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' ' = 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 . 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,, , = - !'. (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 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 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 + 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 +.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*; \. 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, = .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, = ' 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 ( a Z? 3 = aZ? a 4- / D\ IP = aD a + T > '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 ' 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. ' 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'(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(m) (14) If it is developed in powers of m, (tn) = OQ + a i m + (m) = f otjW + 3(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, sin 2 dv D.tiifdv Dto \D sin 2 g g* it sin COS sin cos cos sin sin -4- 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 & + ' *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 ) 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- 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 + 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. " + * 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 2 ex, 2 / 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(/. - 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. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed. This book is DUE on the last date stamped below. 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