
+ 0.079 15
38
+ 0.141 28
0.05683
0.08445 + 0.07925
39
+ 0.14092
0.051 56
0.08935
+ 0.093 57
39
+ 0.14474
0.05372
0.091 01
+ 0.09404
40
+ 0.14439
0.04729
0.097 II
+ O.I 10 00
40
+ 0.14874
0.04966
0.09908
+ 0.11093
4i
+ 0.14843
0.041 77
0.10665
+ 0.12898
41
+ 0.15339
0.04438
0.10300
+ 0.13048
42
+ 0.153 10
0.03467
0.11844
+ O.I5I 20
42
+ 0.15877
0.037 5O
0.121 27
+ 0.15342
43
+ 0.15855
0.02546
0.13311
+ O.I776I
43
+ 0.16505
0.02853
0.13653
+ 0.18072
44
+ 0.16495
0.01347
- 0.15149
+ 0.20947
44
+ 0.17246
0.01677 0.15567
+ 0.213 73
45
+ 0.17257
+ O.OO222
0.17480
+ 0.24852
45
+ 0.181 29
0.00133 0.17997
+ 0.25426
46
+ 0.181 82
+ 0.02290
0.20472
+ 0.29723
46
+ 0.19206
+ O.OI9 II O.2II 17
+ 0.30493
47
+ 0.19336 + 0.05033
0.24370
+ 0.359 18
47
+ 0.205 57
+ 0.04631
0.251 88
+ 0.36951
48
+ 0.208 36 + 0.087 oo
- 0.295 37
+ 0.43967
48
+ 0.223 17
+ 0.082 74
0.30590
+ 0.45358
49
+ 0.22884 + 0.13637
0.36520
+ 0.54676
49
+ 0.24721
+ 0.131 82
0.37903
+ 0.56569
50
+ 0.25860 + 0.20317
0461 77
+ 0.693 12
50
+ 0.282 1 1
+ 0.198 18
0.48029
+ 0.71923
5i
+ 0.305 13 + 0.293 60
0.59874
+ 0.89921
Si
+ 0.33642
+ 0.28773
0.624 15
+ 0.93591
52
+ 0.38393 + 0.41479
0.798 70
+ I.I99I3
52
+ 0.42773
+ 0.40676
0.834 50
+ 1.25184
53
+ 0.52866 + 0.57144
1. 100 II
+ 1.651 13
53
+ 0.59408
+ 0.557 85
I.I5I93
+ 1.72856
54
+ 0.81650
+ 0.75357
1-57007
+ 2.35573
54
+ 0.92198
+ 0.725 14
- 1-64713
+ 2471 57
55
+ 1.43137
+ 0.895 83
2.327 19
+ 348082
55
+ 1.61638
+ 0.8274')
- 2.44383
+ 3.65374
56
+ 2.81766
+ 0.75750
- 3-575 16
+ 5.26294
56
+ 3-16615
+ 0.58408
3.75022
+ 5.50398
57
+ 5.97720
0.36200
5.61520
+ 7.807 13
57
+ 6.641 59
0.78455
5.85704
+ 8.051 79
58
+12.52832
- 3-86693
8.661 40
+ 10.10754
58
+ 13.61205
4.70484
8.90720
+10.018 16
59
+22.18774
10.13334
12.054 39
+ 8.285 33
59
+23.13200
11.051 60
12.08040
+ 7-24047
COEFFICIENTS FOR DIRECT ACTION.
TABLE VII. Continued.
VALUES OF A, B, C, AND D FOR THE ACTION OF VENUS ON THE MOON.
System
8.
System
9-
1
A
B
C
D
I
A
B
C
D
+27.214 80
13.93826
-13.27655
3.50011
+29406 78
14.96887
14.43791
4.12709
I
+ 19.03987
8.02033
11.01953
10.66509
i
+20.202 62
8.251 18
11.95144
11.84365
2
+ 9-631 65
1.99026
7.641 36
10.302 17
2
+ 9.90242
1-721 59
8.18082
11.10640
3
+ 4.31063
+ 0.61744
4.92809
7-32955
3
+ 4.28872
0.911 18
- 5.19988
7.71982
4
+ 1.96054
+ 1.20621
3.16675
4-80935
4
+ 1.89456
+ 140655
3.301 10
4.984 16
5
+ 0.98006
+ 1.10985
2.08990
3.16793
5
+ 0.92598
+ 1.23253
- 2.15852
3.24867
6
+ 0.561 76
+ 0.86813
1.42989
2.155 37
6
+ 0.52426
+ 0.94242
146667
2.19556
7
+ 0.37331
+ 0.64149
1.01479
1-52413
7
+ 0.348 15
+ 0.68736
1.03553
1.54604
8
+ 0.28229
+ 046281
0.74509
Li 17 93
8
+ 0.26537
+ 049196
0.75734
1.13104
9
+ 0.23465
+ 0.32936
0.56401
0.84669
9
+ O.223 12
+ 0.34843
0.571 56
0.855 24
10
+ 0.20740
+ 0.23126
0.43866
0.65889
10
+ 0.19944
+ 0.24406
- 0.44351
0.66492
ii
+ 0.19028
+ 0.159 18
0.34946
0.52446
ii
+ 0.18474
+ 0.16797
0.352 70
0.52899
12
+ 0.17851
+ 0.10591
0.28442
042532
12
+ 0.17464
+ 0.11205
0.28667
042889
13
+ 0.16976
+ 0.06621
0.23597
0.35025
13
+ O.I67O8
+ 0.07053
0.23761
0.353 18
14
+ 0.16285
+ 0.03633
0.19920
0.29205
14
+ O.l6lO5
+ 0.03940
0.20043
0.29452
IS
+ 0.157 17
+ 0.01365
0.17081
0.24598
15
+ 0.15599
+ 0.01580
0.171 81
0.248 10
16
+ 0.15234
0.003 74
0.14861
- 0.20880
16
+ O.I5I 67
0.00223
0.14943
0.21065
17
+ 0.148 19
0.017 17
0.13102
0.17830
17
+ 0.14790
0.016 16
0.131 73
0.17993
18
+ 0.14458
0.02763
0.11696
0.15286
18
+ 0.14458
0.02701
0.11760
0.15430
19
+ 0.14143
0.03583
0.105 61
0.131 33
19
+ O.I4I 69
0.03549
0.10622
0.13262
20
+ 0.13871
0.04229
0.00641
0.11285
20
+ 0.139 16
0.042 17
0.09701
0.11400
21
+ 0.13636
0.04743
0.08894
0.09678
21
+ 0.13698
0.04747
0.08952
0.097 79
22
+ 0.134 36
0.051 51
- 0.08285
0.08260
22
+ 0.135 12
0.051 70
0.08343
0.083 50
23
+ 0.13268
0.05478
0.07792
0.06993
23
+ 0.13357
0.05508
0.07850
0.07072
24
+ 0.131 33
0.05738
0.07394
0.05846
24
+ 0.13233
0.05779
0.07454
0.059 14
25
+ 0.13027
0.05948
0.07080
0.04791
25
+ 0.131 37
0.05996
0.071 41
0.04850
26
+ 0.12951
0.061 14
0.06837
0.038 10
26
+ 0.130 70
0.061 69
0.06901
0.03858
27
+ 0.12904
0.06245
0.06659
0.02882
27
+ 0.13031
0.06305
0.06725
0.02921
28
+ 0.12885
0.06346
0.065 39
0.01993
28
+ 0.13020
0.064 12
0.06608
O.O2O2I
29
+ 0.12894
0.064 21
0.06475
0.01127
29
+ 0.13037
0.06490
0.06546
0.01 1 45
30
+ 0.12933
O.O647I
0.06463
0.00271
30
+ 0.13083
0.06544
0.06538
0.00279
31
+ 0.13002
0.06498
0.06504
+ 0.00589
31
+ 0.131 59
0.065 74
0.06583
+ 0.00591
32
+ 0.131 01
O.O65 O2
0.06599
+ 0.01465
32
+ 0.13263
0.06582
0.06683
+ 0.01478
33
+ 0.13232
0.06482
0.06751
+ 0.02372
33
+ 0.13401
0.06563
0.06839
+ 0.02397
34
+ 0.13397
0.06432
0.06963
+ 0.03325
34
+ 0.135 71
0.065 15
0.07057
+ 0.03362
35
+ 0.13596
0.06352
0.07244
+ 0.04341
35
+ 0.137 77
0.06435
- 0.07343
+ 0.04392
36
+ 0.13834
0.06233
0.07602
+ 0.05441
36
+ 0.14022
0.063 15
0.07706
+ 0.05506
37
+ 0.141 13
O.o6o66
0.08048
+ 0.06647
37
-j- 0.14308
0.06 1 47
0.08 1 59
+ 0.06728
38
+ 0.14440
0.05840
0.08599
+ 0.07989
38
+ 0.14640
0.05920
0.087 18
+ 0.08087
39
+ 0.148 16
0.05541
0.09277
+ 0.09501
39
+ 0.15022
0.056 18
0.09405
+ 0.096 19
40
+ 0.152 52
0.05145
o.ioi 08
+ 0.11229
40
+ 0.15464
0.052 18
0.10246
+ 0.11369
41
+ 0.15756
0.04626
O.I 1 1 29
+ 0.13231
41
+ 0.15973
0.04694
0.11279
+ 0.13396
42
+ 0.163 40
0.03947
0.12392
+ 0.15582
42
+ 0.16561
0.04007
0.125 54
+ 0.15776
43
+ 0.17020
0.03058
0.13961
+ 0.18384
43
+ 0.17245
0.03106
0.141 39
+ 0.186 1 1
44
+ 0.17822
0.01889
0.15931
+ 0.21772
44
+ 0.18048
0.01921
0.161 26
+ 0.22039
45
+ 0.18779
0.00348
0.18430
+ 0.25939
45
+ 0.19004
0.00358
0.18646
+ 0.26254
46
+ 0.19946
+ 0.01695
0.21640
+ 0.31152
46
+ 0.201 66
+ 0.017 16
0.21884
+ 0.31527
47
+ 0.214 10
+ 0.044 19
0.25828
+ 0.37802
47
+ 0.21618
+ 0.04488
0.261 04
+ 0.38255
48
+ 0.233 13
+ 0.08075
0.31387
+ 0.46469
48
+ 0.23409
+ 0.082 10
0.31709
+ 0.47025
49
+ 0.259 12
+ 0.13004
0.389 15
+ 0.58041
49
+ 0.26056
+ 0.13243
- 0.39209
+ 0.58739
50
+ 0.20674
+ 0.10672
0.49344
+ 0.739 12
50
+ 0.29746
+ 0.20073
0.40820
+ 0.748 16
51
+ 0.35509
+ 0.28662
0.641 69
+ 0.06340
Si
+ 0.35463
+ 0.29326
0.64789
+ 0.975 66
52
+ 0.45286
+ 0.40576
0.85863
+ I.2OO97
52
+ 0.450 50
+ 0.41679
0.86729
+ 1.30862
53
+ 0.63050
+ 0.55578
1.18620
+ 1.786 19
53
+ 0.625 31
+ 0.57426
- I.I9957
+ 1.81371
54
+ 0.980 18
+ 0.71766
- 1.69783
+ 2.55944
54
+ 0.971 84
+ 0.74881
1.72064
+ 2.60676
55
+ 1.72092
+ 0.80056
2.52146
+ 3.791 24
55
+ 1.71305
+ 0.851 55
- 2.565 51
+ 3-88150
56
+ 3.37687
+ 0.495 10
- 3-871 98
+ 5-7i6i9
56
+ 3-39986
+ 0.56678
-3-96664
+ 5-903 18
57
+ 7.09411
1.04890
6.04521
+ 8.33290
57
+ 7.271 80
1.00883
6.26296
+ 8.721 65
58
59
+14.50438
+24.307 55
5.33643
-11-95739
9.16795
12.350 17
+ 10.174 19
+ 6.72653
58
59
+15.22832
+26.10678
5.57039
12.841 66
9.65791
13.26509
+ 10.830 84
+ 7-I8745
66
ACTION OF THE PLANETS ON THE MOON.
TABLE Mil. Concluded.
VALUES OF A, B, C, AND D FOR THE ACTION OF VENUS ON THE MOON.
System 10.
System n.
i
A
B
C
D
i
A
B
C
D
+32.96968
16.56997
16.399 71
- 3-66844
+35-831 31
17.896 16
17.935 12
1.71241
i
+22.87840
- 9-29885
13-57954
12.86257
i
+25.08088
10.06956
15.01132
12.871 70
2
+11.01961
1.89825
9.121 36
12.14882
2
+12.64960
2.65575
9-99385
12.89708
3
+ 4-64991
1.01842
- 5.66834
8.32679
3
+ 5-31222
0.803 15
6.II536
8.902 19
4
+ 2.00491
+ 1.52568
- 3.53058
- 5-29455
4
+ 2.27438
+ 148038
- 3-754 76
5.628 18
5
+ 0.961 50
+ 1-31385
2.275 35
- 340808
5
+ 1.08448
+ 1.30802
2.39250
- 3-59521
6
+ 0.53767
+ 0.991 88
- 1-52955
2.28306
6
+ 0.602 79
+ 0.991 77
1-59455
2.39023
7
+ 0.35496
+ 0.71640
1.071 36
1.59699
7
+ 0.39457
+ 0.715 16
1.10973
1.661 64
8
+ 0.270 14
+ 0.50875
0.77890
1.16257
8
+ 0.29706
+ 0.50583
0.80290
1.20341
9
+ 0.22724
+ 0.35797
0.58520
0.875 82
9
+ 0.24704
+ 0.35397
0.60 1 02
0.90267
10
+ 0.20327
+ 0.24927
- 0.45255
0.67894
10
+ 0.21868
+ 0.24478
0.46347
0.697 18
ii
+ 0.18839
+ 0.17058
0.35896
- 0.53889
ii
+ 0.20085
+ 0.16598
0.36683
0.551 59
12
+ 0.178 13
+ 0.11307
0.291 19
0.43609
12
+ 0.18848
+ 0.10859
0.29707
0.44509
13
+ 0.17040
+ 0.07059
0.241 oo
0.358 52
13
+ 0.179 19
+ 0.06636
0.245 53
0.36497
14
+ 0.16420
+ 0.03887
0.20307
0.298 55
14
+ 0.171 75
+ 0.03492
- 0.20668
0.303 18
IS
+ 0.15900
+ 0.01493
0.17392
0.251 17
IS
+ 0.16557
+ o.oi i 29
- 0.17686
0.25449
16
+ 0.15452
0.00331
0.151 19
0.21301
16
+ 0.16028
0.00665
0.153 64
- 0.21535
17
+ 0.15059
0.01736
0.13325
0.181 73
17
+ 0.15570
0.02038
O.I353I
0.18333
18
+ 0.147 15
0.02823
0.11892
- 0.15568
18
+ 0.151 69
0.03098
O.I2O 7O
0.15670
19
+ 0.144 13
0.03672
0.10739
0.13365
19
+ 0.148 17
O.O39 22
0.10896
0.13421
20
+ 0.141 48
0.04340
0.09808
o.i 14 75
20
+ 0.145 09
0.045 66
- 0.09945
0.11495
21
+ 0.139 19
0.04867
0.09050
0.09831
21
+ 0.14243
0.05070
0.091 72
0.09823
22
+ 0.13722
0.05286
0.08436
0.08381
22
+ 0.140 12
0.054(18
0.08545
0.083 50
23
4- 0.135 57
0.05620
0.07938
0.07086
23
+ 0.138 16
0.05781
0.08036
0.07035
24
+ 0.13422
0.058 85
- 0.075 38
0.059 13
24
+ 0.13653
0.060 27
0.07626
0.05846
25
+ 0.133 17
0.06096
0.07222
0.04836
25
+ 0.13522
O.O')220
0.07.1 01
- 0.04755
26
+ 0.13241
0.06263
0.06979
0.03832
26
+ 0.13420
0.063 70
0.070 =;o
0.03741
27
+ 0.131 93
0.06393
0.06801
0.02884
27
+ 0.13348
0.06483
0.06864
0.02783
28
+ 0.131 73
0.06492
0.06681
0.01975
28
+ 0.13305
0.06565
0.06738
0.01866
29
+ 0.131 82
0.06563
0.066 18
0.01090
29
+ 0.13289
O.O6620
- 0.06668
o.oop75
30
+ 0.132 18
0.066 10
0.06608
O.OO2 15
30
+ 0.13302
0.05651
0.06651
0.00096
31
+ 0.13284
0.06632
0.06651
+ O.OO662
3i
+ 0.13344
0.06656
0.06687
+ 0.00785
32
+ 0.13379
0.06631
0.06749
+ 0.01557
32
+ 0.134 15
O.O6637
0.06777
+ 0.01682
33
+ 0.13506
0.06603
0.06903
+ 0.02482
33
+ 0.135 15
0.065 91
0.06923
+ 0.02608
34
+ 0.13664
0.06545
0.071 18
+ 0.03454
34
+ 0.13646
0.065 16
0.071 30
+ 0.035 78
35
+ 0.13857
0.06454
0.07402
+ 0.04489
35
+ 0.138 10
0.06405
0.07404
+ 0.046 ii
36
+ 0.14086
0.063 24
0.07763
+ 0.05608
36
+ 0.14007
0.06254
0.07755
+ 0.05724
37
+ 0.14355
0.061 43
0.082 13
+ 0.06834
37
+ 0.14242
0.060 50
0.081 92
+ 0.06942
38
+ 0.14668
0.05901
0.08767
+ 0.081 06
38
+ 0.145 15
0.05784
0.08732
+ 0.08293
39
+ 0.15029
0.05581
0.09447
+ 0.09730
39
+ 0.14832
0.05437
0.09395
+ 0.098 1 1
40
+ 0.15444
- 0.051 63
0.10280
+ 0.11480
40
+ 0.15196
0.04987
0.10209
+ 0.11540
41
+ 0.15920
0.046 17
0.11305
+ 0.13504
41
+ 0.156 14
0.04406
0.11208
+ 0.13536
42
+ 0.16470
0.03903
0.12568
+ 0.158 78
42
+ 0.16093
0.03652
0.12442
+ 0.15873
43
+ 0.171 06
0.02969
0.141 37
+ 0.18703
43
+ 0.16646
0.02671
0.13973
+ 0.18647
44
+ 0.17850
0.01745
0.161 05
+ O.22I l6
44
+ 0.17285
0.01391
0.15804
+ 0.21993
45
+ 0.18729
o.ooi 31
0.18599
+ 0.26306
45
+ 0.18036
+ 0.00293
0.18328
+ 0.26093
46
+ 0.19790
+ O.O2O 12
0.21801
+ 0.31543
46
+ 0.18932
+ 0.02521
0.214 53
+ 0.31208
47
+ O.2II OS
+ 0.04873
- 0.25978
+ 0.382 19
47
+ 0.20031
+ 0.05497
0.255 27
+ 0.377 IS
48
+ O.22798
+ 0.08726
0.31523
+ 0.469 15
48
+ 0.21431
+ 0.09504
0.30935
+ 0.461 76
49
+ O.25O86
+ 0.13952
0.39037
+ 0.58526
49
+ 0.23309
+ 0.14953
0.38262
+ 0.57453
50
+ 0.283 78
+ O.2I084
0.49461
+ 0.74461
50
+ 0.26004
+ 0.22424
0.48)27
+ 0.729 10
5i
+ 0.33481
+ 0.30836
0.643 17
+ 0.97029
Si
+ 0.301 96
+ 0.32727
0.62923
+ 0.94779
52
+ 0.42084
+ 0.44062
0.861 46
+ I.3OI 21
52
+ 0.37330
+ 0.46914
0.84244
+ 1.26849
53
+ 0.57924
+ 0.614 18
I-I9343
+ I.8(D5 20
53
+ 0.50661
+ 0.66075
I.I6737
+ 1.75769
54
+ 0.89775
+ 0.81982
I-7I755
+ 2.602 39
54
+ 0.77960
+ 0.00268
1.68227
+ 2.53509
55
+ I.5929I
+ 0.08393
2.57684
+ 3-90143
55
+ 1.38811
+ 1.14402
2.532 13
+ 3-81521
56
+ 3-2I360
+ 0.81477
4.02836
+ 6.01562
56
+ 2.84329
+ 1.14104
- 3-98524
+ 5-94655
57
+ 7.07792
0.596 54
6.481 38
+ 9-12585
57
+ 6.43787
+ 0.05782
6.495 70
+ 9-252 71
58
+ 15.48372
5.18993
IO.2Q-58O
+11.00597
58
+ 14.71782
4.16263
IO.W20
+ 12.78088
59
+28.021 10
13.352 48
-14.66860
+ 8.801 20
59
+28.397 74
12.84884
-15.54889
+10.00673
COEFFICIENTS FOR DIRECT ACTION.
6 7
TABLE VIII.
DEVELOPMENT OF A, B, C, AND D FOR VENUS IN PERIODIC SERIES.
Coeff. of
V, g'
^
1
?
<
f~*
1
9
cos
sin
COS
sin
COS
sin
COS
sin
o o
+2.1947
o.oooo
-0.5886
o.oooo
i .606 1
O.OOOO
0.0005
0.0000
0+ I
+0.1742
0.0428
0.0314
+0.0085
0.1428
+0.0343
+0.0080
+0.0774
O+ 2
+0.0523
0.0556
0.0086
+0.0084
0.0438
+0.0471
+0.0124
+0.0131
o+ 3
+0.0046
0.0075
0.0008
+0.0012
0.0038
+0.0063
+O.OO22
+0.0016
o+ 4
+O.OOOI
0.0015
o.oooo
o.oooo
O.OOOI
+O.OOI2
+0.0004
o.oooo
o+ 5
O.OOOI
O.OOO2
0.0000
+O.OOO2
O.OOOO
+0.0002
0.0000
o.oooo
o+ 6
0.0000
o.oooo
o.oooo
o.oooo
O.OOOO
O.OOOO
O.OOOO
o.oooo
i- 7
o.oooo
o.oooo
O.OOOI
+O.OOOI
o.oooo
X O.OOOO
+O.OOOI
0.0000
6
0.0004
+0.0003
0.0003
o.oooo
+O.OOOI
0.0003
0.0000
O.OOOI
5
+0.0004
+0.0014
O.OOO2
0.0005
O.OOOI
O.OOIO
+0.0005
0.0005
4
+0.0049
+0.0084
0.0013
0.0017
0.0038
0.0068
+0.0031
O.OO2I
3
+0.0546
+0.0552
0.0127
0.0115
0x5423
0.0441
+0.0182
0.0199
2
+0.2534
+0.0552
0.0820
0.0156
0.1708
0.0393
+O.O2OI
0.1214
I
+4.0006
0.0006
1.1590
+0.0004
2.8416
0.0000
0.0015
I.I002
1+
+0.0924
0.0288
+0.01 18
+0.0030
0.1036
+0.0255
O.OOIO
+0.0525
1+ I
+0.0501
0.0549
0.0058
+0.0060
0.0445
+0.0490
+0.0063
+0.0068
1+ 2
+0.0038
0.0064
O.OOO2
+0.0008
0.0036
+0.0059
+0.0013
+0.0009
1+3
O.OOOO
0.0014
O.OOOO
+0.0005
O.OOOO
+0.0013
+0.0004
+O.OOOI
1+ 4
0.000 1
O.OOO2
O.OOOO
o.oooo
+O.0002
+0.0004
O.OOO2
0.0000
i+ 5
o.oooo
o.oooo
O.OOOI
+O.OOOI
0.0000
0.0000
+O.OOOI
0.0000
2- 8
o.oooo
o.oooo
O.OOOI
O.OOOI
+O.OOOI
o.oooo
+O.OOOI
0.0000
2 7
O.OOOO
+0.0004
O.OOO2
O.OOOI
+O.OOOI
0.0006
+O.OOO2
O.OOOO
2- 6
+0.0004
+0.0015
O.OOOO
0.0005
O.OOO2
O.OOI I
+O.OOO2
OXIOOI
2- S
+0.0054
+0.0092
o.oo 1 6
0.0025
0.0036
0.0065
+0.0037
0.0027
2 4
+0.0567
+0.0542
0.0174
0.0144
0.0398
0.0308
+0.0223
0.0252
2 3
+0.3093
+0.0648
0.1281
0.0245
0.1826
0.0402
+0.0315
0.1729
2 2
+3.7105
0.0014
1.3562
+0.00 1 1
2.3535
o.oooo
0.0018
1.6716
2 I
+0.0402
-0.0173
+0.0321
+O.OOIO
0.0730
+0.0162
00016
+0.0516
2+ O
+0.0483
0.0536
0.0049
+0.0052
0.0430
+0.0486
+O.OOOI
+O.OOI2
2+ I
+0.0032
0.0054
0.0003
+0.0003
0.0028
+0.0048
+0.0008
+0.0007
2+ 2
+0.0001
O.OOIO
o.oooo
+O.OOO2
.00000
+O.OOII
+O.OOOI
0.0000
2+ 3
O.0002
0.0002
0.0003
O.OOOI
+0.0003
O.OOO2
+0.0002
o.oooo
2+ 4
O.OOOO
O.OOOO
O.OOOI
O.OOOI
+O.OOOI
O.OOOO
+O.OOOI
0.0000
3 9
O.OOOI
+O.OOOI
O.OOOI
O.OOOO
+O.OOOI
o.oooo
+O.OOOI
+O.OOOI
3-8
o.oooo
+0.0006
o.oooo
0.0003
+O.OOOI
0.0003
+O.OOOI
0.0000
3- 7
+0.0004
+O.OO2O
+0.0001
0.0007
0.0003
O.OOI I
+0.0007
O.OOO2
3-6
+0.0062
+0.0093
O.OO2O
0.0032
0.0041
0.0064
+0.0046
0.0033
3- 5
+0.0586
+0.0524
0.0217
0.0168
0.0366
0.0354
+0.0252
0.0305
3- 4
+0.3454
+0.0705
0.1657
0/5317
-0.1789
0.0387
+0.0408
0.2154
3- 3
+3.3553
0.0017
-14697
+O.OOI2
-1.8858
+0.0003
0.0014
1.8630
3 2
+0.0042
0.0079
+0.0456
0.0014
0.0490
+0.0004
+0.0008
+0.0547
3 i
+0.0454
0.0516
0.0044
+0.0049
0.0410
+0.0463
0.0056
0.0043
3- o
+0.0030
0.0045
O.OOO2
O.OOOO
0.0030
+0.0044
O.OOOO
+0.0003
3 i
0.0003
O.OOIO
+0.0004
+O.OOO2
+O.OOOI
+O.OOII
+O.OOO2
OJOOOO
3 2
0.0004
O.OOOO
0.0002
O.OOOI
+.OOOO2
O.OOO2
o.oooo
+O.OOOI
3- 3
O.OOOI
+O.OOOI
O.OOOI
o.oooo
+O.OOOI
0.0000
+O.OOOI
+O.OOOI
4 10
O.OOOI
+O.OO02
+O.OOOI
+O.OOOI
+O.OOO2
o.oooo
+O.OOOI
0.0000
4 9
O.OOOI
+0.0004
+0.0003
+O.OOOI
O.OOOO
O.OOO2
O.OOOO
O.0002
4-8
+0.0003
+0.0018
o.oooo
0.0008
O.OOOI
0.0014
+0.0013
0.0003
4- 7
+0.0064
+0.0096
0.0025
0.0036
0.0038
0.0060
+0.0052
0.0036
4- 6
+0.0584
+0.0408
0.0252
0.0190
0.0332
0.0308
+0.0267
0.0344
4- S
+0.3605
+0.0726
0.1938
0.0370
0.1671
0.0355
+0.0468
0.2442
4 4
+2.9597
0.0013
-1.4783
+O.OOI2
-1.4817
+O.OOO2
0.0014
1.8596
4 3
0.0206
0.0005
+0.0526
0.0043
0.0330
+0.0045
+0.0037
+0.0580
4 2
+0.0424
0.0483
0.0058
+0.0059
0.0369
+0.0420
0.0090
0.0075
4 I
+0.0023
0.0034
o.oooo
O.OOOI
0.0025
+0.0034
O.OOOO
+0.0003
68
ACTION OF THE PLANETS ON THE MOON.
TABLE VIII. Concluded.
DEVELOPMENT OF A, B, C, AND D FOR VENUS IN PERIODIC SERIES.
Coeff. of
V, g'
A
B
C
D
cos
sin
COS
sin
COS
sin
cos
sin
4+ o
O.OO02
0.0010
+O.OOOI
+O.OOOI
+0.0002
+0.0013
O.OOO2
0.0003
4+ i
o.oooo
+O.OOO2
O.OOOI
O.OOOI
O.OOOO
+0.0003
O.OOOI
0.0000
4+ 2
+O.OOOI
+O.OOO2
O.OOOI
+O.OOOI
+0.0004
o.oooo
+0.0001
O.OOOO
s ii
O.OOOO
o.oooo
O.OOOI
o.oooo
+O.OOOI
O.OOOI
O.OOOO
O.OOOI
S-io
0.0004
o.oooo
O.OOOI
0.0004
+O.OOOJ
+O.OOOI
+O.OOO2
o.oooo
S- 9
+0.0004
+0.0018
O.OOOI
0.0008
O.OOOO
o.ooii
+0.0011 0.0003
5-8
+0.0063
+0.0095
0.0028
0.0036
0.0037
0.0056
+0.0057
0.0040
5- 7
+0.0579
+0.0465
0.0284
O.0200
0.0298
0.0268
+0.0276
0.0374
5-6
+0.3602
+0.0712
0.2094
0.0400
0.1501
0.0313
+0.0497
0.2587
5- 5
+2.5539
0.0014
1.4070
+O.OOI2
-1.1467
+O.OOO2
0.0015
1.7422
5 4
0.0350
+0.0047
+0.0564
0.0068
0.0205
+0.0016
+0.0067
+0.0598
5 3
+0.0389
0.0445
0.0064
+0.0069
0.0328
+0.0375
O.OI22
0.0108
5 2
+0.0018
0.0028
+O.OOOI
+O.OOOI
0.0018
+0.0029
+O.OOO2
+O.0002
5 i
+0.0004
O.OOI I
o.oooo
+O.OOOJ
+O.OOO2
+O.OOI2
0.0002
O.OOOO
5+ o
O.OOOI
+O.OOO2
0.0002
+O.OOO2
O.OOOI
+0.000 1
+O.OO02
O.OOOO
5+ I
o.oooo
O.OOOO
O.OOOI
O.OOOO
+0.0001
O.OOOI
O.OOOO
O.OOOI
6-12
O.OOOI
o.oooo
o.oooo
o.oooo
+O.OOO2
o.oooo
+O.OOOI
O.OOO2
6-n
O.OOOI
+0.0003
+O.OOO2
OJOOOO
+0.0004
+O.OOOI
+O.COO2
o.oooo
6io
+0.0003
+0.0018
O.OOO2
0.0008
0.0005
0.0009
+O.OOI2
0.0005
6-9
+0.0061
+0.0091
0.0035
0.0038
0.0033
0.0050
+0.0060
0.0040
6- 8
+0.0557
+0.0428
0.0295
0.0203
0.0264
0.0225
+0.0274
0.0380
6- 7
+0.3455
+0.0684
0.2150
0.0414
0.1312
0.0274
+0.0500
0.2590
6- 6
+2.1645
0.0007
-1.2859
+0.0009
-0.8788
O.OOOI
0.0009
1.5648
6- 5
0.0436
+0.0086
+0.0558
0.0084
0.0128
+O.OO02
+0.0086
+0.0588
6-4
+0.0356
0.0403
0.0075
+0.0078
0.0282
+0.0321
O.OI39 O.OI2O
6-3
+0.0016
0.0022
O.OOOO
O.OOOI
0.0018
+O.O02I
+O.OOO2 O.OOOI
6- 2
O.OOOO
0.0009
+O.0002
+O.OOO2
0.0005
+0.0009
o.oooo
+0.0005
6- i
O.OOOI
+O.OOO2
O.OOO2
+O.OOOI
O.OOOI
+O.OOOI
O.OOO2
0.0006
6+ o
O.OOOI
0.0000
o.oooo
o.oooo
O.O002
o.oooo
O.OOOI
O.OOO2
713
0.000 1
0.0000
O.OOOO
O.OOOI
O.OOOI
o.oooo
0.0002
O.OOOO
712
o.oooo
+0.0005
+0.0003
O.OOOI O.OOOI
O.OOOI
+0.0004
+0.0002
7 ii
+0.0006
+0.0017
+0.0001
0.0009 O.OOO2
0.0009
+0.0007
O.OOOI
7io
+0.0064
+0.0088
0.0033
0.0042 0.0030
0.0046
+0.0058
0.0043
7 9
+0.0532
+0.0392
0.0305
O.O2OO O.0223
0.0184
+0.0260
0.0386
7-8
+0.3227
+0.0632
O.2IIJ
0x1407 0.1116
0.0226
+0.0486
0.2508
7 7
+1.8074
0.0006
-1.1384
+0.0008 0.6689
O.OOO2
0.0009
1.3643
7-6
0.0458
+0.0106
0.0547
O.OIO2 O.OO74
o.ooog
+0.0105
+0.0567
7 5
+0.0315
0.0359
0.0074
+0.0082 0.0240
+0.0274
0.0148
0.0129
7 4
+O.OOI2
0.0015
+O.OOOI
0.0004 0.0012
+0.0016
+O.OO02
+O.OOO2
7 3
O.OOOO
O.OOII
+0.0003
+0.0003 o.oooo
+0.0008
0.0005
+O.0002
7- 2
O.OOOI
0.0002
O.OOOI
+0.0002 0.0000
+0.0001
O.OOOI
O.OOOO
7- i
O.OOOI
o.oooo
o.oooo
O.OOO2 O.OOOI
o.oooo
O.OO02
O.OOOO
814
O.OOOI
O.OOOO
0.0000
O.OOOI O.OOOO
O.OOO2
+O.COOI
o.oooo
8-13
O.OOO2
+O.OOO2
+O.OOO2
O.OOO2 O.OOOO
O.OOOI
+O.OO02
0.0003
8-12
+O.OO02
+0.0014
0.0002
0.0008 0.0002
0.00 10
+0.0013
0.0006
8-ii
+0.0058
+0.0080
0.0034
0.0044 0.0027
0.0037
+0.0050
0.0041
8io
+0.0496
+0.0351
0.0306
0.0190 0.0193
0.0155
+0.0250
0.0374
8- 9
+0.2941
+0.0577
0.1996
0.0385 0.0942
0.0192
+0.0459
0.2335
8- 8
+14892
O.OOOI
0.9842
+0.0003 0.5050
O.OOOI
0.0006
1.1634
8- 7
0.0461
+0.0118
+0.0500
0.0104 0.0048
O.OOI 2
+0.0115
+0.0508
8- 6
+0.0278
0.0316
0.0081
+0.0085 o.o ic
+0.0228
0.0144
O.OI22
8- 5
+O.OOIO
0.00 10
+O.OOOI
0.0004 0.00 T 2
+0.0012
0.0005
.0.0000
8-4
O.OOOI
0.0009
o.oooo
O.OOOI O.OOO2
+0.0009
+O.OOOI
+O.OOO2
8- 3
+O.OOO2
o.oooo
o.oooo
O.OOO2 O.OOOI
+O.OO02
O.OOOI
+0.0005
8- 2
O.OOOI
o.oooo
o.oooo
O.OOOI O.OOOO
0.0002
+O.OOOI
O.OOOO
COEFFICIENTS FOR DIRECT ACTION.
69
TABLE IX.
COMPUTATION OF THE COEFFICIENTS FOR THE HANSENIAN VENUS-TERM OF
LONG PERIOD.
System.
A"
A,
A, 2
A,
A, 2
O
+33-3948
48.0672
15.2486
+ 1.6810
+ 1.6959
I
+30.4226
44.3154
-14-3087
+ 5-2845
+ 5.6929
2
+27.3802
404460
13-3384
+ 6.9388
+ 7-7952
3
+25.7695
3&3437
-12.7915
+ 6.7824
+ 7.8308
4
+25.3715
37-7570
12.6090
+ 5-0267
+ 5.8930
5
+25.2326
37-5000
-12.5055
+ 1-9933
+ 2.3924
6
+24.7525
36-8595
12.3300
- 14599
1.0609
7
+24.2976
36.3246
12.2190
4.35i6
5.1130
8
+24.8273
37-1237
-12.4835
6.2219
- 7.2996
9
+27.0732
-40.1587
13-3322
6.9149
7.9252
10
+30.7124
44.8651
14.5505
5-9534
6.5819
II
+33.6483
48.4814
154035
- 2.7899
3-0391
B"
A,
**
B..
A,
o
19.9136
+27.3868
+ 7-9770
1-3391
1.2601
i
18.5130
+25.9046
+ 7.7612
4.0210
4.0255
2
16.9031
+24.1227
+ 74655
5-0839
- 5.2866
3
-15.8666
+22.8730
+ 7.2028
4-7701
5.0902
4
154249
+22.2499
+ 7-0232
34567
- 3.7608
5
-15.2448
+21.9597
+ 6.9235
14491
1.6516
6
15.0836
+21.7873
+ 6.9008
+ 0.8330
+ O.SIQI
7
15.0404
+21.8370
+ 6.9698
+ 2.9218
+ 3.1445
8
-15-4663
+22.4592
+ 7-1627
+ 4.4869
+ 4.9005
9
-16.6559
+23.9194
+ 74839
+ 5.2206
+ 5.6178
10
18.4569
+25.9344
+ 7.8260
+ 4-5771
+ 4.7687
ii
19.9292
+27.4474
+ 8.0139
+ 2.I2OI
+ 2.1782
C"
c*
c c
Ql
c.*
o
13.4811
+20.6804
+ 7.2716
- 0.3420
0.4358
i
11.9096
+18.4107
+ 6.5475
1.2636
1.6673
2
10.4771
+16.3233
+ 5-8729
- 1.8550
2.0586
3
9.9030
+15-4707
+ 5.5887
2.OI22
- 2.7407
4
9.9466
+15-5072
+ 5-5856
1.5700
- 2.1323
S
- 9.9878
+15.5402
+ 5.5820
0.5442
0.7409
6
9.6689
+15.0722
+ 54291
+ 0.627O
+ 0.8508
7
9.2573
+14.4877
+ 5-2492
+ 14209
+ 1.9685
8
- 9.3611
+14.6645
+ 5-3207
+ I-735I
+ 2.3991
9
104173
+16.2394
+ 5.8483
+ 1.6943
+ 2.3074
10
12.2555
+18.9306
+ 6.7244
+ 1-3762
+ 1.8132
ii
13.7191
+21.0340
+ 7.3897
+ 0.6699
+ 0.8608
A"
A,
A,
A.
A.
o
+ 1.1897
- 1-3484
0.2485
29.8546
38.9716
i
+ 3-5868
4-2373
0.8724
27.5111
-36.6653
2
+ 4.5580
5-5349
1.2117
24.9093
34.0089
3
+ 4-3152
5-3345
I-2I53
23.3292
32.2553
4
+ 3.1531
3-9457
0.9269
22.7393
314609
5
+ 1.3233
1.7080
0.4334
-22.5117
31.0906
6
0.7706
+ 0.8979
+ 0.1689
22.2069
30.7822
7
2.6659
+ 3-3202
+ 0.7653
22.0282
30.7403
8
4-0554
+ 5-1025
+ 1.2108
22.6267
-31.5788
9
-4-6896
+ 5.8349
+ 1.3576
24.5324
33.7503
10
4-1057
+ 4.9831
+ 1.1038
274981
36.8202
ii
1.9060
+ 2.2816
+ 04940
29.9252
39.1222
70 ACTION OF THE PLANETS ON THE MOON.
TABLE X.
COEFFICIENTS OF cos iSL AND SIN i8L FOR A, B, C, AND D IN EACH OF
12 SYSTEMS (L = V "')
System.
3oA c
3^.
30^ e
30^.
3oC.
30 c.
3oA
3oA
o
+6.2049
+0.6020
4.9971
-0.5329
1.2077
0.0692
+0.5720
-54864
I
+5-1524
+1.6797
4.2291
14580
0.9233
0.2217
+1-5716
4-6133
2
+4.0908
+2.0174
3-4091
1.7277
-0.6816
0.2896
+1-8673
3-7002
3
+3-5721
+1.8476
2.9712
1-5447
0.6009
0.3028
+1-6835
3.2281
4
+3-5031
+1.3168
-2.8674
1.0770
-0.6358
0.2398
+1.1840
3-1341
5
+3-5273
+0.4896
-2.8577
04074
0.6697
0.0821
. +0.4448
3-1352
6
+3-3871
04122
2.7680
+0.3160
0.6191
+0.0962
0.3565
3-0267
7
+3-1874
1.1332
2.6537
+0.9305
-0.5337
+0.2028
1.0208
-2.8813
8
+3-2562
1.6267
2.7313
+ 1.3868
0.5250
+0.2401
-1.2991
2.9577
9
+3-8775
1.9181
3-2097
+1.6630
-0.6677
+0.2552
-1.7883
34938
10
+5.0768
1-7932
4.1114
+I-550I
0.9655
+0.2431
1.6730
-4-5098
ii
+6.2050
0.8671
4.9641
+0.7360
1.2409
+0.1311
0.8012
-54650
a.
+4-2534
+0.0169
-3-4808
0.0138
0.7726
0.0031
+0.0154
3.8026
at
+1-3404
+04237
1.0631
0.3618
0.2773
0.0620
+0.3915
-1.1728
.
0.2170
+1.9499
+0.1657
-1.6577
+0.0512
0.2923
+1.7968
+0.1855
at
+0.5358
+0.0646
0.3961
0.0834
0.1397
+0.0188
+0.0796
04480
0.3789
+0.2585
+0.2565
0.2410
+0.1224
0.0174
+0.2547
+0.3019
ai
+0.0682
+0.0800
0.0512
0.0602
0.0170
0.0107
+0.0698
0.0569
0.
0.0679
+0.0673
+0.0488
0.0540
+O.OIO2
0.0133
+0.0611
+0.0554
The coefficients A c , A,, etc., have a separate value for each ol the 12 systems.
These special values are developed in a periodic series proceeding according to the
sines and cosines of multiples of g"', in the form (a) 36 with results shown in the
last seven lines above. The final development is then shown below in the form (b}.
TABLE XI.
COMPUTATION OF A- AND K-COEFFICIENTS FOR THE HANSENIAN INEQUALITY OF
LONG PERIOD.
Arg.
30^
3oA t
30^ c
302?.
30 c c
30 c.
y>*>.
3oZ>.
iSv-iSg 7
i8v 17^
i8v 16^
iSv-is?'
+4-2534
0.3048
+0.1387
+0.0005
+0.0169
+0.1034
0.1572
+0.0061
-3.4808
+0.2973
0.0775
+0.0014
0.0138
0.0981
+0.0865
0.0057
0.7726
+0.0075
0.06 1 1
0.0019
0.0031
0.0054
+0.0706
0.0003
+0.0154
+0.1030
o.nu
+0.0072
-3.8026
+0.3120
0.0966
+O.002I
Arg.
30*;
30*;
\<?MK e
i<?MK t
io*MC c
io 3 MC t
io 3 AfD c
ioWZ>.
i8v 18/
i8v 17^
i8v 16^
i8v-is'
+3.8671
0.3010
+0.1081
0.0004
+0.0104
+0.1008
0.1218
+0.0059
+0.5469
0.0426
+0.01528
0.00006
+0.0015
+0.0142
0.01723
+0.00083
0.1093
+0.00 1 1
0.00864
0.00027
0.0004
0.0008
+0.00998
0.00004
+O.OO22
+O.OI46
O.OI57I
+O.OOI02
-0^5376
+0.0441
0.01366
+0.00030
COEFFICIENTS FOR DIRECT ACTION.
37. Coefficients E and F for Venus. Some preliminary computations ren-
der it doubtful whether the planetary coefficients E and F would lead to sensible
inequalities in any case. But, in order to leave no doubt, they are computed for
six ot the twelve systems and thirty alternate values of the index for Venus. The
separate numerical results are shown in Table XII. The general development
will be, so far as it seemed useful to use it, found in Part IV.
TABLE XII.
SPECIAL VALUES OF E AND F FOR THE ACTION OF VENUS ON THE MOON.
Coefficient E.
Coefficient F.
i
System
O
System
2
System
4
System
6
System
8
System
10
l
System
System
2
System
4
System
6
System
8
System
IO
O
+3.528
+6.997
+3.567
2.569
6.300
-4.684
o
+0.071
+0.617
+0.231
+0.039
+0.532
+0.345
I
+2.2 16
+3-683
+ 1.277
-1.664
2.541
1.248
i
-I.I54
1.613
0.556
+0.836
+1484
+0.750
2
+0.643
+0.872
+0.160
0.545
0.620
0.150
2
0.571
+0.701
0.126
+0.459
+0.574
+0.144
3
+0.218
+0.233
0.006
0.198
0.179
o.ooo
3
0.229
0.226
0.005
+0.198
+0.192
o.ooo
4
+0.094
+0.078
0.024
0.089
0.064
+0.019
4
O.IOO
-0.077
+0.023
+0.090
+0x169
O.02I
5
+0.049
+0.030
O.022
0.046
0.026
+0.018
5
0.048
0.028
+O.02O
+0.045
+0.026
0.019
6
+0.028
+O.OI2
O.OlS
0.027
0.024
+0.015
6
0.025
O.OIO
+0.015
+0.024
+O.OIO
0.014
7
+0.017
+0.004
O.OI4
0.017
0.004
+0.013
7
0.014
+0.003
+O.OI I
+0.013
+0.003
O.OIO
8
+O.OII
0.000
0.012
O.OII
o.ooo
+O.OII
8
0.007
0.000
+0.007
+0.007
o.ooo
0.007
9
+0.007
0.002
0.009
0.007
+0.002
+0.009
9
0.004
+O.OOI
+0.005
+0.004
0.001
0.005
10
+0.004
0.003
0.008
0.004
+0.003
+0.008
10
0.002
+O.002
+0.003
+O.OO2
O.OOI
0.004
ii
+0.003
O.OO4
O.OO7
0.003
+0.004
+0.006
ii
O.OOI
+O.OOI
+O.OO2
+O.OOI
O.OOI
O.OO2
12
+O.OOI
O.OO4
0.005
O.OOI
+0.004
+0.005
12
0.000
+0.001
+0.001
0.000
0.001
0.002
13
o.ooo
0.005
0.005
0.000
+0.004
+0.004
13
o.ooo
+O.OOI
+O.OOI
o.ooo
O.OOI
O.OOI
14
0.00 1
0.005
0.004
+0.001
+0.005
+0.004
14
0.000
0.000
0.000
o.ooo
O.OOI
o.ooo
IS
O.OO2
0.005
O.OO3
+O.OO2
+0.005
+0.003
IS
0.000
o.ooo
0.000
0.000
0.000
0.000
16
0.003
0.005
O.OO2
+0.003
+0.005
+O.OO2
16
o.ooo
O.OOI
o.ooo
o.ooo
o.ooo
o.ooo
17
0.004
0.005
0.001
+0.004
+0.005
+0.001
17
O.OOI
O.OOI
o.ooo
+O.OOI
+O.OOI
o.ooo
18
0.005
0.005
O.OOO
+0.005
+0.005
0.000
18
O.OOI
O.OOI
0.000
+O.OOI
+0.001
0.000
19
0.006
0.005
+O.OOI
+0.006
+0.005
O.OO2
19
O.OO2
O.OO2
o.ooo
+O.OO2
+O.O02
O.OOI
20
0.008
0.004
+0.003
+0.007
+0.005
0.003
20
0.004
O.OO2
+O.OOI
+0.003
+O.OO2
O.OO2
21
0.0 10
0.004
+0.005
+0.009
+0.004
0.006
21
0.006
O.002
+0.003
+0.005
+0.002
0.003
22
0.012
0.003
+0.009
+O.OI2
+0.002
O.OIO
22
0.008
O.OOI
+0.006
+0.008
+O.OO2
0.007
23
0.016
o.ooo
+0.014
+0.015
o.ooo
0.017
23
O.OI2
o.ooo
+0.01 1
+O.OI2
o.ooo
O.OI2
24
O.O2I
+0.005
+0.024
+O.020
0.007
0.029
24
0.019
+0.005
+0.022
+0.017
0.006
0.025
25
0.028
+0.018
+0.043
+0.027
0.023
0.053
25
0.028
+0.018
+0.043
+0.026
O.02I
0.050
26
0.039
+0.050
+0.084
+0.037
0.065
0.108
26
0.042
+0.055
+0.090
+0.038
0.064
0.109
27
0.050
+0.152
+0.194
+0.045
0.208
0.264
27
0.053
+0.167
+0.205
+0.044
0.198
0.260
28
o.ooo
+0.569
+0.561
0.017
0.799
0.819
28
0.000
+0-543
+0.507
0.014
0.623
0.672
29
+0.730
+2.564
+1.885
-0.681
3.325
2.919
29
+0.406
+ 1-557
+1.061
0.325
1.402
1.331
B. ACTION OF MARS.
38. For Mars the coefficients A, B, C, and D were developed much in the same
way as for Venus. But, owing to the supposed absence of terms having a high mul-
tiple of the mean longitude of Mars, it was considered sufficient to divide the mean
orbit of Mars into 24 parts for the special computations of the yl-coefficients. The
adopted number of systems was 12, as in the case of Venus.
The following statements, with the diagram, will make clear the method of carry-
ing out the computation. In system o the Earth remains at rest at its perihelion
7 2
ACTION OF THE PLANETS ON THE MOON.
and is, therefore, in longitude approximately TT O ' = 99.5. Mars starting from this
same mean longitude, TT O ', takes the twenty-four consecutive mean longitudes TT O ',
IT O ' -f- 15, TT O ' -j- 30, etc., to if 1 ' -f- 345. These twenty-four positions are designated
by the twenty-four indices o, i, 2, 3, ... 23.
In system i the Earth is in mean anomaly 30. Then, as before, Mars takes the
successive mean longitudes TT O ' + 30, TT O ' -\- 45 , . . . up to TT O ' -f- 15.
The same plan is carried through; the constant mean anomaly of the Earth in
the tth system being i X 30, while Mars, starting with the same mean longitude,
goes through its twenty-four consecutive mean positions, the indices which express
the mean longitude of Mars always starting with the value o when Mars is in mean
conjunction with the Earth.
As in the case of Venus, the elements were taken with their values for 1800, in
order to correspond to the mean of the period during which the longitude of the
Moon has been observed. The numbers and data for computing the longitude of
Mars are, then, as follows:
TT O ' ; long, of 's perihelion for 1800; .... 99 30' 7". 6
7T 4 ; " " Mars' " " ";.... 332 22 42 .9
7r '-7r 4 ; initial mean anom. of Mars for 1800; 127 7 24 .7
Initial mean anomaly of Mars in system j
x/
M
For system j and index i
Equinox
7'2 4 ".7 + 3o x/+ 15 x i
From the numbers found in Tables
of MarS) page 397, it is found that to
this initial mean anomaly corresponds
Fund. Arg. N= 243^.0948
and that the increment of JVfor 15 of
mean anomaly is
Arrangement of Coordinate Axes in Systems o, i, etc.,
for Mars. have :
For the numbers arising from the
inclination of the orbit of Mars we
Long, of node, 1800; . 6 = 48 o' 52". 5
Node from e's perihelion ; 308 30' 44". 9
Inclination, 1800; .... /=i5i' 3". 6
COEFFICIENTS FOR DIRECT ACTION.
73
The results of the main steps in the computation of the coordinates of Mars are
shown in the following table. The first column corresponds to the indices of
system o. In they'th system they are diminished by 2;.
The second column shows the value of N actually used in entering the tables.
The discrepancy of two units in the fourth place results from using two computa-
tions of N. Columny gives the mean anomaly as taken from the tables, reduced
by the secular variation to 1800. Column u is formed by adding to^ the distance
from the node to the perihelion of Mars and applying the reduction to the ecliptic.
This reduction was applied in order to use for x and y simple formulae for the
ecliptic longitude. Actually, through a misapprehension, the rectangular coordi-
nates were computed on the supposition that u was counted along the orbit, as in
the case of Venus. There is therefore an error in the last figures of the coordinates,
the amount of which can readily be determined, but which has been deemed too
small to need correction for the present problem.
TABLE XIII.
COMPUTATION OF HELIOCENTRIC COORDINATES OF MARS.
i
N
/
U
log. r
X
y
z
o
243.0946
i it
135 3 17
59 24 20
0.208 749
1.601 16
O.222 32
+0.04497
i
271.7194
148 7 I
72 28 19
0.214918
1.53086
0.58691
+0.050 53
2
300.3442
160 52 ii
85 13 52
0.219 143
1.37661
-0.919 51
+0.053 32
3
328.0600
173 26 7
07 48 12
0.221 310
1.14904
1.20324
+0.053 27
4 357-5939
185 56 13
no 18 38
O.22I 364
0.86 1 21
-1.42386
+0.05043
5
386.2187
198 29 47
122 52 26
0.219 303
0.528 30
-1.56981
+0.04495
6
414.8435
211 14 12
135 36 56
0.2I5I79 O.I6745
1.63228
+0.03708
7
4434683
224 16 54
148 39 31
O.209 102 +0.202 38
1-60553
+0.027 19
8
472.0932
237 45 02
162 7 24
0.201 264 +0.560 38
148737
+0.015 76
9
500.7180
251 45 55
176 7 52
0.191962 +0.88448
1-27995
+0.00339
10
529.3428
266 25 53
190 47 23
O.lSl 634
+1-151 73
0.090 76
0.009 19
ii
557-9676
281 49 50
206 10 58
0.170884
+1.33964
-0.633 72
O.O2I 13
12
586.5024
298 o 12
222 21 9
0.160502
+142836
0.230 06
0.031 49
13
615.2172
314 55 23
239 16 26
O.I5I4I9
+ 1.40360
+0.191 54
-0.03935
14
643.8421
332 28 46
256 50 12
0.144609
+1.26023
+0.50686
0.043 88
15
672.4669
350 28 12
274 50 II
O.I40 9IO
+1-00534
+0.94909
0.044 52
16
14.0061
8 37 13
292 59 42
O.I408I7
+0.65923
+ 1.21507
0.041 12
17
42.7209
26 37 36
311 oo 19
0.144343
+0.253 23
+ 1.37065
0.033 99
18
71-3457
44 12 35
328 35 12
O.I5IOI4
0.17528
+ 1.40475
0.023 84
19
09.9705
61 9 53
345 32 8
O.I6OOO6
-0.58917
+1.31988
0.011 66
20
128.5954
77 22 34
I 44 20
0.170346
0.956 46
+1.12979
+0.001 45
21
157 2202
92 48 50
17 10 10
O.lSl O98
1.25306
+0.85563
+0.01447
22
185.8450
107 30 54
31 Si 56
O.I9I 465
1.46347
+0.522 13
+0.026 50
23
214.4608
121 33 36 45 54 32
0.200829 1.57994
+0.154 73
+0.036 84
74
ACTION OF THE PLANETS ON THE MOON.
TABLE XIV.
G-COORDINATES OF MARS REDUCED TO THE DIFFERENT SYSTEMS.
System o.
System i.
System 2.
System 3.
jr
r
X
r
X
r
X
r
o
I
2
3
4
0.617 95
-0.54765
0.393 40
0.165 83
+O.I220O
0.22232
0.586 91
0.919 Si
1.203 24
142386
0.666 54
0.61133
047237
0.25703
+0.024 24
0.091 17
0450 67
0.785 65
1.07849
1-31303
0.672 31
0.632 25
0.505 93
0.297 85
0.016 52
+0.063 05
0.298 23
0.641 98
0.948 87
1.19984
0.632 56
0.605 8 1
0487 65
0.280 23
+0.00896
+O.2OI 03
O.l688o
O.526 8O
0.850 go
1.11815
5
6
8
9
+0.45491
+0.8I5 76
+ I-I85S9
+ 1-54359
+ 1.86769
-1.56981
1.63228
-1.60553
-1.48737
-1-27995
+0.35789
+0.727 02
+ I.III 41
+ 1.48744
+ 1.82870
147477
I-35I 44
-1.53386
1.41703
1.201 79
+0.325 iC
+0.709 23
+1.11239
+ 1-50633
+ 1-85907
-1.37681
146366
144788
1.32288
1.09063
+0.366 oo
+0.76966
+1.191 26
+ 1-59658
+1.94881
1.30606
1.39478
1.37002
1.22665
0.971 76
10
ii
12
13
14
+2.134 94
+2.322 85
+2.41157
+2.38681
+2.243 44
0.99076
0.633 72
0.230 06
+0.191 54
+0.59686
+2.107 36
+2.29671
+2.375 22
+2.330 58
+2.16383
-0.89657
0.51907
0.096 37
+0.336 12
+0.739 52
+2.13839
+2.31600
+2.373 28
+2.305 02
+2.I203O
0.763 82
0.366 96
+0.065 77
+0495 16
+0.883 32
+2.214 79
+2.370 37
+2404 47
+2.31960
+2.129 51
0.625 65
0.219 65
+0.20886
+0.622 75
+0.990 04
IS
16
17
18
19
+ 1.98855
+1.64244
+ 1.23644
+0.80793
+0.39404
+0.04900
+ 1.21507
+1-37065
+ 1-40475
+1-31988
+1.89002
+1.53596
+1-13509
+0.721 96
+0.32801
+ 1.07726
+ 1.32104
+ 1.45449
+ I-4735I
+ 1-38438
+ 1.83987
+I-49I 59
+ 1.10586
+0.71184
+0-335 42
+I-I9933
+ 1.42236
+ 1-542 IS
+i.5576i
+ 1.47478
+ 1-85535
+ 1.52185
+I-I5445
+0.777 40
+0.41281
+1.28664
+1-49705
+1.61352
+1.63474
+1.56444
20
21
22
23
+0.026 75
0.26985
0.480 26
0.596 73
+1.12979
+0.85563
+0.522 13
+0.154 73
0.020 95
0.305 52
0.51242
0.63383
+ 1.20076
+0.040 82
+0.624 89
+0.274 oo
o.oo i 73
0.282 32
0.493 23
0.625 17
+ 1.30464
+ 1.06147
+0.761 58
+0.422 63
+0.08021
0.203 52
0424 14
0.570 09
+1.410 19
+1.18262
+0.894 79
+0.561 88
System 4.
System 5.
System 6.
System 7.
O
I
2
3
4
0.560 10
0.542 52
0.425 69
0.210 45
+0.094 77
+0.28739
0.09700
0473 03
0.814 29
1.092 95
0478 34
0462 56
0.337 56
0.105 31
+0.221 50
+0.29889
0.104 27
0.498 21
0.850 95
1.13027
-0.4II57
0.38681
0.243 44
+0.0 1 1 45
+0.357 56
+0.230 06
0.191 54
-0.59686
0.94909
1.21507
0.37S 36
0.330 72
0.16397
+0.10984
+0.463 90
+0.09649
0.336 oo
0.73940
1.07714
1.32092
S
6
8
9
+0472 27
+0.89497
+1.32746
+1.73086
+2.06860
1.282 30
1.36081
1.316 17
1.14942
0.875 61
+0.618 36
+1.05109
+148048
+1.86864
+2.18465
-1.30788
1.365 16
1.29690
1. 112 l8
-0.831 75
+0.763 56
+1.19207
+ 1.60596
+ 1.97325
+2.26985
1.37065
1.40475
-1.31988
1.12979
0.855 63
+0.864 77
+ 1.27790
+ 1.67185
+2.O2O 8l
+2.305 38
1-454 37
1-473 39
1.38426
1.20064
0.040 70
10
ii
12
13
14
+2.31238
+2.445 83
+2.46485
+2.375 72
+2.192 10
0.521 55
0.12068
+0.292 45
+0.68640
+1.03536
+2.407 68
+2.52747
+2.54293
+2.460 10
+2.289 96
0483 47
0.097 74
+0.296 28
+0.672 70
+ 1.00985
+2.480 26
+2.596 73
+2.61795
+2.547 65
+2.323 -10
0.522 13
0.15473
+O.222 32
+0.586 91
+0.9I95I
+2.51228
+2.633 69
+2.666 40
+2.61 1 19
+2.47223
0.624 77
0.273 88
+0.091 29
+0.450 79
+0.785 77
11
17
18
19
+1.932 16
+1.61623
+1-26534
+0.900 17
+0.54067
+I.3I993
+ 1.52683
+ 1.64824
+ 1.68095
+1.625 74
+2.04679
+ 1.74690
+1.40795
+1.04837
+0.687 09
+1.29044
+ I-50I 35
+ 1.63329
+ 1.68043
+1.64037
+2.16583
+ 1.87800
+ 1-54509
+ 1.18424
+0.81441
+ 1.20324
+ 142386
+ I.5698I
+ 1.63228
+1-605 53
+2.256 89
+1.97562
+ 1.64197
+ 1.27284
+0.88845
+ 1.07861
+ I.3I3I5
+ 1.47489
+ I.55I 56
+ 1.53398
20
21
22
23
+0.205 69
0.087 15
0.321 69
048343
+148678
+1.271 44
+0.090 17
+0.656 52
+0.343 34
+0.03645
0.214 52
0.391 49
+1-51405
+ 1-30597
+ 1.02464
+0.682 96
+0.456 41
+0.13231
0.134 94
0.322 85
+148737
+ 1-27995
+0.990 76
+0.633 72
+0.51242
+0.171 16
0.107 50
0.296 85
+ I.4I7I5
+ I.2OI 91
+0.89669
+0.519 19
COEFFICIENTS FOR DIRECT ACTION.
75
TABLE XIV. Concluded.
G-COORDINATES OF MARS REDUCED TO THE DIFFERENT SYSTEMS.
System 8.
System 9.
System 10.
System n.
^
f
X
r
X
r
X
r
o
I
2
a
4
0.373 70
0.305 44
0.12072
+O.I597I
+0.507 99
0.065 64
0495 03
0.883 19
1.19920
1.422 23
0.405 03
0.320 16
0.13007
+0.14409
+0.47759
0.20886
0.622 75
0.090 04
1.28664
1497 05
0.465 27
0.376 14
0.192 52
+0.067 42
+0.383 35
0.29258
-0.686 53
1.035 49
1.32006
1.52696
0.54307
0460 24
0.290 10
0.046 93
+0.252 96
029640
0.672 82
1.00997
1.20056
1.501 47
5
6
8
9
+0.893 72
+ 1.28774
+ 1.664 16
+2.001 31
+2.281 90
1.54202
-1.55748
147465
I.3045I
1.06134
+0.84499
+ 1.22204
+1.58663
+ 1.91923
+2.2O2 96
1.613 52
1.634 74
1.56444
1.41019
1.18262
+0.734 24
+ 1.09941
+145891
+ 1.79389
+2.08673
-1.64837
1.681 08
1.62587
148691
-1.271 57
+0.591 91
+0.95149
+1.31277
+1.65652
+1.96341
1-63341
1.680 55
1.640 49
-I.5I4 17
1.30609
10
it
12
13
14
+2.49281
+2.624 75
+2.671 89
+2.631 83
+2.505 51
0.761 45
0.422 50
0.062 92
+0.298 36
+0.642 1 1
+2.423 58
+2.56953
+2.632 oo
+2.605 25
+2.487 09
0.804 79
0.561 88
O.2OI O3
+0.l688o
+0.52680
+2.321 27
+2483 01
+2.55968
+2.542 10
+2.425 27
0.99030
0.656 65
0.287 52
+0.096 87
+0472 90
+2.214 38
+2.391 35
+2478 20
+2.462 42
+2.33742
1.02476
-0.68308
0.29901
+0.104 15
+0.49809
15
16
11
19
+2.29743
+2.016 10
+1.67442
+1.29035
+0.887 19
+0.94900
+I.I9997
+1.37694
+ 1.46379
+144801
+2.27967
+1.09048
+1.63344
+1.22978
+0.808 18
+0.850 oo
+ I.II8I5
+ I.3O6o6
+ 1.39478
+ 1.37002
+2.21003
+1.00481
+ 1-52731
+ 1.10461
+0.672 12
+0.814 16
+1.09282
+1.282 17
+1.36068
+1.31604
+2.105 17
+ 1.77836
+1.381 50
+0.94877
+0.51938
+0.85083
+M30 15
+1.30776
+ 1.36504
+1.20678
20
21
22
23
+0.493 25
+0.14051
0.13881
0.316 42
+I.3230I
+1.09076
+0.763 95
+0.36709
+0.402 86
+0.050 63
0.215 35
0.37093
+ 1.22665
+0.971 76
+0.625 65
+0.21965
+0.268 72
0.069 O2
O.3I2 8O
0.446 25
+1.14929
+0.87548
+0.521 42
+0.120 55
+O.I3I 22
0.18479
0407 82
0.527 61
+ 1.11206
+0.831 63
+0483 35
+0.097 62
TABLE XV.
SPECIAL VALUES OF A, B, C, AND D FOR MARS.
System o.
System i.
i
A
B
C
D
i
A
B
C
D
I
2
3
4
+ 1.92041
+0.250 33
0.17822
0.175 IS
O.I 1 1 SO
0.768 57
+0.382 65
+0.507 19
+0.35960
+0.225 06
1.15195
0.63303
0.32897
0.18443
0.11356
+i.iii 16
+0.954 82
+0.358 95
+0.075 13
0.029 05
O
I
2
3
4
+2.08860
+0.705 52
-0.08866
0.204 72
0.14686
1.02497
+0.038 62
+0.515 53
+0.447 51
+0.293 Si
1.06403
0.744 19
0.426 83
0.242 79
0.146 66
+043399
+1.07690
+0.56893
+0.16480
0.008 13
6
8
9
0.058 53
0.021 97
+0.002 45
+0.01880
+0.029 90
+0.13450
+0.076 72
+0.039 44
+0.01503
o.ooi 19
0.076 06
0.054 76
0.041 89
0.033 83
0.028 72
0.06 1 07
0.065 74
0.060 08
0.050 72
0.040 17
5
6
8
9
0.079 42
0.030 47
+0.001 61
+O.O22 O2
+0.034 83
+0.17465
+0.096 73
+0.04744
+0.016 43
0.003 03
0.095 23
0.066 25
0.049 04
0.038 44
0.031 80
0.065 51
0.076 39
0.069 91
0.057 60
0.043 79
10
it
12
13
14
+0.037 54
+0.042 79
+0.046 23
+0.048 05
+0.04795
0.01198
0.01892
0.02281
0.023 80
0.021 35
0.025 57
0.023 87
0.023 43
0.024 25
0.026 60
0.029 29
0.018 19
0.006 65
+0.005 80
+0.01984
10
ii
12
13
14
+0.042 73
+0.04731
+0.049 46
+0.049 46
+0.04697
0.014 99
O.O2I 8 1
0.024 68
0.023 96
0.019 12
0.027 72
0.025 51
0.024 77
0.025 49
0.027 83
0.029 08
0.01646
0.003 01
+0.0108 1
+0.025 58
15
16
17
18
19
+0.044 92
+0.036 65
+0.01830
0.019 93
0.006 22
0.013 81
+0.002 37
+0.034 61
+0.098 16
+0.223 72
0.031 10
0.03901
0.052 92
0.078 25
0.127 50
+0.036 30
+0.056 01
+0.079 oo
-fo.ioi 49
+0.10486
11
17
18
19
+0.040 91
+0.029 03
+0.007 '9
0.031 63
0.097 28
0.008 56
+O.OI I O4
+0.045 88
+0.10709
+0.212 09
0.032 35
0.040 06
0.053 06
0.075 45
0.11569
+0.041 77
+0.05944
+O.077 22
+0.08944
+0.077 88
20
21
22
23
0.230 56
0.33621
+0-347 58
+2.549 oo
+0461 52
+0.797 25
+0.580 43
1.148 74
0.23095
0.461 05
0.927 87
1.400 16
+0.016 40
0.30696
1.39090
1.02700
20
21
22
23
0.192 14
0.245 44
+0.12735
+ 1.52082
+0.384 18
+0.58762
+0495 62
0.53246
0.192 04
O.342 21
0.622 oo
0.98831
o.oio 06
0.302 42
0.922 oo
1.091 63
7 6
ACTION OF THE PLANETS ON THE MOON.
TABLE XV .Continued.
SPECIAL VALUES OF A, B, C, AND D FOR MARS.
System 2.
System 3.
i
^4
B
C
D
i
A
B
C
D
i
2
3
4
+2.IOI 41
+ I40O 17
+0.08945
0.247 49
0.192 78
1.045 67
0.44244
+0.515 34
+0.585 27
+0.385 57
1.055 73
0.957 75
0.604 76
0.337 75
0.192 79
0.297 76
+1.11788
+0.885 66
+0.289 97
+0.007 96
o
i
2
3
4
+1-94747
+2.37680
+0.345 55
-0.32752
0.238 32
-0.823 38
1.04803
+0.55368
+0.791 16
+047668
1.12408
1.32882
-0.899 15
0.463 59
0.238 32
0.979 52
+1.03457
+1.34603
+0.41324
0.005 73
6
8
9
0.09908
0.033 27
+0.00621
+0.028 67
+0.04097
+0.21683
+0.1 10 73
+0.048 52
+0.012 65
0.007 72
O.H774
-0.077 45
-0.054 73
0.041 33
0.033 24
0.079 02
0.091 19
0.07934
0.06 1 50
0.043 55
6
I
9
0.104 37
0.024 71
+0.01621
+0.036 14
+0.045 21
+0.237 83
+0.10704
+0.039 39
+0.00461
0.013 oo
0.13343
0.082 30
0.055 58
0.04075
0.032 22
0.10406
0.10454
0.082 65
0.059 12
0.038 63
10
ii
12
13
14
+O.O47 22
+0.049 75
+0.04971
+0.04748
+0.042 80
0.01880
0.023 94
0.024 83
0.022 06
0.015 30
0.02843
0.025 8 1
0.024 87
0.025 41
0.027 49
0.027 03
0.01198
+O.O02 O7
+0.015 66
+0.02929
10
ii
12
13
14
+0.048 58
+0.048 76
+0.046 87
+0.043 26
+0.037 75
O.O2I 27
O.024 O7
0.023 1 8
O.OI9 21
O.OI2 O2
0.027 30
0.024 68
0.023 70
0.024 06
0.025 74
0.021 44
0.006 8 1
+0.006 13
+0.01807
+0.029 52
IS
16
17
18
19
+0.034 78
+0.021 75
+O.OOOO2
0.031 96
0.082 07
0.003 32
+0.016 33
+0.04785
+0.09826
+0.178 16
0.031 46
0.038 07
0.048 76
0.066 29
0.09609
+0.043 18
+0.05704
+0.06929
+0.075 22
+0.062 41
15
16
19
+0.029 7O
+0.017 96
+0.00066
0.025 10
0.063 25
0.000 75
+0.016 27
+0.041 06
+0.081 14
+0.141 60
0.028 96
0.03424
0.042 62
0.056 05
+0.078 35
+0.040 68
+0.05137
+0.060 55
+0.065 29
+0.058 10
20
21
22
23
0.149 84
O.2OI 17
0.051 76
+0.807 50
+0.299 14
+045032
+0491 04
0.049 04
0.14931
0.249 17
043933
0.75848
O.OOO6O
0.18647
0.605 51
-1.06638
20
21
22
23
0.11689
-0.175 77
0.15423
+0.334 45
+0.23443
+0.366 06
+0493 56
+0.306 39
O.II753
0.19120
0.33931
0.640 8 1
+O.O2O OS
0.096 25
0.306 04
0.96723
System 4.
System 5.
o
I
2
3
4
+ 1.832 10
+3-797 02
+0.44209
0.454 66
0.246 47
0.50077
1.806 10
+0.849 47
+ 1.01321
+049794
I-33I 96
1.99000
1.29202
0.558 66
-0.251 49
1.62480
+ 1.03488
+ 1.92787
+0.406 52
0.065 O4
O
I
2
3
4
+2.148 15
+5-764 33
0.089 37
0.503 27
0.19356
0.293 03
-2.665 97
+ 1.600 16
+1.02700
+0.410 28
-1.855 19
-3-098 75
1.51072
-0.523 75
0.21673
2.502 41
+2.0O2 IO
+2.II620
+0.192 32
0.12306
5
6
8
9
0.083 76
0.00731
+0.026 10
+0.040 16
+0.045 38
+0.213 91
+0.084 19
+0.024 79
0.003 05
0.01600
0.130 15
0.076 89
0.050 89
0.037 10
0.02938
O.I2683
O.IO605
O.076 42
0.051 33
0.031 65
6
8
9
0.040 74
+0.007 54
+0.030 46
+0.039 37
+0.042 28
+0.15937
+0.057 46
+0.013 21
0.006 08
0.016 18
0.10963
0.065 oi
0.043 66
0.032 40
0.020 09
0.12733
0.00438
0.064 96
O.O42 72
0.026 O3
10
ii
12
13
14
+0.046 40
+0.045 24
+0.042 68
+0.038 98
+0.033 97
0.021 39
0.022 53
0.02089
0.016 95
0.010 59
0.025 oi
O.022 70
O.02I 80
O.O22 O4
0.023 38
0.016 ii
0.003 35
+0.007 65
+0.017 63
+0.027 09
10
ii
12
13
14
+0.042 40
+0.041 09
+0.038 92
+0.035 97
+0.032 ii
0.019 89
0.02051
0.019 06
0.015 89
o.oio 87
O.022 51
O.O2O 60
0.01985
O.020 07
O.O2I 23
O.OI3 03
0.002 39
+0.006 85
+0.015 33
+0.023 53
IS
16
H
19
+0.027 18
+0.017 72
+0.004 14
0.015 94
0.046 43
o.ooi 19
+0.012 55
+0.032 90
+0.063 85
+0.11241
0.025 99
0.030 27
0.037 05
0.04791
0.065 99
+0.03551
+0.045 37
+0.053 75
+0.05991
+0.059 39
15
16
17
18
19
+0.026 95
+0.01975
+0.009 24
0.006 86
0.032 71
0.003 45
+0.007 46
+0.023 92
+0.049 63
+0.091 80
0.023 48
O.0272I
0.033 15
0.042 78
0.059 09
+0.031 81
+0.040 39
+0.04924
+0.057 70
+0.063 25
20
21
22
23
0.092 87
0.15848
-0-357 45
+0.032 96
+0.19096
+0.31861
+ 1.20062
+0.579 17
-0.09808
0.160 13
0.843 21
0.612 03
+0.040 04
0.032 86
1.00468
0.87868
20
21
22
23
0.075 96
0.14903
-0.253 88
0.17633
+0.16481
+0.208 19
+0.544 16
+0.859 Si
-0.08886
0.149 19
0.290 26
0.683 21
+0.057 56
+0.012 49
0.17474
0.884 34
COEFFICIENTS FOR DIRECT ACTION.
77
TABLE XV '. Continued.
SPECIAL VALUES OF A, B, C, AND D FOR MARS.
System 6.
System 7.
i
^
B
C
D
1
A
B
C
D
o
+4-029 84
-0.912 73
-3.11674
4.018 27
o
+9.99063
4.58621
5404 24
4.012 20
i
+5-687 73
-1.695 28
3.992 42
+4.843 56
i
+145390
+1.601 76
-3.055 56
+4.666 10
2
0.709 S3
+ 1.92811
1.2187.)
+1.29050
2
0.656 79
+1.41386
-0.757 IS
+0.48293
3
-0.38833
+0.774 20
0.385 94
0.014 03
3
0.254 14
+0.515 57
0.261 46
0.07932
4
0.12466
+0.287 92
0.163 25
0.13292
4
0.081 45
+0.202 75
O.I2I 32
0.11385
5
0.024 99
+O.III 12
O.O86 12
0.10995
5
0.014 79
+0.083 51
0.068 73
0.09063
6
+0.013 63
+0.039 63
0.053 26
0.07885
6
+0.012 94
+0.032 oo
0.04493
0.066 72
7
+0.029 34
+0.007 77
0.037 10
0.054 60
7
+0.025 42
+0.007 18 ! 0.032 59
0.048 03
8
+0.035 71
0.007 35
0.028 35
0.03668
8
+0.031 23
0.00558 . 0.02565
0.033 80
9
+0.037 98
0.014 63
0.023 35
O.023 12
9
+0.033 91
0.012 35 0.021 58
O.O22 65
10
+0.038 32
0.017 86
O.O2O 46
0.012 38
10
+0.035 04
0.015 85 0.019 18
0.013 49
ii
+0.037 65
0.018 73
O.OI892
0.003 37
ii
+0.035 30
0.01736 ' 0.01792
0.005 54
12
+0.03633
0.017 98
O.OI83S
+0.004 65
12
+0.035 oo
0.01748
0.017 52
+0.001 80
13
+0.034 45
0.015 82
O.OI862
+0.012 23
13
+0.034 24
0.016 35
0.017 88
+0.00900
14
+0.031 88
0.012 14
0.01973
+0.019 84
14
+0.032 oo
0.013 83
0.019 06
+0.016 52
IS
+0.02828
0.00641
O.O2I 87
+0.027 88
IS
+0.030 68
0.00942
O.O2I 27
+0.024 84
16
+0.023 oo
+0.002 40
O.O25 42
+0.036 73
16
+0.02696
0.002 02
0.024 94
+0.034 5i
17
+0.014 84
+0.016 27
O.03I II
+0.046 70
17
+0.020 46
+0.010 52
0.030 99
+O.046 22
18
+0.001 39
+0.039 19
0.040 59
+0.05791
18
+0.008 52
+0.032 70
0.041 23
+0.06066
iQ
O.O22 OS
+0.079 14
0.057 07
+0.069 1 1
19
+0.008 53
+0.032 71
0.041 24
+0.06067
20
0.065 65
+0.154 13
0.08847
+0.074 45
20
0.063 61
+0.16008
0.097 38
+0.093 42
21
0.154 98
+0.315 09
0.160 14
+0.04951
21
-0.175 09
+0.361 07
0.186 02
+0.07793
22
-0.315 17
+0.648 46
0.333 30
0.133 72
22
0432 50
+0.882 46
0450 02
0.15994
23
-0.353 82
+1.27669
0.922 91
1. 121 87
23
0.40883
+1.93710
1.528 17
-1.99273
System 8.
System 9.
+ 11.25307
5.464 71
-5.78803
+3.030 oo
+4.78203
1.30327
-347885
+4.27464
i
0.29587
+ 1.96385
-1.66747
+2.25142
I
0.362 oo
+I-33I 70
0.96966
+I.I8354
2
0.44425
+0.913 37
046909
+0.189 10
2
+0.31773
+0.652 46
0.334 77
+0.12970
3
0.17840
+0.366 62
0.18819
0.073 90
3
0.14787
+0.301 39
0.153 52
0.050 95
4
0.06393
+0.16071
0.096 78
0.091 97
4
0.062 08
+0.14789
0.085 80
0.074 57
S
0.01447
+0.073 33 0.058 87
0.076 63
5
0.019 55
+0.074 61
0.055 06
-0.067 95
6
+ 0.00879
+0.03156 0.04036
0.059 48
6
+0.002 94
+0.036 19
0.039 12
0.056 34
7
+ 0.02061
+0.009 67
0.030 29
0.045 12
7
+0.015 67
+0.014 40
0.030 06
0.045 14
8
+ 0.02700
0.002 58
0.024 40
0.033 52
8
+0.023 35
+O.OOI 26
0.024 62
0.035 28
9
+ 0.03062
0.009 76
O.O20 87
0.023 97
9
+0.028 30
0.007 oi
O.O2I 28
0.026 63
10
+ 0.03279
0.014 oo
0.018 79
0.015 76
10
+0.031 67
0.012 37
O.OI9 3O
0.018 82
ii
+ 0.03409
0.016 39
0.01771
0.008 34
ii
+0.034 10
0.015 80
O.OlS 29
O.OI I 46
12
+ 0.03485
0.017 42
0.01743
o.ooi 23
12
+0.035 9i
0.01780
0.01811
0.004 13
13 + 0.035 17
0.017 25
0.01791
+0.006 02
13
+0.037 22
0.0 1 8 50
0.0 1 8 72
+0.003 63
14
+ 0.03495
0.015 70
0.019 25
+0.013 89
14
+0.03706
0.017 67
O.O2O 28
+0.012 31
IS
+ 0.03390
O.OI22I
O.O2I 69
+0.022 97
IS
+0.037 78
0.014 65
0.023 14
+0.022 74
16
+ 0.031 36
0.005 56
0.025 8 1
+0.03403
16
+0.035 87
0.007 85
0.028 oi
+0.035 88
17
+ 0.02684
+0.00688
0.032 72
+0.048 15
17
+0.030 24
+0.006 33
0.036 58
+0.053 56
18
+ 0.01399
+0.03088
0.044 86
+0.066 76
18
+0.016 16
+0.035 63
0.051 78
+0.077 10
19
0.01233
+0.080 37
0.068 02
+0.09094
19
0.018 70
+0.101 31
0.082 60
+0.10859
20
0.07503
+0.193 18
0.11815
+0.1 16 14
20
0.10951
+0.263 59
0.154 08
+0.13735
21
0.23787
+048697
0.249 14
+0.094 95
21
0-357 72
+0.716 16
0.358 39
+0.056 10
22
0.64095
+ 1.34280
0.701 91
0.372 76
22 0.781 89
+1.91271
1.13082
I.O52 12
23
+ 0.77527
+2.043 47
-2.818 93
-4-2S3 90
23 +4-977 43
-0.931 5i
-4.045 08
-5.38850
8o
ACTION OF THE PLANETS ON THE MOON.
C. ACTION OF JUPITER.
39. The action of Jupiter being computed on the same general method as
Venus and Mars, but being much simpler, no detailed explanation seems
necessary. Six systems, which suffice to carry the coefficients to terms of the
third order in the eccentricities, were deemed enough.
The principal numbers used or derived are shown in the following tables. The
fundamental data in the first table were derived from Hill's Tables of Jupiter.
TABLE XVII.
ECLIPTIC COORDINATES OF JUPITER FOR THE 12 POINTS OF DIVISION.
_/"+ Red. to
log. T
Arg. i.
Ecliptic
(Table 37).
(Table 60).
1
/ a.
X
y
z
o
1062.6047
ota
93 49 10
0.716 624
JOS I 44
185 31 36
-5.1832
0.5015
+0.0137
I
I423-6537
123 o 50
0.726 798
134 13 23
214 43 16
4-3816
3.0364 +0.0714
2
1784.7027
151 3 o
0.733 92
162 15 33
242 45 26
24808
4.8184 , +0.1114
3
2145.7517
178 26 SS
0.736 696
189 39 28
270 9 21
+0.0148
5-4537 +0.1248
4
2506.8007
205 48 41
0.734514
217 I IS
297 31 7
+2.5072
4-8125
+O.IO9I
5
2867.8497
233 44 42
0.727 800
244 57 16
325 27 8
+4.4010
3.0300
+0.0675
6
3228.8987 262 47 7
0.717865
273 59 41
354 29 33
+5.1982
0.5012 +0.0092
7
3589.9477 : 293 16 53
0.707018
304 29 26
24 59 19
+4.6167
+2.1517 0.0512
8
3950.9967
325 13 29
0.608 343
336 26 2
56 55 55
+2.7242
+4.1841
0.0969
9
4312.0457 ' 358 7 4
0.694 769
9 19 38
89 49 30
+0.0151
+4.9518
0.1 134
10
340.5067
31 3 59
0.697 634
42 16 32
122 46 24
-2.6083
+4.1912
0.0948
ii
701.5557
63 9 2
0.705 860
74 21 36
154 Si 28
-4.5987
+2.1583
-0.0474
TABLE XVIII.
JUPITER; DIRECT ACTION; SPECIAL VALUES OF THE A-COEFFICIENTS FOR 6 SYSTEMS.
System o.
System i.
i
A
B
C
D
i
A
B
C
D
+0.008 62
0.004 22
0.004 40
+0.001 56
+0.007 65
0.003 80
0.003 83
+0.00060
i
+0.002 35
o.ooi 17
0.003 52
+0.005 25
I
+0.003 28
+0.000 13
0.003 40
+0.004 86
2
o.ooi 91
+0.004 50
0.002 59
+O.OO2 2O
2
0.001 51
+0.004 27
0.002 76
+0.002 97
3
o.oo i 76
+0.003 72
o.ooi 95
o.ooi 04
3
O.O02 13
+0.004 33
O.OO2 20
o.ooo 70
4
+0.000 05
+0.001 53
o.ooi 58
O.002 26
4
o.ooo 13
+0.001 95
o.ooi 82
O.OO2 52
5
+O.OOI 81
o.ooo 39
o.ooi 41
o.ooi 8 1
5
+0.002 06
o.ooo 46
O.OOI 6l
0.00206
6
+0.002 77
0.001 37
o.ooi 40
0.00034
6
+0.003 II
0.001 55
o.ooi 56
o.ooo 19
7
+0.002 49
0.00095
0.001 54
+0.001 55
7
+0.002 43
0.000 76
o.ooi 67
+0.001 93
8
+O.OOO 61
+0.001 30
o.ooi 91
+O.OO2 84
8
+O.OOO IO
+0.001 87
o.ooi 97
+0.00281
9
O.OO2 28
+0.004 86
O.OO2 58
+0.001 SO
9
0.002 39
+0.004 87
O.OO2 48
+0.00081
10
0.002 O4
+0.005 63
0.003 58
0.003 77
10
o.ooi 40
+0.004 56
0.003 15
0.00368
ii
+0.005 41
o.ooo 95
0.004 46
0.005 89
II
+0.004 49
0.00075
-0.003 73
0.004 95
System 2.
System 3.
O
+0.007 67
0.003 8 1
0.003 85
0.00069
o
+0.008 73
0.004 27
0.004 46
0.001 58
I
+0.004 63
o.ooo 83
0.003 8 1
+0.005 02
I
+0.005 47
o.ooo 95
0.004 51
+0.005 97
2
0.001 42
+0.004 67
0.003 25
+0.003 80
2
0.002 06
+0.005 67
0.003 60
+0.003 79
3
0.002 45
+0.005 oo
0.002 55
0.00084
3
O.OO2 28
+0.004 86
O.O02 58
o.ooi 50
4
+O.OOO 12
+0.001 88
O.OO2 OO
O.OO2 87
4
+0.000 61 +0.001 29
o.ooi 90
0.002 82
5
+O.O02 46
o.ooo 78
o.ooi 68
o.ooi 92
5
+0.002 47
o.ooo 94
o.ooi 53
o.ooi 54
6
+0.003 09
0.001 54
0.001 55
+O.OOO 21
6
+0.002 74
o.ooi 36
0.001 39
+0.000 33
7
+O.OO2 01
0.00043
o.ooi 58
+O.002 04
7
+0.001 79
o.ooo 39
0.001 40
+0.001 80
8
o.ooo 15
+0.001 92
o.ooi 76
+O.OO2 46
8
+0.00006
+0.001 52
o.ooi 58
+O.OO2 25
9
O.OO2 08
+0.004 23
O.OO2 15
+0.00068
9
o.ooi 76
+0.003 7i
0.001 95
+o.ooi 04
10
o.ooi 49
+0.004 19
O.OO2 70
O.OO2 OO
10
o.ooi 92
+0.004 52
O.O02 OO
O.OO22I
ii
+0.003 18
+0.000 19
0.003 36
0.004 82
II
+O.OO2 36
+O.OOI 19
0.003 55
0.005 30
COEFFICIENTS FOR DIRECT ACTION.
8l
TABLE XVIII. Concluded.
JUPITER ; DIRECT ACTION ; SPECIAL VALUES OF THE A-COEFFICIENTS FOR 6 SYSTEMS.
System 4.
System 5.
i
A
B
c
D
i
A
B
C
D
o
+0.010 47
O.OOS 21
0.005 26
0.00094
o
+0.01036
0.005 IS
O.OO5 21
+0.001 06
I
+0.004 29
+0.00044
0.004 73
+0.00684
I
+0.002 55
+0.001 57
O.OO4 12
+0.006 16
2
0.002 58
+0.005 91
0.003 33
+O.OO2 64
2
0.002 34
+0.005 16
O.OO2 82
+0.001 95
3
o.ooi 75
+0.004 oo
0.002 25
o.ooi 77
3
0.001 58
+0.003 55
0.001 97
0.001 48
4
+0.00068
+0.00096
o.ooi 65
O.OO2 46
4
+0.00041
+O.OOI II
o.ooi 51
O.OO2 24
5
+0.002 l6
0.00080
o.ooi 35
0.001 39
5
+0.001 88
0.000 57
0.001 30
0.001 52
6
+0.00251
o.ooi 25
o.ooi 25
+0.000 14
6
+O.002 51
0.001 25
0.001 26
0.000 16
7
+0.001 90
0.000 59
o.ooi 31
+0.001 52
7
+O.OO2 2O
0.00083
0.001 37
+0.001 39
8
+O.OOO 43
+O.OOI II
o.ooi 54
+O.OO2 28
8
+0.000 72
+0.00096
0.00168
+0.002 52
9
o.ooi 6 1
+0.003 62
O.OO2 O2
+O.OOI 52
9
0.001 79
+0.004 10
O.OO2 31
+O.OOI 82
10
0.002 41
+0.005 30
0.00290
O.O02 OI
IO
O.O02 64
+0.00606
0.00342
O.OO2 72
ii
+0.002 68
+0.001 55
0.004 23
0.006 32
II
+0.00444
+0.00036
0.004 80
0.006 91
TABLE XIX.
JUPITER ; DIRECT ACTION ; DEVELOPMENT OF THE A-COEFFICIENTS.
Arg.
ufA
10* 8
io C
ioZ>
LJ, g'
cos
sin
cos
sin
cos
sin
cos
sin
O O
+1347
o
+1237
O
-2583
O
I
O
O I
4
-58
i
19
+ 4
+ 76
- 18
+ 3
12
7
- 77
+ 17
+ 86
12
IO
+ 82
13
I I
+ III2
o
+ 358
o
1470
o
o
+ 539
I O
II
211
- 16
-184
+ 28
+395
o
+ 2
I+I
3
O
__ j
+ 2
+ 5
o
+ I
23
3
- 82
+ 8
+ 82
4
i
+ 83
5
2 2
+3878
+ 2
-3526
u , C
- 355
+ 2
3
+3699
21
7
-165
5
S3
+ 12
+217
+ 53
2
2 O
- 16
+ 2
IS
+ 3
+ 30
5
+ i
II
o
O
o
o
o
o
O
34
+ 13
40
12
+ 40
3
o
+ 42
+ 12
33
+ 1779 + I
I7OO
3
- 79
+ I
2
+ 1737
32
28
-6 7 2
+ 24
+S9S
+ 5
+ 77
+633
- 25
3-1
- 18
+ 3
- 6
+ i
+ 24
4
O
- 6
4-5
+ 10
IO
- 8
+ 10
+ I
+ 21
+ 9
4-4
+ 587
3
570
+ I
- 18
+ I
+ 2
+ 575
43
- 18
434
+ 16
+412
+ 2
+ 22
+423
17
42
- 80
+ 7
+ 70
6
+ H
- 6
- 76
5-6
+ 2
+ 16
3
14
+ I
I
+ 26
+ 6
5-5
+ 176
2
172
+ 2
3
+ 3
+ 154
5-4
- 8
-185
+ 7
+180
+ 5
+184
- 8
53
- 66
+ 6
+ 64
- 6
+ 4
i
- 6
- 67
82 ACTION OF THE PLANETS ON THE MOON.
D. ACTION OF SATURN AND MERCURY.
40. The inequalities due to the direct action of Saturn are so minute that an
approximate development will suffice. I have therefore used the development of
A~ 3 and A~ 5 by spherical harmonics. We put a^ for the mean distance of Saturn,
Z, as usual, for the difference of mean heliocentric longitudes of the planet and
Earth (L s g'}, and a for the ratio of the mean distances. With this notation
the developments to a 4 are
J = i + fa* + -Yi 6 - 4 + + (3 + -V- 2 + ) cos L
+ ( + -Vg 5 -" 2 )* 2 cos 2 ^ + Y* 3 cos 3 Z + W* 4 cos 4 Z
^ o, i + -Y- 2 + HI 40 * + + (5 + J F 2 + ) cos Z
* cos 2Z + 1 - a ' cos Z + 1la4 cos Z
This development is valid when the eccentricities are taken account of, provided
we use the true radii vectores and true longitudes instead of the mean ones. But
this is unnecessary in the present case. For Saturn we have
a = 0.1070
Reducing to numbers this gives
s = 1.0262 + 0.328 cos L + 0.044 cos 2 -^ + -o5 cos 3^
TT= 1.0741 + 0.557 cosZ + o.no cos 2Z + .016 cos 3Z +
For the geocentric coordinates X, Y, Z, of Saturn we have
X= a' a, cos Z = a l (a cos Z) Y= a l sin Z Z = o
Then
2^= ^.(X* - Y 2 ) = a 3 ^ (a 2 - 2<z cos Z + cos 2 Z)
/- /'^ a3 i*
C. = ' C = --
3 &
Z>, = a' 3 /? = o|i- (} sin 2Z - a sin Z)
Reducing to numbers, and performing the necessary multiplications we find
= -f .0027 + .031 cos Z + .620 cos 2Z
io s C, = .419 .135 cos Z .018 cos 2Z
io 3 Z>, = .029 sin Z -(- .607 sin 2Z
COEFFICIENTS FOR DIRECT ACTION. 83
Then, the principal terms are
io 3 MK= + .0013 + .015 cos L + .307 cos zL
loU/C, = .208 .067 cos L .009 cos 2Z,
io s MD l = .014 sin L + .302 sin 2.Z,
41. The mass of Mercury is so minute that its action upon Venus, the only
planet whose motion it can sensibly affect, has never been determined with cer-
tainty. There is every reason to believe that the uncertain determinations of the
mass which have been made were too great by 2 or 3 times their entire amount.
From Hill's estimate, based on the volume and probable density of the planet, it is
very probable that the mass is less than i -=- 10000000 that of the Sun. From the
results of 30 it is inferred that its secular effect on the motion of the lunar elements
is proportionally yet smaller than its mass.
The only periodic inequalities that could become sensible are those of compara-
tively long period. Their probable limiting values are considered in Action,
p. 273, from which it appears that the largest inequality is that depending on the
argument
/ + TT + T,M' + g'
and that the limiting value of the coefficient was estimated at o".i. For another
argument the limiting value was o".o4. These estimates rest on a mass double of
what may now be considered the most probable value. For these reasons it was
intended to leave the action of Mercury entirely out of consideration in the present
investigation. But, for the sake of completeness, and to leave open as few questions
as possible, it was at length decided to compute the action in the same way as that
of Venus. Twelve systems and twelve indices were used. With 144 special
values, it is easy to compute not only the secular, but the principal periodic terms.
Among the results are the following constant terms and terms depending on the
above argument, the form being
A = A + A e cos (3M' -|- ') + A, sin (3*1' + g')
A = + 0.867 A c = .00059 A. = o
B 0.381 B c = + .00035 B. = + -00008
C = 0.486 C c = + .00026 C t = .00006
Z> = -f 0.0022 D c = + .0005 D t = .0023
K^ == + 0.624 K e = .00047 K t = .00004
42. K-coefficients. From the preceding developments of A, B, C, and D for
the four disturbing planets the coefficients K = y z (A B) are formed, and
K, 0^ and D t are multiplied by M.
This special set of coefficients, containing the factor M, are designated as K-
coefficients. Their values are tabulated for Venus, Mars, and Jupiter as follows.
The values for Saturn are found at the end of 40 preceding.
82 ACTION OF THE PLANETS ON THE MOON.
D. ACTION OF SATURN AND MERCURY.
40. The inequalities due to the direct action of Saturn are so minute that an
approximate development will suffice. I have therefore used the development of
A~ 3 and A~ 5 by spherical harmonics. We put a 1 for the mean distance of Saturn,
Z, as usual, for the difference of mean heliocentric longitudes of the planet and
Earth (L = s g'}, and a for the ratio of the mean distances. With this notation
the developments to a 4 are
J- = i + fa* + *-a< + . . . + (3 + _45 a 2 + . . . )a CQS L
+ ( + 1 i<r a *) af cos 2L + -^ cos 3^ + - 3 e- a4 cos 4 Z
J - I + -V- 2 + iff** 4 + . . . + (s + IJia* + . . .) a cos L
' cos 2L + 1Aa * cos 3^ + JL - a< cos Z
This development is valid when the eccentricities are taken account of, provided
we use the true radii vectores and true longitudes instead of the mean ones. But
this is unnecessary in the present case. For Saturn we have
OL = O.IO7O
Reducing to numbers this gives
a 3
A = i .0262 + 0.328 cos L + 0.044 cos 2j L + 0.005 cos 3-
Ty = 1.0741 + 0.557 cos L + o.uo cos iL + .016 cos 3-Z, +
For the geocentric coordinates X, Y, Z, of Saturn we have
X '= a' a, cos L = a l (a cos L) Y= a, sin L Z =o
Then
'* a, 5
- F 2 ) = a?-~ (of 20. cos Z -f cos 2Z)
1 3 A
Z>, = ' 3 Z> = of i- (i sin 2Z - a sin Z)
Reducing to numbers, and performing the necessary multiplications we find
io 3 /if = + .0027 + .031 cos Z + .620 cos 2Z
lo'C 1 , = .419 .135 cos Z .018 cos 2Z
io 3 D 1 = .029 sin Z -f .607 sin 2Z
COEFFICIENTS FOR DIRECT ACTION. 83
Then, the principal terms are
io 3 MK= + .0013 + .015 cos L + .307 cos iL
loWC, = .208 .067 cos L .009 cos 2L
ioWZ>, = .014 sin L + .302 sin iL
41. The mass of Mercury is so minute that its action upon Venus, the only
planet whose motion it can sensibly affect, has never been determined with cer-
tainty. There is every reason to believe that the uncertain determinations of the
mass which have been made were too great by 2 or 3 times their entire amount.
From Hill's estimate, based on the volume and probable density of the planet, it is
very probable that the mass is less than i -r- 10000000 that of the Sun. From the
results of 30 it is inferred that its secular effect on the motion of the lunar elements
is proportionally yet smaller than its mass.
The only periodic inequalities that could become sensible are those of compara-
tively long period. Their probable limiting values are considered in Action,
p. 273, from which it appears that the largest inequality is that depending on the
argument
/ + TT + $M' + g'
and that the limiting value of the coefficient was estimated at o".i. For another
argument the limiting value was o".o4. These estimates rest on a mass double of
what may now be considered the most probable value. For these reasons it was
intended to leave the action of Mercury entirely out of consideration in the present
investigation. But, for the sake of completeness, and to leave open as few questions
as possible, it was at length decided to compute the action in the same way as that
of Venus. Twelve systems and twelve indices were used. With 144 special
values, it is easy to compute not only the secular, but the principal periodic terms.
Among the results are the following constant terms and terms depending on the
above argument, the form being
A = A + A e cos (3f' + g') + A t sin (311' -f g')
A a = + 0.867 -A c ~ ~ -oooSP A t = o
B^ = 0.381 B e = + .00035 B, = + -00008
C = 0.486 C c = + .00026 C t = .00006
D a = + 0.0022 D c = -f .0005 D t = .0023
Jf Q = + 0.624 K e = .00047 K t = .00004
42. K-coefficients. From the preceding developments of A, B, C, and D for
the four disturbing planets the coefficients K '= ^4 (A .?) are formed, and
K, C\, and D^ are multiplied by M.
This special set of coefficients, containing the factor M, are designated as K-
coefficients. Their values are tabulated for Venus, Mars, and Jupiter as follows.
The values for Saturn are found at the end of 40 preceding.
8 4
ACTION OF THE PLANETS ON THE MOON.
TABLE XX.
K-COEFFICIENTS FOR DlRECT ACTION OF VENUS.
v, g'
10 s MK e
io 3 MK,
% io s MC c
^io 3 MC,
io 3 MD c
io 3 MD t
li
a
m
tl
o, o
+ 5-903
o.oo
3.406
0.000
o.ooo
o.ooo
0, I
+ 0.44
O.I I
0.30
+0.07
+0.03
+0.33
0, 2
+ 0.13
0.14
0.09
+0.10 +0.05
+0.06
I, -2
+ 0.71
+0.15
0.36
0.08 +0.08
0.52
I, I
+10.95
o.oo
6.03
o.oo
O.OI
4.92
I.
+ 0.17
0.07
O.22
+0.05
O.OO
+O.22
2, 4
+ 0.16
+0.14
0.08
0.08
+0.09
O.I I
2, 3
+ 0.93
+0.19
0.39
0.08
+0.13
0.73
2, 2
+10.75
o.oo
4.99
o.oo
0.01
7.09
2, I
+ O.O2
0.04
0.16
+0.03
O.OI
+0.22
3, -6
+ O.O2
+0.03
O.OI
O.OI
+O.O2
O.OI
3, -5
+ O.I7O
+0.147
0.078
0.075
+O.II
0.13
3, -4
+ 1.08
+0.22
-0.38
0.08
+0.17
0.91
3, -3
+ 10.24
O.OO
4.00
O.OO
O.OI
-7.91
4, 6
+ 0.18
+0.15
0.07
0.07
+0.1 1
0.15
4, -5
+ 1.17
+0.24
0.36
-0.08
+0.20
1.04
4, -4
+ 9-42
o.oo
3.14
o.oo
O.OI
-7.89
5, -8
+ 0.019
+0.028
0.007
O.OI2
+0.024
0.017
S, -7
+ 0.19
+0.14
0.06
0.06
+O.I2
0.16
5, -6
+ 1.21
+0.24
0.32
0.07
+O.2I
-1.05
S, -5
+ 8.40
O.OO
2.43
0.00
O.OI
7-39
6, -7
+ 1.19
+0.24
0.28
0.06
+O.2I
O.I I
6, -6
+ 7-32
o.oo
-1.86
o.oo
o.oo
-6.64
TABLE XXI.
K-COEFFICIENTS FOR DlRECT ACTION OF MARS.
M, g'
IO 3 MK C
io 3 MX,
i^lO 3 MC C
y 2 io> MC,
io 3 MD c
io 3 MD,
0,
0, I
+ 0*0465
O.O2O
H
O.OOO
0.024
0.1006
+0.028
0.000
+0.029
O.OOO
0.008
O.OOO
+O.OIO
I, -2
I, I
I, O
O.OI2
+ O.IIO
O.O25
+0.014
+0.001
0.029
+0.017
0.169
+0.042
0.0 18
0.000
+0.045
0.008
o.ooo
O.OOO
O.OI 2
+0.054
0.000
2, 3
2, 2
2, -I
2,
O.O02
+ 0.194
0.035
o.ooo
o.ooo
+O.O02
0.039
+O.OI I
+0.009
0.129
+0.049
+0.003
O.OIO
+0.001
+0.053
0.016
+0.006
o.ooo
+0.013
+O.O02
O.OO2
+0.165
0.014
O.002
3, 3
3, 2
+ 0.233
0.064
+0.004
0.070
0.093
+0.049
+O.OOI
+0.053
O.OO2
+0.050
+0.217
0.048
4, 4
4, 3
4, -2
+ 0.229
0.094
O.OO2
+0.004
0.103
+0.030
0.064
+0.044
+0.003
+0.001
+0.049
0.028
0.003
+0.000
0.019
+O.22I
0.083
0.003
5, -4
5, 3
5, -2
0.115
0.005
+ 0.009
0.126
+0.051
0.004
+0.037
+0.003
0.006
+0.042
0.030
+0.005
+0.119
0.042
+0.003
0.107
0.004
+0.007
6, -5
6, -4
6, -3
O.I22
0.009
+ O.OI5
0.134
+0.074
0.009
+0.030
+O.OO2
O.007
+0.034
0.029
+0.006
+0.131
0.068
+0.008
0.117
0.007
+O.OI I
COEFFICIENTS FOR DIRECT ACTION.
TABLE XXII.
K-COEFFICIENTS FOR DlRECT ACTION OF JUPITER.
J, g 1
io 3 K e
ioAT.
tfio'AfC.
^jo 3 MC t
ioWZ> c
ioWZ>.
lt
tt
H
tt
II
//
0,
+ 0.091
O.OOO
2.135
OJOOO
O.OOO
0.000
0, I
O.OO2
0.032
+0.003
+0.063
0.030
+0.005
I, 2
O.O2O
0.135
O.OIO
0.008
+O.I3S
O.02I
I. I
+ 0.623
O.OOO
1.215
O.OOO
O.OOO
+0.593
I, O
+ 0.004
O.O22
+0.023
+0.326
O.OOO
+0.003
I, +1
O.O02
O.OOI
+0.004
o.ooo
O.OOO
+O.OO2
2, 3
0.009
0.135
0.003
O.OOI
+0.137
0.008
2, 2
+ 6.II9
+0.006
0.293
+0.001
0.005
+6.114
2, -I
O.OOI
0.093
+O.OIO
+0.179
+0.087
0.003
2,
O.OOI O.OOI
+0.025
0.004
+O.OOI
0.018
3, 3
+ 2.875
+0.003
0.065
+O.OOI
0.003
+2.875
3, 2
0.043
1.048
+0.004
+0.064
+1.046
0.041
CHAPTER V.
PLANETARY COEFFICIENTS FOR THE INDIRECT ACTION.
43. Our next step is to form the coefficients G, J, and / which are the planetary
coefficients for the indirect action, and correspond to K, ^C, and D. These we have
found to be linear functions of the perturbations in the motion of the Earth around
the Sun produced by the action of all the planets. From the way in which they
are formed it will be seen that they should include all deviations in the motion of
the Sun from the actual formulae adopted for the expression of fl as used in deter-
mining the action of the Sun itself. It would therefore be necessary, in strictness,
to include the effect of any corrections that may be necessary to the elements of
the Sun's motion employed by Delaunay. But as the eccentricity of the Earth's
orbit enters as a symbolic quantity into the theories of both Delaunay and Brown,
it will not be necessary to apply any correction on this account. The same remark
applies to the position of the Earth's perihelion. But as the solar elements are
assumed to be constant in the first integration it is necessary to take into account
the eftects of their secular variations, as well as of the periodic inequalities.
Moreover, in developing the action of the Sun upon the Moon for the first inte-
gration, it is assumed that the mean distance of the Earth's orbit is strictly connected
with its mean motion by the fundamental relation
It is therefore necessary to include in 8p' the constant correction arising from the
action of the planets.
We may conveniently classify the various terms of 8v' and 8p which are to be
used in the expressions (60) as follows:
1. The terms arising from the secular variation of the eccentricity of the
Earth's orbit.
2. Constant and periodic terms independent of the mean longitude of the dis-
turbing planet.
3. Periodic terms containing that mean longitude.
44. Secular terms arising' from the variation of the eccentricity of the
Earth's orbit.
The action of the Sun upon the Moon being a function of the eccentricity of the
Earth's orbit it follows that the indirect action will vary with that element. The
variation may be taken account of by assigning to 8v' and 8p' the increments of the
87
88 ACTION OF THE PLANETS ON THE MOON.
Earth's polar coordinates due to the variation of the eccentricity. It is not necessary
to take into account the variation of the solar perigee, because this element is
retained in its general form in the final expressions of all the perturbations.
To find the required values of oV and 8p' we differentiate the expressions for v'
and p' in terms of the eccentricity, thus obtaining
=-7 = (2 %e' ) sin^' + (|-' ^-e' ) sin 2g' + *e' sin %g' + Vr^' s ' n 4"'
(84)
- Hk' V***) cos zer' yc' 2 cos 3^-' 14-c' 3 cos $g'
Putting Ae' for the increment of e' due to secular variation, the values of 6V and
Sp' to quantities of the first order are found by multiplying these derivatives by Ae '.
To determine what terms of higher order are necessary we remark that for an
interval of 1000 years before or after 1900 we have
A*' = .000418 = 86".o
whence
(A*') 2 = o".o 35
This quantity is so small that the powers of Ae' above the first order may be dropped.
But Ae' will contain terms in T 2 which it will be well to include for the sake of
approximation to rigor in the theory.
Substituting in the values of the differential coefficients just found the numerical
value of e' for 1850,
e' = .0167711
we shall have
8v'
5-7 = 1.999 79 sin^-' -f .041 92 sin 2g' + .000 91 sin T,g' -\- .000 02 sin
e p >
, = .008 39 .999 68 cos^-' .025 15 cos 2g' .000 60 cos 3^-' .000 01 cos $g'
The value of Ae' by which these expressions are to be multiplied is that used in
the author's Tables of the Sun:
Ae' = - 8".595 T- o".026o T 2
T being counted in centuries from 1850.
The corresponding portions of G, J, and /are found by substituting for oV and
Sp' in the expressions (60) the quantities
e
COEFFICIENTS FOR INDIRECT ACTION. 89
If we suppose that G, J, and /are expressed in the form
G=G
with similar forms for J and /, we find by developing to e' 2
- -* cos *g - i - e cos
/, = - K - (I + i^'") cos^' - ff' cos 25-' - ^' 2 cos 3^'
7 i = (3 ~ W*'*) ' + -V-*' sin 2g' + *&e' 2 sin $g'
The following numerical values have not been formed from these, but by multi-
plying the numerical values of the factors given in (60) and (85), which are derived
from developments to e' .
<9, = -f .06238 2.24517 cos^' .21296 cos 2g' .01115 cos 3g'
/, = .01257 0.75039 cosg-' .03771 cos 2g' .00140 cos 3g' (86)
/, = + 2.96280 sin g' + .21366 sin 2g' + .04342 sin $g'
45. Terms independent of the mean longitude of the disturbing planet.
These terms arise from the terms of Sv ' and 8/>' which are either constants, or func-
tions of g' alone. In the case of the longitude the eccentricity and perihelion of
the Earth's orbit are so adjusted that both the constant terms and those dependent
on Arg. g' shall vanish, leaving the only terms of 8v' to be considered those depend-
ing on Arg. 2g' etc. Both these terms themselves and the factors by which they
are subsequently multiplied to form G, f, and / are so minute that the results are
assumed to be insensible; we have, therefore, only to consider the terms of 8p'
which remain after the adjustment of the eccentricity and perihelion just mentioned.
These might be derived from the numbers in Tables of the Sun; but the author
finds that the results have not been carried out with the precision desirable in
the present problem. He has, therefore, computed these terms independently from
theory, using the method of variation of elements, and carrying the results to terms
of the second order in the eccentricities and mutual inclination. The general for-
mula are as follows.* The accented quantities refer to the outer planet.
Action of an outer on an inner planet.
Bp = m'a{p a + jOj cos II + (/ 0>c + ft,,, cos II) cos^ + ft,, sin TI sing-}
where
*The derivation has appeared in the Astronomical Journal, vol. xxv.
90 ACTION OF THE PLANETS ON THE MOON.
Action of an inner on an outer planet.
V = m{pj + pj cos IF + (p a / 4- p liC ' cos IF) cosg-' 4- p lt , sin IF sing'}
where
IF = TT' - TT
, r
Pl= 2J
e e
Pi c C 2 -^ + -^Mi ft / = ~
4 4
The two actions are mutually interchanged by replacing Z> by (i + D) in
either. They were, however, developed independently in order that this relation
might serve as a test of the accuracy of both.
The coefficients A and A are functions of the mutual inclination of the orbits
(<r = sin -j/) and of the coefficients b^ defined by the development
(i 20. cos L 4 a 2 )' = JS^ cos iL
D n means the nth derivative as to log a, or the symbolic value of [a(5/5a)] n .
If it be desired to use the usual successive derivatives as to a itself, we may do so
by the substitutions
D = Z> a Z? 3 = aZ? a 4- / D\ IP = aD a + T > <D\ 4- a?D\
From the numerical values of the coefficients A , A^ and their Z>'s we have the fol-
lowing results:
Action of Venus io g &p' = + 1443.0 + 31 cosg-' 17 sing-'
Mars 30. 4- ncosg-' Ssing-'
Jupiter 1183.1 + 90 cosg-' 4- 50 sin g-'
Saturn 55.4
Uranus i.o
Total + 173.5 + 132 cos g-' + 25 sing'
Additional to these we have, for Mercury, with mass io~ 7
io 9 V = + 38.0 7 cos g' + 3 sing-'
which I treat separately.
COEFFICIENTS FOR INDIRECT ACTION.
From these (60) gives the following coefficients for G,J, and /, these quantities
being expressed in the torm
G=G,+ G e cos g' + G, sing'
Action of
ioG
10 9 <? c
ioG.
lo'/o
icPJc io*J, io 9 /o io 9 / c
I0 9 /.
Venus
+3 2 4 6
.8
+ 233-3
- 3 8.2
+ 1083.3
+ 77-
8
-12.8
+ 1
3
-217.7
Mars
- 66
.8
+ 21.4
- 18.0
- 22.3
+ 7-
2
- 6.0
+
.6
+ 4-5
Jupiter
-2655
.0
+ 68.8
+ 112.3
- 886.0
+ 22.
8
+ 37-5
3
7 I
+ 178.2
Saturn
- 124
.6
- 6.3
o
41.6
2.
i
o
+ 8.4
Uranus
2
.2
.1
- .8
o
o
+ .2
Total
+ 397
.6
+316.8
+ 56.1
+ 132-7
+ I05-
5
+ 18.7
i
.9 -.1
- 26.6
Mercury
85
.1
- 11.4
+ 6.7
+ 28.4
- 3-
9
+ 2.2
o
.2 O
- 5-7
(87)
The totals here given are not formed by addition, but by an independent com-
putation of the entire amount. Hence small discrepancies between the totals and
the sums.
46. Periodic perturbations of the -point G, containing the mean longitude
of the disturbed planet.
These are taken from Astronomical Papers^ Vol. Ill, Part V, where they are
found :
For Venus, on pp. 486-488
For Mars, " " 527-530
For Jupiter, " " 550-551
The perturbations by Venus are diminished by the factor .015 for reduction to the
adopted value of the mass.
The expressions thus found are shown in tabular form below. In the original
the constituents of the arguments were the mean anomalies alone, but, in the present
work, the longitudes of the disturbing planets are reckoned from the Earth's perihe-
lion. In order that the arguments for the direct and indirect inequalities may coin-
cide, these perturbations have been transformed so that the planet's mean longitude
shall be reckoned from the Earth's perihelion. The following are the numbers used:
Earth
Venus
Mars
Jupiter
129 27
333 18
ii 56
o o'
29 6
232 57
271 35
Then, if any pair of terms in 8v' or 8p' be represented by
v, cos (t'g t + i'g') + v, sin (ig^ + t'g ')
9 2
ACTION OF THE PLANETS ON THE MOON.
we have to compute
to transform them into
b c = v e cos t(-7r t IT') v t sin t(ir t TT')
b t = v c sin t(v t IT') + v t cos i(ir 4 TT')
6, cos(/7 4 + i'g') + b, sm(t7 t + i'g')
/ 4 being the mean longitude of the planet from TT', designated by v, M, j, and s in
the cases of the individual planets.
The original and transformed values of &V' and 8p' are shown in Tables XXIII
to XXVI.
The subsequent steps are shown in Tables XXVII-XXXIV in the following
order:
The values of G,J, and / given in Tables XXVII-XXX are formed from the
expressions of 8z>' and Sp in terms of g by the formulae (60). These are then sub-
jected to the transformation of 46 and multiplied by the constant coefficient
io 3 w 2 = 5.595. The factor io 3 is introduced in order to have the most convenient
unit in subsequent computation. As a check upon the work the values ofy, G, and
/were also computed using the transformed expressions for &V and Sp', and the
results compared with the others. It has not been deemed necessary to set forth
the steps of this simple duplicate computation.
TABLE XXIII.
ACTION OF VENUS ON THE EARTH.
Arg.
do'
dp'
Arg.
d'j'
iff
"4 g'
COS
sin
COS
sin
V, g'
cos
sin
COS
sin
JM
tl
//
u
u
//
//
a
-I,
+0.03
O.OI
0.041
0.018
I,
o.oo
0.07
-0.045
+0.004
I, I
+2.35
-4-23
0.980
0.544
i, i
O.OI
-4.84
I.I2I
+O.OOI
I, 2
0.06
0.03
+0.032
0.006
I, 2
006
O.OO
+0.025
O.O2I
-2, I
O.IO
+0.06
+0.041
+0.065
2, I
o.oo
+O.II
+0.077
O.OOI
2, I
4.70
+2.00
+1.709
+2.765
2, 2
O.O2
+5.52
+3.251
+0.004
2, 3
+1.80
-1.74
0.282
0.300
2, 3
0-53
-2.45
0404
+0.082
2, 4
+0.03
0.03
+0.018
+0.016
2, 4
O.OI
0.05
+0.023
0.007
3, 3
0.67
+0.03
+O.O2O
+0.406
3, 3
o.oo
+0.67
+0497
+0.003
3, 4
+I.5I
0.40
0.181
0.689
3, 4
0.33
-1-53
0.697
+0.149
-3. 5
+0.76
-0.68
+0.059
+0.069
3. 5
0.64
-0.79
+0.072
0.056
-3, 6
+0.01
O.OI
+0.006
+0.006
-3, 6
O.OI
O.OI
+0.006
0.006
-4, 4
0.19
0.09
0.079
+0.160
4, 4
O.OO
+0.21
+0.178
0.000
4, 5
0.14
0.04
0.024
+0.089
-4, 5
+O.02
+0.15
+0.091
0.018
4, 6
+0.15
0.04
O.OI2
0.04 >,
-4, 6
O.I I
O.I I
0.034
+0.030
-S, 6
0.03
O.O2
0.018
+O.020
-5, 6
+O.OI
+0.04
+0.026
0.006
-5, 7
O.I 2
0.03
0.018
+0.065
-5, 7
+0.08
+0.09
+0.052
0.044
-5, 8
+0.154
0.001
o.ooo
0.013
-5, 8
0.128
0.086
0.007
+O.OI I
-8, ii
o.ooo
O.OO2
O.OOI
o.ooo
-8, ii
+0.002
+0.001
+O.OOI
O.OOI
-8, 12
0.008
0.041
0.0205
+0.003 9
-8, 12
+0.038
+0.019
+0.0093
0.018 7
-8, 13
+1.268
+1416
+0.004 23
0.003 66
-8, 13
-1.895
+0.153
+0.000 36
+0.005 58
-8, 14
+O.02I
+0.024
+0.0106
+0.004 3
-8, 14
0.032
+O.O02
0.0098
+0.0058
COEFFICIENTS FOR INDIRECT ACTION.
93
TABLE XXIV.
ACTION OF MARS ON THE EARTH.
Arg.
oV
w
Arg.
<V
<y
g< s 1
cos
sin
COS
sin
M, g'
cos
sin
cos
sin
I, I
I, O
0.216
0.008
0.167
-0.047
0.043
+0.013
+0^056
0.003
i, -I
I,
0.003
0.033
+0*273
+0.034
+o!o7l
O.OIO
o'.000
0.008
2, -3
2, 2
2, I
2,
+0.040
+1.963
-1.659
0.024
O.OIO
0.567
0.617
+0.015
0.006
0.272
+0.030
0.008
0.024
0.937
0.065
O.OI2
2, -3
2, 2
2, I
2, O
o.ool
+0.007
+1.048
0.007
+0.041
+2.043
1427
0.027
+0.025
+0.977
+0.054
+0.014
+O.OOI
0.005
+0.047
0.005
3, -3
3, 2
+0.053
+0.396
0.118
O.IS3
-0.073
0.037
0.032
0.006
3, 3
3, -2
+0.006
+0.314
0.129
0.286
0.080
0.070
0.004
0.077
4, 4
4, 3
4, 2
+0.001
0.131
+0.526
+0.032
+0.483
0.256
+O.O22
+O.2I9
+O.O2I
0.008
+0.059
+0.045
4, -4
4, -3
4, 2
+0.008
+0.366
-0.582
0.033
-0.342
0.059
0.023
-O.I5S
+0.006
0.005
0.165
0.049
S, -4
5, 3
+0.049
0.038
+0.069
+O.20O
+0.041
+0.041
0.029
+0.008
S, -4
5, -3
0.064
O.2O2
+0.055
O.02O
+0.033
0.004
+0.038
+0.042
6, -S
6, -4
6, -3
O.O2O
0.104
O.OII
O.002
O.II3
+O.IOO
O.OOI
O.O48
0.013
+0.014
+0.045
O.OO2
6, -5
6, -4
6, -3
0.016
0.153
+0.059
+O.OI2
O.OI4
+0.08 1
+0.008
0.006
O.OII
+O.OII
+0.065
+0.008
IS, -9
IS, -8
+0.018
+O.2OI
0.023
0.030
O.OOS
O.OOO
O.007
O.OO3
IS, 9
IS, -8
0.027
0.083
O.OII
0.184
0.005
0.003
+0.010
+O.OOI
TABLE XXV.
ACTION OF JUPITER ON THE EARTH.
Arg.
3*
/
*/
9?
Arg.
to
/
<*
o'
"4 g'
cos
sin
COS
sin
J, g'
cos
sin
COS
sin
//
//
//
II
H
H
a
3
0.003
O.OOI
0.001
+O.O02
I, -3
O.OOI
+0.003
+O.OO2
+O.OOI
2
0.155
0.052
0.037
+O.O92
I, 2
0.056
+0.154
+O.O9I
+0.040
I
7.208
+0.059
+0.026
+3.3S6
I, I
0.140
+7.207
+3.356
+0.067
O
0.307
-2.582
+0.108
0.042
I,
-2.589
+0.236
0.039
0.109
+ 1
+0.008
0.073
+0.037
+0.004
I, +1
0.073
O.OIO
+0.005
0.037
2 3
+O.OII
+0.068
+0.049
-0.008
2, -3
0.008
0.069
0.049
+0.005
2 2
+0.136
+2.728
+1.910
0.097
2, 2
+0.014
2.731
1.912
0.008
2 I
-0.537
+1.518
+0.654
+0.231
2, I
+0.619
1486
0.640
0-267
2,
O.022
0.070
O.OOO
0.004
2,
+0.018
+0.071
O.OOO
+0.004
3, 4
0.005
+O.OO2
+O.OOI
+0.004
3, 4
O.O02
0.005
0.004
+O.OOI
3, 3
0.162
+0.027
+O.O2I
+0.132
3, 3
0.014
0.164
0.134
+O.OIO
3, 2
+0.071
+0.551
+0.378
0.049
3, 2
-0.555
+0025
+0.018
+0.381
3, -I
0.031
+0.208
+0.082
+O.OI2
3, -I
0.205
0.048
0.019
+0.08 1
94
ACTION OF THE PLANETS ON THE MOON.
TABLE XXVI.
ACTION OF SATURN ON THE EARTH.
Arg.
3v' io'<J/>'
Arg. 3v' dp'
g* g'
COS
sin
cos
sin
*, g'
COS
sin
cos
sin
I, I
I,
-0.077
0.003
+0412
0.320
+972
+ 18
+ 182
2
I, I
I,
0.003
0.060
+0419
0.314
+0.204
+0.004
+O.OOI
O.OOI
2, 2
2, I
+0.038
+0.045
O.IOI
O.IO3
-350
-236
-132
IOI
2, 2
2, I
O.OOI
+0.006
0.108
0.113
0.077
0.053
0.000
O.OO2
TABLE XXVII.
PLANETARY COEFFICIENTS FOR THE INDIRECT ACTION OF VENUS.
Arg.
G
J
/
gl g'
sin
cos
sin
COS
sin
cos
I.
0.190
0^359
0.024
0.049
-0.33
+o.'i75
I, I
1.220
2.203
0.408
0.735
6.34
+3-52
I. 2
+0.074
+0.229
0.014
+0.005
0.125
0.04
-2, I
+0.533
+0.330
+O.IOI
+0.062
+0.32
-0.53
-2, 2
+6.1 13
+3-737
+2.069
+1.278
+4.26
-6.96
2, 3
0.755
-0.683
0.173
0.180
-2.63
+2-73
2, 4
+0.108
+O.IIO
+0.007
+0.009
O.IO
+0.09
-3. 3
+ I.OOI
+0.014
+0.359
+O.OI2
+0.016
-0.895
3. 4
1.590
0439
0.507
0.133
0.62
+2.30
3. 5
+0.191
+0.143
+0.039
+O.O4I
i. 02
+ I-I45
-3, 6
+0.056
+0.051
+0.005
+O.005
0.045
+0.05
4, 4
+0.372
-0.181
+O.I22
0.059
0.13
0.30
4, 5
+0.100
0.057
+0.069
0.019
0.06
O.2O
-4, 6
0.099
0.026
0.030
0.009
0.06
+0.23
-5, 6
+0.061
0.042
+O.OII
0.013
0.03
-0.055
5, 7
+0.136
0.040
+0.049
0.013
0.04
0.17
-5, 8
0.031
O.OOO
0.009
O.OOO
O.OOO
+0.231
8, 12
0.0550
+0.025
+O.O02 9O
0.015
0.007
+0.036
-8, 13
-8. 14
0.0093
+0.073
+O.OI22
-0.047
0.002 6O
+0.003
+0.00298
+0.008
+2.126
+0.093
+1.903
+0.079
COEFFICIENTS FOR INDIRECT ACTION.
95
TABLE XXVIII.
PLANETARY COEFFICIENT FOR THE INDIRECT ACTION OF MARS.
Arg.
/ f
&< g'
sin
COS
sin
COS
sin
cos
I, I
I,
u
+O.I26
+O.OIS
o!oo8
+0.035
M
+0.042
O.OOI
o!c>33
+0.009
0.251
0.073
0^324
0.016
2, 3
2, 2
2, I
2,
0.209
2.026
O.IOO
0.113
0.057
0.641
+0.082
+0.015
0.036
0.704
0.067
O.OIO
0.009
0.203
+0.017
0.005
0.058
0.871
0.924
0.003
+0.208
+2.884
-2485
ox>io
3, 3
3, 2
0.097
0.215
0.174
0.08 1
0.026
0.072
0.056
0.029
-0.186
0.227
+O.IOI
+0.594
4, 4
4, 3
4, 2
0.009
+O.III
+0.097
+0.085
+0480
+0.035
0.005
+0.045
+0.035
+O.O2O
+0.164
+O.O2O
+0.083
+0.714
-0.383
+0.008
0.180
+0.788
5, -4
S, -3
0.063
+0.018
+0.104
+0.091
O.O22
+O.O06
+0.032
+0.032
+0.114
+0.299
+0.071
-0.057
6, -5
6, 4
6, -3
+0.039
+O.IO2
O.O06
O.OII
0.104
0.026
+O.OI2
+0.034
O.OOI
O.O02
OX36
O.OII
o.on
-0.166
+0.150
-0.037
0.156
0.017
IS, -9
15, -8
0.026
O.006
O.O2O
+0.001
0.005
0.003
O.O06
O.OOO
0.036
0.046
+0.035
+0.301
TABLE XXIX.
PLANETARY COEFFICIENTS FOR THE INDIRECT ACTION OF JUPITER.
Arg.
G
/
/
"4 g'
COS
sin
COS
sin
COS
sin
i, 3
0.007
+0.039
//
O.O02
//
+0.006
o'.039
O.OO6
I, 2
0.084
+0-759
0.028
+0.132
0.757
0.079
I, I
0.064
+7-554
+O.O2O
+2.518
10.813
O.OOO
I,
+0.238
-0.266
+0.082
+0.032
0479
-3*70
I, +1
+0.218
O.OI2
+O.O29
+O.OO4
0.007
0.214
2, 3
+0.360
O.O30
+0.073
0.008
+ 0.026
+0.352
2, 2
+4.407
0.177
+ 1-447
0.069
+ 0.167
+4-104
2, I
+ 1437
+0.522
+0.527
+O.I7I
0.808
+2.231
2,
0.039
0.023
+0.013
+O.OOI
0.036
-0.097
3, 4
+0.006
+O.O24
+O.OOI
+0.005
0.023
+0.008
3, 3
+0.096
+0.290
+0.023
+0.008
0.236
+0.090
3, 2
+0.865
O.IO9
+0.286
0.035
+ 0.109
+0.839
3, I
+0.177
+0.028
+0.069
+0.008
0.047
+0.304
9 6
ACTION OF THE PLANETS ON THE MOON.
TABLE XXX.
PLANETARY COEFFICIENTS FOR THE INDIRECT ACTION OF SATURN.
Arg.
G
/
/
*' J
s, g'
cos
sin
COS
sin
COS
sin
Ii I
I, O
+0-443
O.OOI
+o"oo5
O.OO2
+o.'iS3
+0.007
+O.OOI
O.OOI
0.006
0.090
+0.616
0.471
2, 2
2, I
0.182
0.1 18
o.ooo
0.004
0.059
0.041
O.OOO
O.O02
+O.OOI
+0.009
0.171
0.167
TABLE XXXI.
G-COEFFICIENTS FOR VENUS.
Arg.
itfntG
lo'w^y
IO 3 ? 2 /
V, g'
cos
sin
cos
sin
COS
sin
I,
a
2.272
a
+0.048
0.304
+0.016
II
0.042
2!o89
I, I
14.09
+0.031
4-703
+0.005
0.043
40.572
I, 2
+ 1.320
0.261
0.014
-0.082
0.536
0.502
2, I
2, 2
+ 3-508
+40.100
+O.O02
+0.253
+ 0.663
+13.764
+0.003
+O.I2I
0.071
0.263
+ 3-510
+45-655
2, 3
5.607
+ 1.024
1-354
+0.346
4-457
-20.735
2, 4
+ 0.838
0.205
+ 0.060
O.O22
0.209
0.723
-3. 3
3, 4
+ 5-002
9.004
+1.186
+2.036
+ 2.OIO
- 2.869
+0.028
+0.609
0.146
- 2.855
+ 5.006
13-017
3, 5
+ 1.106
0.749
+ 0.229
0.219
5400
6.668
*" f
-3, 6
+ 0.326
0.270
+ O.029
O.O27
0.239
0.292
4. 4
+ 2.314
0.018
+ 0.759
O.OOS
+ 0.095
+ 1-827
4, 5
+ 1*94
0.180
+ 0.393
-0.077
+ 0.197
+ 1.152
^
4, 6
0431
+0.378
0.128
+O.I2I
- 0.873
1.003
-5, 6
+ 0.389
0.148
+ 0.095
O.OIO
+ 0.159
+ 0.312
Si 7
+ 0.616
0.500
+ 0.215 0.185
+ 0.657
+ 0.723
-5, 8
0.098
+0.143
0.029 +0.042
1.065
0.732
-8, ii
+ 0.016
O.OI I
+ 0.005 0.006
0.788
+ 1-039
-8, 12
+ 0.1611
+0.2094
+ 0.039
-0.077 7
0.091
+ 0.184
-8, 13
+ 0.0045
+0.0661
+ 0.00145
+O.O22 OS
15-912
+ 1.289
-8. 14
- 0.166
-0.4588
0.040
+0.024 6
0.681
+ 0.037
COEFFICIENTS FOR INDIRECT ACTION.
97
TABLE XXXII.
G-COEFFICIENTS FOR MARS.
Arg.
io'w 2 <9
lO'/M 2 /
io 3 w 2 /
M, g'
cos
sin
cos
sin
cos
sin
I. I
+ 0.893
+O.OI2
+0.300
+0.006
0^)65
+ 2.293
I,
0.185
O.IO5
0.035
0.036
0.272
+ 0.318
2, -3
+ I.2II
+0.013
+0.207
+0.007
0*06
+ 1.208
2, 2
+ 12.777
0.089
+4.100
0.014
+0.073
+17.531
2, I
+ 0413
+0.595
+0.335
+0.194
+1.161
+14.821
2, O
0.309
o.ooo
+0.062
O.OI2
0.019
+ 0.129
3, 3
1.205
0.155
0.344
O.O22
+0.151
1-175
3, 2
- 0.857
0.958
0.296
O.3l8
+2.641
- 2.384
4, 4
0431
0.208
0.109
0.035
+0.207
0418
4, 3
I-956
-1-943
-0.638
-0.695
+2.961
- 2.864
4, 2
+ 0.119
-0.565
+0.009
0.224
-4.877
0.502
5, -4
+ 0404 +0.548
+0.139
+0.167
0.600
+ 0.452
5, -3
- 0.054
+0.516
0.018
+0.181
I.69S
0.166
6-4
0.047
+0.815
O.O2I
+0.276
1.271
o.ioo
6, -3
0.130
+0.073
0.05O
+0.038
+0495 + 0.685
IS, 9
0.109
+0.147
0.018
+0.041
0.246 0.135
IS, -8
0.035
+0.003
0.016
+0.005
0.704
- I-55I
TABLE XXXIII.
G-COEFFICIENTS FOR JUPITER.
Arg.
IO 3 i
rfG
10 s *
V
I0 3
; 2 /
J, g'
cos
sin
cos
sin
COS
sin
It
H
H
H
m
n
, 3
+ 0.2 18
+0.039
+ O.O22
+O.OII
0.039
+ 0.217
, 2
+ 4.240
+0-594
+ 0.738
+0.173
0.553
+ 4.241
, -I
+42.237
+ 1-534
+ 14.088
+0.274
1.672
+60471
,
1459
1.376
+ 0.196
-0458
21.709
+ 2.097
, +1
0.040
1.225
+ 0.017
-0.168
1.203
+ 0.006
2, 3
2.OI2
+0.050
O408
+O.O22
0.061
I-99I
2, 2
-24.668
-0.358
8.096
0.061
+ 0.363
-23488
2, I
- 7.872
3-363
- 2.887
1. 119
+ 5-192
12.225
2,
+ 0.218
+0.140
O.O67
O.OII
+ 0.173
+ 0.542
3, 4
O.I4O
+0.017
0.034
+0.006
0.017
0.129
3. 3
1.677
+0.403
- 0.559
+0.078
0.397
- 1.365
3, i
+ O.2I2
+474
+ 0.06 1
+1.611
4-733
+ O.2I2
3. I
0.24O
+0.984
0.078
+0.380
1.672
O402
TABLE XXXIV.
G-COEFFICIENTS FOR SATURN.
Arg.
icfufG
icfm 2 j
IO 3 #/ 2 /
s, g'
cos
sin
cos
sin
COS
sin
I, -I
I,
+248
+O.OI
+0.03
O.OI
+0.86
+0.03
//
+0.01
+O.OI
0.03
0.50
+345
+2.64
2, 2
2, I
I.O2
-0.66
o.oo
O.02
-0.33
0.23
o.oo
O.OI
0.00
+0.05
0.96
0.94
t
PART III.
FUNCTIONS OF THE COORDINATES
OF THE MOON.
CHAPTER VI.
FORMATION OF THE LUNAR COEFFICIENTS.
47. In attacking the problem before us it has been assumed that we have
expressions of the Moon's coordinates relative to the centre of the Earth as func-
tions of the six arbitrary constants introduced through integrating the differential
equations in these coordinates. Moreover, the constants in question enter the
expressions for the disturbing function R only through these coordinates. It
follows from the general expression of R that if c represents any one of the six
lunar elements, the partial derivatives of the disturbing function may be derived
from the form
A n ^
= A -sr + B-i: + C-.-+ z> -..-+ (i)
oc dc dc dc
It is therefore necessary to have such expressions tor the squares and products of
the coordinates of the Moon that each of the required derivatives can be found as
easily as may be.
When the present work was commenced it was intended to make use of the
developments of the powers and products x 2 , y 1 , etc., as derived from Delaunay's
theory, and found in Action, pp. 154-172 and 213-224, where the processes by
which these quantities may be expressed are fully set forth. But, before the work
was put into final shape, Brown's work on the Lunar Theory was completed and
published so far as the action of the Sun was concerned; and it therefore became
a question whether to use Brown's expressions instead of those of Delaunay, or to
go on with the latter. Each course was found to have its drawbacks. The former
developments from Delaunay's theory being intended mainly to make an exhaustive
search for possible terms hitherto unknown in the Moon's motion, were not com-
pleted beyond the third order, though the constant term was carried to the sixth
order. To speak more exactly, the development was carried to such a point that
the square of each coefficient would be correct to the sixth order.
It was found, however, that the use of Brown's more rigorous theory would be
quite convenient except in a single point. In this theory the coordinates are explicit
functions of all the lunar elements except the Moon's distance, which enters into /,
and of which Brown used only the numerical value in his developments. Brown
has shown how it is possible from the data and methods of his theory to form the
complete derivatives as to this element without using an analytic development in
powers of m. But as the application of this method would require a longer and
more laborious study of the subject than the author was prepared to enter upon, it
was decided to use the Delaunay developments for obtaining the derivatives as to
102 ACTION OF THE PLANETS ON THE MOON.
> \
log a. The outcome of these considerations has been that, for the sake of trial and
3onipa r Json, both Delaunay's and Brown's developments have been to a large extent
independently used, and the results compared with a view of facilitating an esti-
mate of the errors to which the analytic development is subject.
48. Reduction of coordinates to the radius vector of the Mean Sun as
X-axis.
In the developments in Action the Sun's perihelion was taken as the origin
from which longitudes were measured. When the present work was undertaken, it
being found that the development of the vl-coefficients would be most easily effected
by taking the direction of the mean Sun as the axis of X, the same origin had to
be taken for the lunar coordinates. This has been done throughout the work; and
it must be understood in the subsequent developments that x and y are referred to
the radius vector of the mean Sun as axis of X except in terms arising from the
motion of the ecliptic.
In the use of either theory we 'take, as the initial data of the problem, the rec-
tangular coordinates of the Moon referred to the mean Moon as the axis of X,
which coordinates we represent by x l and y^ These coordinates are those which
Brown's theory gives in the first instance, and they are also those which I have
developed in powers of /, etc., from Delaunay's theory in Action, pp. 167 and 169.
The notation of arguments from the latter paper is:
gi the Moon's mean anomaly; g' ', the Sun's mean anomaly;
X, the mean elongation of the Moon from the Moon's ascending node,
equivalent to Delaunay's f, or the mean argument of latitude ;
X', the same for the mean Sun ;
0, the longitude of the Moon's node;
/ = g -\- IT -f- 6, the Moon's mean longitude; /' = g' + X' + 0, the Sun's
mean longitude ;
D = / /' = X X', as in Delaunay, the mean Moon's departure from the
mean Sun.
Putting Sv for the excess of the true longitude over /, we then have
x l = r cos ft cos f = aS,k cos N
y l = r cos ft sin &v = d2,k' sin N (2)
z = r sin ft = aLc sin N'
where k, ', and c of dimensions o are developed in powers of e, e',y, and m.
The general form of the arguments TV^and N' is
N or N' = ig + i'g' +j\ +/X' (3)
where g = I - v , g' = /' - TT', X = / - 0, X' = /' - 0.
The equations for transforming x l and y L into x and y are
,v = x l cos D y l sin D y = x 1 sin D + y l cos D (4)
FUNCTIONS OF LUNAR COORDINATES. 103
If we put
h = \(k + V) h' = $(k - k')
the substitution of the development will give
- = 2A cos (D + N) + 2/*' cos (D - JV) =
(5)
y - = 2A sin (D + JV) + 2A' sin (D - N} = r,
49. There are now two ways of proceeding in order to form the squares and
products. We may either form the last expressions for x and y and square them,
or we may torm the squares and products of x and y ly and transform them from
the mean Sun to the mean Moon. Following the latter method we have
x? = %( x ? + y?) + \(x? - y?) cos 2D - x l y l sin 2D
/ = K*, 2 + y*) - \(x? - j, 2 ) cos 2D + Xl y, sin 2D
(6)
x 1 J 2 = (x* jKi 2 ) cos 2D 2x^y l sin
2xy = 2x l y l cos 2D + (^ 2 jy, 2 ) sin
The three junctions required in the work being
(x 2 y 2 ), r 2 3^ 2 and 2xy
we see from the preceding equations that the first and third can be formed at once
from the corresponding functions of x* and y* by a transformation through the angle
2D. If we have, for any argument ^V,
*, 2 y? = h y cos N 2x l y l = h 2 sin N (7)
the corresponding terms referred to the mean Sun are
** _ / = J(* + A) cos (iD + N) + $(*, - /y cos
(8)
2xy = J(A, + /4 2 ) sin (2Z> + ^V) 4- i(A, - A,) sin (2.O - TV)
There are some cases in which a reference to a fixed axis is convenient. Let us
put,
x , y , coordinates referred to any fixed axis.
So long as this axis is unrestricted the coefficients for x and y will be equal, as
is seen from (5). Hence, if we write for any term of x and of y depending on
any argument N
x a = A, cos N y = A t sin N (9)
this term will be transformed into the corresponding term of AT and of y, and vice
versa, by means of the equations
104 ACTION OF THE PLANETS ON THE MOON.
x = x t cos /' + y sin /' y = y a cos /' x sin /'
or
* = x cos I' y sin /' y g = y cos /' + x sin /' (10)
The special term (9) will, therefore, transform into the terms of x and of y
l') (n)
For the special functions required in the lunar theory we shall have the follow-
ing transformations of the same form as (6)
x 2 - / = (* 2 - jr 2 ) cos 2/' + 2x y v sin 2/'
(12)
2xy = 2x y Q cos 2l' (x* jy 2 ) sin 2/'
^ 2 y 2 = (x* jK 2 ) cos 2l' 2Ary sin 2/'
(13)
2x y Q = 2xy cos 2!' + (x? y 2 ) sin 2l'
The transformation of any one term may be made by the equations (6) by writ-
ing + 2/' or 2/' for 2D.
If, as in most of the present work, the solar perigee is taken as the fundamental
fixed Jf-axis, we write g' instead of/' in these equations.
An important remark to be made on these transformations of terms from
one axis to the other is that the equality of coefficients expressed in the equations (9)
and (n) is true only when the fixed axis of J^is unrestricted. If, as will sometimes
be more convenient, we take the direction of the so'ar perigee for this axis, some
values of argument N in (9) will be equal with opposite signs. By combining the
terms depending on these arguments the equality in question will cease to hold.
If, however, the Sun's eccentricity is dropped, the general equations will remain
valid for the Sun's perigee also.
It thus happens that, in the developments given in Action, pages 213-215, the
coordinates are quite general, while the expressions for their squares and products
given on pp. 217-223 are not general, because the solar perigee is here taken as the
fundamental axis.
50. Recalling that throughout the work we use the symbol D to represent the
logarithmic derivative as to a of any function, a serious question is that of determin-
ing the value of this derivative with the necessary precision in each special class
of terms. In actually performing the work so many tentative combinations have
been made, as better and better methods were found, that it is difficult to present
any one process as the definitive one. The following method was at length seen
to be the best under the circumstances I have described. Let
u = a'<f>(i)
be any function of the coordinates of which D is to be formed. Practically i will
FUNCTIONS OF LUNAR COORDINATES.
be equal to i or 2 according as the expression we are dealing with is of the first or
second degree in the rectangular coordinates. If we can compute the value of
D(j)(m} with sufficient precision the complete value of Du will be
Du = /*<(*) -f a l D<f>(m) (14)
If it is developed in powers of m, <f>(tn) = OQ + a i m + <V 2 + and we shall have
><j>(m) = f otjW + 3<x 2 7 2 +
the coefficient of each term being f of the exponent of m. (v. 12, Eq. 23.)
If we have the numerically accurate value of any tf>(m) from Brown's theory
and an approximate one from the analytic development, the comparison of the two
will furnish a rude index to the probable value of the omitted powers of m in the
development. It follows that the nearest approximation to the value of Du will
be obtained by using in the first term of the second member of (14) the numerical
value of a { (j)(m) = u, the analytic development being used only for the second term.
Moreover, having an approximate estimate of the value of the omitted terms of the
analytic development of the second member of (14), we may use it to correct the
last term of this member. I conceive that no lack of theoretical rigor pertaining
to this process will lead to an error of the slightest importance in the present work.
51. Formation of the D's from Delaunay's Theory.
In the final formation of the /^-derivatives I have extended the developments
given in Act to*, by the aid of Delaunay's results, as follows. Delaunay expresses
the reciprocal of the Moon's radius vector in a form which we may write
a
-
where tt\, is put for the sum of an infinite' series of terms, each developed in powers
of ?, as well as of e, e' ', and y. This quantity 77^ is related to the Moon's parallax TT
by the equation
sin TT = (i -f TT.)
a ^
#! being the Earth's equatorial radius.
It is to be remarked that Delaunay's expression for the parallax was only carried
to terms of the fifth order, so that it does not suffice for all theoretical purposes.
It is indeed fairly probable that it would suffice for the object now in view. In
order, however, to lessen the danger of any insufficiency in this respect I have, in
forming the value of TT^ compared each coefficient in the expression of Delaunay's
parallax found in my transformation of Hansen's lunar theory with the more accu-
rate value derived from Hansen's or Brown's expression. We may conceive that
the correction necessary to reduce Delaunay's coefficient to Hansen's value is of
the form
STT, = ajn* + <+l w' +1 -\ ----
106 ACTION OF THE PLANETS ON THE MOON.
in which i is the power of m next above the highest to which Delaunay has carried
his coefficient. From what we have already shown it follows that the corresponding
correction to Dtr^ is
an approximate value of which is
In order to make this correction rigorously exact we should know the values of
the coefficients of the omitted powers of i. This being unknown, the minute cor-
rection is to a certain extent a matter of estimation. I do not conceive, however,
that the uncertainty is at all important in the present investigation.
We have next to consider the Z>-derivatives of the three functions
p 2 3? 2 ; fi* '"li > an d 2 fi 7 ?i
Starting with the equations (2) the values of 8v and ft, developed in powers of m,
are given at the end of Delaunay's TAeorie, Vol. II. The values of Dv are
formed from these with great lacility by means of the form (23) of 12, because
Delaunay gives the numerical value of each part of every term of the longitude.
The steps of the subsequent process consist in simple trigonometric multiplica-
tions, and are presented in tabular form on the following pages. The fundamental
quantities are
a a
7T. = i, Sz', ft and D -, Dov. Dfj
r r
which are formed from Delaunay's numbers in the way just shown.
The following functions are then formed by trigonometric multiplication
P = I TTj + 7T, 2 7T, 3 + p~ = I 27T, + 37T, 2 4^,' + p*= I 377-, + 67T, 2 IO7r, 3 +
In the final work, however, p has been formed from Brown's theory. Then
p> = ? + r,* + t;*
2 \/3' C os 2 /3 = i - S in 2 #
sinSz; = 8-|8^ cos &v = i - iSz; 2
%? V? /* cos 2 /8(i 2 sin 2 8z>) ^,17, = p* cos 2 ft cos Sv sin Sv
S=ps'mft ? 2 = /) 2 sin 2 y3 D p 2 = - 2p^hr l
> sin = cos ftDft D sin 2 ft = 2 sin ftD sin ft Z>p 2 cos* ft = cos 2 ft Dp 2 + pW cos 2 ft (16)
Z7 sin 2 Sz> = 2 sin Sz^Z? sin Sy = 2 sin 8>v cos SvDSv
%? - *)*) = (i - 2 sin 2 fo) Dp- cos 2 /9 - 2/> 2 cos 2 ftZ> sin 2 w
? Stfi = sin Sy cos BvDp 2 cos 2 y3 + p 2 cos 2 /9(i 2 sin 2 Sv)2)&v
D C = p*D sin 2 ft + sin 2 ftZ) p 2
FUNCTIONS OF LUNAR COORDINATES. 107
The same method might be used to form the derivatives as to the e and y, but
this has been deemed unnecessary, as they can be formed with entire precision
from Brown's Theory, and probably with all necessary precision from the develop-
ments found in Action with some extensions in special cases. As a matter of
fact they have been formed by both methods.
52. Derivatives from Brown's theory. To form the partial derivatives as to
Delaunay's e and y from Brown's expressions it is to be noted that Brown uses
instead of e and y two constants e and k which, omitting unimportant terms, are
expressed thus in terms of the Delaunay elements:
e = (2.000543 + ,o^e n )c .3668^ 2.oi2ey*
k = (1.000128 .O004e' 2 )7 .4967* 0.499^7
A distinction is to be made between the a of the present work, defined by the con-
dition a 3 n? = p, and Brown's a, used in his work. Brown's e is defined as the
coefficient of sin g in the development of jj/a, or, using the notation of the present
paper, in the development of
a r . a
cos p sin 0v = i}
a a a
This will enable us to make a comparison of the preceding value of e with that
to be derived from the analytic development in Action, p. 168, from which we find
- e = (2 - ^' + ^ '' + *firW> 4 + V''> - (| + i7 2 - T\VV - (2
Brown's 2k is the coefficient of sin X, X being the mean argument of latitude, in
the development of r/a sin ft, tound on p. 159 of Action. From the coefficient as
developed in Action we find
a 2
Brown also gives
^=.999093; ^=1.000908
The two results are as follows, B indicating those from Brown's formulae, A those
Irom the analytic development.
B; e = 2.000557^ .367^ 2.oi2^y 2 B ; k = 1.0001287 .499^7 .4967*
A ; e = 2.000426^ .371^ 2.OO4C7 2 A ; k = 1.0001087 .501^7 .5007*
The difference, arising from the dropping of higher powers of m in the analytic
development, is too small to affect the solution of our present problem.
To find the partial derivatives of any function u of e and k with respect to e and
y we have from the preceding expressions
io8 ACTION OF THE PLANETS ON THE MOON.
du dude dudVi du du
d~e = dede + dkfo = I '9 9 3 2 fe~ 99d]i
(17)
du du de du 5k Bu du
a~ = 5-=- + 5r=-= o.ggSo 5l .0025; -5-
dy de dy T 3k 7 /yo 3k a de
These equations enable us to find the derivatives as to e and y from Brown's as to
e and k.
53. Tables of the functions and derivatives of the Moon's coordinates.
The numerical processes by which the required functions of the coordinates
were developed may be followed and tested by the aid of the following tables.
The notation of the arguments, expressed by the indices in the first column, has
been defined in 48.
Owing to the circumstances mentioned in 46, and to the widely different degree
of precision required in the coefficients of different arguments, the numbers of these
tables are not always consistently continuous. The terms of many, perhaps more
than half the arguments, lead to no sensible inequalities; with these pains were not
always taken to reach a higher degree of precision than was required to show the
order of magnitude of the results. In the preliminary steps of the investigation it
was deemed sufficient to carry the expressions for the Moon's coordinates to the
5th place of decimals, and those for the derivatives to the 3d or 4th place. But
when the inequalities of the elements themselves were reached by integration, it
was found that this degree of precision, while more than sufficient for the periodic
terms in general, was not sufficient either in the terms related to the evection, or
in those determining the secular variations and accelerations of /, IT, and 6. A
number of successive revisions was found to be necessary, in which the coefficients
depending on the argument g' were carried to the yth place of decimals. As the
last place was always more or less doubtful only the sixth place has been included
in the printing.
It may also be remarked that in commencing the tables it was supposed that the
analytic development in Action would suffice for the work. This expectation
proving ill-founded, the developments of the Moon's coordinates given by De-
launay, then those by Hansen, and finally those by Brown were successively
used in the case of those terms in which greater precision was needed. Finally
the /^-derivatives were, in their important terms, recomputed by formula; proposed
by Dr. Ross, which were much briefer than those already given in (16) of 51.
The want of homogeneity thus arising in the tables could be cured by a fresh
development from the fundamental data of Brown and Delaunay, but I do not think
any important change would thus result in the expressions for the inequalities of
the Moon's elements found in Part IV.
FUNCTIONS OF LUNAR COORDINATES.
TABLE XXXV.
FUNCTIONS OF THE LATITUDE AND THEIR DERIVATIVES.
109
Arg.
ft
4
sin/9
Z>sin/9
C=/> sin ft
zc
dflde
W
******
sin
sin
sin
sin
sin
sin
sin
sin
2 O I O
I O I O
O O O
I O O
2O O
.000154
.004842
+.089 503
+.004898
-j-.ooo 301
+.00009
+.00032
.00000
.00005
.000 01
.000 154
.004 840
+.089413
+.004896
+ 000301
+.OOOOO
+.00032
.OOOOO
.00005
.000088
.007 240
+.089474
+.002 466
+.00008
+.00044
.00018
+.00003
.0049
.1304
.0049
+0454
0.0030
0.161 5
+1.998
+0.055
O I O
I I O
+.000023
.000038
+.000023
.OOO 038
+.OOOO29
.000 017
+.000052
+O.000642
I I O
O I O
0030
.000 032
.000 030
+.000013
+.OOO OI
.000 030
ooo 030
+.000013
+ OOO
.OOOO24
+.000031
.000079
0.000 535
I 032
I I ^ 2
+.000968
+.OO2O2
+.000968
+.O02 OI
+.000545
+.00124
+.0097
+O.OI2 2
O 03 2
+ 000568
+ ooi 80
+ 000568
+ 001 80
_L nm 58
I I I 2
O I I 2
+.000 144
+.OOO3O
4-.OOO idd
+ ooo 30
+.000 165
+ ooo 18
I O I 2
O O I 2
I O I 2
Oil 2
+.000808
+.OO3 O22
+.000 161
.000060
+.001 75
+.00542
+.00028
.OOO 12
+.000808
+.003 020
+.000 161
000060
+.001 74
+.005 40
+.00028
.OOO 12
+.001 179
+.003308
+.000080
.000061
+.00244
+.00647
+.OOOI2
+.000 18
+.02 1 i
.0007
4-0.0261
+0.0739
+0.0017
TABLE XXXVI.
FUNCTIONS OF THE RADIUS VECTOR AND LATITUDE.
Arg.
DK,
/) 2 =r 2 /a 2
c
f>* cos 2 /3
D. ff
D.C
D. f> 2 cos *p
g g* I *
cos
cos
COS
COS
COS
COS
COS
O O O
+.002 638
+1.002866
+.004 038
+.908828
-.00443
+.OOOOI
-.00444
I O O O
.OOI40
-0.10858
.00042
.108 17
+.00480
+.00006
+.00474
2 O O O
.OOOl6
0.001 52
.00002
.001 51
+.00008
+.OOOOI
+.00008
I I O
+.000917
0.00070
.OOOOO
.00070
.ooi 77
.OOOOO
.00176
O I O O
.OO0267
+0.000 266
+.OOOOOI
+.000266
+.000 713
+.000001
+.000712
I I O O
.000 637
+0.000 54
.OOOOO
+.00054
+.001 27
.OOOOO
+.001 26
I O 2 O
+.OOOIO
+0.000 32
+.00065
.00033
.000 19
.00003
.00015
O O 2 O
+.00006
O.OOOO2
.00398
+.003 97
.00010
+.000 O2
.OOO 12
1020
+.OOOOO
+O.OOOOO
.00022
+.OO022
.OOOOO
.OOOOO
+.OOOOI
2 O 2 2
.00032
+0.001 83
.OOOOO
+.001 82
+.00380
+.OOOO2
+.00377
I O 2 2
+.019 66
0.018 91
.00003
.01888
.035 22
.00006
-.035 15
O O 2 2
+.02643
0.014 86
.00029
-.014 57
.049 46
.00054
.04892
I O2 2
+.002 91
o.ooo 48
.00002
.00046
.00154
.00003
.001 50
2 O 2 2
+.00027
0.00004
.OOOOO
.00004
.00007
.OOOOO
.00007
I I 2 2
+.000216
+0.00006
.OOOOO
+.00006
.00048
.OOOOO
-.00047
O 2 2
.000441
+0.000 144
+.000005
+.000 139
+.00084
.000016
+.000856
I 2 2
.000 058
0.00000
.OOOOO
.OOOOO
+.OOOOI
.OOOOO
+.OOOOI
I 2 2
+.000844
0.00080
.OOOOO
.00080
.001 40
.OOOOO
.OOI 40
O 2 2
+.OO2 O27
o.ooi 043
.000015
.001 028
-.00386
.00002
.00384
I 2 2
+.000260
O.OOOOO
.OOOOO
.OOOOO
.00015
.OOOOO
.00015
I O O 2
.00002
0.000 05
.00002
.00003
.00002
.00004
+.OOOOI
0002
.00008
+0.0005
+.000287
.00023
+.00016
+.00056
.00040
1002
.00004
+0.0006
+ .OOOII
.00005
.00007
+.00023
.00030
O 2 O O
.000 018
+.000 036
+.000 036
O 2 2 2
+.000 106
.OOO 212
.OOO 212
O 22 2
.000 004
+.000 008
+.000008
no
ACTION OF THE PLANETS ON THE MOON.
TABLE XXXVII.
FUNCTIONS OF THE LONGITUDE AND THEIR DERIVATIVES.
Arg.
sin Sv
cos Sv
isin 2<?Z>,
sin 2 dv
D.tiifdv
Dto
\D sin 2<Jz>
g g* it
sin
COS
sin
cos
cos
sin
sin
-4- <v/i8i
I O O O
2000
I I O O
O I O O
I I O O
I O 2 O
O O 2 O
+.IOQ58
+00371
.00072
.003 233
.00053
.00019
.00199
.00035
+.00300
+.00017
.00002
.00018
+XXXJII
.00001
+.10903
+.00365
.00071
.00320
.00053
.00019
.001 97
-(-.OUO33
+XWO7O
.00597
+.00033
+.00004
+.00036
.00022
+.000 02
+.001 55
+.000 20
.000 19
+.000 249
+.00036
+.00003
XXX) OI
.000000
.000138
.002 052
.002 8437
.001 199
+.000215
+.000091
.OOO 320
.000314
.002008
.002 7902
.OOI 2OO
+.OOO2I2
+.OOOOOO
2 O 2 2
I O 2 2
O 2 2
I O 2 2
2 O 2 2
I 2 2
O 2 2
I 2 2
+.001 03
+.O22 17
+.OII45
+.OOO96
+.00009
OOOI4
.OOOI2
JOOI 25
.00058
+.001 17
+.00067
+.00007
XXX) 03
.00003
+.001 O7
+.02203
+.01135
+ 00008
+.000 15
.000 14
.00012
+.00249
+.001 16
.00234
.00134
.00015
+.00007
+.00006
+.00531
+.003 77
.00483
.00427
.00048
+.000 20
+.000148
+.00008
+.002 545
+.047405
+.037838
+.003 056
+.000228
+.000482
.0003608
+.OO2 7O5
+.046777
+037 341
+.003 273
+.000469
+.000400
.000 3662
xxx) 075
I 2 2
O 22
I 2 2
I O O 2
+.OOIOO
+.00080
+.00007
.0000 1
+.00007
+.00005
+.00009
+.00080
+.00008
+.000 02
.00016
.000 IO
+.00007
.000451
.00039
+.O02 274
+.002 9357
+.000271
+.002 223
+.0020079
+.000305
O O 2
I O O 2
.00027
-J-.OOOOI
.00027
.00002
.OOOO8
.000204
.000300
+ ooo 008
O 2 O O
ooo 036
O 2 2 2
+ 000039
TABLE XXXVIII.
COORDINATES REFERRED TO MEAN MOON AND THEIR DERIVATIVES.
Arg.
f,
Brown.
z* t
Anal.
%J3e
Brown.
*Ar
Brown.
f,
Brown.
A,
Anal.
fyjde
Brown.
W*T
Brown.
g g' i v
cos
cos
cos
cos
sin
sin
sin
sin
O O O O
+995 47
.00302
0^8
080/4
I O O O
JOU 4O
-L- OOI 27
One e
2 O O O
+.001 51
+.OOOOI
4- OSS 01
I I O O
.000 174
+.000 04
OOt 17
O I O O
I I O O
+xxx>iO34
-KOODOOS
+.00033
00004
.000346
xxx) 063
.003 2283
.003 12
O.OO2 508
+.OOI5II
I O 2 O
O O 2 O
+.00004
+.001 09
.00000
+.OOO 01
+.0008
+xx>i8
4- nnS i
.00025
.00000
0.004 4
.OIO9
OO78
I O 2 O
2 O 2 2
+.00005
.000 so^
.00000
ooo 63
+.OOIO
+.0025
.OOOO5
XXX) OO
O.OOIO
.OO25
I O 2 2
O O 2 2
I O2 2
.010 16
XXX) 36
+xxi8
x>i93i
.022 85
+.00029
-.1844
+.0237
+.0048
+.0019
+XXX58
+.022 24
-j- .OIO 32
+XH488
+.03521
+04037
+0.010 6
4-OOO4. 8
0047
jooy?
2 O 2 2
+.000 02
I 2 2
+.000028
+.000 19
+ ooo 52
O 22
I 2 2
+.0000478
.000002
+.000 20
.000262
ooo 04
.OOOIlS
.0000042
.00031
0.000 169
+XXXM32
I 2 2
.000400
.00044
O 2 2
I 2 2
.000 4506
+.000013
.001 55
+.001 256
-j- ooo 24
+.000337
+ .000 7327
+.00225
+0.000 504
+o 00026
.000406
O O O 2
.000 10
.OOO 17
+ 00008
.OO45
FUNCTIONS OF LUNAR COORDINATES.
TABLE XXXIX.
FUNCTIONS OF COORDINATES REFERRED TO MEAN MOON.
in
Arg.
*,'
Brown.
Brown.
Brown.
2 ^i ! ?i
Brown.
7~\fl; 2 w 2\
.Z-/1 V. ~~Yl )
Delaunay.
2-Z?(C l !0 I )
Delaunay.
f g' * V
cos
cos
cos
sin
cos
sin
O O O
i o o o
2 O O O
i i o o
O I O O
I I O O
1020
O O 2 O
I O 2 O
2 O 2 2
I O 2 2
O 022
I O 2 2
2 O 2 2
I 122
122
I 122
I I 2 2
O I 2 2
I I 22
I O O 2
O O 2
1002
2 044
i 044
004 4
+.992 529
.10850
+.004504
.000 35
+.000 217
+.000 19
.000 ii
+.003 97
.OOOOOO
.000 637
.01990
.012 13
+.00069
.00002
+.00006
+.000092
.OOOOO
.00080
.000875
.OOOOO
.00005
.00021
.OOOOO
+.000 187
+.00006
+.OOOO2
+.006299
+.00033
.005989
.00035
+.000049
+.00035
.00022
.OOOOO
+.OOO22
+.002444
+.OOI O2
.00244
.001 15
.OOOO2
+.986230
.10883
+.010493
.OOOOOO
+.000168
.000 16
+.000 II
+00397
.00022
.003081
.02092
.00969
+.00184
.OOOOO
+.00006
+.000036
.OOOOO
.00080
.000721
.00000
.00005
.00019
+.00003
+.000440
+.00029
+.00008
-.007345
+.00195
.00018
.001 36
+.000198
+.00054
.00021
.000 ii
+.00003
.00670
.04209
.03863
+.00632
+.00012
.00091
+.000587
.OOOI2
.001 50
.002889
+.00055
.000 14
.00024
+.00006
.000 ii
+.00264
+00074
+217 74
.004501
.OOI IO
-.006386
.00070
.OOO 12
.003 95
+.OOO 22
+.001 843
+.04434
+.018 19
.00082
+.00004
.00024
.000154
.OOOOO
+.001 97
+.001336
.OOOOO
.00005
.00056
.OOOOO
.000 175
.00016
.00006
.00070
+.OOO 12
.00384
.005 552
.002 10
+.00041
+.00016
.00003
+.00432
+09472
+.065 56
.00303
+.00019
+.OOI 12
.000 553
+.00007
+.00441
+.005 097
.00026
.000 1 1
.00060
+.00008
.00102
.00051
+.00004
+ .O00056
.OOOOOO
.OOOOOO
.000154
.00000
JOOO O2
.00003
.coo 253
.00023
.00006
Arg.
a A 1
d.tf
**J&
dj
d.tf
23 A?, ! 5-c s
d-c
de
de
de
dr
dr
d r
de
d r
g g' X X'
COS
cos
sin
COS
COS
sin
cos
cos
o o o o
I O O O
2000
I I O O
O I O O
I I O
I O 2
2 O
I O 2
2 O 2 2
I O 2 2
022
I 022
O I 2 2
I I 2 2
O I 22
O 2
0.058 598
1.9784
+O.I63 69
O.OO64
0.000419
+0.001 8
+.227 700
+.0084
21735
.178059
+.01 1 7
+.0006
.000 281
.0247
.000695
.0076
+.179588
.0192
+3.040 o
0.16464
0.023 5
0.003 478
0.015 9
0.002
+000040
+.001 537
.000067
.000014
+.001786
+.000004
.0039
+.0107
.0054
.1948
+.016 I
+.01 1 5
+.0018
-0039
+.0290
.1781
.0098
+.195 6
+.0039
+.08881
+.017 7
.087766
.021 7
+.000877
+0.005 8
+0.0263
+0.7969
0.066 474
0.0087
+0.000 230
0.036 2
O.OO2 870
.010 7
0.023 02
0.360 o
+0.067 295
+0.014 3
o.ooo 620
0.014 7
+0.003 395
+.0053
+.014 604
.013 7
.016 720
+.000597
+.000 i
.013 2
.OOO 121
.000044
+.000 106
.004482
+.000375
.0090
+.000049
+.000432
XG44
+0132
112
ACTION OF THE PLANETS ON THE MOON.
TABLE XL.
FUNCTIONS OF COORDINATES REFERRED TO MEAN SUN.
Arg.
P-lf
/>'-3C 2
2651
D(?-f)
>(/> 2 -3C 2 )
2Z>,-jy
g g' * *
COS
cos
sin
cos
cos
sin
o o o o
I O O O
2000
I I
O I O O
I I O O
I O 2 O
O O 2 O
I O 2 O
2 O 2 2
I O 2 2
O O 2 2
I O 2 2
2 O 2 2
112 2
O I 2 2
112 2
I I 2 2
O I 2 2
I I 22
I O O 2
O O O 2
I O O 2
O 2 O O
.01396
.031 26
.00250
+.00013
.000933
-.00135
.00004
.00038
.00002
+.007 809
.163 02
+.086I9
+.05444
+.00300
.00055
.003 18
.00042
+.00027
+.OO3 22
+.00054
.00020
+.00396
+.OOO IO
.000 059
+.99074
-.10860
+.00096
.00071
+.000264
+.00055
.00032
+.01196
+.00066
+.001807
.018 60
01399
.00042
.00002
+.00015
+.00003
.00000
00087
.00006
.00000
.00032
.00084
.00004
+.000005
.033 92
.00243
.00008
.001 123
.00134
+ .000 10
.00038
.00002
+.007 185
163 54
+.98611
+.05444
+.00300
.00056
.003 17
.00042
+.00028
+.003 14
+.00054
+.00024
.00396
+.OOO IO
.000059
.052095
.063 73
.005 55
.00058
.003 423
.003 04
.00001
.00042
.OOO 12
+.00030
+.00291
.00700
+.OOO62
.00003
.00260
.00276
.00077
+.001 33
+.00289
+.001 24
+.00003
.000 13
.00031
.000 258
.004447
+.004 62
+.00007
.001 76
+.000 710
+.00126
.00009
.000 15
+.000 01
+.003 73
.035 03
.04784
.001 53
.00007
aoo 47
+.00089
+.OOO 01
.001 40
.00398
.00015
.00076
.001 52
+.00008
+.000036
-.07308
.005 47
+.001 37
.004 563
.00289
.00001
.00042
.OOO 12
.00060
.00025
.O07 70
+.00062
.00003
.00260
.00276
00077
+.001 33
+.00289
+.001 23
.00003
+.00013
.00031
.000266
Arir
a(-/,')
^-ac 2 )
#3
a^-tf)
3(/> 2 -3C 2 )
aej
Arg.
de
de
2 de
dr
^
df
g 1 * *
cos
cos
sin
cos
cos
sin
_.. ct7 r^r
I O O O
-0.565 7
1-955 i
-0.6088
+.0094
+.050 2
+.0095
O I O O
+0.004 519
+0.001 109
+O.OO6 22O
+.000 287
.000 161
+.000469
058 o
o 003 6
OI71
-4-.cei6
.CIS I
_j_ QIQ 6
I O 2 2
O O 2 2
2.964 i
0.2863
0.343 1
0.021 5
2.964 2
0.2864
if) rfjfi Q
+.0182
.1782
0066
+.0052
+ .0410
+/>l83
.1780
006 4
-f-OOO88
TOt 2
PART IV.
DERIVATION OF RESULTS.
CHAPTER VII.
CONSTANT AND SECULAR TERMS.
We recall the arrangement of the present work. In Part I, the general equations
have been formed, the theory outlined, and the methods developed so far as could
be done. Nearly all the fundamental quantities were developed as sums of prod-
ucts of two factors, one factor of each pair being a function of the Moon's coordi-
nates, the other a function of the coordinates of the planets. The latter functions
are developed in detail in Part II, one chapter of which is devoted to the develop-
ments of the coefficients of the direct action, the other to the coefficients of the
indirect action. In Part III, Chapter VI, have been developed the numerical
functions for the lunar coefficients. These are the same for both actions. The
present concluding Part is devoted to the combination of these factors and the
derivation and discussion of results.
We may divide the matter of this part into three chapters. In the first chapter
we consider the terms not purely periodic. By a purely periodic term is meant
one of which the coefficient of the sine or cosine is constant. We may, therefore,
define the terms to be first considered as constant and secular, two classes which
need not be considered separately.
54. The arguments on which the planetary and lunar factors depend are all
distinct except g', which is common to both. It follows that no constant or secular
term in the variations of the elements can arise by the multiplication of factors
depending on any other variable argument than^''. In all cases in which another
argument than this enters into either factor, the results will be periodic in form, the
coefficient, however, having, in the general case, a secular variation. Since no
terms of the class in question contain /, TT, or 0, they give
D. = o D = o Z>7 = o
To form the constant and secular terms we begin by collecting those planetary
factors which are either constant or depend on the argument g'. We shall con-
sider the direct and indirect actions separately. The planetary factors for the
direct action, as collected from 42, with some revision of the numbers there
found, are shown on the next page.
"5
Il6 ACTION OF THE PLANETS ON THE MOON.
FACTORS FOR DIRECT ACTION.
Action of Venus.
' = + 5". 9045 + o".44 cosg-' o".n sing-'
= 3 .4072 .30cosg-'+o .07 sing-'
== + o''.33 sing-' + o".O3 cosg-'
Action of Mars.
= + o".0468 o".O2O cosg-' o".O24 sing-'
= o .1006 + .028 cosg-' + o .029 sing-' (18)
= + o".oiosing-' o".oo8 cosg-'
Action of Jupiter.
\o*MK= + o".O9i3 o".oo2 cosg-' o".o32 sing-'
%io 3 AfC 2 .1348 + .oO4Cosg-' + o .062 sing-'
= + o".oo5 sing-' o".O3O cos g 1
Action of Saturn,
= + p".ooi3 ioWC' = o".i040 io*J/Z> = o
The corresponding factors for the indirect action have been combined for the
five disturbing planets, Venus to Uranus. From the combined values of G, J,
and 7, reached in 44, we find, including Uranus, but omitting Mercury:
io*m*G = + o".459 + o".36 cosg-' + o".o6 sing-'
io s /w 2 /= + o .153 + .12 cosg-' + o .02 sing-'
io'wz'/= o .03 sing-'
55. Lunar Factors. If, for brevity, we put F for any one of the three lunar
factors, say
F=?-f F'={t-tf* F" = 2fr (19)
the terms of the fundamental equations (42) or (57) corresponding to each F will be :
From the tabular values of the functions of the coordinates and their derivatives
in Table XL, p. 112, noting that symbolically, D' D-\-2, we have the following
values of the terms of these functions which are independent of the lunar arguments
F= 2 ?? 2 = .013 96 .000 933 cosg-' .000 06 cos 2g-'
F' = p* 3? 2 = .990 74 + .000 264 cosg-'
F" 2^rj = .001 123 sing-' .000 06 sin 2g-'
CONSTANT AND SECULAR TERMS. 117
D'F = D'(? r) 2 ) = .080 02 .005 288 cosg' - .000 38 cos 2g'
D' F' = -D'(p 2 3? 3 ) = 1-9770 + .001 239 cosg-' + .000 04 cos 2g'
D' F" = iD'ty) = .006 809 sing' .000 39 sin 2g'
de ' de
= + .1705 + .001 109
dF"
-^ = = = + .006 220 sin g'
de de
= + .0154 + .000 287 cos g'
<5375 ~ ' C S
= + .000469810^-'
The factors a,, e,-, etc., are derived in 14, and found in (26). From the preceding
scheme we find by using the preceding values and their derivatives in (20)
FI = 0.1641 .010 92 cosg-' .000 77 cos 2g'
F t = 2.1194 .086 40 COS-' + .OOO OI COS 2g'
FS = 0.0859 - 001 63 cos g'
FI = + 4.0086 + .002 49 cosg-' + .000 08 cos 2g'
FJ = -3- 3 J 4 2 -.021 36 cos g' (20)
FJ = + 3.0093 .000 90 cosg"'
F" = .013 89 sin g' .000 79 sin 2g'
FJ' = .118 94 sing-' + .000 01 sin 2g'
F" = .002 64 sing-'
56. Secular motions of I, TT, and 6. The function Aflf,as defined in 20, may
now be written
MH= MKF- \MCf + MD.F"
and introducing the linear functions of its derivatives which we have just formed
we have from (42)
DJ, = - MKF, + IMC^F' - MDF>'
Djf^ = - MKFi + \MCFJ - MDF? (21)
DJ = - MKF, + \MCFj - MDFl'
of all which factors we have just given the numerical values. For the indirect
action the second members are
..-(1=1,2,3) (22)
Il8 ACTION OF THE PLANETS ON THE MOON.
Performing the multiplications we find the following secular motions of /, TT O , and
arising from the terms of direct and indirect action under consideration.
Direct Action of io 5 /? n( /o
Venus 12.66 + 23.846 9-747
Mars 0.41 + 0.433 0.300
Jupiter 8.54 -}- 7.269 6.416
Saturn 0.42 + 0.346 0.318
Uranus o.oi -f- 0.007 0.006
Sum 22.04 + 3 1 -9 I 16.786
Indirect + 0.54 J-495 4- 0.422
Total 21.50 + 30.406 16.364
Taking the Julian century as unit of time n = 8400. The centennial motions
arising from the factors here employed are therefore:
Centennial motion of / = -- i8o".6, of TT O = + 255.41, of = 137.46. (23)
From the vanishing of D t a, D t e, and D t y we have
Sti = const. STTJ = const. B0 l = const.
the constants being functions of the arbitrary constants of integration, determined
at the end of this chapter.
57. Terms arising from the secular variation of the earth's eccentricity,
Both the direct and indirect actions contain, in rigor, terms of this class. They
enter into the direct action because the direct action of the planet on the Moon varies
with the variation of the orbit of the earth around the Sun. But the effect of this
variation is found to be so slight that it will be left out of consideration in the
present work. We therefore begin with the indirect action. The terms of the
coefficients G, /, and J, on which the action depends, have been developed in
Chapter V, 44.
Our fundamental quantity for the indirect action is H' of 25, of which the
only terms required are
H' = - G(? - O -J( P * - 3 O + 2/f, = - GF-JF' + IF" (24)
The terms of G, /, and / required for the present purpose are
G = Ge' I = 7,Ac' / = J^e'
G lt /i, andyi being found in 44 and
be' = - 8".595 T o".o26oT 2 = - 8". 595 T(i + .00302 T)
The secular terms of these coefficients thus become
+ i".8^ cos 2g-')T(i + .00302 T)
/=(4-o".io8o4-6".44cos-' +o".32cos 2g')T(i + .00302 T) (25)
/= (- 25". 4 5 sin^-' - i".8 4 sin 2g')T(i + .00302 T)
CONSTANT AND SECULAR TERMS. 119
Using these values in (56) we find
If we put G',J', and /' ior the coefficients of Tin (25) we shall have from (20)
and (22) the following computation for the secular accelerations from the funda-
mental equations (57), in which only the non-periodic terms are to be used :
[/] = G'F, + J'F{ - I'F," = + o".2 4 67
M = G'F, + J'Fj - I'F," = - i .6378
Iff] = G'F, + /'/7 - f'F," = + o .3246
Then, postponing the terms in Z 12
DJ=m*[l]T Djr=m*\*]T DJ = m 2 [0]T (26)
The terms in T 2 in (25) are only those arising from the term of e' in T 2 . To find
the complete values we note that all the terms of [/], [TT], and [0] contain e' as a
factor, and may therefore be expressed in the form e'k, k being a quantity which,
though containing minute terms in e' 2 , may be regarded as a constant. Then
D e'
/?,[/] = kDff = [/] -^- = - .002495 [/]
and the actual values of [/], [TT], and [0] may be written in the form
IX] = [/](! .00250 T) (I +.00302 T)= [/](! + . 0005 2 T)
[/], etc., being the values computed above. Multiplying by T we find that the
terms of D t l, D,TT, and D t O in T 2 are found from those in T by multiplying the
latter by the factor -|- .00052 T.
Taking the Julian century as the unit of time, w 2 = 46.998, whence
D l /= + ii".6oT + o".oo6oT 2 D t ir= -76". 98 T-o". 040 T 1 Z>,0= + i5. 25^+0.00797^
Then by integration
8/=5".8or 2 + o".oo2or 3 S 7 r=-38".497 12 -o".oi3r 3 W = f .62 T* + o" .0026 T* (27)
This value of the secular acceleration of the mean longitude is, I believe,
markedly smaller than any heretofore found. Delaunay's last result was 6".n,
which, reduced to the now adopted value of the secular diminution of e', would
become 6".O2. The necessity of using Delaunay's development of the parallax in
forming the Z>'s of some of the coefficients leads to some uncertainty in the present
result. But my rough estimate would lead to the conclusion that the uncertainty
should be less than one per cent, of the whole amount. The question of the pre-
cision of the value here reached I must leave to other investigators.*
* As this work is going through the press the author notices that Brown's value found in Monthly Notices Royal
Astronomical Society, vol. LVir, is reduced from $".<)i to 5".8i when the now adopted Die' is used.
120 ACTION OF THE PLANETS ON THE MOON.
58. We have next to consider the secular variations of the periodic terms in
general. Taking any set of such terms depending on any argument N
?* i;*= 2p cos JV p* 3%* = 2q cos JV 2%i) = K 4 sin JV
we shall have the terms of H' in (24)
(i".o72/-o".2i6?) Tcos A'-(i9".3/+6".4sr+ 12".7 4 ) Tcos (W-g 1
Forming the partial derivatives of these terms of H' as to /, TT, 0, a, e, and y, and
carrying them into the fundamental equations (64) and (65) by the processes of
22 and 23 we shall be led to
D ,a=m\i".o'j2ap+o".2i6ag)Tsin 7T+ w 2 ( I 9"-3 a /+ 6 "-4 a ?+ I2 "-7 a *<)^ sin (Wg 1 )
(28)
with similar equations for e and y formed by writing e and g respectively for a.
Also, we shall have
(20)
-7 l (i9". 3 Z' + 6". 4 Z"- i.2".>]L t ) ' v
with similar equations for D^ ir and D, u 6 Q formed by writing P and R respectively
forZ
59. The special values of N of most importance in the present connection are
on which depend, respectively, the constant term, the annual equation, the equation
of the center, and the evection
CASE I; 7V=o.
The factors for Z^a, D nt e, and D nt y all vanish. The values of the Z-coefficients
are found in the first line of Table XLIX, p. 147. The first or purely secular term
of (29) has already been computed. The remaining terms give
Z>/ = w s ( 3 8".6Z' + i2". 9 L")Tcosg-'
Z? ( 7r = 2 2 (38 .6P' +12 .<)P")Tcosg'
Dfa = m 2 n (38 .6ft' + 1 2 .9^?") T cos g'
Substituting the numerical values of Z, /"*, /?, and 7 2 = 47.oo;
/?,/= + 1062" T cos g' Z> ( TT O -- 2 9 6i"Tcosg' >&= +82f'Tcosg-'
We cite, for convenient reference, the following indefinite integrals
/it C if
t sin titdt = , sin N/ -- cos N I / cos wtdt = t cos N/ + - sin N/
N N J N N
CONSTANT AND SECULAR TERMS. 121
The unit of t in these equations being 100 years, N is the motion of g' in this
period, for which we may take 200 TT, or N =628.
Integration by the above formulas then gives
S/ = + i". 69 T sin g' + o".oo3 cos^'
87r = 4 .71 Ts'mg-' .ooScosg-' (30)
S0 = -f i .32 T sing' + .002
CASE II; N=g'; the annual term.
Here also the variations of a, e, and y vanish, so that only those of / , ir , and
are affected. Carrying into the equations (29) the numerical values of the lunar
coefficients for the Arg. g' we find, dropping the constant terms, which have been
already computed,
m^n T(o".oo6i cosg-' o".o82 cos 2g') = o".2gTcosg' 3".8o Tcos 2g'
m 2 nT(o .0485 cos g' + o .131 cos 2g') = 2 .28 T cos g' + 3 .^Tcos 2g'
nfn T(o".ooio cosg-' + o".O22 cos 2g) = o".tf Tcosg' + i .03 Tcos 2g'
Then, integrating, and dropping insignificant constant coefficients
8/ = -f o".ooo47 Tsing' o". 0036 T sin 2g'
8-7r = + 0.0037 Tsin g-' 0.0028 7"sin 2g'
S0 g = + 0.0008 T sin g-' o. 0016 T sin 2g'
CASE III; N-ff.
For this argument I have used the following preliminary values of the lunar
coefficients, differing from those of Tables XLVIII and XLIX by amounts here
unimportant
L' = 0.1197 L" = 0.1962 Z. 4 = 0.2648
P'=+ 5.403 P" = + 18.734 PI = + "-688
7?'== 0.0028 R" = 0.1431 7? 4 = 0.0560
ap =0.03223 &q = 0.11049 a* 4 = 0.06970
e/ = 0.300 45 6^= 1.03340 e* 4 = 0.650 66
gp = + o.ooo 09 g?= + o.ooo 31 g*4= + o.ooo 10
Carrying these values into the equations (28) and (29) we find, for the terms depend-
ing on the argument g alone,
D t e= +
, = + ^".o^Tcosg- Z?,7r = - 82". 2 Tcosg-
For the motion of^f, N =8329.
Integration then gives
8ot= o".ooo 062 T cos -+(7 4"-r- io')sin^- Be = o".oo$6Tcosg-
S/ 9 = + o".ooo 485 Ts'm g- b* =- o".oo9 87 Tsin g-
122 ACTION OF THE PLANETS ON THE MOON.
We drop the terms with constant coefficients, owing to their minuteness, and
find, with # = 8400;
Sn = - |Sa = + o"-78 T cosg
Then by integration
= + o".ooo
This, added to 8/ , gives for the entire term in S/
8t= + o".ooo 578:Tsin- (31)
60. In order to determine the complete expressions for the coordinates them-
selves, the terms computed in the present section, together with those which may
be found in a similar way for the other periodic terms, are to be carried into the
expression for the Moon's true longitude in terms of the elements. I have not,
however, deemed it necessary to do this in the case of the secular variations of
the periodic terms, because these can be most readily determined by varying the
value of e' in the Delaunay or Brown expressions for the Moon's longitude.
I have, however, computed the preceding variations of some terms owing to the
theoretical interest which attaches to the relations implied by the equality of the
result of the present method to those of the other method. The two methods
correspond to the two methods by which the secular acceleration ma}' be deter-
mined. In Action, p. 191, it is shown that the secular acceleration of /, TT, and 6
may be derived from the secular change of e' by determining the corresponding
secular changes in a, e, and y. This theorem has been discussed and extended by
Brown in his paper on Transmitted Motions and Indirect Perturbations.*
By this method the secular variations in question appear as variations of , rr^ and
&i, the latter being functions of the variables a, e, and y. But, in the present theory,
a, e, and y remain constant so far as the secular change of e' is concerned, and the
changes are thrown wholly upon / , TT O , and 6 .
There is therefore a seeming contradiction in that the lunar elements a, e, and y
are affected by a secular variation in one theory, while in the other they are prac-
tically constant. Referring to Brown's paper for the theory of the subject it will be
instructive to show the relation between the two methods.
In what I have, for brevity, called the Delaunay solution of the problem, the
Moon's coordinates appear as functions of the lunar elements, introduced as arbi-
trary constants, and of the Sun's eccentricity, which is regarded as a quantity given
in advance. But, when the action of the planets is introduced, the solar element
e', as well as the lunar elements a, e, and y, become variable. In what I may call
method A of treating the planetary action, which was that adopted in Action,
the final values of the coordinates as affected by planetary action are determined
by introducing the simultaneous variations of all four elements into the Delaunay
* Transactions of the American Mathematical Society, vol. vi, p. 332. See also, Monthly Notices, Roval Astro-
nomical Society, vol. LVII.
CONSTANT AND SECULAR TERMS. 123
expressions. But in method J3, adopted in the present work, the entire variations
have been thrown upon the lunar elements, the solar elements being regarded as
constant. In the case of the periodic perturbations this course is practically a
necessity, owing to the extreme complexity introduced into the formulae if we sup-
pose the coordinates expressed in terms of the value of e' affected by periodic
inequalities. But it is different in the case of the secular motion of e'. Here it
is more logical to consider that at any epoch the action of the Sun is computed
with the actual eccentricity at that epoch, and so to use method A.
Not having done this in the present work, but having regarded the value of e' at
the epoch 1850 as a fundamental constant, the values of G, J, and /, though func-
tions of e', and therefore variable, have appeared in the theory as constants.
In the present investigation the author has not, for want of time, investigated the
modifications which would be made in the problem if these coefficients were taken
as affected by their secular variations. One reason for refraining from this course
was that the determination of the secular acceleration from the equations given in
Action, page 191, require a much more extended development of the canonical
elements in terms of e' than it was practicable to undertake in the present paper.
The question is therefore left to others, reference being made to Brown's paper on
the variation of given and arbitrary constants.*
A comparison of the secular variation of the coefficient of sin g' with that found
by Delaunay's value of this term will, however, be of interest. With the eccen-
tricity of 1850 the coefficient of this annual term is 670". It contains e' as a
factor, the portion arising from higher powers of this element being unimportant in
the present case. It follows that the secular variation of the coefficient of sin g'
in 8v is
-670"^ = + i".6>jT
e'
The term found in (30) for 8/ is i".69 T. I have not computed 8v itself.
The two methods of treating the effect of the motion of the ecliptic are related
to each other in the same way as this just discussed. Had the method of the
present paper been strictly followed throughout, the coordinates of the Moon would
have been referred to a fixed ecliptic, because the ecliptic remains fixed when
planetary action is omitted. But it was seen that by a very slight and easily deter-
mined change, the coordinates could be referred to the actually moving ecliptic, and
and the work was carried on accordingly. In concluding the work, it is a matter
of regret to the author that he did not investigate the question whether the Moon's
coordinates could not, on the same principle, be expressed in terms of a varying
solar eccentricity, ab initio, thus simplifying the problem in conception at least.
Owing, however, to the theoretical interest attaching to the relation between the
two methods, the effect of the motion of the ecliptic might be treated by both methods.
* L. c., vol. iv, p. 333.
124 ACTION OF THE PLANETS ON THE MOON.
61. Adjustment of the Arbitrary Constants. The problem before us may be
outlined thus. The preliminary solution of the problem of three bodies leads to
expression of the Moon's coordinates as functions of six arbitrary constants, through
the intermediary of three other functions of these constants /, IT, and 6. The solu-
tion in terms of the six elements a, e, y, /, TT, 6 takes the form:
the functions <jt being of a form not necessary to specify at present. As already
mentioned, n, TT^ and 6 l are functions of a (or ), e, and y. The solution of our
problem is now completed by adding to the expressions for the Moon's coordi-
nates = v, r, and /8, the quantities
dv - dv - dv . dv
with similar forms for r and /8, which we need not write. For our present purpose
it will be necessary and sufficient to consider the following terms in v, the true
longitude.
v = / + 2e sin (/ TT)
We then have
dv dv dv
dv . dv dir dv dv dl dv dir
de~ dir de da~ dl da dir da
Substituting in (32) and emitting unimportant terms
Bv = fyit(\ + 2e cos-)8a 2c COS^TT + (i + ze cos^)8/ + 2 sing-Se
We put
H' 8< *o' ^o' 8/ o'
the arbitrary constants to be added to the perturbations 8a, 8e, STT, and 87. We then
have the following perturbations depending on the purely lunar arguments
Sa = Sa o".ooi6 sing- Be = Sc o".oi$o cosg
&I= &t g ",02i2nt + ".0059 sin- STT = STT O + ".o$i$nt ".272
Substituting in the derivatives we have the result that the mean sidereal motion
of the Moon is
nt(i o".O2i2 |S a)
We now determine 8 a by the condition that the mean motion shall be repre-
sented by n. Thus
S a = o". 0141 8 = o".o2i2 = + 178" (33)
CONSTANT AND SECULAR TERMS. 125
Also, the coefficient of sin g- in the expression for the longitude becomes
2e -f o
We now determine S e by the condition that the expression for the coefficient shall
remain unchanged. This gives
V = o".oo3
The longitudes, perigee, and node being given by the equations
ir = w, + V e = e a + oj
the introduction of the perturbations of the elements will give rise to the increments
*,-'&, + >& + ' *y W, = f' Bn + % Be + %**,
on ffe cy dn de dy
The eftects 8 e and 8 y are inappreciable. Taking only 8 from (33) we have
STT I .014 8oS n = o".ooo 3i4 &0 t = .001 oiS a n = ".ooo O2i
Taking the century as the unit, the adjustment gives
S7r 1= -2".6 S0 = -o".i8 and Sir -- 2".6T S6 -- o".i8T
Adding thereto the secular terms of IT O and already found, we have the following
results, tor the entire secular effect of the action of the planets on TT and 6
D,v D&
Direct action of the planets Venus to Uranus + 267". 97 141". oo
Indirect action of the planets Venus to Uranus 12 .56 + 3 .54
Total action of Mercury (/ == io~ 7 ) + o .45 o .21
Adjustment of elements 2 .64 o .18
Sum + 253 .22 137 .85
This motion of the perigee, greater by 5" than that found by Brown, goes to
confirm his conclusion that the gravitation of the Earth does not deviate from
Newton's law of the inverse square.
62. As the reason for the last correction may not be quite clear, it may be of
interest to state in a general way how it enters into the theory. The action of the
planets on the Moon is found on the supposition of what we may call an undisturbed
orbit of the Moon, meaning thereby an orbit in which the action of the Sun is com-
pletely taken account of, on the supposition that no other extraneous action enters.
We thus have a certain mean motion n determined from observations, and a certain
undisturbed mean distance, a, determined by the relation a s n 2 = /i, which requires a
constant A of correction to the mean distance computed from the action of the
Sun, giving rise to an expression for the constant of the Moon's radius vector
a -)- A^ = #! completely representing the action of the Sun on the supposition of
no planetary action.
126 ACTION OF THE PLANETS ON THE MOON.
It is with this mean distance a, that the actions of the planets, both direct and
indirect, are computed. But, as a matter of fact, the action of the planet modifies
the relation between a^ and n, so that we must change either the mean motion or
the mean distance according to what values of the elements we assume. If we
take the arbitrary constants so that the mean motion remains unchanged, then the
actual mean distance will require a constant correction on account of the action of
the planets. If we regard the mean distance as an invariable quantity, then there
will be a correction to the mean motion.
It follows by either method that when we compute the motion of the perigee and
node under the action of the Sun alone, we must make one or the other of these
modifications produced by the action of the planet, and determine the effect upon
the motion of ir and 0. If we regard the actually observed mean motion as that
due to the Sun alone then we must introduce a correction to the mean distance, and
determine its effect upon -n v and 6^ But if, which is the more natural method, we
regard the mean distance of the Moon as the given actual element, then we must
compute that part of the motion of the perigee and node due to the Sun alone with
a different n from that given by observation; that is, with a value found by sub-
ducting the planetary effect from the observed value.
We ma)' therefore regard the corrections -- 2" .64 and o".i8 to 77^ and l as
reducing ir^ and 0, to their true values under the action of the Sun alone.
63. Secular Variation of e. If we require, as we should, that the coefficient
of sin g in the Moon's true longitude should be represented by a function of e then
the expression (31) shows that this element will be affected by the secular variation
This being less than o".oi in a thousand years, is of no practical importance, though
of theoretical interest.
It may also be remarked in the present connection that the existence of this
variation, and the approximate algebraic expression for its amount, was first made
known by Adams.*
* Monthly Notices, Royal Astronomical Society, vol. XIX, p. 207.
CHAPTER VIII.
SPECIAL PERIODIC INEQUALITIES.
64. Reduction to the moving ecliptic. Since when the Sun is the disturbing
body the plane of the ecliptic remains fixed, the inequalities of the coordinates so
lar reached are referred to the ecliptic of any date regarded as fixed. The only
way in which they are affected by the motion of the ecliptic is through the secular
variations of the coordinates of the planet arising from that motion. The effects of
these are supposed to be too small to need consideration at present. It is, however,
necessary to refer the elements to the moving ecliptic. I have shown in 4 how
this may be done by the simple device of adding to the perturbative function the
terms
AT? = 2z(pD i x l - qD t y^) + 2 (qy - j>x) Dft (33)
and then integrating the portions of the differential equations thus arising. In this
expression p and q are the coefficients expressing the speed of rotation of the
ecliptic around the axes of y and x respectively, and are found by putting
II, the longitude of the ascending node of the moving on the fixed ecliptic;
K, the speed of rotation.
Then
p = K sin II q = K cos II (34)
It is to be noted that K is here used as the speed of rotation, and not as the actual
angle rotated through. It is, therefore, of dimension Z 1 " 1 and the expression for
AT? is of dimensions Z* 2 T^~~, which, by introducing the dimensions of mass, become
identical with the dimensions of P as hitherto used.
The partial derivatives of AT? as to the lunar elements are to be taken only as they
enter through x, y, and z, so that the Z>, of the Moon's coordinates, the latter being
called for this purpose x lt y v , and z lf are to be regarded as numerically given
quantities.
To form the partial derivatives of x, y, and z we use the developments of these
coordinates in terms of the lunar elements already given, substituting in x, y, and z
the values of , TJ, and . But in this part of the work it will be convenient to refer
the coordinates x and y to a general fixed X-axis, instead of the mean Sun, as here-
tofore. When this is done the expressions for the ratios of the coordinates to a
take the torm
7, = 2/&sinyV =2csinJV' (35)
127
128 ACTION OF THE PLANETS ON THE MOON.
where N and N' are of the general form
N= il + /> + i,0 +j'P +/X N 1 = i 7 / -f ,V + i t '0 +//'
the indices satisfying the conditions
* + *, + *; + j + /, = o i r + *,' + 1/ +/ + // = i
Informing the Z>,'s of these expressions we put n, n', the ratios of the motion of
the arguments N or N' to n, that of the Moon. We then have
sin TV -^(Ji = anS.kn cos TV -O^, = an"S.cn' cos TV'
The values (34) of ^> and q then give
//>,#, - gD t y l = anicZkn cos (TV - II) (36)
qypx = aiCLk sin (TV II) (37)
Our next step is to form the derivatives of z and qypx as to the lunar elements.
The partial derivatives as to z are found from the last equation (35)
Dz = alD'c sin TV' ~=2^sinTV' f = S^sinTV' (38)
cte fo dy dy
dz dz dz
=-. = aZt'c cos TV' a~ = aSz'.V cos TV' ~ = a2/c cos TV' (39)
Ol CTT VO
By differentiating (37) on the same system we have
D(qy -PX) = aKLD'k sin (TV- H)
., { ^_n) ( 4 o)
cos y_
= a 2^ cos (TV- H)
We next have to form the products of (36) by the derivatives (38) and (39) and
of D t z by (40) and (41), and form their several sums. We thus find that the
MOTION OF THE ECLIPTIC. 129
combination of any term of argument TV 7 " with any term of argument N' gives rise
to the following terms in the partial derivatives as to e and /:
-^ 2 w/c I nV^ nk^ \ sin (N -f N' II) + 2 /c I n'c^r + nk^ \ sin (TV 7 " N' II)
cte , oe oe ) oe Be }
d ~~^ = a>nKck{m' - t'n} {cos (N+ N'- H) + cos (N - N'- II)} (42)
The derivatives as to log a and y are formed from the first of these equations by
simple substitution. Those as to ir and 6 are formed from the last equation by
writing t\ and / 2 for z, and z'^ and z'' 2 for i' .
The derivatives thus formed being substituted in the fundamental equations the
integration of the latter will give the inequalities of the elements. It will be con-
venient to use the following formulae of substitution. We first put, in the combina-
tion of any term of argument JV with any term of argument JV' :
k a = n'cD'k + nkD'c k' = n'cD'k - nkD'c
, dk Be , dk dc
K = n'c -^ (- n ^- k ' = n'c^ -- nk =-
oe oe de oe
*,
, dk 7 dc , dk ,dc
k = rf c jr- + nk jr- t'**n'c-s -- n^-
oy dy ey oy
The quantities k a , /&', etc., will then be the coefficients of the constant factor
crnK in the expressions for the derivatives of the elements. Substituting AT? for
P 1 in the differential equations (27), p. 18, the latter will reduce to the form
Dp = (a/-, + OLJI, + a/,)* {cos (JV+ N' - II) + cos (N-N 1 -
D f = (eft + e 2 k, + e 3 k,} K { cos (N +JV'-U) + cos (N- N' - II) } (44)
ft = (7,*, + 7A + 7s*f) {cos (JV+ N' - H) + cos (N- N' - II)}
sin (^~ N> - n ) + (,*.'+ ?,*.'+ 7,V) sin
sm (N- JV'-U) + (of.' + e&+ t&')e sin
sin (N- N' - H) + (a/ a '+ '&+ 7^')* sin (^+ N> ~ D )
By integrating these equations we shall have, in the case of each argument, a
divisor which we may call N, equal to the motion of the argument in the unit of
time. The quotient K -^- N expresses the angular motion of the ecliptic during the
time required for the argument to move through the unit radian.
In the above differential equations we substitute for a,, e t , and y h their numerical
values and write, for brevity,
C t = 2.023^ 0.017^ 0.0229^
C, = - 0.0301^ - 19.153*. - o.o2o y (45)
C t = 0.0075^,, + o.oo26 e 5.570^
ACTION OF THE PLANETS ON THE MOON,
with similar expressions for the accented quantities, and
C a = 2.023-6, - 0.0301^ + 0.0075^
C e = 0.0168^, 19.153^, + O.0026/6,
C v = 0.0229/6, 0.0200/6,, 5.570/6,
(46)
We also put for brevity
A = TV- N' -
A' = TV+TV' -n
The values of N and N', the coefficients of the time in A and A' respectively take
the form
N = (n n') N' = (n + n')
and the differential variations become
Dp. = Cjc (cos A + cos A') Z>,/ = C> sin A + C/K sin A'
Df = Cjc (cos A + cos A')
D t y = C y x(cos A + cos A')
We shall then have by integration
K 1C
: = - C a sin A + , C sin
if if
Se = - C e sin A H -, C t sin A*
N N
5 7 = ~ C y sin ^ + ~ C y sin ^'
Z> ( TT O = C> sin .,4 + C> sin ^4'
D t = C> sin A + C t 'ic sin A'
/, = C, cos ^4 + C/ cos A'
K K
i = N ' C S + N"' ' C S
> <=-C t cosA + ^Qcos
(47)
(48)
The largest terms which enter into the theory are shown in Table XLI, for
Arg. TV, and Table XLI for Arg. TV'. The coefficients of the principal terms of
each have been derived from the numbers given in Part III.
TABLE XLIa.
COEFFICIENTS FOR FORMING pxqy ; ARG. TV.
NA
/ JT /' ff'
k
D'k
dkldf
dklBf
nt
Z ! '* ^1 ^
VKjUf
i
I O O O O
+ I.OOOOO
+-995S
+.9925
0.0588
+9955
+9955
2
2 I O O O
+ 1.091 55
+.0275
+.0275
+04980
.0000
+0547
3
O I O O O
+0.008 45
.0820
.0815
14934
.0000
.00069
4
I 2 O O O
0.983 i
+.0004
+.0004
+0.0142
xxwo
.0004
5
2 I O I I
+1.9168
+.O002
+.000 1
+0.0038
.0000
+.0003
6
O I O I I
+0.083 2
.0004
.0000
0.0070
.0000
.0000
7
I 02 O O
1.0080
+.OO2O
+.OO2O
+O.OOO2
+.0980
.0020
8
I 0020
0.850 4
.0083
.0081
+0.0052
+.0072
+0071
MOTION OF THE ECLIPTIC.
TABLE XLI3.
COEFFICIENTS FOR ; ARG. N' .
7 a it ^.t
No.
n'
c
D'c
Be/Be
dc/d r
n'c
i
2
O I I O O
I O I O O
0.0125
1.00402
.0072
+.0895
-.0068
+.0893
131
.0050
0.165
+1.989
.0001
+.0899
3
4
211 00
I O 120
1-995 56
0.8464
+.00247
+.0033
+.0025
+.0098
+.045
.000
+0.055
+0.068
+.0050
+.0028
It should be added that the coefficients of the smaller terms show only the order
of magnitude in each case, and not the precise numerical value. The latter will be
required only in the case of terms found to be sensible.
The theorem that all the inequalities have, as a coefficient, the motion of the
ecliptic during nearly one-sixth the period of the argument will enable us to limit
the combinations of the arguments to be considered.
In the case of any argument N N' containing the Moon's mean longitude,
one-sixth the period will never materially exceed 5 days, for which we have
- = o".oo64
N
In none of the terms of this class is there a factor C so large as to bring the coefficient
up to o."o5. It follows that all the combinations N =t N' which contain the Moon's
mean longitude may be omitted.
Of the terms which remain none can have a period several times greater than
that of the node, for which the ratio K : N = i".5. It follows that no combinations
of arguments giving products of coefficients less than o.oi need to be computed.
Numbering the arguments N as in the first column of the tables, these two rules
will be found to leave the following combinations as the only ones to be considered:
A; + Ay = e jv t + Ay = 2-* - A; - Ay = a/' - e A; + Ay = 2/' -
Using these numbers the computation of the formulae (43) gives the following
values of the coefficients , k M etc., for the argument 9.
Arg.
A; - AV
A;-A?
A; + A;'
Sum
h
*.
fc
*a
*.
*Y
.000358
O
+ 0.0891
+ 0.1779
O.OIO2
+ 1.972
I
O
I
+ .0002
+ .0050
+ .003
O
oooo
+ .0002
.000
I
O
+ .0002
+ .0004
+ oooo
+ .013
O
.0002
oooo
oooo
.001
.000359
O
+ 0.0889
+ 0.1785
0.0050
+ 1.987
132 ACTION OF THE PLANETS ON THE MOON.
It will be remarked that the only accented coefficients are those of the last two
lines; and that, as the combined argument 6 II is the only one included in sum-
mation, the accented and unaccented a , etc., may all be combined.
For these numbers we derive by (45) and (46)
C,= + 0.3157 C, = + 0.0506 C, = - 11.067
C a = + 0.001393 C t = + 0.000187 c -,= - -49S 2
We have from the adopted elements of motion of the ecliptic:
* = o". 4 7i 4 n = 173 30' + o'.59(/ - 1850)
The following are then the results for the argument 0:
8/ -- o". 44 i cos (0 - II) $7r = - o".oi cos (6 - II) 80 = + i5". 4 5 cos (6 - II)
(4Q/)
8a = - o".ooi945 sin (6 II) Se = o".ooo By = + o".6^i sin (0 - II)
To these expressions for 8/ , Sir , and 80 are to be added the respective increments
fSndt f&v^t and JX7
arising from substituting the values of 8(= f8a), 8e(= o), and 8y in the ana-
lytic expressions for n, TT,, and 0,.
The value of 8a gives the inequality of n
BH = |w8ot = o".oo292 sin (6 II)
This adds to the mean longitude the inequality
/= o". 002921; cos (6 II)
where v is the ratio n:(6 1 Z> ( H) = 248.7.
The complete inequality of the mean longitude thus takes the coefficient o".285.
We have from 27, (74)
BTT I = (.02283 . 0043387) n
The substitution of the preceding values of Sot and Sy gives the increments
&ir l = .00304 sin (6 H) and 8?r = o".'j6 cos (0 II)
We find, in the same way, the increment
80= + o".n cos(0-II)
The inequalities of /, TT, and 6 now become
87= +o". 285 cos (6- H) 87r = -o".77cos(0-n) 80 = + is".s6 cos (6 - II) (50)
The coefficients of the arguments 2-rr 6 and 2/' 6 seem so small that we leave
them out of consideration.
NODAL TERMS. 133
65. Inequalities arising from the coefficients E and F.
These inequalities have been considered separately on account of their minute-
ness, and on their depending on arguments different from those of the other in-
equalities. Some special values of the coefficients E and /''for Venus are given in
tabular form in Table XII. In these expressions the axis of X passes through the
mean Sun, as in the case of the inequalities depending on the mean longitudes.
But, on essaying the computation of the principal inequalities arising from E and
F, it was found that a fixed axis of X would be more convenient to use. The ex-
pressions were therefore transformed so as to refer them to the Sun's perigee as the
initial axis. From the form of the expressions the equations of the transformation
for x and y are readily found to be
x' = x cos g' y sin^' y' = x sing-' + y cos g'
where the accents refer to the fixed solar perigee. It follows that if
E = a cos N + b sin N F= a' cos N+ b' sin N
be any pair of the terms E and F depending on the argument A, the correspond-
ing transformed terms, which we represent by E' and F', will be
E' = l(a+ b') cos (N+g r ) 4- 1( - b') cos (JV-g f )
+ l(b- a') sin (N+ g'} + %(b + a') sin (JV- g')
F' = \(a' - b) cos (N + g 1 ) + \(a' + b} cos (IV -g')
The transformed expressions thus arising are shown subsequently in Table XLII.
As a check against any large accidental error in the development of the coeffi-
cients, their approximate values, neglecting the small eccentricities of Venus and
the Earth, were also computed by analytic development as follows: Taking the
mean radius vector of the Earth as the unit of distance, and putting a for the corre-
sponding numerical expression for the radius vector of Venus, the Laplace-Gauss
form of development will give
A- 5 = J23w cos t'L
L being the difference of the heliocentric longitudes of Venus and of the Earth
which we represent for the present by / and /' respectively.
The expressions for the rectangular geocentric coordinates of Venus will then
be, when powers of the eccentricities and inclination are dropped in the development
X = cos I' + a cos / Y = sin /' 4- a. sin / Z = a sin /sin (/ V )
where / is the inclination of the orbit of Venus, and 6 V the longitude of its node,
reckoned from an arbitrary fixed origin. Forming the product of the several fac-
tors which form E and F, noting that the summation changes from positive to
134 ACTION OF THE PLANETS ON THE MOON.
negative, changing and transforming the indices so as to reduce the summation to
its simplest form, the values of E and F take the following general form:
- a 2 6 fi+2) ) sin (iL zl' 4- V )}
sin I
F= 2{ -(a6 f <*+ l > - a 2 6 >) cos (iL + <? v ) 4- W +1) - 2 6 ( '+ 2) ) cos (iL - zl' + V )}
If we put, for brevity
fi. = J(ot3 e (l ' +1) 2 5 (<) ) sin / /8/ = i(a^ 5 ( ' +l) o 2 3 s (<+2) ) sin /
we shall have
E = 2/3. sin (iL 4-0)4- 2/3.' sin (iL zl' + V )
F= - 2/3 4 cos (iL 4- V ) 4- 2/3.' cos (iL - zl' 4- 6> v )
The numerical values of the coefficients b (i} may be taken from any one of various
publications. In Astronomical Papers of the American Epkemeris, Vol. V, Pt.
IV, p. 343, are found values of c 5 (i) = 2 (i) for Venus and the Earth, as follows:
*= o i 2 3 4
c 5 ( "= 44.88 43.64 40.61 36.52 31.99
From these we find:
i = 2
fi<5 6 ( '-*- 1) = 15.10
ft -f- sin 7= 4.95
ft'-*-sin/= 3.88
We thus have the following general expressions for E and /% the axis of X, in the
ecliptic, being arbitrary.
We use
sin /= .0592
Then
E= + .293 sin ( zL 4- V ) 4- .229 sin ( zL zl 1 4- V )
+ .273 sin (- L 4- V ) 4- .273 sin (- L zl' 4- <? v )
4- .229 sin V + .293 sin ( 2/'4- V )
4- .186 sin (L 4- # v ) 4- -291 sin ( L zl' 4- V )
4- .148 sin (zL 4- V ) 4- -274 sin (zL zl' 4- V )
F= .293 cos ( zL 4- V ) 4- -229 cos ( 2Z zl' 4- V )
.273 cos ( L 4- V ) 4- .273 cos ( L zl' 4- V )
.229 cos V 4- .293 cos ( zl' 4- V )
.186 cos (L + V ) + .291 cos (L zl' + V )
.148 cos (zL + V ) 4- .274 cos (zL zl' 4- V )
Measuring # v from the solar perigee, in longitude 279. 5, we have
V = 1550.4 P=g' 4-180
I
o
4- i
4-2
15-53
15.10
14.05
12.64
10.91
11.22
10.91
10.15
4.62
3 .88
3-H
2-51
4.62
4-95
4.92
4.64
NODAL TERMS.
135
The results both of this computation and of the analytic development are shown
in tabular form as follows:
TABLE XLII.
E AND F FOR THE ACTION OF VENUS.
Arg.
1
?
Arg.
j
F
T rfl
CC
s
si
!
T rrl
C<
>s
s
n
L g
Num.
Anal.
Num.
Anal.
L S
Num.
Anal.
Num.
Anal.
o o
+.088
-4- OO7
+095
.OOO
+ 008
.OOO
O I
+.197
-4- oiJ.
+.208
.OOO
.005
.OOO
O 2
I 2
+.124
+.I2O
+.122
+.121
+.259
.252
006
+.266
-.264
O 2
I 2
j i
.255
.252
+.Q2O
-.266
-.264
+.116
.113
+ on
+.122
.121
I
+.183
+.IQI
+.078
+.078
I O
I I
+.398
-4- oio
+417
.034
008
.036
I 2
2 2
2 I
2 O
2 I
2 2
+.U3
+.112
+.OIO
+.174
+.OIO
+.096
+.114
+.114
+"183
+.095
+.237
-.236
.002
+.129
+.014
+.198
+.248
249
+Ti32
+^208
I 2
2 2
2 I
2 O
2 I
2 2
28
.236
+.022
+.382
+.006
.200
.248
249
+401
-^08
+.104
.104
-.008
-.058
-.008
+.087
+.114
.114
-!o6o
+.095
The largest terms arising from E and F are those whose arguments are inde-
pendent of the mean longitude of the Moon, Sun, and Earth. These arise from the
constant terms of E and F, which are, when referred to the solar perigee
E= + .088
= + .197
The computation of the inequalities arising from this pair of terms will be yet
further simplified by taking the node of Venus as the axis of A. By transforming
to this axis we shall have
.ZT= .002 F
We may regard this value of E as evanescent, thus confining the terms we have
to determine to the expression
From the expressions for , 77, and we find the largest terms of the products
and their derivatives to be :
2 = .0895 sin .0039 sin (2/' 6) + .0006 sin (ZTT 6)
2Z>'^= .1786 " .0135 " +.0012 "
^ = .0137 " +.0006 " +.0218 "
Be
- . 089
+ -014
136 ACTION OF THE PLANETS ON THE MOON.
27;?= + .0895 COS + .0039 COS (2/' 0) .OO06 COS (27T 0)
yrj% = + .1786 " + .0135 " .0012 "
= + .0137 " .0006 " .0218 "
Brit
" +.089 .014
The resulting terms of H, heretofore omitted, are
Taking the node of Venus as origin, we have, as shown on p. 135, the following
terms of E and F
E = .285 sin 2/' F= -.216+ -285 cos 2/'
With these numbers we find for argument 6
H = .0182 cos D'H= .0348 cos -=r- = .0032 cos
- d Ji- d Ji- 5H _
These derivatives are to be substituted in the fundamental equations (41) and (42),
21, and each equation integrated. For the latter process the factor of integration is
j=- 248.8
i
The product of this into M for Venus ( 17) is i".o55
We thus have the following results:
= + .o6iMcos D^FQ = .o'joMcos -A.A = 2. 24^" cos
S/ = o".o64 sin 8ir = + ".074 sin 80 Q = -f 2". 36 sin
~) , = + .oooi36.fl/"sin D.e= + .ooood.'jMs'm0 D .7 = .loi^Ms'm '
*t( v nt it nt * i
Sa= + o". 000144 cos Be = -f ".000050 cos By = o".io7 cos
To find the complete inequalities in /, TT, and 6 we must add the respective quantities
f&ndt
of which the expressions in terms of 8, 8e, and 8y are formed by 27, Eq. 74.
We thus have, dropping unimportant terms,
= !8ot = o".ooo2i6w cos STT, = o.oi488w o.oo4387 = + o".ooo46w cos
&0 l = + .00388^ -f .ooo66&y = .00007 1 M cos ^
NODAL TERMS. 137
The completed values of 8/, 8?r, and 80 thus become
S/ = -f o".054 sin + 8/ = o".oio sin 6
STT = 0.115 sin & + ^""o = ~ o".O4i sin 6 (52)
80 = + 0.018 sin 6 + 80 = + 2".38 sin
in all which expressions 6 is reckoned from the ascending node of Venus.
The coefficients of the term in 2ir 6 are, for 8y and 80, less than one hundredth
those for 0, and the integrating factor v is less than 0.3 as great. The coefficients
in 2/' 6 are but a fraction of those in 0, and the integrating divisor is nearly 40
times as great. We therefore conclude that the inequalities depending on these
arguments are inappreciable.
66. Action of Mars and Jupiter. In Mars the product J/sin /is about .08
that for Venus. I have therefore not computed the terms.
In the case of Jupiter the largest quantities which enter into the constant part
of F are
-^-=1.26 YZ = \a*ya\I sin 7=0.0231
Hence
a'*jF= 1.26 x .02310'= + .000103
The product io 3 J// r is, approximately,
For Venus o"-92 For Jupiter + o".i7o
The inequalities depending on 6 are proportional to this product. We conclude
that the inequalities arising from the action of Jupiter may be derived from those
of Venus by multiplying the coefficients by 0.185. We thus have, from the
action of Jupiter,
8(9 = - o".43 sin (0 - 0,) 87 = + 0.020 cos (0 - 6,) (53)
where 0j is the longitude of the ascending node of Jupiter on the ecliptic. The
inequalities of the other elements are unimportant.
67. Combination of terms depending' on the longitude of the Moorfs Node.
The inequalities (49), (50), (51), (52), and (53), all depending on the same argument
0, may now be combined. We shall do this for the two epochs, 1800 and 1900.
The value of II which I have derived in Elements and Constants, p. 186, there
called Z', is
n = i732 9 '.7 -f S4'-4^ ( fr om 1850)
Taking approximate values of the nodes of Jupiter and Saturn, and this value of II,
we have
1800 1900
n 173 2' 1 73 57'
138 ACTION OF THE PLANETS ON THE MOON.
We take the nodes of Venus and Jupiter as constant, using the values for 1850
0y = 75-3 9, = 98. 9
Carrying these values into the inequalities of the elements in question and combin-
ing them, we find:
SI = + o".029 sin o".27i cos
STT= o.io sin + 0.80 cos
50 = + 2.55 sin 17.33 cos (for 1800)
80 =+ 2.31 sin 17.34 cos ^ ' ' ' (* or 1900)
&y = o". 1 14 cos 0.769 sin (for 1800)
&y = 0.103 cos 0.770 sin (for 1900)
68. Special computation of the Hansenian Venus-term of long period.
The following are the planetary and lunar arguments whose differences make
up the argument
i8v i6g-' g
of the term in question.
Planetary Lunar
(1) l8v l&g-' g 2g'
(2) 18 17 g g'
(3) 18 -16 g
(4) 18-15 ff + g'
The coefficients h^, h^, etc., are computed by 22 and 23. The planetary
coefficients MK, MC, and MD are found in Table X. The lunar coefficients ap,
etc., are given in the next chapter, Tables XLVIII and XLIX. For the argument
g g' we change the signs of a, e, g and k, as given for the argument g-\-g'.
I have not computed the coefficients for the argument^ 2g' believing their
effect to be insensible. Their characteristic is ee'" 1 = .000050, and, in the principal
term of 17, this is in Brown's theory multiplied by a factor of the order of magnitude
.04. The largest planetary coefficient being 0.5 -=- io 3 , the value of h^ will be of
the order of magnitude i" -=- 10, which would result in a term in 8/ of the order of
magnitude o".o2. Actually, the computation shows that the combinations (2) and
(4) are also much smaller than (3).
We have now all the data for computing the coefficients h^, h^,', etc., from
the formulae of 22. The results are:
*.'-- "-5597 - 10 " A., .' = + ".4880 -4-io 6
h ec ' = .0052 -4- io 3 k e ,'=+ .0047 -4- io s
h liC ' = .00081 -T- io 3 k, / = .00094 "^ IC>3
h, c = + .084 -T- io 3 k^ > ,'= + .095 -T- io 3
The coefficients for y and are much smaller, and are omitted. The coefficients
we have given correspond to the argument
N A r 4 = g + i6g' i8v = A
HANSENIAN INEQUALITIES. 139
of which the annual motion is
N = - 4 747".8
giving
" = - 3649
We therefore have the following inequalities in I M IT, and e
S/ = o".oo3 sin A + o".oo3 cos A
STT + o .31 sin A o .35 cos A
Be = o .019 sin A o .017 cos A
The term of 8/ is so minute as to be unimportant. For the term in the mean
longitude arising from 8 we have
which gives
8/= - n".i8cos^ + 9".7Ssin^ = i4".83 sin (A 4855'.2)
It will be convenient to use the negative of this argument in order that its motion
may be positive. We shall therefore write
8/= i4".83 sin (i8v i6g' g+ 2285s'.2)
where v is the mean longitude of Venus measured from the earth's perihelion.
It will be of interest to compare this result with those reached by other investi-
gators. The following are arranged in the order of time. Putting
Z, the mean long, of Venus that of Earth
M= i8Z + ig' g
and reducing all results to the mass 1-^408,000 of Venus, there has been found, for
the direct action, by
Hansen* 8/ = 15". 34 sin (M + 229. 2)
Delaunayf = 16 .34 sin (M+ 228 .5)
NewcombJ = 14 .80 sin (M -f 229 .5)
Radau = 14 .14 sin (M+ 229 .o)
Newcomb (above) = 14 .83 sin (M '+ 228 .9)
To judge the precision of this value we have to estimate the error to which the
development by mechanical quadratures is liable. The circle being divided into
60 parts, any coefficient which we have taken as A 18 is really the sum of an infinite
series of which the first two terms are A a A i . We have dropped all the terms
after the first. From the progression of the coefficients it would seem that the
* Tables de la Lune, p. 9. t Conn, des Temps, 1862, App., p. 58.
^Action of Planets, p. 286. \Inegalitts Planitaires, p. 113.
140 ACTION OF THE PLANETS ON THE MOON.
ratio A ( : A i+ i is approximately i : 1.26, whence the ratio A w : A& would be about
250. The error of the computed term may therefore well be o".o6. It has
been only as this work is in press that the author has looked into the possible effect
of the slow convergence ; and while it seems likely that the error entering through
the coefficients J*C and C will not exceed that just stated, the same may not be true
of the coefficient D.
A quantitative estimate of the correction may be made in various ways; but
the author is unable to enter upon the subject in the present work.
It is also to be noted that the term as above computed contains the effect, what-
ever it may be, of the mutual perturbations of Venus and the Earth. A separate
computation has been made of the fundamental numbers due to these perturbations,
but as the final result of the coefficients amounts to only a fraction of a second, the
computation has not been completed. The effect being included in the computed
term, a knowledge of its amount is necessary to compare the result with that reached
by the ordinary method of development.
The change in the term as computed is too minute to account for the observed
variation of long period in the Moon's mean motion. As the period of this varia-
tion seems to be nearly the same as that of the inequality under consideration, the
question naturally arises whether the effect of the indirect action may be appreciable.
This being the most important question in the lunar theory, a computation of the
principal part of the indirect term has been made. The result being altogether un-
important, it seems unnecessary to do more than present such a brief statement of
the method as will enable the subject to be taken up by another in case the author's
conclusion is not well founded. The required perturbations of the Earth by Venus
are most easily computed for the case in question by using, instead of the Lagrangian
brackets, the corresponding functions of the coordinates. The formulae necessary
for the purpose are found in Moulton's Celestial Mechanics, p. 291. The eccen-
tricities have been dropped as unnecessary, and attention was confined to the longi-
tude elements. The terms dependent upon the action of the planet on the Sun are
also dropped, being appreciable only in terms depending on small multiples of mean
a' 3
longitude. The development of -g- used in the computation is that in Action,
pp. 248-251. The result for the indirect action is
81= + o".O44 cos A o".036 sin A.
This, being added to the terms already found, gives for the entire term
S/= i4"-77 sin (i8v \6g' g + 228 54')
which is the definitive result of the present investigation.
RADAU'S TERMS. 141
69. The Radau terms oj long -period, Radau has computed certain addi-
tional terms of long period due to the action of Venus, with the following results,
the arguments being reduced to those adopted in the present work:
Sv = + o".i40 sin (ITT +g 2ov -f 19^-' + 171) Per. = 34^.8
-f o .no sin (g 26v + 29g-' + 62) 127.2
+ o .056 sin (g 2iv + 2ig) 8.35
+ o .019 sin (ir.+ g- 23V -f 24^' + 295) 55.
+ O .016 sin (TT + g I5v + ng' + 219) 71.
o .012 sin (27r g + 24V 26g' + 159) 58.
+ o .012 sin (g 23v + 24g-' + 14) 7.6
+ o .008 sin (tr Q+g 23V + 24^+ 101) 28.2
+ o .004 sin (20---f 23V -24^-+ 183) 42.
+ O .003 sin (TT g+ 2IV 2lg' + 288) 148.
The first three of these terms are the only ones that need be considered for the
practical applications of the lunar theory. The third might also be omitted, but is
easily computed in connection with the first.
For all the terms except the second the planetary coefficients A, B, C, and D
may be derived with all necessary precision from the special values of these coef-
ficients given in Table VII, by the following process. Putting
let the value of the planetary arguments for which we desire the coefficients be
N= hL + kg'
Recalling that the 720 special values of each coefficient, say A, are arranged in
12 systems of 60 indices each, the special value of N~ corresponding to they'th sys-
tem and the index i will be
7Vy = 6 x hi+ 30 x kj
We may mark each special value of A in the same way. The values of the
coefficients A c and A, will then be given by the equations
= 2 AU cos N^
360^4, = 2 Afj sin JV itJ
The terms of A for the special argument TV will then be
A = A c cos N+ A t sin TV
In most cases the computation may be simplified, as in the usual method of
executing periodic developments, by adding together in advance the special values of
A which are to be multiplied by the same sine or the same cosine. Another method
142 ACTION OF THE PLANETS ON THE MOON.
may be used in computing these terms by the developments found in Action,
Chapter III, 18. Some modification is, however, necessary owing to the circum-
stance that in that work the rectangular coordinates are reckoned from a fixed axis
passing through the earth's perihelion or the solar perigee, while in the present
case the axes pass through the mean sun. It is therefore necessary to use the ex-
pressions for the geocentric coordinates of Venus referred to this moving axis, a
development which may readily be made from the special values already given
for the coordinates of Venus and the sun. It is necessary to transform the table
so that the arguments shall be the mean anomaly of Venus instead of its mean
longitude because the development for A" 5 which are tabulated on p. 25 of
Action have the mean anomaly of Venus as an argument.
I have applied the method of development from special values to the first term
with the following results:
Planetary Coefficients for Arg. 2OV 2ig' '.
A C = + -03562 A.= + .00694
B c =- .02999 B t = - .00583
C c = .00563 C t = .001 1 2
D c = + .00634 D . = -3 2 S4
K e = -f .03280 K t = + .00638
C e = .00282 C t = .00056
The lunar portion of the argument is equivalent iD g, of which the indices in
Table XL are ( i, o, 2, 2). From the numbers in this table we find for the
direct action
7r 2ov + 19^ + 10)
IT being measured from the earth's perihelion.
This coefficient is less than that found by Radau ; but the lunar argument is one
to which the present method is not well adapted and a redetermination is desirable.
None of the other Radau terms are completely computed in the present work.
Such computations as I have made seem to indicate even smaller coefficients than
those found by Radau.
CHAPTER IX.
PERIODIC INEQUALITIES IN GENERAL.
70. For convenience we mention the formulae derived in Part I, giving
them the special form adopted in the actual numerical work. We recall that the
combination of any lunar argument N with a planetary argument N gives rise to
two arguments G, N+N^ and N JV 4 . For each argument there are two terms
in the D nt of each of the elements, one a cosine term; the other a sine term. We
represent the coefficients of these terms for the element a by
&a,c, ^,. A *,c', and h^
with a similar notation for the remaining elements,
e, 7 / *> and e o
except that the coefficients for the angular elements have the negative sign.
The expressions of these coefficients for the direct action are given in extenso in
Part I by the equations (46), (47), (48), (50), and (51). For the indirect action the
coefficients are given in (64) and (65), but we may use the equations for direct
action by making the substitution indicated in 25 (66), which gives the expres-
sions for the sum of the two actions.
For convenience in computation the coefficients are so used as to give the result
in terms of o".ooi as the unit. The numerical values of the planetary coefficients
practically used for the purely periodic inequalities are these
K t ' = \MK c - m*G c ) CJ = io\(\MC c + </J A' = 3 (MD e + VJ
with corresponding values of Kj, C s ', and D,'.
Since each combination of a lunar with a planetary argument gives rise to two
combined arguments, one equal to their sum the other equal to their difference,
the coefficients relating to the latter are distinguished by accents.
The numerical values of the planetary coefficients, as derived from the numbers
of Part II, and just defined, are shown in the following tables.
i 4 4
ACTION OF THE PLANETS ON THE MOON.
Values of the Planetary K' -Coefficients, Combining Direct and Indirect Action.
TABLE XLIII.
ACTION OF VENUS.
Arg ^
v, g 1
K:
c;
D:
*:
c. f
D:
12
0.61
o'tf
O.O2
o!ii
It
.00
0.46
I I
+25.04
10.73
+35-66
+0.03
.01
0.05
I O
+ 2.44
0.52'
+ 2.31
O.O2
+.03
0.04
2 4
- 0.68
O.O2
+ 0.61
O.O6
-.06
0.12
2 3
+ 6.53
- 1-74
+20.01
+ I.2I
47
4-33
22
29-33
+ 8.77
52.76
+0.25
.12
0.27
21
349
+ 0.50
3-29
O.O2
+.03
0.08
3-6
0.31
+ O.O2
+ 0.28
0.24
+.02
O.22
3-5
0.94
+ 0.15
+ 6.54
0.61
+.15
5.29
34
+10.08
3.25
+ 12.11
+2.25
-.69
2.69
3 3
+ 4-64
1-99
12.92
+0.19
03
0.16
4-6
+ 0.61
O.2O
+ 0.85
+0.53
-.19
0.76
4 - 5
+ 0.68
+ 0.03
2.19
+0.05
.00
+ 0.40
4 4
+ 7-1 1
2.38
9-72
O.O2
+.01
+ 0.08
5 -8
+ O.I2
0.04
+ 0.71
+0.17
-.05
1.04
5-7
043
+ 0.16
- o.8S
0.36
+.12
+ 0.78
5 -6
+ 0.82
0.23
- 1.36
+0.09
-.06
+ 0.37
5-5
+ 840
2.43
7-39
6-9
+ O.O2
O.OI
O.O2
+0.03
.01
+ 0.03
6-8
+ 0.18
0.06
0.16
+0.14
.05
+ O.I2
6-7
+ 1.19
0.28
O.I I
+0.24
-.06
+ 0.21
6-6
+ 7.32
1.86
- 6.64
o.oo
.00
0.00
7-9
+ 0.18
0.05
0.16
+O.I2
.04
+ O.II
7-8
+ 1.13
0.24
O.II
-j-O.22
-.05
+ 0.21
7-7
+ 6.25
1.42
- 5-79
0.00
.00
0.00
8 -13
0.0045
+ 0.003
1.289
+O.o66l
.022
15.92
NOTE. The units in these tables are o".ooi.
PLANETARY COEFFICIENTS.
TABLE XLIV.
ACTION OF MARS.
Arg.
M . g*
*7
c>
D:
K:
CY
D:
I I
I
0.78
+ 0.18
+0.13
-0.31
+ 2.34
+ 0.32
a
O.OI
+0.08
+.01
+.01
II
0.05
0.27
2 2
2 I
11.70
- 0.45
+3-97
+0.38
+17.04
-11.08
+0.34
0.63
.01
+.24
+0.26
+8.80
3 3
3 2
+ 1.38
+ 0.8 1
0.43
0.25
0.95
2.44
+0.16
+0.89
J02
27
+0.15
+2.69
4 4
4 3
4 2
+ 0.66
+ 1.86
0.12
0.17
0.60
+O.OI
O.2O
2.Q4
0.50
+O.22
+ 1.84
+0.59
.04
-.64
-25
+O.2I
+3-05
4.90
5 4
5 3
0.52
+ 0.04
+0.18
0.02
+ 0.34
0.19
-0.68
0.47
+.21
+.15
0.48
-1-73
6 -5
6 -4
6 -3
O.22
+ 0.04
+ 0.15
+0.07
O.O2
0.05
0.03
O.I I
+ 0.69
0.33
0.74
0.08
+.09
+.25
+.05
0.06
1-34
+0.50
IS -9
IS -8
+ O.I I
+ 0.04
O.O2
O.O3
0.13
1-55
0.15
O.OO
+.04
+.01
0.25
0.70
TABLE XLV.
ACTION OF JUPITER.
Arg.
J, S'
K:
c:
*>:
A'/
c-:
D:
+ 1 -2
4-25
+ o'-72
+ 4-20
072
+o."i9
043
+ 1 -I
-41.87
+12.87
+61.07
1.53
+0.28
1.67
+ 1
+ 1.45
+ O.2I
+ 2.08
+1.35
0.12
21.72
+2 -3
+ 2.OI
0.41
- 1.98
O.2O
+O.O2
+ O.IO
+2 -2
+30.81
- 8.39
-17.37
+0.38
O.o6
+ 0.35
+2 -I
+ 7-86
- 2.88
12.22
+3.26
0.94
+ 5-29
+2
0.21
0.04
+ 0.53
O.I4
O.OI
+ 0.17
+3 3
+ 4-53
0.63
10.71
0^0
+0.08
0.39
+3 -2
0.15
+ 0.06
+ 0.18
5-93
+1.67
3-69
+3 -i
+ 0.24
0.08
0.40
0.97
+0.38
- 1.67
TABLE XLVI.
ACTION OF SATURN.
Arg
s, g 1
K:
c:
D;
K:
c.'
D:
I I
I
246
+O.OI
+o&3
+ox>3
+348
2.64
0.03
+O.OI
+o'.oi
O.OI
0.03
0.50
2 2
2 I
+1-33
+0.66
0.33
0.23
-I-S7
0.94
O.OO
+O.O2
O.OO
0.01
O.OO
+0.05
146
ACTION OF THE PLANETS ON THE MOON.
71. The lunar coefficients fall into two classes, one determining the elements
a, e, and y and called, for brevity, the a-coefficients; the other determining /, TT,
and 0, and called the Z-coefficients. Those of the first class are computed by
the formulae of 20 and 22 ; those of the second class by the formulae of 23,
Eq. (50). In the computation we write k for 4 .
The a-coefficients are the nine products of the factors a, e, and g defined in 22,
Eq. 43, into p, y, and k.
From 22, (46) to (48), it will be seen that by using the planetary factors in the
form just given and taking the a-coefficients
a<7,
etc.
the coefficients of the terms of D Rt (a., e, and y) will each be the sum of three prod-
ucts of two factors each. But the quantities we actually compute are the values
of 2e and zSy. We therefore double the coefficients for 8e and Sy, using
2eg,
and gk
We have also multiplied the inequalities of TT and 6 by the factors 2e and 2y,
required to reduce them to inequalities of the actual longitude and latitude. To do
this we take for the nine Z-coefficients
L', L",
2cP", eP
t
Each of the coefficients to form a term of D nt l w 2eD, tl ir or 2yD nl 6 t> will then be
the sum of three products formed by taking one factor from one of the Tables XLIII
to XLVI, and the other from Table XLIX, the product Z>'Z 4 being divided by 2.
TABLE XLVII.
DATA FOR a-coEFFiciENTs.
Arg.
***.*
Arg.
i i' i"
a
e
g
o o o o
000
0.0000
oo.oooo
00.0000
O I O O
g'
O O
o.oooo
00.0000
oo.oooo
I I O O
g-ff
I I O
+2.0529
+19-137
00.0029
I O O O
g
I I
+2.0529
+19.137
00.0029
I I O O
g+e'
I I
+2.0529
+19.137
00.0029
2 O O O
2g
2 2 O
+4.106
+38.274
00.0058
2 O 2 2
2n2g
O 2
0.0602
38.307
00.0400
I O 2 2
2Dg
I I O
+1.9927
19.170
00.0429
O 022
2D
2 O O
+4.0456
00.0336
00.0458
I 022
2D+g
31
+6.098
+19.103
00.0487
O I 2 2
20 g"
200
+4.046
00.0336
00.0458
O I 2 2
20+g'
2 O O
+4.046
00.0336
00.0458
I O 2
2\-g
I I 2
+1.978
19-175
+11.0971
O 2 O
2\
2 O 2
+4.0306
00.0388
+11.0942
1020
2\+g
3 i 2
+6.083
+19.098
+11.0913
O O O 2
2\'
O O 2
+0.015
0.005
+11.140
VALUES OP THE LUNAR COEFFICIENTS.
TABLE XLVIII.
LUNAR H-COEFFICIENTS FOR a, e, AND 7-.
Arguments.
a/
zq
y^k
e#
e?
y^k
st
g?
y*&
g, g'^ *
/, 1C, 9, g'
o o o o
O O O
o
o
o
o
o
o
o
I O O
+ 1 I
0.032 09
.11147
0.034 8 1
0^299 i
1.039 i
0.324 6
+.00005
+.00016
+.00005
2 O O
+22
0.005 13
+.00197
0.004 99
0.047 84
+0.018 37
0.046 50
+.OOO OI
o
+.OOOOI
I I O
-I +1 +1
+0.000 13
.00073
+0.00008
+0.001 24
0.006 79
+0.00077
o
o
o
I I O O
+ 1 -I +1
0.001 39
+.00056
0.001 38
O.OI2 92
+0.005 26
0.01282
o
o
o
r 020
+ 1 +1 2
o.ooo 04
.00032
+O.OOO IO
+0.000 38
+0.003 17
0.00096
.00022
.00178
+.00056
O O 2 O
+2 O 2 O
o.ooo 77
+.024 10
0.00077
+O.OOO OI
o.ooo 23
+0.000 01
.002 II
+.06634
.002 1 1
I O 2 O
+3 I 2
0.00006
+.00199
0.00006
o.ooo 19
+0.00630
0.000 19
.000 1 1
+.00366
.000 1 1
2 O 2 2
0+2 O 2
o.ooo 235
.000 054
o.ooo 216
O.I49 20
0.034 28
0.13790
xioo 16
.00004
.000 14
I O 2 2
+ 1 +1 2
0.16243
.018 53
0.162 95
+1.562 6
+0.1783
+1.5675
+.003 So
+.00040
+.003 51
O O 2 2
+2 O O 2
+1.9949
.028 26
+1.99470
0.016 56
+0.000 23
0.016 56
.02258
+.00032
-.02258
I O 2 2
+3 +1 -2
+0.1660
.001 28
+0.1660
+0.519 98
0.004 01
+0.51998
.00132
+.OOOOI
.00132
I 22
+2 O O I
0.00643
+.00006
0.006 41
+0.000 05
35 * 10"
+0.00005
+.0^007
68y-!-io
+.00007
O I 22
2 O O 3
+0.00651
.00012
+0.006 35
0.00005
708 -s- io 9
0.000 05
.oo< > 07
i37-;-io 8
.00007
O O 2
O O 2 +2
+297-t-io 7
63 -no 1
297-j-io 7
99-t-lo'
+ 2I-S-IO 7
+99-*-io 7
+.02206
.00468
.02206
TABLE XLIX.
LUNAR Z.-COEFFICIENTS FOR /, n, AND 0.
Arguments.
L'
L"
L>
*eP'
2eP"
eP,
ifR
2 r J?"
r^
g, s' i, i
/, ff, e, ff '
o o o o
I O O O
2 O O O
I I O O
O I O O
I I O O
I O 2 O
O O 2
I O 2
2 22
I O 22
O O 2 2
I O 2 2
2022
I I 2 2
O 22
I 22
I 2 2
22
I 22
I O O 2
O O 2
I O O 2
2 I 2 2
2 I 2 2
o o o o
+ 1 I
+2 2 O O
-I +1 4
O O O +1
+ 1 -I +1
+ 1 +1 2
+2 O 2 O
+3 I 2 o
0+2+02
+ 1 +1 O 2
+2 O O 2
+3 I 2
+4 I 2
+ 1 +1 -I
+2 O O I
+3 -i o -i
+ 1 +1 3
2 O O 3
0.08204
0.123 05
0.010 05
0.000 32
0.005 454
-0.005 59
0.00009
0.00099
o.ooo 16
+0.013 67
0.302 23
+1.09222
+0.10262
+0.005 12
0.003 614
0.00922
0.001 58
+0.001 810
+0.00943
+O.OO2 27
O.OOO 1 1
+0x05 65
0.00037
o.ooo 019
+O.OOO IOI
+2.004 31
0.199 14
+O.OO2 48
0.003 16
+0.001 241
+O.OO2 42
+0.000 1 1
+0.01775
+0.001 OS
+0.00684
O.070 22
0.076 97
0.002 34
O.OOOII
o.ooo 172
+0.000 94
+0.000 OI
0.003 176
0.004 14
o.ooo 15
0.00007
o.ooi 69
0.000 95
+0.000 130
0.000006
0.11635
+0.595 03
+0.077 07
o
0.004 743
+0.02651
0.181 96
+2.05609
+0.057 30
+0.006 73
o.ooi 172
0.00039
+0.024 25
+0.003 15
0.008 24
0.06908
+0.36088
+O.O22 69
+O.OO7 90
GO"? 8^
+135 12
.012 85
0.27507
0.019 74
+0.002 44
0.014 128
0.01086
+0.00038
0.00203
o.ooo 32
+0.02351
0.612 74
+3-984 14
+0.205 24
+0.010 24
0.007 273
0.018 40
0.003 16
+0.003 66 1
+0.01866
+0.004 54
O.OOO 22
o.on 29
+0.00090
o.ooo 207
+0.000 224
+0.6404
+0.072 8
.00246
.000 oi
.00250
.00004
0.00653
+0.027 o
.000073
+.000041
.00012
+.014 50
13790
.00490
+XXJOOI
.001 36
.010 28
+O.OOO 02
+O.O0002
+.00433
+.00378
0.29875
+3-11720
+0.298 oo
1.027 38
0.11462
+0.015 989
+0.000 02
+0.006 73
0.009992
o.ooo 02
0.008 927
-0.275 19
+3-1 17 4
+0.2981
1x1273
0.1146
+0.015 785
+XWOO4
.00500
+.045 17
+.00180
+.OOO OI
.000003
+.00004
.005 03
+045 14
+.001 75
+.00001
.000002
+0.0063
O.OIO I
O.OOOO2
0.0088
+O.OI5 462
+O.OOOOI
0.00464
+.000 002
XXX) OO3
+.000 ooi
3 i o 3
I +1 2 +2
02+2
+ 1 I 2 +2
0.000 23
+0.000 04
+O.OOO 22
-.04875
+.00885
+.04880
+0.000 915
0.002 797
0.002766
0.000042
+0.002 976
0.002966
o
o
o
o
o
o
148 ACTION OF THE PLANETS ON THE MOON.
72. From these two tables the four coefficients for each element are formed by
the following computation, an adaptation of (46) to (51)
The inequalities of e have received the factor 2, and those of TT the factor 2e in
order to transform them into the principal terms of the true longitude without
further multiplication.
Two other points which may be recalled are these: (i) We use k instead of K t
in the formulae; (2) it is to be recalled that C c ' and C,' contain only ^C, as that
symbol is used in Part I.
Element a.
, = *>.# - C,'ag 2 = - JT'a/ + C^q
Element e
y *, = - 2/T/e/
Element y
Element /
\ = K c 'L'-C c 'L" \ = X i
h lt c = \- \D:L, h lt / = x,
+ \ *i. / = i
Element TT O
- \
- 2 C'eP"
= TT, - D.'eP t
= D c 'eP t + 7r 2
Element 6
In the exceptional cases when one of the constituent factors of either class,
planetary or lunar, is a constant, there will be a merging of the accented and unac-
cented arguments and terms.
PERIODIC INEQUALITIES IN GENERAL. 149
For the case JV = o, a, e and g all vanish, and we have
A ( = A/ = A,7 = o = o
while (49) of 23 may be written
- DJ. = (tK.'LJ -2C c 'L a ") cos 7V 4 + (iK.'LJ - 2C.'L 9 ") sin N t
We have, therefore, in this case, only to double the values of the Z-coefficients
for argument o.
In the combination of a constant planetary factor (7V 4 = o)Vith a periodic lunar
factor we may use, instead of (46)
Then
Sa = vh^ , cos N
with similar equations for e and y, formed by writing e and g for a. We also have,
instead of (51)
Then
8/ = - vh lc> sin N
with similar equations for IT and 6.
As neither D nor /has a constant term, there are only cosine-terms of this class
in a, e, and y, and only sine-terms in /, TT, and 0.
From these coefficients for the D nt of the elements we have those for the ele-
ments themselves by multiplication by the integrating factor v. The motion of the
lunar argument is
in + i'ir l + t"0 l + jn' = N
and that of the planetary argument
k'n' + kn t = N 4
We compute
I
v =
Then the coefficients which we compute are
. = Vk a,c = - "/',, .' = "'^, ' / = - V 'kJ
2e,' = 2v'h etC ' 2e c ' = - 2v'/i ti , f
27, = 2v/l yiC 2J C = 2V/l y> , 2y,' = 2v'k yc '
r s = i/ x
T r = -f- 1/ x
with similar forms for when required,
150 ACTION OF THE PLANETS ON THE MOON.
The inequalities of the elements are then
SI = l c cos (N+ N,} + 1. sin (N + N^ + // cos (N- JV t ) + // sin (N - 1VJ
STT = 7T C " + TT. " +7T/ " + IT.' "
$0 = e c " + e. " + <?' " + e.' "
A similar computation was made for y and 0; but the results were unimportant in
all but one of the arguments.
73. The motions of the arguments from which the integrating factors v or v
are to be computed are the following. The sidereal motion for a Julian year is
given in revolutions for the lunar, and in seconds for the planetary arguments.
Then follows the ratio of each to the mean motion of the Moon.
Motions of Arguments.
Mot. in 365 d .25 n
;".= i3'-255 523 0.9915452
/; = 13 .368 513 i.
TT ;?!,= 0.112990 0.0084518
; 1 = o .053 765 0.004 021 8
Venus 2106 64i".38 0.121 5913
Earth 1295 977 .43 0.074 8013
Mars 689 050 .9 0.039 777
Jupiter 109 256 .6 0.006 3061
Saturn 43 996 .2 0.002 5394
The elemental inequalities computed from these formulae are shown in tabular
form on the following pages. On making the computation it was found that the
coefficients for a were so minute that no terms in the parallax would need to be
considered, and only in some exceptional cases, generally terms of long period, did
the inequality of y affect the longitude. The coefficients for these elements are
therefore omitted in the tables of longitude elements. The given coefficients are
those for the mean longitude, 8/, 280, e8ir. It must be remembered that the accented
e' and ir' do not refer to solar elements, but designate only the coefficients depending
upon the differences between the lunar and the planetary arguments, while the
unaccented coefficients depend upon their sum.
It was also found that the inequalities of y and 6 were insensible in nearly all
cases. The few terms of these elements found to be sensible are therefore given
separately.
74. Terms -with purely Lunar Arguments. We here make a single com-
putation for the combined action of all the planets. To include the effect of the
indirect action, we have only to modify the values of MK, etc., as indicated in (66).
INEQUALITIES OF ELEMENTS. 151
Then, from the values of the constant term already given for the four principal dis-
turbing planets in 54 we find
io'2y]/^T = + 6".o7o io s I.MC -- 5".76
- io 3 w 2 = - 0.459 loW/, = + 0.153
lo 3 /^' = + 5.611 io 3 C ' = 2.727
For the terms in question we now have, for each lunar argument
and
the terms in y and 6 being omitted as unimportant.
The inequalities of / , e, and IT may now be computed as in 26 and 27. The
most condensed formulae of computation are
ioV --
The elemental inequalities then are
SI = / t sin TV e&ir = eir i sin TV Se = e e cos TV
The results of this computation for the only terms which I have found to give
any appreciable result are, in units of o".ooi;
Arg.
g
?,D2g
2Z>*
The only corrections of the true longitude to be considered are the following
to the evection and variation.
I0 3 /.
I0e c
icfeir.
3-8
- 9 .0
- 9 .0
+ 1.5
+ 14.0
-I 3 .6
+ 8.2
+ 21.6
-21.5
22.
O.I
- 0.9
+ 0.4
+ 2.O
+ 2.0
sn 2
o".O2i sin 2D.
75. Elemental Inequalities. The miscellaneous inequalities of the mean
longitude, the eccentricity and the perigee, as given by the preceding formulae and
data, are tabulated in the following pages.
It may be repeated that the mean longitudes, v, M, j, and s, are measured from
the solar perigee.
ACTION OF THE PLANETS ON THE MOON.
Periodic Elemental Inequalities in Units of o,"ooi.
TERMS INDEPENDENT OF THE LUNAR ARGUMENTS. (1V= o.)
Action of Venus.
Action of Mars.
Arg.
V
I.
I.
2<?7T t
1<?7T.
Arg.
V
I.
'.
2r.
2r.
V
+ 8.22
- 14
i
+ 6
O
M
+ 25.1
30
i
+ 4
o
v-/
+ 21-37
-831
+ i
+208
M-Z-
28.6
II
+ I
+ 7
V 2/
35-70
+ 56
i
I
2*1 g"
+2II.O
+308
-181
-48
+49
2V 2^
+ 10.69
+324
+ 5
107
I
2VL2&
- 14.27
200
+60
2V 3^
+ 53-25
-314
+ 90
+114
-24
3M 2g
33-o
+ 29
- 3i
9
+ 10
2V 4^
- 17-85
+ 3
4
+ 3
O
3M-3^
9-5
+ 14
4
o
3V 3^
+ 7-12
- 5i
+ i
+ 13
o
4M 2g'
+ 105-5
+ 2
+ 95
3
-24
3V-4S'
+ 15.25
173
+ 36
+ 54
12
4M 3^
- 15-3
+ 32
35
10
+ 10
SV-Sg"
-108.3
-48
+ 54
+ 30
21
SM 3g'
39-1
+ 23
20
i
- 6
SV-Sg"
+105
15
+ 18
+ 4
- 6
ISM-Sg'
540.5
+ 39
+ 22
9
2
8v-i3/
+310-4
+ 16
+246
- 5
-74
Action of Saturn.
Action of Jupiter.
Arg.
V
I.
'.
ar.
2/r.
Arg.
V
t.
1.
2en g
27r c
S
+ 39-4
+ 48
+ 16
4
o
J
+1586
+ 171
+ 41
+ 4i
57
s-s-
- 13-8
40
+ ii
o
1-*
- 14.6
652
+ 13
+2IO
7
25 g'
14-3
+ "
i
- 3
1-of
7-0
15
+ 4
+ 9
2
252?
- 6.9
+ 8
o
- 3
o
2}~g'
16.1
+165
- 52
-46
-18
2}2g'
7.30
+208
i
- 74
i
2J3g'
4-7
+ 6
o
3
o
3J-S 1
17-9
+ 5
24
2
- 6
31 2g"
- 7-6
2
+ 42
+ I
-15
3J3S"
4.9
+ 9
+ I
- 6
+ i
LUNAR ARGUMENT N=.
Planetary
Argument.
V
I/
/.
I,
/;
c
.
a*.
*;.
2C c >
2 ex,
2<? ' T c
2C7T,'
2ex c '
V-/
+0.9631
+ 1.0585
+ I
O
+16
o
o
-13
o
-63
13
-63
o
2V-3^
+0.9898
+ I.028I
4
+1
+ 6
+ I
+ i
+ 5
+ 5
22
+ 5
I
22
- 5
2V2g r
+O.92I6
+I-II37
I
o
22
o
o
+ i
o
+77
+ i
o
+77
o
3V Sg"
+ 1.0180
+0.0993
+ i
+1
+ I
+ 1
+ 3
+ 5
+ 3
- 3
+ 5
- 3
3
^_ ij
3V-4S"
+0.9460
+1.0800
o
o
+ 4
o
o
- 5
+ 4
23
5
-23
4
3V 3^
+0.8814
+I.I750
+ 3
o
I
o
o
-13
o
+ i
13
o
+ 1
O
2M 2g"
+1.0853
+0.9420
- 6
o
+ i
o
o
+29
o
+ 5
+29
+ 5
o
2M g'
+ 1.0038
+ 1.0134
+ 2
o
2
tj
- 6
+ 9
I
o
+ 7
+ 5
+ 9
+ 5
J-f
+ 1.0834
+0.9434
23
o
O
o
+ 2
+98
I
+ 10
+97
2
+ 11
o
J
+ I.OO22
+1.0150
O
+5
O
+13
+ 5
o
+15
2
13
2
-IS
2J2g'
+ I.I703
+0.8861
+ 150
o
+40
o
o
-54
o
21
-54
21
o
2Jg'
+ I.076I
+0.9490
+ 40
I
oo
- 8
- 8
21
+ i
3
21
+ 8
3
I
INEQUALITIES OF ELEMENTS.
LUNAR ARGUMENT N= iD 2g = in ig' (EVECTION-TERMS).
'53
Planetary
Argument.
V
i/
I,
1.
//
V
2e,
2C ,
**:
*;
2 en.
2en e
2671,'
*/
V-g'
11.64
5-572
i
+ 6
o
o
- 19
o
+ 101
+ 18
O
101
o
2V 3^
- 8.779
- 6.602
2
o
+ 3
+ I
^_ ty
30
II
+ 50
+ 30
- 8
- 51
ii
2V 2^
- 2S-S7
4419
+ 9
o
- 6
o
O
+ 134
__ J
106
-130
o
+ 107
i
3V Sg'
7-046
8.100
i
+ I
+ i
+ I
12
- IS
IO
+ 12
+ IS
12
13
ii
3V 4^
14-90
5-045
+ 2
o
O
i
7
+ 33
o
I
33
7
3V 3g"
+ 129.9
- 3-66
+72
o
2
-661
7
+659
2
+ 7
o
6v-8/
625.0
- 3-79
+50
37
+49
+ 64
-64
+49
o
o
SV-8^
+737-7
3-43
o
o
o
o
- 5
- 24
o
o
+ 24
- 5
o
H-*
- 5-062
10.24
o
o
o
o
- 5
o
+ 4
+ 5
4
o
2M 2g"
4-932
- 15-96
2
o
+ i
o
+ i
- 42
o
+ IS
+ 42
o
15
2M g"
- 7-815
- 7-276
+ I
2
5
2
+ 17
+ 25
+ 19
25
- 25
+ 17
+ 25
+ 19
3M 3g'
- 4-205
36-23
O
o
o
+ 3
o
+ 6
_ .j
o
6
3M 2g'
- 6.135
- 9-766
o
o
O
+ 6
+ 6
+ 4
4
- 6
+ 6
+ 4
+ 5
4M 3^
5.050
- 14.84
o
O
+ 7
+ 7
+ 4
3
7
+ 7
+ 4
+ 5
4M 2^
8.116
7-034
o
+ 9
+ I
ii
i
i
IO
+ i
ii
iStf-Sg-
7429
- 7.64S
o
o
o
o
I
+ 3
i
3
i
29
o
o
6M-5S 7
3-73
+3704
o
i
- 8
i
22
+ 29
+ o
i
29
23
J 2^
- 3-6232
+ 94-3396
o
o
i
i
- 9
IO
+ 15
+ 9
i
- 16
9
J-S'
4.9702
- 15.5763
- 8
o
+ 3
5
-151
O
+ 54
+I5i
5
- 53
o
J
7-9II4
- 7-1942
o
+ 2
O
+ 3
44
I
- 4 6
+ 7
+ i
44
- 7
- 46
2J 3^
2.9028
+' 12.6422
o
o
o
O
o
+ 3
I
i
- 3
O
o
I
2J 2^
- 3-7078
+232.720
+ 3
o
+256
+ I
+ i
+ 54
+ 5
-1158
- 54
+ I
+1164
+ 3
2Jg"
- 5.1308
14.1844
+ 2
I
I
o
+13
+ 30
+ 6
12
31
+13
+ IS
+ 6
3J-3/
2.9568
+ 13.7363
+ I
o
+ I
o
+ 11
+ 13
o
+ 22
13
I
21
3J-2^
3.7965
497.760
o
+ I
- 8
-258
ii
o +429
14
o
II
+ 15
+43i
SJ-g'
5-3022
13.0208
o
o
o
o
- 4
+ i
2
i
4
+ i
2
LUNAR ARGUMENT N= 2D g=g'+ 2n 2g-'.
Planetary
Argument.
V
!/
I,
/
//
V
2*.
2g e
M.>
/
2 en,
2en c
2 en,'
2en c '
V-g'
1.1042
I-23I
- 6
o
+47
o
- 33
o
+239
+ 32
o
237
o
2V 3^
1. 1395
I.I9O
IO
+ 4
+20
+ I
IO
-48
20
+ IOO
+ 47
ii
99
21
2V 2^
1.0500
1-307
+10
o
-69
o
+ 75
3
342
74
o
+341
3
3v se"
I.I770
I.I52
- 4
+ S
+ 4
+ 4
21
- 28
-16
+ 21
+ 27
22
20
-17
3V-4S"
1.0828
I.26I
i
o
+18
+ 5
I
- 5
19
+ 89
+ 5
I
-89
-19
3V 3^
1. 008
1.392
+ 10
o
- 7
o
+ I
+ 57
i
35
- 55
+ I
+ 35
i
2M 2^
1.2679
1.0765
23
o
+ 3
o
+ 3
-116
o
+ 17
+ 121
O
- 16
o
2M g'
1.1581
I.I708
+ 9
6
9
7
+30
+ 42
+35
-46
- 42
+29
+ 44
+34
4M 3g"
1.2602
I.082I
+ 2
2
o
o
+20
+ 19
+ 4
3
- 19
+19
+ 3
+ 4
1-2S"
1.3976
0.9979
+ 9
- 6
9
o
- 5
- 38
+ i
i
+ 36
5
+ i
J-/
1.2653
1.0784
-8 1
+ 10
o
-13
416
o
+ 59
+415
13
- 59
o
J
I-I559
I.I730
i
+16
+ i
+16
-74
2
-84
+ 14
+ I
-74
13
84
2J 3g~
I-S456
0.9340
+ 3
o
o
2
+ 19
+ i
19
2
o
+ i
2J2g'
1.3854
1.0042
+44
o
+ 8
+ 3
+215
o
+ 45
211
+ 3
45
o
zjg"
1.2553
1.0858
+15
5
4
4
+34
+ 81
+ 8
13
- 81
+34
+ 13
+ 8
3J 3g"
1-5307
0.9395
+17
3
o
3
+ 73
o
- 19
72
- 3
+ 18
3J 2g"
1-3734
i. 0106
+ 10
o
I
42
3
+ 7
o
+ 3
42
o
+ 7
M-i*
i. 2455
1.0933
o
o
o
IO
+ 2
2
o
2
10
o
2
ACTION OF THE PLANETS ON THE MOON.
LUNAR ARGUMENT zD = ig + in 2g' .
Planetary
Argument.
V
V>
I.
(i
I.'
?
2e,
2e c
*.'
2*.'
2<r7r (
2OT C
2C7T/
2en c '
v-S
+527
+554
+ 21
122
o
o
o
O
I
+ 2
10
o
2V 2?
+.514
+-569
43
o
+ 172
T
O
+2
4
+14
o
2M 2^
+.562
+521
+ 59
+ 2
II
o
+i
o
I
+ 5
i
o
J-e"
+.56I
+521
+212
- 6
-36
o
+2
I
+ 17
o
3
o
J
+539
+542
+ I
39
7
45
+1
O
+1
! + I
-4
I
-4
212?
+.584
+-503
106
+ 2
23
o
2
o
o
9
o
3
o
LUNAR ARGUMENT
2n 2g' .
Planetary
Argument.
V
v'
I
I
i:
V
26,
2C c
*'.'
2eJ
2en t
2en ,
Mr/
2r/
v-f
+.346
+.358
+1
O
-4
o
O
3
+22
4
o
+23
o
2V 2^
+341
+.364
I
o
+6
o
+ 8
o
31
+ 8
o
31
o
2tt2g'
+.361
+343
+2
i
o
o
ii
o
+ 2
ii
o
+ 2
J-f
+.36I
+344
+7
o
i
o
I
-39
o
+ 7
-38
+1
+ 7
o
2J2S"
+370
+.336
3
o
i
o
+18
o
+ 5
+18
o
+ 5
o
LUNAR ARGUMENT i\ 2D.
r.
r c
r/
r/
r.
r c
r/
r/
v-g"
+13
- 5
4X3S'
_!
T
I
+ I
2V Sg"
+1
+ 7
+1
4
4M 2g"
+ 1
o
+1
+14
2V 2g"
3V 4^
3V Sf
4V 4g'
o
+1
+1
o
14
+ 4
+ 2
o
o
o
+2
o
+17
i
2
24
1-21?
1-8?
I
O
+6
o
+11
+ I
o
+1
+6
I
19
2 J 3 /
o
o
o
M g'
o
+ I
o
I
2J 2^
+25
+ 7
2M2g'
o
+ 3
o
5
2j e"
I
2
2
+ 4
2V. g
o
3
o
- 4
3J 3^
o
- 6
+ 2
3M 3^
o
o
o
3J 2^
-3
o
+1
o
3M 2g"
I
i
I
+ 1
3J-S-
o
o
+1
o
76. Reduction to inequalities of true longitude and collection of results.
To complete the work it is necessary to transform the elemental inequalities into
inequalities of the coordinates. As already remarked, the parallax appears to
contain no sensible terms arising from the action of the planets; only inequalities
of longitude and latitude are therefore considered. In the case of terms of very
long period the transformation to true longitude is unnecessary, because these terms
can best be used and compared as elemental inequalities. A precise classification
can not, however, be made between the terms which are to be transformed and those
which are not. What has actually been done is to retain as elemental inequalities
those depending on the longitude of the Moon's node, because though they may
INEQUALITIES OF ELEMENTS. 155
ultimately be transformed for use into the inequalities of the coordinates, they are
to be combined with terms arising from the compression of the Earth having the
same argument. The two Venus-terms of very long period have not been trans-
formed because, as already remarked, they can be most conveniently applied to the
elements. To transform the other terms put Sv, the perturbations in longitude in
orbit. Then
v = I + 2e sin g + %# sin 2g
Sv = 8/ -f zSe sin g- + $e8e sin 2g + 2eSg cos g -f \ e 2 Sg cos 2g
Substituting
81= l c cos G + /, sin G STT = TT. cos G + ir t sin G Be = e c cos G + e t sin G
SI$Tr=8g=g- t cos G+G,sin G
we shall have
Sv = S/ + 2<? f sin G sin g + 2e e cos G sin g
+ 2eg c cos G cos g + 2eg t sin G cos g
+ \ee a sin G sin 2g + ^ce c cos G sin 2g
+ \ e 2 g c cos G cos 2g + f 2 g", sin G cos 2^
= g/ _ (e, - eg c ) cos (6! + g) + (e c + eg) sin (G + g)
+ (. + eg.) cos (G-g)- (e c - eg,) sin (G - g)
- \e(e, - eg) cos (G + 2g) + \e(e c + eg,) sin (G + 2g)
cos ^ - 2 - . - eg'. sin - 2^-)
In nearly or quite all cases we may drop terms of the second order in e and use
Sv = l c cos G + l.sinG + [_e(l c - TT,) - .] cos (G + g) + [>(/. - TT,) + e e ] sin (G + *)
T) + .] cos (^ - ^) + [(/. - TT.) - e,-] sin(G- g)
The subsequent processes are so simple and familiar as to scarcely need statement.
All terms of 8v depending on the same argument are combined into two, one
depending on the sine, the other on the cosine of the argument. Their values are
shown for each argument in the following table. The two terms are then combined
into a monomial satisfying the equation
v t sin G + v c cos G = Sv sin (G + A)
Terms of which the coefficient St> was less than o".oo3, have generally, but not
always, been dropped. It will be seen that even exceeding this limit there are
more than 150 periodic inequalities. These are so arranged that any one argument
can, it is hoped, readily be found on a system which will be evident by a little
examination.
The constituents of the arguments, including ir, are all measured from the
Earth's perihelion (7r = 99.5). The secular variations of the coefficients of the
periodic terms are omitted, because they can better be derived by varying the
eccentricity of the Earth's orbit in the expressions for the inequalities due to the
Sun's action.
156
ACTION OF THE PLANETS ON THE MOON.
PERIODIC INEQUALITIES OF THE TRUE LONGITUDE.
ACTION OF VENUS.
Argument.
v.
i
9f
A
Argument.
z> c
v,
dv
A
V 2g
II
.001
+.055
n
.055
359o
g+2TT 3V+g"
.000
.014
.014
180.0
V-g-
+.OOI
-.882
.882
179.9
g+2ir3V+2g r
+ .012
+.051
053
13.2
V
.000
.014
.014
184.0
g+2TT3V+3g'
+.014
+.016
.021
41.2
2V 4^
.004
+.003
.005
306.8
g+2TT2V
+ .OOI
-.174
.174
180.0
2V Sg*
+.096
340
354
164.3
g+2ir2V+g'
+ .OII
+.070
.071
9.0
2V 2^
+.005
+401
401
0-7
g+2ir v g"
.000
+.146
.146
O.O
3V -Sg"
+.060
.056
.082
133-0
g+2ir+ V Sg"
.000
-.025
.025
180.0
3V-4/
+.040
.191
.197
168.3
g+2ir+2VSg'
+ .011
.040
.041
164.5
3V 3^
+.001
.037
037
178.4
g+2ir+2V4g'
.000
+.142
.142
O.O
Sv-Sg'
+.018
-.015
.023
129.8
g+2ir+3V-7g'
+.017
.019
.026
138.2
g+2w+3v-sg'
+.OO2
-.646
.646
179.8
g-Sv+Sg"
+.004
+.003
.005
53-2
g+2Tr+Sv-8g'
+.005
.024
.025
168.3
e-sv+sg'
.000
+.008
.008
O.O
g+2ir+6v log'
.051
+.066
.083
322.3
g3V+4g'
+.008
+.040
.041
II-3
gSV+Sg*
+.015
+.018
.023
39-8
g2V+2g"
+.OOI
-.093
093
180.0
2g+2ir3v+g'
.000
035
035
180.0
g-2V+3g'
+.018
+.080
.082
12.6
2g+2lT3V+2g'
+.019
+.090
.092
11.9
2g+2ir3V+3g'
+.016
+.020
.026
38-7
gv
.000
.004
.004
180.0
2g-\-21T2V
+.003
-.142
143
178.8
eV+f
.000
+.166
.166
O.O
2g+2ir2V+g'
+.O2O
+.IOO
.102
1 1-3
g- V +2g"
.000
.003
.003
180.0
2g+2lT V g"
.000
+096
.096
0.0
g+ V 2g-
.000
+.003
.003
O.O
2g+2ir+ v -tf
.000
.008
.008
180.0
g+V g'
.000
.149
149
180.0
2g+2lT+2V5g'
+.OIO
.048
049
168.3
g+v
.000
-.004
.004
180.0
2g+21T+2V-4g'
.000
+.024
.024
O.O
2g+2ir+3V7g r
+.O22
.027
035
140.8
g+2V3g'
+.018
.078
.080
167.0
2g+2ir+3V6g'
+.OOI
.005
.005
168.7
g-\-2V2g"
+.001
+.071
.071
0.6
2g+2ir+3V5g'
.001
+.056
.056
359-0
e+SV-Sg"
+.015
.016
.022
136.9
+3V-4 /
+.008
.036
037
167-5
g+3V-3g"
.000
-.006
.006
180.0
3g+2ir2V
.000
+.009
.009
0.0
g+SV-Sg-
+.004
.003
.005
126.8
3g+2ir- V -g-
.000
-.006
joo5
180.0
27T 2V
.000
.009
.009
180.0
21T 2V+g'
+ .001
+.003
.003
184
4g+2TT2V
.000
+.003
.003
O.O
27T V ff
.000
+.007
.007
O.O
4g+2TT V g"
.000
.002
.002
180.0
27T+2V 4g"
.000
+.008
.008
O.O
27T+3V $g"
.000
+.072
.072
O.O
2ir- g +3V-5/
.000
+.003
.003
O.O
27T+6V lOg'
-.037
+.050
.062
323-5
2TT g +6V IO/
.002
+.002
.003
315-0
INEQUALITIES OF THE TRUE LONGITUDE.
'57
ACTION OF MARS.
Argument.
v c
v.
8v
A
Argument.
V c
f.
8v
A
H
n
II
II
II
o
M-g-
+.OOI
.Oil
.Oil
174-3
2ir 2M g'
.000
+.OO2
.002
O.O
M
.OOI
.030
.030
l8l.9
2M 2g"
.OOO
.224
.224
180.0
27T 2M 4^
.000
.004
.004
180.0
2Mg'
-IQ3
+.317
372
328.7
3M3S'
.OOO
+.014
.014
0.0
S+21T 4M
+ .011
.OOI
.Oil
95-2
3M-2g'
.031
+.029
.042
313.1
g+2ir 4M+/
.004
.003
.005
233.2
4M3g'
.035
+.032
.047
312.4
+2?r 3M
.004
.004
.006
225.0
4M 2^
+.095
+ .OO2
095
88.8
g+27T-3M+g /
.000
+.006
.006
o.o
SM 3^
.020
+.023
.030
319.0
2+27T 2M+^
.026
.034
043
217.4
+27T 2M
.000
+.017
.017
O.O
f 4M + 22'
+.017
.002
.017
96.8
g+2lT M g"
.000
+.004
.004
o.o
S-4M+3/
.007
.007
.010
225.0
g+2TT+ M 3^
.000
.005
.005
180.0
2 3ii+2g-
-.006
-.006
.009
225.0
g+21T+2M 4g'
.000
-.066
.066
180.0
g3M+3g'
.000
.003
.003
180.0
g+2ir+2U3g'
.023
+034
.041
325.9
g2tt+g-
.035
.O42
055
2194
g+2ir+3U5g r
.000
+.003
.003
0.0
g2U+2g-
.000
+043
043
o.o
g+2ir+3U4g'
-.006
+.006
.009
315.0
gM
.000
+.003
.003
0.0
g+2ir+4USg'
-.009
+.009
.013
315.0
g-M+g~
.000
+.004
JOO4
0.0
g+2ir+4U4g'
+.009
+.OOI
.009
83.7
g+ M ^
.000
.004
.004
180.0
g+M
.000
.003
.003
180.0
2g+2ir4M+g'
.004
.003
.005
233.2
+2M 2g~
.000
.048
.048
180.0
2g+2TT2TAg'
035
.046
.058
217.3
g+2U-g'
.036
+.042
055
3194
2g+21T2M.
.000
+.006
.006
0.0
+3M 2^
.006
+.006
.009
3i5.o
2g+21T+2ll4g'
.000
.050
.050
180.0
2+3M 3g'
.000
+.003
.003
o.o
2g+2lT+2M3g'
.030
+.042
.052
324.5
g+4U 2g"
+.017
+.002
.017
83.2
2g+2ir+4V5g'
.020
+.019
.028
313.5
g+wss"
.007
+ .007
.010
315.0
3g+2TT+2M4g'
.000
+.004
.004
o.o
'58
ACTION OF THE PLANETS ON THE MOON.
ACTION OF JUPITER.
Argument.
.
*,
8v
A
Argument.
*.
v,
9v
A
'j-2g-
//
+.004
!ois
.016
165.1
g+2TT-3J
445
tl
0.015
0445
26&JO
j-e'
+.015
.741
741
178.8
g+2TT-3]+g'
.000
+0.018
0.018
O.O
J
+.070
+.169
.183
22.5
g+2ir2]g'
.010
0.016
0.019
2I2.O
2J-3S'
.000
+.006
.006
O.O
S+21T2J
.005
1.140
1.140
180.2
2J 2?
.001
+.242
.242
359-8
21 g?
-059
+.183
193
342.2
g+2TT ] 2g-
+.062
+0.008
0.062
82.7
3J-3^
+.001
+.000
.009
6.4
S+2TT J g"
.000
+0.064
0.064
O.O
3J 2^
+042
002
.042
177.2
g+2lt J
+.OIO
+0.015
0.018
33-7
3J -e"
-.024
+.005
.024
281.8
g+2TT+ J 4g"
+ .001
0.018
0.018
176.8
g+2TT+ J 3g"
+.004
0.230
0.230
179.0
g+2ir+ j 2g"
+.060
0.003
0.060
92.9
e 3J+g r
+.OO2
.002
.003
135-0
g+2ir+2j-4g'
+.OOI
+0.098
0.098
0.7
g3J+2g /
+.OIO
.000
.010
90.0
g+2V+2J3g'
.018
+0.045
0.048
338.2
?-3J+3^
.OOO
.004
.004
180.0
g+2TT+3J-Sg'
+.001
+0.030
0.030
1.9
ff-2J+^
+.005
.032
.032
179.1
g+2ir+3J4g-
+.021
0.001
O.O2I
92.8
21+2^
.000
.045
045
1 80.0
g+2ir+3J3g'
+.004
+O.OOI
0.004
75-9
t-J+i'
+.004
+.140
.140
1.6
e J +2g'
+.001
+.006
.006
9-6
2g+2ir-3J
.007
o.ooo
0.007
270.0
* j
+.036
+.OII
.038
73-0
2g+2ir-3J+g'
.000
0.018
0.018
180.0
*+ J 2?
+.OOI
.006
.006
1704
2g+2TT2Jg'
-.008
0.013
0.015
21 1.6
g+j-g'
+.004
-.163
.163
178.6
2g+2lT2J
.000
+0.018
0.018
0.0
e+'j
+.036
JOll
.038
107.0
2g+2lT J 2g"
+.040
+0.006
0.040
81.5
g+2J2g'
.000
+.064
.064
O.O
2g+2ir J g"
.000
+0.016
0.016
O.O
+21 g 1
+.005
+.036
.036
7-9
2+27T+ J 4 /
+.005
0.038
0.038
172.5
g+SJSg*
JOOO
+.003
.003
O.O
2g+2ir+ J Sg"
+.006
-0.168
0.168
177.9
g+3J2g'
+.OIO
.000
.010
90.0
23+21T+ J 2g-
+.036
O.OOO
0.036
90.0
g+SJ-g'
+.OO2
+.002
.003
45-0
2g+2ir+2]Sg"
+.OO2
+0.019
0.019
6.0
2g+2TT+2] 4g"
.001
+0.092
0.092
359.3
2g+2ir+2J3g'
-.035
+0.082
0.089
336.9
21T3J
-.258
-.008
.258
268.2
2g+2TT+3J5g'
+.003
+0.074
0.074
2-3
2TT2J
.000
+.256
.256
O.O
2g+2TT+3J4g'
+.042
0.003
0.042
94.0
2ir J 2g"
+.004
.000
.004
90.0
22+27T+3J 3g"
+.OIO
+O.OO2
0.010
78.5
2*+ J 3^
.000
.on
.on
180.0
35+21T+ J Sg"
.000
+O.OII
O.OII
O.O
27T+ J 2^
+.004
.000
.004
90.0
3g+2ir+2J4g'
.000
0.006
0.006
180.0
2TT+2]4g'
.000
+.004
.004
O.O
27T g 2J
.000
+O.OIO
O.OIO
0.0
21T+2J 3^
.001
+.003
.003
341.6
2ir g3J
-.015
O.OOI
0.015
183.8
ACTION OF SATURN.
Argument.
v <
to
A
Argument.
V ,
,
8v
A
II
//
II
o
II
It
a
S
+.016
+.048
x>5i
184
+S
+.001
+0.004
0.004
14.1
s-gf
XXX)
.040
.040
180.0
g+S-g-
.000
0.008
0.008
180.0
asg'
.001
+.011
.on
354-8
252?
.000
+.008
.008
0.0
g-s
+.OOI
0.004
0.004
165.9
4T-S+/
.000
+0.008
0.008
O.O
INEQUALITIES OF LONG " "PERIOD. 159
77. Inequalities of the elements -which have not been reduced to inequali-
ties of the longitude,
Mean longitude.
sin(8v 13^' + 86. 4) + o".O3O sin0 o".273 cos
Longitude' of Perigee.
TT = TT O -f vj + 253". 22 T 38".49r 2 - o".oi3 T 3 + o".tf sin (i8v - i6g'g + 228.5)
o".67 sin (8v 13^'+ 86.4) o".io sin + o".8o cos
Longitude of Node.
=, + #,/ - i37".8s T+ f'.62 T z + o".oo262 rs + 2".55 sin - i7"-33 cos (1800)
-f 2". 31 sin 17". 36 cos (1900)
Sin y 2 Inclination.
By = o".ii5 cos o". f ]6c) sin (1800)
o".iO4 cos o".77o sin (1900)
Hence:
Inclination.
87= o".230 cos i".S39 sin (1800)
o".2o8 cos i"-54i sin (1900)
It may be found advisable, in the construction of new lunar tables, to include
also the term
S/ = o".2S6 sin (27r 2/)
in the mean longitude. The effect of including this term in the preceding trans-
formations is that the Jovian evection, and the coefficient of the term of argument
27r ^J g, have each received the increment -|-o".oi4. Hence, if the term
were included in the mean longitude, the coefficient of the Jovian evection would
be i". 154, and of the other term named o".oo4.
78. Remarks on the Possibility of Unknown Terms of Long Period. In his
Researches on the Motion of the Moon, published in 1878,* the author found that
the representation of the Moon's mean longitude during the period from 1650 to
1875 showed a discrepancy between existing theory and observation which might
be represented by a term having a period of two or three centuries, and a coefficient
of about 15". This coefficient may be somewhat reduced by the introduction of
the improved values of the terms of short period now available, but it does not
seem likely that the deviation can be brought below 10". One hypothesis on
which the discrepancy might be explained is that of minute fluctuations in the
* Washington Obseri'ations for 1875, App. II, p. 268. See also Monthly Notices, Royal Astronomical Society,
vol. i.xiri. March, 1903, p. 316.
160 ACTION OF THE PLANETS ON THE MOON.
Earth's diurnal rotation, which might be produced by the motion of solids and fluids
on its surface. Observations of transits of Mercury leave scarcely more than a
possibility of changes in the measure of time having the magnitude required to
explain the deviation. The observed phenomena, therefore, point very strongly
to the inference that there must be some term of long period still undiscovered
in the actual mean motion of the Moon. The preceding researches seem to
remove the possibility that there can be any undiscovered term in the action of the
planets. It is true that there are two possible classes of inequality which are not
considered in the present work. One of these has the solar parallax as a factor,
and may arise from two sources; one the development of the potential to terms of
higher order than the principal ones; the other to the parallactic terms in the
Moon's coordinates. The author had intended to carry the development of R and
fl p one step further, so as to include these terms. But, on examining the periods
of the inequalities that might thus arise, none were found that could lead to any
important term.
Yet another class of terms comprises those of the second order arising from the
action of the planets being modified by their mutual perturbations. An examina-
tion which I believe to be exhaustive was therefore made for terms of long period
of this class. None have been found, and the writer believes that none can exist
more important than one of o".oi8 computed by Radau. This term has the argu-
ment $S ij of the great inequality between Jupiter and Saturn. In this connec-
tion it may be again remarked that, in determining the action of Venus in the
present work, the mutual perturbations of Venus and the Earth have been taken
account of. But no change is thus produced except in the Hansenian term of long
period.
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