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D 32X 1 2 3 1 2 3 4 5 6 9iim>«Bmiaimmimtmmmtm wmmimmmmmm s CONSIDERATIONS OK THB APPARENT INEQUALITIES OP LONG PERIOD If IM THE MEAN MOTION OF THE MOON. Bt SIMON NEWOOMB. 'f ii i-. [Froh tbb Auebioam Joubnal of SoiBKOB AitD Ahts, Vol. L, Sept., 1870.] n«>w- Hc)-'^^^^ I > CUtt •CTit fflno ;S**i^ [Fkom tub Ambuioan Journai. op Scibhob and AuTti, Vol.. L, Sbit., 1870.J ^5^ k U -< CONSIDERATIONS I ON THE APPARENT INEQUALITIES OF LONG PERIOD IN THE MEAN MOTION OF THE MOON. By SIMON NEWCOMB. -♦♦♦- [Read to the NationHl Academy, April, 1870.] The problem of determining the motion of the moon around the earth under the influence of the combined attraction of the sun and planets has, more than any other, called forth the efforts of mathematicians and Mtronomera Nearly every great geo- meter since Newton has added something to the simplicity or the accuracy of the solution, and, in our own day we have seen it successfully completed in its simplest form, ia which the earth, the moon, and the sun are considered as material points, mov- ing under the influence of their mutual attractiona The satis- factory solutions are due to the genius of Hansen and of De- launay. Working independently of each other, each using a method of his own invention more rigorous than had before been applied, they arrived at expressions for the longitude of the moon which, being compared, were found to exhibit an av- erage discrepancy of less than a second of arc. No doubt could remain of the substantial correctness of each. The solutions here referred to exhibit only inequalities of short period in the motion of the moon. But, it has long been known, from observation, that the mean motion of the moon is subject to apparent changes of very long period, and especially to a secular acceleration by which it has been gradually increas- ing, fix)m century to century, since tlie time of the earliest re- corded observations. If we inquire into the problem of these inequalities of long period, we shall find it seemingly no nearer a final solution than it was left by La Place, observation having since added more anomalies than theory has satisfacto- rily shown to exist The first inequality in the order of discovery was the secular acceleration. This was discovered about the middle of the laat century by a comparison of ancient eclipses with modem ob- * See Buff's paper in the Annalen d. Chem. u. Pbarm., 4th xuppleniieat vol., 1866-6. Or, see hia " Grundlelirea dcr tlieoretiat^hen Chemie." i m 2 «V. Neivcomh on the apparent ineqtiah'tfes servationa Its (iause was first discovered bv La Place, who showed thatr it was due to the effect of the action of the planets in changing the eccentricity of the earth's orbit The results of his computations agreed substantially with ob- servations, and was therefore received with entire conflclence until less than twenty yeiirs ago. The question being then taken up by Mr. John C. Adams, this eminent mathematician was led to the conclusion that La Place's result was nearly twice too large. The same conclusion was reached independently by Delaunay, and gave rise to a remarkable discussion, the history of which is too familiar to be now recounted. It is now conceded that the value found by Adams and Deluimay is theoretically correct. The new result no longer agreeing witii observation, the dif- ference is now accounted for by an increase in the length of the day. That this length is increasing is also known from theoret- ical considerations, but the data for its accurate determination are wanting.* In the third volume of the Mecantque Celeste (Seconde Partie, Livre vii, Chapitre v) La Place discusses an apparent inequal- ity of long period in the motion of the moon. The discussion is mainly empirical. The existence of the inequality is inferred from obsei-vation*, these showing that the mean motion of the moon during the half century following 1756 was less than dur- ing the half century preceding. He then assumed that the in- equality wfs due to the fact that twice the mean motion of the moon's node, plus the motion of its perigee, minus that of the sun's perigee was a very small quantity, less than two degrees per annum, and determined the coefficient of the varying angle solely from the observations. The result was that these might be satisfied by supposing the inequality of mean longitude Si=4r-5l [or l5"-39] sin (2 ^ > -f-»» D — an©) If, in this expression, we substitute Hansen's values of the elements, it becomes J/=15"-39 sin [173° 26'-}-(l° 57''4) (<— 1800)]. When in 1811 Burckhardt constructed his tables of the moon, * The time and place when the diaoordance referred to waa first distinctly attrib- uted to the tidal retardation of the earth having been a subject of discussion, tlie following extract fiom an article on " Modem Theoretical Astronomy" in the North American Review for October, 1S61 (voL 93, p. 380), may not be devoid of interest. " It seems to be well established that the new theory is inconsistent with the ob- se 'Viitions of ancient eclipses, and if it should prove to be correct, we may be driven to the conclusion, tliat a [lortion of the acceleration proceeds iVom some otlter cause than the att-ac ion of gravitntion, or that the lei gth of the day is gradually increas- ing to nn extent which ha'< become perceptible from the cause to which we have already referred [the tiilal retardation, p. 314]. If, as centuries roll by, the day should gradually mcreaHo, the moon would move a little farther in the course of a ■lay than if no such increase should take plai-e. Since, in our calculationx, we sup- pose the day constant, the apparent acceleration would be greater than the real — precisely the effect observed. The difference can be entirely accounted for b^ sup- posing an increase of something less than one thousandth of a second per century in tlie length of the day, and a corresponding diminution in the lunar month." of hmj perimi in tlie menu motion of tlie Moon, 8 ja Place, who of tlie planets ,ially with ob- iufind per century Eir month." he omitted the sun's perigee from this argument by the author- ity of La Place, himself, who now attributed the inequality to a difference of compression between the two hemispheres of the earth. The function was also changed from sin to cos and the coefficient altered. The adopted term thus became dl-=i-\2"S cos [291° 67'4-(2° 0'-45)(«- 1800)1 = 12"-6 sin [201° 67'+(2°0"45)(«- 1800)] Succeeding investigators have regarded the theoretical coeffi- cients of botli of these terms as insensible. It does not seem likely that there is any such difference between the two terres- trial hemispheres as could produce the second, but I am not aware that the coefficient of the first has ever been shown to be insensible by any published computation. This coefficient is of the ninth order and the argument is, In Delaunay's notation, 8D-2F-;-f8;'; In Hansen's, w—8w'. The period is 184 years, and the large value of the ratio of this period to that of the moon itself might render the coefficient sensiole. Both Hansen and Delaunay pronounce it insensible, but neither publish their computations of its magnitude. These tenns have ceased to figure in the theory of the moon since Hansen announced that the action of Venus was capable of producing inequalities of the kind in question. So i'ar as I am aware, Hansen s first publication on this subject is that found in No. 597 of the Astronomische Nachrichten (B. 25, S. 325.) Here, in a letter dated March 12, he alludes to La Place's coeffi- cients, and says he has not been able to find any sensible coeffi- cient for La Place's argument of long period. But on examin- ing the action of Venus on the moon he found, considering only the firet power of the disturbing force, the following term in the moon's mean longitude : 81=: 1«"'01 sin (-^— ]6gr'+18iy"-f36° 20'). g, g' and g" being the mean anomilies of the moon, the earth and Venus respectively. As this expression still failed to ac- count for the observed variations - * the moon's longitude he continued the approximation to tl. )urth power of the dis- turbing force, ana found that the tenvxi of the third and fourth order increased the coefficient to 27""4, the angle remaining un- changed, so that the term became 27"-4 sin (— jr-16flr'4.18^"4-.36° 20'), But this increase made the theory rather worse, and the temi depending on the argument of Airy's equation between the earth and Venus was then tried with the result — 91 = 23"-2 sin (8«7"- 13^'+;U5° 30'). The introduction of this term seemed to reconcile the theory with observation. i 4 iS. Newromb on tiie apparent inetjualities KnnBcn finally romarkH that thene values of the coefficients aro Htill Hubjet having fin- 11 tht! Montlily ^v. 1864, Ui.ii- II hJH tiihloH of [ualities in the JCfjualities hy s itHolf in the asicMiH and by es, but I have other. I am nination by a ng the opera- te my tables ^m but which nt the ancient etneas, and it ture obaerva- nuH and the ns become «red to what i first power )f the second ? but slight- 61, Feb. 2d, lansen a^ain ts coefficient time he has nation of it, that in the ?yond Brad- the ancient ks is the ce- cal calcula- ! has never Jxcept that at power of oj long /term/ in the mean motion of tiie M(Hm. 6 the dicturbing fiii-ce. Thi8 computation is found in vol. xvi of the Memoirs of the Royal Astronomical Society. In the sec- ond part ofhis "DarU'gung" we Hnd a general metlwxl of treat- ing ineqimiities of long periotl, but — unless I have overlnii)ariM<>n ot'tliu two tables, t'roui wliu^li it apiu'ai's that Hiirck- liui'Mt's inttan longitiKlu was tlu>n greater tlian ilanscn's l>y alx)ut 14"'2. The general aKrciMncnt bctwocn 1700 and 1W()<), whon hotli tabloM agreed with observations, sliows thnttho dirtbrcnco of mean motion is ucrtuinly atVwted with no Hcnsiblc error. i Diirckhnnlt. llanMii. H,-B. Yi-ar. ^. Her. V«r toDH Von. Mi'mi I'vrloil. I.oi>i|UaU( . I'. V»r. I.ODH Herlixl. Mi'iin l.onKllnclo. 1630 100 1 II :g 28() + 4-9 II 1 II - 80 100 10 240 18 14-4 II + 88-5 -214 l8o 18 s'lS II -53-4 40 347 45-4 + 3-6 -10 8 347 6 38-2 5 3';s + 34-I -20 347 r. 60-4 -47'8 50 233 04 2 7 + 2-5 -12-3 233 53 52'9 52 58'3 4 30 -17-2 223 53 111 -318 (iO|l30 41 20 1 + 16 -12 3 120 41 4 40 20-3 + 261 -131 120 40 333 -36-8 70! 7 28 374 + O-O -10-8 7 28 27-5 27 422 + 22-5 - 81 7 27 66(1 -309 80 2&4 16 54'8 + 04 - 80 254 15 47-2 15 4-2 + 102 - 23 264 15 31-1 -261 no 141 3 121 + 01 - 4-2 141 3 7 8 2 26' 1 + 161 + 3-9 141 2 461 -21-7 17iiOi 27 50 295 + 00 + 2 27 50 297 40 48' 1 + I3'3 + 100 27 50 11-4 -183 lo!274 37 468 + 01 + 4 4 274 37 513 37 100 + 108 + 150 274 37 30-4 -140 20101 25 4-.> + 0-4 + 8-3 1fll 25 12-9 24 3'2 + 8-5 f 20-6 161 26 10 -11-0 80 i 48 12 215 + <>9 + 110, 48 12 33-4 11 69'9 + 6-6 + 24 2 48 12 247 - 8-7 401204 fiO 38'9 + 1-6, + 12-4 294 59 620 69 16'9 + 4-8 + 26-4 294 50 47 1 - 5-8 50 181 47 66-2 + 2-6 + 12-2 181 47 100 46 37-9 + 3-3 + 20 9 181 47 81 - 2-9 60 68 34 136 + 36 + 10-6 68 34 27-8 33 69-S + 2-1 + 25-7 CH 64 27-6 - 0-2 70 315 21 30-9 + 49 + 7-8 316 21 43" 21 21-8 + 1-2 + 220 315 21 45 + 22 80 202 8 4H'3 + 6-4 + 39 202 8 58'6 8 437 + 0-6 + 18-6 202 9 2-7 + 4'1 90 HH 56 5-6 + 8-1 - 0-4 88 56 13-4 56 57 + 01 + 12-8 88 60 18-6 + 5'2 1800 335 43 230 + 100 - 4-7 336 43 28-4 43 27 7 00 + CI 335 43 33-8 + 6-4 10 222 30 40-4 + 121 - 83 222 30 44-2 30 49-6 + 01 - 11 222 30 480 + 4-4 20 100 17 67-8 + 141 -110 109 18 1-2 18 11-6 + 0-6 - 84 109 18 37 + 2-5 30 356 6 152 + 100 -12-4 366 6 10-7 6 33-5 + 1-2 -16-4 356 6 19'3 - 0-4 40 242 52 3i-8 + 190 -122 242 62 39-9 62 66 6 + 2-1 -21-6 242 52 36 - 39 50 129 39 490 + 22-5 -10-6 12» 40 1-8 40 176 + 3-3 -26-5 129 39 64-3 - 7-6 60 16 27 7-2 + 26« - 76 16 27 25-2 27 30 4 + 4-8 -29-8 10 27 14-4 -108 70 263 14 24-6 + 28-9 - 3-8 263 14 40-7 !6 1-4 + «'5 -31-3 263 14 36-6 -131 Burckhardt's tables have been selected for this comparison because they have been extensively compared with observations made before 1700. The additions to the Connais.sance des Temps for 1824 contain a paper by Burckhardt himself giving a com- parison of his tables with observations of occultations ma^le by Flanistoad, Hevelius and others, between 1637 and 1700. The general result of this comparison is that the mean longitude of his tables could hardly have been more than a very few seconds in error in the year 1670. But, the preceding table shows that ft)r this epoch Hansen's mean longitude is 80 ' less than Burck- hardt's. Therefore, unless we sappose Burckhardt's investi- gation to be aliected with some egregious systematic error we must admit that the mean longitude of Hansen's tables for the epoch 1670 is about 30" too small. Desiring an independent test of this conclusion T have select- ed certain observations which, with the data available, seemed ties rticiiliu"r|MM'liH. lHfi2 is lomid ii tiVH tliiit Uiirck- risen 'h 1»v ulxnit nd !«()(■), wlioii ttlio (lirt'orencu liblc error. m. Mc'un I'OnKUadc. H-B 100 347 2-^3 120 7 254 141 27 274 161 48 37 26 12 4 294 9|t81 71 68 9315 202 88 3:)s 222 109 1356 5:242 >ll29 i\ 10 11263 IH 31 '5 Ti 60'4 63 11-1 40 33-3 27 66'6 15 211 2 461 50 11-4 364 10 24-7 59 471 47 81 64 27-6 21 459 2-7 50 18-6 43 33-8 30 48» 18 3-7 6 19'3 52 36 39 64-3 27 14-4 14 36-6 -53 4 -47-8 -318 -36-« -309 -261 -217 -18-3 -149 -110 - 8-7 - 68 - 20 - 02 22 41 5-2 64 4-4 2-6 0-4 39 7-6 -108 -131 is comparison I observations ice des Temps giving a corn- ions made by i 1700. The I longitude of f few second's ie shows that 1 than Burck- ii-dt's investi- atic error we tables for the r have select- lable, seemed o/ hug period in the menu inotion ojUif MtHni. 7 well flttetl to iiiiMwtir tliis purpoHc and coiupHrt'd them directly with IIiuiHen'M Tables. Tliey are 1. Occnitation of Aldebaran, KtHO, Sept. 18, observed at Greenwich by Fhinistead. 2. Oceidtation of the same star KWO, Nov. 7, ol)served at Greenwich by Fhinistead, and at Lundon by llalley. 8. Total eclipse of the sun 1715, May 8, observed at Lon- don, Grijenwicii and Wanstead by llalley, Klanistead and Pound. To compute the oeeultations of Aldebaran the mean position for 1680'0 was derived from Lt? Verrier's Tattles (Annulcs de rObservatoire, Tome 11) correcting the right ascension by -j-0""Ul, and was us follows: «(1«80) = 4'' 17'" ST'-OI «J -f-U)" 49' 11"-H The corrections for reduction to apparent place are for Sept. 1.1, a« =+2-90 ; Aa=+1"-1 Nov. 7, Sa =-\^^'\i^ Ji3=-f2-4 The following geocentric jxisitions ftf the moon were derived from Hansen's Tables. Date (Jullnn 0*1.) Gr. Moan Time, ( 'b LoDKitude, " Latitude, " Parallax, 8«pt. 18. NOV. 7. h m « 1 h m ■ 7 50 39 1 8 48 15 64° 33' 11 "•6:65° 9' 49 "-6 -4 39 2«-9 1-4 40 480 1 1 185 1 I 1 1T8 h m ■ 15 63 64" 64' 24"3 -4 46 jo-a 59 30(» h m II 16 12 63 05° 37' 20"'4 -4 48 106 59 28-8 From these data we derive the following times for the im- mersion and emersion of Aldebaran for the dates in question. The observed times have been concluded from the observeil altitudes and clock times given by Flamstead in the Ilistona Celestis,'kmd\y furnished me by I*rof. Winlock. They dill'er but little from the result- of Flamstead himself, when the latter are corrected for the equation of time. Compnted. Obierved. 0-0. h m I Sept 13, TmineraioD, 15 2 49 la 53 + 116 KmerBion, 16 10 6 16 9 12 + 53 Not. 7. Immersion, 7 61 47 7 60 4» + 64 Kmersioii, 8 48 16 8 47 12 -f 64 The great diflference between the results of the two phases of the first oecultation gives rise to a suspicion of error in the ob- servations or the data of reduction. The second observation is confirmed by that of Halley in London, he having observed the immersion at 7'' 50'" 9", and noticed that the star was " new- ly emerged " at 8'' 47"' 1". His place of observation was prob- ably twenty-five or thirty seconds west of Greenwich, and there- 8 S. Newcqm no the apparent inequalities fore his ob^rvation agrees well with that of Plamstead. The discordance between the observed and computed times, of tliis second occultatic; indicates a correction of about +82" to Hansenis mean longitude at the epoch 1680, and tho first may be considered as confirming this correction in direction, if not in amount For the eclipse of May 8, 1715 we have the following com- puted and observed times. I have assumed Hnlley's station to be in latitude 61° 81' and longitude 25" west. Pound's is taken in accordance with his own statement to be in latitude 51° 34' ana longitude 8« west These agree pretty well with Flam- stead s statements that Wanstoad is seven or eight miles N by B. from Greenwich,* and that Crane Court is half a minute of time West of Greenwich. HaUey at London. First ooutact, Beginning of Totality, End of " End of Eclipse, Oompnted. Observed. O-O 8 — 2 + 13 + 1 + 18 h m B h m 8 " 20 2 35 20 2 37 21 6 62 21 5 .39 21 9 3 21 9 2 22 16 65 22 16 37 Pounit at Wanstead. Eclipse first perceived, Tlie total iiiiiiiersioii, The emersion, The justend of the eclipse, Oompnted. O'jserved. O-O 11 m B 20 3 18 21 6 38 21 9 48 22 17 42 h 20 21 21 22 m K 3 15 . 6 6 9 26 17 10 B + 3 + 32 + 22 + 32 The only information I have respecting Flamstead'.s observa- tions IS contained in a letter of his found in Daily's ' Life and Correspondence of Flamstead, p. 315, from which it tippeare that his times differ only a few seconds from Halley's, instead of differing by the half minute required by the difference of meri- dians. An obvious slip of the pen, {later being written instead of earli^) makes it doubtful in whicli way the " few seconds " are to be counted. It can, however, be fairly inferred from his statement that his observations diverge from the tabular times as much or more than Pound's. The discordance of the results of first and last contact may be attnbuted to this cause: that with their imperfect telescopes the observers did not begin to see the moon until several seconds alter the actual commencement of the eclipse, and lost sight of It a few seconds before the actual end. The discordance in the duration of totality indicates with a high pi-obubility that the computed shadow path falls a few miles too far north In this case the mean of the results for beginning and end of totality • Baily's Flamstead, p. 316 p. 328. I '*'** **"""'~^""" ' Ti — mmw ii u twi 5S instead. The times, of tliis )ut +82" to the first may rection, if not llowing com- jy's station to ind's is taken itude 51° 34', 1 with Flam- t miles N. by P a minute ot O-O 8 a 3? — 2 39 + 13 2 + 1 37 + 18 O-O K B 16 . + 3 6 + 32 26 + 22 10 + 32 id's observa- 's * Life and appeal's that J, instead of ace of meri- itten instead w seconds" 'cd from his ibular times ontact may ;t telcsc()|)es eral seconds lost sight of iance in the ty that the th. In this of totality ^•^^i^fif mj t t t m mmm of long period in the mean motion of the Moon. 9 will be about right, and we have for the excess of computed times Halley's observations, + 7' Pound's, + 27 Flamstead's, + 80 d= I infer from these results that the correction to Hansen's mean longitude at the epoch 1715 is about +11". Comparing the corrections thus found for the epochs 1680 and 1715, we find they are substantially those required to reduce Hansen's mean longitude to Burckhardt's. 1 conclude, there- fore, that no egregious systematic error has crept into the re- searches by which Burcknardt sought to show that the epoch of his tables was substantially correct during the latter half of the seventeenth century, and that the difference between the meain longitude of Hansen and Burckhardt during that period repre- sents approximately, at least, errors of Hansen's mean longitude. The observations of the moon made at the observatories of Greenwich and Washington during the last ten years, indicate a tabular deviation of a remarkable character. From 1850 to 1862 we find the moon slowly running ahead of the tables, until the latter required a correction of plus two seconds in lon- gitude to make them agree with observation. But this correc- tion, instead of continuing to increase as all analogy would have led us to anticipate, suddenly began to diminish, so that since 1862 the moon seems to have been falling behind the tables at the rate of a second a year. This is shown by the fol- lowing exhibit of the corrections to Hansen's mean longitude, or right-ascension, deduced from the meridian observations of the two observatories. Correction given by Tesr. Qreenwlcb. Wuhington. Hesn. Corr. mean. // // // II 1860 + 0-3 -1-3 00 + 1-0 51 + 1-5 + 0-6 + 1-3 + 2-7 52 + 0-9 + 0-9 + 2-4 56 + 1-0 .... + 1-0 + 1-4 57 + 1-5 - - . - + 1-5 + 1-4 68 + 20 + 1-5 + 1-8 + 13 62 + 2-4 + 2-4 + 2 4 + 0-9 63 + 2-2 + 1-2 + 1-7 + 0-5 64 + 01 -10 -0-4 -1-2 66 -11 -2-4 -1-7 -21 66 -2-2 -2-6 -2-4 -24 67 -3-9 -41 -40 -3-6 68 -4-4 -4-5 -4-5 -36 69 -56 -5-6 -4-3 The corrections here given as those of Greenwich are, previ- ous to 1859, derived from the comparison found in the Green- vl r 10 S. Newcoml on the apparent inequalities wich observations for 1869. From 1863 forward they are deriv- ed from a paper by Mr. Dunkin in the Monthly Notices of the Koyal Astronomical Society for April, 1869. The work of only the four principal observers is therefore included in the comparison. The object of this comparison being not so much to determine the absolute correction to the epoch of the tables as to show the changes of this correction, it is better to reject the results of the observers whose labors were discontinuous. In the case of the Washington observations, such a selection could not be made : the results given are therefore an indis- criminate mean of all. The systematic personal differences are however found to be very smalL That these corrections are real will not, I conceive, be dispu- ted. To suppose them due to errors of observation, would be to suppose that six or eight long practiced observers divided between the two hemispheres, all progressively changed their habits of observing in the same way, and to nearly the same amount, through a period of seven or eight years. A portion of the observed discordance may arise from a small error in Hansen's value of the coSflScient depending on the ellipticity of the earth, which is more than a second greater than the values derived by previous investigators, either from theory or observation. The last column of the preceding table shows what the correction would be if Hansen's coefficient were 1"'5 smaller than it is. From all these comparisons it would appear that the problem of the inequalities of long period in the moon's mean motion is really no nearer such a solution as will agree with observation, than when it was left by La Place. By a partially empirical correction, Hansen has succeeded in securing a very good agree- ment during the period 1760-1860, but, if the results ^the preceding examination are correct, this has been gained only by sacrificing the agreement for the century previous to 1750, and for the years following 1860. This failure to reconcile theory with observation must arise from one of two sources. Either : (1) The concluded theory does not correctly represent the mean motion of the moon. Or : — (2) The rotation of the earth on its axis is subject to inequal- ities of irregular character and long period. The first hypothesis admits of two explanationa We may suppose either that the mean motion of the moon is subject to change from some other cause than the gravitation of the known bodies of the solar system, or that the effect of this grav- itation is incorrectly calculated, and that theory and observa- tion will be reconciled by a correct calculation. There are difficulties in the way of accepting either of these explanations. In reference to the first it may be remarked that H.-«„ MM IMW es they are deriv- Nntices of the The work of Bluded in the g not so much of the tables »etter to reject discontinuoa& ich a selection fore an indis- differences are jive, be dispu- ion, would be rvers divided 3hanged their irly the same e from a small nding on the econd greater 5, either from receding table oefficient were t the problem can motion is b observation, illy empirical "y good agree- results of the ained only by I to 1750, and oncile theory xjCvS, Either : represent the 3ct to inequal- aa We may I is subject to tation of the t of this grav- and observa- ther of these emarked that fifhmg period in the mean motion of the Moon. 11 anomalies of mean motion cannot be accounted for by a devia- tion from the received law of gravitation inversely as the square of the distance, because the anomalies produced by such deviation would be regularly progressive, and would be most sensible in the secular motion of the moon's perigee. The com- parison of the theoretical and observed values of this motion is, perhaps, the severest test to which the Newtonian law has yet been subjected. That the anomalies proceed from the attrac- tion of unknown bodies passing througn the system seems ex- tremely improbable, since, if they were distant, they would affect the earth and planets more than the moon, while the clo- ser passage of bodies could scarcely escape detection. Still, this explanation does not admit of being mathematically dis- proved. If we attribute the deviation to the impact of mete- oric matter, we must suppose the moon to have encountered such matter in quantities nearly incredible. These three causes exhaust those on which we can base the first explanation, unless we invalidate the third law of motion. For, by that law, matter moves only by the influence of other matter. Other matter can affect the motion of the moon only by impact and gravitatioa The gravitation of known bodies, the gravitation of unknown bodies, and the impact of matter is therefore an exhaustive enumeration. We pass now to the second explanation of the first hypothe- sis, namely, errors or omissions in the theoretical computation of the effect of gravitation. The wide difference between the conclusions of Hansen and Delaunay suggests the possibility that there may be inequalities sti?,l overlooked. We nave how- ever the assurance of Hansen that there are none, and we shall find it extremely difficult to introduce any periodic terms what- ever which will represent the observed deviation of the moon from the tables during the past ten years, without discordance during the century previous, when the agreement of Hansen's tables with theory is believed to be quite clos& It is however hardly worth while to dwell upon this explanation until we have a more rigorous theory of the inequalities of long period produced by gravitation. C asidenng that the reconciliation of theory and observa- tion is not very probable, the second hypothesis may become worthy of serious consideration. If we accept \% we must ad- mit that between the years 1860 and 1862 the rotation of the earth was so accelerated that our reckoning of time is already eight or ten seconds ahead of what it would have been had the day remained invariable. Such an acceleration could proceed only frt>m a change in the arrangement of the matter of the earth. The possibility of this effect being produced by changes in the quantity of ice accumulated around the poles nas, I be- HMfBES f 12 & Newcomb on the apparent inequalitiea, etc. lieve, been pointed out by geologists. But the effect of this cause could scarcely be sensible. But, if we admit that the interior of the earth is a fluid, and also admit that general changes in the arrangement of this fluid are possible, we have all that is necessary to account for considerable changes in the rotation of the outer crust That this fluid, admitting its ex- istence, is not in a state of entire quiescence is rendered proba- ble by the phenomena of volcanoes and earthquakea If we suppose a large mass of it to move from the equatorial regions to a position nearer the axis, a mass from the latter position taking its place, the following effects will follow : — 1. A diminution in the angular velocity of the surface of the fluid, accompanied by a corresponding increase in the velo- city of the axial portion. The velocity of the outer crust will then be gradually retarded by friction. , 2. The gradual transmission of the increased rotation of the central mass to the surface by friction and viscosity. The motion of the crust will then be gradually accelerated. The velocity of rotation finally attained will be greater or less than the original velocity, according as the radius of gyration of the fluid mass is diminished or increased by the change in the arrangement of the fluid. I conclude, from this discussion, that we have reason to sus- pect that the motion of rotation of the crust of the earth is subject to inequalities of an irregular character, which, in the present state of science, can be detected only by observations of the moon. This suspicion can be neither confirmed nor remov- ed until we have more positive knowledge than we now have of the possible inequalities which may be produced in the mean motion of the moon by the action of gravitation. The operation of calculating these inequalities, though com- plicated and difficult, is certainly within tne powers of analysis. When it is completely and thoroughly done, we may ascertain whether the result can be made to represent observationa If so, well; the length of the day is not variable, and the future positions of the moon can be safely predicted. If not, it will follow either that the motion of the moon is affected by other causes than the gravitation of the known bodies of tlie solar system, or the day is irregularly variable. By the end of the present century, if not sooner, we shall have an independent test of the latter hypothesis, in the agree- ment of the observed and theoretical times of the transits of Mercury and Venus. If the hypothesis is a true one, the irreg- ularities may ran^e over half a minute of time in the course of a century, and this quantity might be detected even by merid- ian observations of the planets in question. 1 \' i imww.rtnK i^iiK^m t-HfUlUV* Wi JAW^.ti^ < »--'.iSrf*w ;