192 4MNRAS. .85. . .IIS observations havebeenreducedonauniformplan,theplacesof diameter, eccentricityoforbit,andcoefficient oftheprincipaltermin It thereforeappeareddesirabletodiscussthe observationsuptothe be usedinthereductionofCapeObservations ofOccultations. restricted totheperiod18801922. the yearsbefore1880wereveryfew;discussion hasthereforebeen the parallacticinequality.Thenumbersof observations inmanyof end of1922withaviewtoderivingthe Moon’slongitude,semi- rections. Thetabularplacesforthefirstseriesweresuppliedby Moon beingderivedfromHansen’sTables,withoutNewcomb’scor- to 1922areinmanuscriptreadyforpublication.Thewholeseriesof Annals, 2,part6.Thedetailsoftheobservationsforperiod1907 Boyal Observatory,CapeofGoodHope,togetherwiththeresulting Newcomb. in theCapeAnnals,1,part4;forperiod1881to1895 equations ofcondition,havebeenpublishedfortheperiod18351880 effect ofearlycloserpackingistoincreasethetime-scalebyafactor of aseculardecreasemassjustaboutaccountsfortheobservedorbits 1923 onwardsarebaseduponBrown’sTables, whichwillhenceforth so that,farasregardstheeffectsofencountersbetweenstars, Cape Annals,2,part3;andfortheperiod1896to1906in planets asomewhat,althoughnotexcessively,rareevent. of thestarsandtheirvelocities,whileleavingtidalgenesis of binarystarsandfortheobservedcorrelationbetweenmasses The Moon'sMeanLongitude,LongitudesofPerigeeandNode,Semi- mentioned in§6bythefactorof6*4or10*3,wefindthatconception M/M =6,itisequalto10*3.Ifwemultiplyalltheexpectations Nov. 1924.TheMoon'sMeanLongitude,n 0 The equationsofconditionhavebeenpublished intheform The positionsoftheMoongiveninNauticalAlmanacfrom The detailsoftheoccultationsstarsbyMoonobservedat If wetakeM/M=4,thisfactorbecoinesequalto6*4;with 0 4^-? -S=(a)Aa+(S)A8+ (a')Aa'+(S')AS'+(T)AT Hope, 1880-1922.ByH.SpencerJones,M.A.,B.Sc.,H.M. tions ofStarsobservedattheRoyalObservatory,CapeGood Diameter andParallacticInequalityderivedfromOcculta- Astronomer. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System M(M +M)* 0 2M2 0 + {p)p +(*)», 12 Mr. H. Spencer Jones, LXXXV. I,

where Aa, AS are the corrections to a, S, the Moon’s tabular R.A. and Dec. in seconds of arc. Aa', AS' are the corrections in seconds of arc to the R.A. and Dec. of the star occulted. AT is the correction in seconds of time to the mean local time of observation. AZ is the correction to the assumed longitude, Z, in seconds of time. A0' is the correction to the geocentric latitude, ) is the sine of the true horizontal equatorial parallax of the Moon, the tabular value being sinTr. sino-(i+s) is the sine of the true semi-diameter of the Moon, the tabular value being sin or.

The longitude and latitude of the transit-circle adopted for the reductions are - ih i3m 55s,oo and - 33° 56' 3"'4.

The values of ', log p, and e are adopted as

4>'= -33° 45' 26"-26 log p = 9-9995512 e= 1/300.

The parallax of the Moon is taken from Hansen’s Tables, and the semi- diameter is deduced from it by the formula

O43559]x ^ The longitude of the transit-circle derived by telegraphic operations is - ih 13111 54s72, requiring a correction of +o8,28 to the adopted longitude. Wireless time-signal comparisons indicate a slightly larger correction. The adopted correction is + o8*30. For <£', p Hayford’s spheroid of 1909 has been adopted, giving

^=-33° 45'i9"-8 e= 1^2 97» log P = 9*9995469

As regards the correction to the horizontal equatorial parallax, observations at one place can only give the value of the local constant, TT + dir', where d(p7r) = pd7r'; dir, dir being the corrections to the true and local constants respectively. Hence dir == dir + Trd log p.

The value of dir has been derived by the method used by Newcomb,* but with revised values of the constants. Newcomb pointed out that the parallax of the Moon referred to the mean radius of the Earth may be regarded as a known quantity not subject to correction. Assuming * Researches on the Motion of the Moon, part ii. p. 40 {Astronomical Papers of the American Ephemeris, 9, parti., 1902).

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 192 4MNRAS. .85. . .IIS 2 of theMoon,partii.,p.42. American Ephemeris1,78). Newcomb hasusedthevalue3422"*09,*and alsothevalue3422"*o7.t constant actuallyemployedbyHansenforthe constructionofhistables. so thattheconstantofsineequatorial horizontalparallaxis radius p,andtheellipticity,e,isgivenby Tr beingtheconstantofMoon’shorizontalparallax,andpradius and forthemeanradius,6371270metres,correspondingtoahorizontal i parallax tt^wefind used byBrown,itfollowsfromthemassofEarthpreviously The meansiderealmotionoftheMooninaJuliancenturybeing of theEarthcorrespondingtoit. where aisexpressedincms. found that Moon, nthemeanmotioninarconesecond, may becomputedandpeliminatedfromtheequationsofcondition. the value1/297forellipticity,correctiontolocalparallax Nov. 1924.TheMoon'sMeanLongitude.13 x of themassEarthexpressedincm.-sec.unitsisfoundtobe metres, whence,usingthevalueofgravityjustderived,logarithm we obtain,atmeanlatitude(sin=1/3), 1336*85136 revolutions,logi/rc=5*574841;adoptingE/M=81*530,as 20*600572. t CompendiumofSpherical Astronomy,p.136.AlsoResearchesontheMotion * “TransformationofHansen’sLunarTheory”{Astronomical Papersofthe There hasbeensomedifferenceofopinionas tothevalueofthis The radius,p,oftheEarthinanylatitude<£',termsmean If therefore,pisexpressedinmetres, But If aisthesemi-majoraxisofrelativeorbitEarthand Adopting Helmert’svalueofgravity, Hayford’s spheroidgivesforthemeanradiusofEarth6371270 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 2 2 Apparent gravity.979*767cm./sec. Actual gravity.982*020„ Centrifugal force.2*253,, 978*046[i +*005302sin-*0000072], sin ir"=341S"71(i+Je)." 0 sin 7r=[1*415211—io]p, a sinTr=i-0009068p, sin tt/'=3418"*71. loga= 10*585183 z2 an =E+M. 14 Mr. H. Spencer Jones, lxxxv. i,

Battermann* has discussed the question thoroughly, and concludes that the value 3422"* 12 is the correct one. This value has been adopted in the present investigation. With Hansen’s parallax and an ellipticity e0) the horizontal parallax for latitude <£' will be 2 3422"*i2(i - e0 sin <£').

The value resulting from the value of the equatorial horizontal parallax derived above is 34i8"‘7i[i ^ “ sin2 SÍO]-

The difference between these two expressions gives the correction to the adopted parallax. The correction to the local horizontal parallax is therefore A(p sin 7t)= - 3"*41 + ii4o"e + sin2 - 3il9"(e ~ eo)]' Putting

A(p sin tt) = + o''32. Also A(log p)= - *0000043 sin 7rA(log p) — -"‘01.

The correction to be applied to the assumed value of the horizontal parallax is therefore +o",3i. The quantity p used in the reductions of the Cape occultations is consequently + o‘3i sin P'/sintt or + 0*090-7-103. A correction of this amount has been applied to the equations of condition. The star-places for the first and third series are taken, for the most part, from Newcomb’s Catalogue of 1098 Standard Clock and Zodiacal Stars and Catalogue of Zodiacal Stars. Those for the second series are derived from Cape Observations, and to these mean corrections of + o"*32 in R.A. and +o,,‘i4 in Declination have been adopted as a reduction to Newcomb’s system. With these corrections the equations of condition as published are reduced to the form

(a)Aa + (8)A8 + (s)i>* = 4^-^ - S = w • • (i*) x 7 w sin i

For further discussion it is necessary to transform the equations. If X, ß are the ecliptic longitude and latitude of the Moon, we may write (a) Aa + (S)AS = (X)AX + (ß)Aß . . . (ii.) where (À) = [l + (». a)](a) + («. 8)(S)

(ß) = (ß ■ a)(a) + C1 + (ß ■ 8)](8)'

* Beobachtungs-Ergebnisse der Königlichen Stermvarte zu Berlin, No. 13, p. 11.

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 192 4MNRAS. .85. . .IIS Nov. 1924.TheMoorísMeanLongitude.15 The quantities(v.a),£),(ß.B)areextractedfromthetable in theorbit,itiseasytoshowthat À, ßarederivedfromtheNauticalAlmanacforG.M.T.of observation oftheoccultation. given byNewcomb,*withargumentsX(=v)andß.Thequantities also ascending node,itheinclinationoforbittoecliptic,andwhere ,,, where ldenotesthetruelongitudeinorbit,Oof and where,inNewcomb’snotation, (F.Z), (F./2)aretabulated,withargumentw=A-ß,byNewcombin gation ofCorrectionstoHansen'sTablesthe Moon. form coefficient oftheprincipalterminparallacticinequality,P, table x.,p.50,InvestigationofCorrectionstoHansen'sTablesthe is tabulatedbyNewcomb,withargumentp, in tableix.,p.50,Investi- expressed intermsofcorrections,AZ,tothemeanorbitallongitude, Moon. v where gdenotestheMoon’smeananomaly. Ae, totheeccentricity,AlllongitudeofperigeeandAP Also Com'pendiumofSpherical Astronomy,tablexxi.,p.432. W = -I25i4[sinD-*144sin(D+g)*149sin(D-p) +*068sin(Dg)\ - I25"*i54sinD+i8"*023sin(D+g)i8"'6o9 sin(D-g) Transforming fromeclipticlongitudeandlatitudetotrue f TablesoftheMotion Moon,sect,i.,chap,Listio.. * InvestigationofCorrections toHanserisTablesoftheMoon,tablexi.,p.51. The correctiontothetruelongitudeinorbit,AZ„,maybe Brown tgivesfortheprincipalparallacticterms The quantity © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System A^ =(1+F)A¿+2sin^Aß-2cos^.eAII+(F.P)AP(iv.) W (A)AA +(/3)A/3=LAZ(O)sin¿AÍ2(¿)A¿.(iii.) V (i') =sin(A-ß)(/3), L =(A)[+(F.0](/3)(F.^) 2 I (F. /?)=sinicos(X-O). (F .Z)=sin¿{sin(X-ß) 2 F =2ecos<7+fa2g Sill 1/ + •£)(/?)] # - 8"46sin(D+p) 192 4MNRAS. .85. . .IIS / and derived byBattermann,*isadopted,sothat Investigation ofCorrectionstoHansen’sTablestheMoon. this termtheconstantcorrectionAß'=+i"*ooisapplied. a meansfortheremovalofconstantcorrection-i'^ooapplied been reduced. L'A^i +Lsin¿7.2Ae—cosg.2eAHL(F.P)AP +(O)sin¿AH coincidence ofthecentresgravityandfigureMoon,whichhas (À)AÀ +(ß)Aß=L'AZLsin£.2Ae-cosg.2eAn with butsmallweightfromtheoccultations.Thevaluederivedby no justificationontheoreticalorobservationalgrounds.!Toremove Hewcomb {isthereforeadopted:Ai=- by HansentothelatitudeofMoononaccountasupposednon- used intheconstructionofHansen’sTables,value-i26'*45, (F. P)=sinD-‘144sin(D+g)-149-<7)*068(v.) Iq denotingthelongitudeofSun,üqSun’s which istheformtoalloriginalequations ofconditionhave where perigee. m where I)=l-,g^/n i6 Mr.H.SpencerJones,lxxxv.i, mQ0 * Beobachtungs-Ergebnisse der KöniglichenSternwartezuBerlin,No.11,p.19. t Newcomb,Researcheson the MotionofMoon,partii.,p.38. J Newcomb,Investigationof CorrectionstoHansen'sTablesoftheMoon,p.36. The correctiontotheinclination,Ai,issmall,andcanbedetermined The quantity8ßistabulatedbyNewcombintablex.,p.50, As thecoefficientofprincipalterminparallacticinequality We thenhave Introducing thevalueofAlfrom(iv.)into(iii.),itfollowsthat The equationmayforconvenienceberewritten intheform The equationsofcondition(i.)nowassumetheform The lasttermontheright-handsideisincludedinordertoprovide We maythereforeadopt 1 v © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System / (¿c)A^ +(2/)Ai/4-(^)A^(P)AP(0)sintA0 (s)s =7i.(vii.) (i)Ai =-o"*15sin(X0>)(ß)Sßxiß) (OAi+^A/î'^i+SftX/?). f + L(F.P)AP(O)siniAQ(i)Ai{ß)Aß(vi.) P =i26"*45+AP. I/-L(i+F). + («)*=■«-(i8)/?), /1 192 4MNRAS. .85. . .IIS 3 /, , and wheresheredenotestheoriginalmultipliedby10. from thedatainNauticalAlmanac. Nov. 1924.TheMoon'sMeanLongitude.17 accuracy aspossible,correctionshavebeenappliedfortermsinthe puted foreachoccultation,thevaluesofargumentsbeingderived where tion ofhistablesandtheplanetarytermsaccordingtoBrown.He between thetermsofplanetaryoriginusedbyHansenincompila- Tables torequirecorrection.Battermann*hasmadeacomparison of revolution.’Theinclusionthesmalltermswiththeseperiodsis anomalistic periodofrevolution,andtermswithcoefficientsor coefficients oforgreaterthano"'04,whoseperiodsapproximatetothe angewandten Störungen.Brown’scomparisonthereforerequirestobe terms withcoefficientsofatleasto'^io,and,inaddition, gives alistofthedifferences,BrownminusHansen,whichincludesall Moon’s motionomittedbyHansenorfoundoncomparisonwithBrown’s date. Thetermshavebeennumberedtoagree withBattermann’slist. negligible. longitude ofperigee,andtheparallacticinequality. this valuethecorrectiontoinclinationarising fromthesetermsis Battermann. greater thano'2,whoseperiodsapproximatetothesynodicalperiod longitudes oftheMoon’sperigee,Venus, Earth,Mars,andJupiter applied toHansendependsuponthevalueassumedforellipticityof necessary inordertoderiveaccuratevaluesoftheeccentricity, the DarlegungdertheoretischenBerechnungindenMondtafeln the distancesfromMoon’sascendingnode totheMoon’sperigee the Earth.Thevalue1/297isadoptedinpresentdiscussion:for ing onthefigureofEarthandmotioneclipticaccording the DarlegungandTablesdelaLune.Theresultingterms node, ß,andtheeclipticlongitude,À,arereferred totheequinoxof respectively, referredtotheequinoxofi85o o. Thelongitudeofthe Table I.y,y'denotethemeananomaliesof MoonandSun,wN to histheoryandasusedinHansen’sTables.Thecorrectionbe corrected bytheadditionoftermsrepresentingdifferencebetween which needtobeappliedHansen’stabularpositionsaregivenby between histheoryandNewcomb’sTra?isformationofHanserisLunar and theSun’sperigeerespectively.H,V,E, M,Jdenotethemean Theory. ThebasisofthelatterwasnotHansen’sTablesdelaLunebut (x) =L(i+F),(y)Lsing,(z)'Leo^g(P)L(F.P), i Ax =AZ^,Ay2Ae",A^=-20AII, i M.N.R.A.S.,68,148. î M.N.R.A.S.,74,392. * Beobachtungs-Ergebnisse der Königlichen-SternwartezuBerlin,No.13,p.16. The variouscoefficientsenteringintoequation(vii.)havebeencom- In ordertoderivethequantitiesAx,Ay,.etc.,withasgreatan As regardstermsofsolarorigin,Brownfhasgivenacomparison The listofthetermsappliedtoHansenis reproducedbelowin Brown Jhasgivenadetailedcomparisonbetweenthetermsdepend- © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 2 192 4MNRAS. .85. . .IIS 18 Mr.H.SpencerJones,lxxxv.i, I. PlanetaryTermsinLongitude. (c) LongPeriod. (b) Middle-lengthPeriod. (a) ShortPeriod. 0, 0 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 0 0, 0 0, 0 0 0, 0 o 0, 0, 0 17. E0*85sin[—(18Y -16E—^)4-334°*4] 16. Eo-24sin[-(8Y- 13E)4-3i3°‘9] 15. Eo*i3sin[^-(E2II)2o(YE)E273 o] 14. E0*48sin(a)-a/E180)(solarterm) 12. Eo’iosin[(E—J)E174] 13. E0*17sin[(E-J)JE2985] 10. E0-35sin[E-2(Y-E)E272*9] 11. Eo*i8sin[E-3(Y-E)4-27i°7] 34. +0-05sin[D(IT-J)i8o9] 33. E0*02sin[De(H—J)—JE116] 32. +o‘i6sin[D-(II-J)E359°‘o] 30. Ecrissin[D+(E-II)—(Y—E)180] 29. +0*44sin[g-J2(11-J)7°-5] 28. +r14sin[(?+2(n-J)i8o3] 31. +0*14sin[D-(EH)E2(Y—E)180] 27. +0*04sin[^+2II—3(2M—E)-l-162] 25. +0*07sin[^(2M-E)2233] 24. +o'o8sin[g-2(EII)6(YE)2Ei626] 26. 4-0*07sin[¿7-(2M—E)+3o6°'3] 23. +o’óósinf^-2(EIT)+3(YE)180] 0, 0 0, 0, , 9. Eo*i2sin[2(Y—E)4-i8o°] 8. E0*28sin(Y-E) 4. +0*13sin[¿7—(E-2II)-Y] 7. -ho^isin[^+2(11-J)-3(E-J)-t-X784] 6. +0*19sin[y—2(E-J)+180] 5. +

18. + 0*35 sin [-(18V - i6E-£) + (8V- i3E) + 33'] 19. +o*ii sin [ — (18V- 16E —i/) - (8Y ~ I3E)+ ii20,o] 20. 4-o‘24sin[2(II-J) + o0,i] 21. 4-o‘28 sin [J-2(11 —J) + i720,5] 22. + 0*13 sin (J+286°)

IT. Solar Terms in Longitude.

35 4- cr 31 sin (D - g) 36 + 0*23 sin [2(D —<7) — 2o>] 37 + o*i6 sin [4(D — g) — 2(0) - to')] 38 + o*i i sin [4(D — — 2a> - 2(0) - co')] 39 4-0*28 sin [^4- 2(o4- i8o°] 40 — o*X2 sin (2D - g) 41 — 019 sin 2D 42 + 0*25 sin g

III. Terms due to the Motion of tho Ecliptic and the Figure of the Earth. // // 43. A¿m= — 0*97 sin O -b 0-42 cos ß 44. Az = 4-0*42 sin ß - 0*05 cos ß 45. (sin ^Aß) = 4-0*05 sin ß - 0*04 cos ß

The short-period terms Nos. 1-34 and terms Nos. 35-42 were calculated for each occultation. Terms Nos. 8-13 were computed for 10-day intervals in the years in which occultations were numerous and interpolated for the time of occultation : in the years with comparatively few occultations these terms were computed for each occultation. The short-period terms represent corrections to the true longitude in the orbit and their sum after multiplication by L was applied as a correction to the residual in equation (vii.). The long-period terms Nos. 14-22 were computed at intervals of 100 days and interpolation made for each occultation. These terms represent corrections to the longitude in the orbit, and therefore require to be multiplied by L(i4-F) and applied as a correction to the residual of the equation of condition. The term No. 43 was treated in the same manner. The term No. 44 is a long-period disturbance of the quantity z=—2eAII. This correction was not applied in the formation of the equations of condition, but was introduced in the subsequent solution. Two further corrections were applied, (i) In forming the original equations of condition the Hansen-Newcomb correction had been removed from the tabular places in order to secure a direct comparison with Hansen’s Tables. This correction was reinserted ; for the year 1880 the values were specially computed, (ii) A correction was intro-

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 192 4MNRAS. .85. . .IIS ro,,2,,,3 2,IOr03 ,/ / ,,,2 acceleration foundbyFotheringham,andwiththeconstantterm Hence theHansen-Newcomblongitude(includinggreatempirical duced toreducetheHansen-Newcombmeanlongitude(including for eachobservation.AftermultiplicationbyL(i+F)ithasbeen Hence thecorrectiontobeappliedHansen-Newcomblongitudeis modified byDysonandGrommelin,is term) asusedintheNauticalAlmanacis Fotheringham,t andfurthercorrectedbyDysonCrommelin.J great empiricalterm)tothevalueadoptedbyBrown,*ascorrected where tdenotescenturiesfrom1800*0. Brown’s value(Astron.Journ.,No.799),correctedforthesecular 20 Mr.H.SpencerJones,lxxxv.i, cordant equationswereexamined.Inafewinstanceserrors applied asacorrectiontotheresiduals. 335° 43'25"*56+i33630753'io-4^-9"*54f+oi35 traced intheoriginalreductions;othersanerrorofoneortwo 335° 43'5'+367°53'i3"82/i"-9^oo6B^ groups ofonetofouryears,accordingthe numberofobservations, rejected. Normalequationswereformedfrom theremainderinsmall which theobservedtimewasapparentlyconsiderablyinerror,were tions, whichweremarkedbytheobserveraspoororuncertainin minutes intheobservedtimewasassumed.Asmallnumberofobserva- 5T ischangedfrom-27i to-2*5. 0 each unknownbeingdeducedintheusualmanner fromthesquaresof the unknowns,probableerrorofoneequation ofconditionand giving eachoccultationequalweight.These equationsweresolvedfor known, aregiveninTableII. the residuals. with theprobableerrorsofoneequationcondition andofeachun- c c - o"-46+3"38í27^"*o6i3"-6osin(139X+io4°*2) c 7 f Fotheringham,A.12.^.¿S'., 81,125. t DysonandGrommelin,M.N.R. A.S.,83,361.TheconstanttermofBrown’s * Brown,TablesoftheMoon,sect.1,chap,i.,p.28; Astron. Joum.,No.799. ro Hansen’s meanlongitudeis Newcomb’s correctionasusedintheilis This termhasbeencomputedatyearlyintervalsandinterpolated After theformationofequationscondition,obviouslydis- The valuesofthevariousunknownsderived inthisway,together 335° 43'26"7o+i33Ó.30753'39"'6it+ c © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 2, + i"-i4-29'-ijtf-yôtç15"49sin(132%93°-9). c /,0 + i3*6osin(i39¿i42). 0 0 0 + i5"*49sin(i32%+93-9). - i5"-49sin(i32%+93*9). 192 4MNRAS. .85. . .IIS - # [ 96 -318 88 -2*41 87 -1-87 86 -1*95 85 -3o6 84-1 *6o 83 -3'36 82 -2*04*24] 81 -2'35'35h 80 -2*i8+o*I9^ ar. 97 -3'°8 95 -371 93 -3'28 89 -278 92 -3*57 91 -3*12 98 -333 94 -3'9° 90 —2*26 99 —2’62 00 -23i 01 -i*8i 02 “i*39 □3 -0*64 □4 +0-17 35 -o*i8 36 +0-23 37 +0-95 38 39 +2-11 10 +2-20 11 +279 12 +3*04 t3 +377 t4 +3*54 5 +3*34 [6 +4*01 Nov. 1924. x=Al. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System , , , , -i*i6± *io-0*13+-12+o*29±*19+178+*i8+0*04+’140-57 -i*i8± *13-o74±-17+i*i9±'20+1-44+-19-0*36+*120*56' -o‘4o±o*24 -o'99+o*2i+i*39+o*26+i*024;0'27-o*39o'I9+o*86 -0S9+ *16-o*8i±*13+1"22+*25^-i^i*19-o37±*200*89 -o39± ’19-o*37±*2i+0*90+‘27+o’o8+*30-0*30+*15o*86 -0-65+ *15+oi5±*i6+i*40±-29+o*93±'24-o^4±'20076 — <3*32+*i8-072+‘17+0*62+*28+0*83+'26+o*o6+*19075 -049+ *20-o*93±*22+1*65+’29+o*39±*24-0*41+’iS0*85 -0-06+ ’16-i’07±‘13+2*04+*26+i'oi+’I7-i*38±*21076 -o’33± ‘17-o*95±*21+i89±*43+i'i7±'28-0*87+*340*85 — i’36+*20-o*57±’24+i*i7±*390*21*27-029+*250*80 -o*62± *i8—0’20±+i'33±*44+o*o7±'26-o'44±*270*67 -0-93+ *15-0*69+-i6+i'i5±'28+1’63+'i8 -0-15+*17072 -o'39± *17-ro4±*18+i'8o±*31+i'i6±*28 -o74±*21078 , y =2Ae".2=-26AIT.AP. i2+ '17-075+*2541'20+*33-0*63+‘26-0*16+*180*95 // nu 105+ *25-ro9+*23+i'43± *28+1*85+*26-o*43±‘21+o*8i The MaorisMeanLongitude. Table II. Sin iAO. 21 P.E. of Equa- tion. One 192 4MNRAS. .85. . .IIS Year. x=Al. 1920 +3-67-23' 1917 +475±•■24' 22 +378+*21 21 +374*27 19 +4*68*24, 18 +3-46-33 to deriverevisedvaluesforthelongitudecorrectionsfromnormal further withtheapproximation. corrections wereallsmall,sothatitwasnotnecessarytoproceedany solution madeforthesequantities.Thevaluessoobtainedwereused group intothenormalequationsfory,z,AP,sin¿AO,ands,a preliminary valuesofy,z,AP,sin¿Aß,andswereobtained.These would beexpectedfrom theassignedprobableerrors,exceptin the during thethreeperiods isverygood,andinagreementwithwhat the longitudecorrectionsderivedinTableI.weresubstitutedeach during thisperiodarepublishedintheCapeAnnals,Thevaluesof from afinalsolutioncombiningthethreegroupstogether.Theprobable and semi-diameteraregiveninTableIII.Thelongitudecorrections to theeccentricity,longitudesofperigeeandnode,parallacticinequality, preliminary solutionwerethencombinedforthethreeperiods1880to by theapplicationoftermNo.44. probable errorforoneequationofconditionwas±o"‘88theperiod are notgiven,asdefinitivevaluesofthelongitudeerrorswerederived equations forthex’s,andwiththeserevisedvaluescorrectionsto addition oftheyear1880—tothreegroupsinwhichoccultations errors giveninthistablehavebeenassignedbyassumingthatthe 1895, 1896to1906,19071922,corresponding—exceptforthe 1880 to1895,and±o"‘8ofortheperiods189619061907 1922. 22 // © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System No. ofoccultationused The quantitiestabulatedas2intheabovetablehavebeencorrected The valuesderivedfromthesethreesolutionsforthecorrections The partialnormalequationsforthesmallgroupsusedin The agreementofthevalues resultingfromtheoccultationsobserved Z= -26ATI y =2Ae" sin iAii AP s - -0*22+ *21-Io6+*24+1*63+*26+1-26+*25-0*38±*i8 , i*24± *19-I‘19+’18+1*04+*41+i'6i+*34+oo8± // u y =2Ae".z=~zeAU.AP.SiniAß.8. + 0*24*115 + i*289±'in -0*63 ±o’o77 -0*30 ±*075 -077 ±*078 Mr, H,SpencerJones,lxxxv.i, 1880 to1895. Table ll-^continued. // 379 Table III. + i*278±*103 + i*ii±*075 -070 ±0*051 -0*48 ±*075 -o*66 ±*050 1896 to1906. // 743 -0*69 ±0*077 -0*38 ±*088 4-1 *50±*109 +1’4511 ’153 -o*86 ±*077 1907 to1922. a // 316 P.E. of ±073 ±0-83 tion. Equa- One 192 4MNRAS. .85. . .IIS for itseffectonthe solar parallaxtobepracticallynegligible. To produce analteration in thesolarparallaxofo'^ooi,ratioE/M mass oftheMoon,but this ratioisnowknownwithsufficientaccuracy upon thevalueassumedforratioof mass oftheEarthto the resultingvalueis8"8040±"*0048,or,with sufficientaccuracy, fundamental constantsinBrown’stheoryare:— three periodsare:— is thesolarparallax,Moon’sequatorialhorizontal revolution. Cisaconstantwhichcanbederivedtheoretically.Ifit a, a'arethesemi-axesmajoroforbitsEarthandMoon,which for undisturbedellipticmotioncorrespondtothesiderealperiodof can beexpressedintheform term intheparallacticinequalityforthreeperiods:— where E,MdenoterespectivelythemassesofEarthandMoon, discordant. case ofthequantitysin¿AO,forwhichvaluesaresomewhat Nov. 1924. - 126"*450givethefollowingvaluesforcoefficientofprincipal The valueofthesolarparallaxderivedin this wayisdependent Weighting thethreevaluesinaccordancewith theprobableerrors, With thesefiguresthederivedvaluesforsolarparallax The coefficientoftheprincipalterminparallacticinequality,P, The valuesderivedforAPwhenappliedtothebasicvalue — sin,, © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System lunar parallax’E+M00251287 à>= 1-00090768^o=34i9-596 E-hM =8i-53 -, =sin7T7t=8*80549 -P=i25"-i54 solar parallaxE--M 1907 to1922 1896 to1906 1880 to1895 1907 to1922 1896 to1906 1880 to1895 The Moon'sMeanLongitude. aEM _ - Pr 8"*8o4 ±o"'oo$. ° 'a'E+M 87944 ±*0109 8*8o68± *0074 8’8o6o ±0*0079 I2 “ 4‘9 - 125*161 - 125*172 n2 s>Po=34"‘54o 2 Po=34"7o C =[4-697301] 23 192 4MNRAS. .85. . .IIS ' variationinthededucedvaluesofsin¿AÍ2.Investigationhasfailedto to allowforthis,andalso toderivetwoadditionalnormalequations for two years,itisnecessary tointroducetermsintothenormalequations secular variationarisingfromthecorrectionsrequired byHansen’smean motions, andthelengthofwholeperiod under discussionisforty- together toderiveonegeneralsolution.AsAH andAÛaresubjectto- series. Themeanvaluederivedfromthe wholesolutionmaybe into approximateuniformity.Thederivedvaluesofsin¿AOaretoa accepted asgivingthecorrectiontolongitude ofthenodeforan to anaccidentalaccumulationofsmallvalues intheearlypartoftho be concludedthatthesmallvaluefoundduring thefirstperiodisdue those for1880,1896to1901,19151916arelarge.Theterm range theruninvaluesofsin¿A0isnot greatlyaltered.Itmust- values forthelattermaybetakenas1/293and 1/298,andwithinthis being alteredbychangeinthevalueofellipticity.Butextreme the Earth’sfigure,coefficientofsinOintermHo.45TableI. certain extentdependentuponthevalueofellipticityassumedfor period ofrevolutionthenodehavebeenincludedinforming of thenode,providedthatalltermsinMoon’smotionwith sin¿AO intheexpressionforAlwillgiveerroroflongitude epoch approximately1900*0. reveal anytermwhichwouldbringtheresultsforthreeperiods tabular places:theomissionofanysuchtermwillresultinaperiodic instance, thevaluesfor1883to1886and19021906aresmall; errors ortotheomissionofaperiodicterm.Thereappearsbesome arises whetherthedifferencesinthreevaluesaredue(apartfrom evidence ofatermwithperiodaboutnineteenyears:thus,for the errorinHansen’smeanmotionofnode)entirelytoaccidental the discordancebetweenthreeseparatevalues.Thequestion the threeperiods.Itismuchsmallerthanwouldbeanticipatedfrom usual mannerfromtheprobableerrorsassignedtomeanvaluesin three solutionsareasfollows:— from thelunarinequalityisinverycloseagreementwithHinks’value. deduced fromtheparallacticinequalitybyasmucho"'ooo5. It isthereforeveryimprobablethatanysubsequentrevisionofthe value oftheratioE/Mwillchangesolarparallax adopted byBrownwasderivedHinksfromtheobservationsof 24 Mr.H.SpencerJones,lxxxv.i, Eros, andwasassignedaprobableerrorof±0*05.Thevaluederived would requiretobechangedby0*39.Thevalue81*53whichis The normalequationsobtainedforthethreegroups canbecombined The probableerrorassignedtosin¿AOhasbeenderivedinthe The valuesderivedfortheotherquantitiesbycombiningabove © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System — zeAU sin ¿AO 2A6" AP + i*oi±*054 -o*39± *045^ + 1*32±*068> — o*68±0*037^ - 0*73±*037 // Solution I. 192 4MNRAS. .85. . .IIS The probableerrorsare notappreciablyaltered. Astronomy, 1895,pp.49-52. solution is:—„ in thenormalequationsfory,z,AP,sin¿AO, ands.Theresulting of themeanmotionstothoseadoptedby Brown weresubstituted normal equations,thecorrectionsrequiredto reduceHansen’svalues several smallvaluesintheearlypartofseries,towhichreference motion ofthenodeisduelargelytoaccidentalaccumulation has alreadybeenmade. this investigationbeingtooshorttoenablereliablecorrectionsbe derived. ThelargevalueobtainedforthecorrectiontoHansen’smean of perigeeandnodehaveonlyasmallweight,theperiodcoveredby residuals, is±(/‘Si. applied tothemeanvalueofz,whensubstitutinginoriginal one occultationandgivenunitweight.Theprobableerrorofasingle equation ofcondition,derivedintheusualway,fromsquares observations ofthesameoccultationbydifferentobserversaretreatedas the truevalueof2;meaninequalityforeachyearwas equations ofcondition. the separateyearsinequalityinzwasremovedordertoobtain of theunknownsthusderivedhavebeensubstitutedintooriginal additional computation,usingthemethoddevelopedbyNewcomb.* equations ofconditioninordertoderivetheprobableerrorone were thenusedtocorrectthelongitudeerrors,andsoon.Thevalues equation ofcondition.Incombiningthepartialnormalequationsfor to derivepreliminaryvaluesfortheotherunknowns.These values ofthelongitudeerrorsobtainedinpreviousworkbeingused Nov. 1924.TheMoon'sMeanLongitude.25 the secularvariations.Thiscanbedonewithoutagreatdealof * TheElementsoftheFour InnerPlanetsandtheFundamentalConstantsof Instead, therefore,ofderivingthesetwocorrectionsfromtheir The correctionsAzandA(sin¿AO)tothecentennialmeanmotions The valuessoderivedare:— The solutiondependsupon1438occultations,inwhichtwoormore The equationswereagainsolvedbysuccessiveapproximation,the © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System z= -2eAH A (sin¿AO) 2/ =2Ae" sin ¿Aß sin ¿Aß=^+i*oi±*053 AP AP Az s s =-0*39±044 V z ; = -0*71±*036 = +1*296± = --0*65±0*037' + 3*45±*54^ + i*oi±*053« — 0*64±*36 -0*39 ±*044 + 1*291±*o6i -0*71 ±*036 -0*65 ±0*037' // *061 ^-Solution III. Epoch 1900*0. Solution II. 192 4MNRAS. .85. . .IIS adopted semi-diameteroftheMoonareclosely boundupwithone from systematicerror.Theparallacticinequalityandthevalueof parison ofitsextremeobservedeffectsonthe Moon’s position,thevalue Ñewcomb,* theaccurate determinationofthisinequalityundoubtedly inequality havethereforebeengenerally regarded assubjectto small, andthevalueisinalmostexactagreementwithErosresult. systematic errorsofunknownamount.But, aspointedoutby The questionariseswhetherthedeterminationcanberegardedasfree observations was as thededucedvalueofsolarparallaxmean The previoussolutionresultedinthevalue it isnegative;followsthat,iftheinequality isdeducedfromacom- ment withintheassignedprobableerrors,andwemaythereforeadopt larger forthefirstperiod.Thethreesolutionsare,however,inagree- same weight.InSolutionI.,ontheotherhand,eachequationofcon- quantity cannotbesatisfactorilydeterminedfrom meridianobservations. in theadoptedsemi-diameter.Owingto effectsofirradiationthis so determinedwillbeaffectedpracticallybythe fullamountoftheerror observed whentheinequalityispositive,and the otherlimbonlywhen deduced fromtheinternalaccordanceofresults,issatisfactorily values derivedfromthesmallgroupsforprobableerrorofone dition receivedaslightlydiminishedweightinthefirstperiod,as Determinations ofthesolarparallaxfrom valueoftheparallactic another. Inthecaseofmeridianobservations onelimbcanonlybe equation, andgiveninthelastcolumnofTableI.,wereonaverage the principaltermofparallacticinequalityare:— values ofthesolarparallaxare:— The latterisidenticalwithBrown'sadoptedvalue.corresponding 26 Mr.H.SpencerJones,lxxxv.i, The probableerrorofthisdeterminationthesolarparallax,as The valuederivedbyHinksfromthediscussionofEros In SolutionsII.andIII.eachequationofconditionreceivesthe The valuesresultingfromthesetwosolutionsforthecoefficientof © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System The ParallacticInequalityandtheSolarParallax. * ResearchesontheMotion of theMoon,partii.,p.13. - 125-154±*061„III. - 125*159±o'oóiSolutionII. 8*805 ±*0044„III. 8-8o6 ±0*0044SolutionII. // // v ,,, 8"*8o6 ±o^-ooq. 8"*8o5 ±o*5. 8*o4 ±5. 192 4MNRAS. .85. . .IIS ,/ ,/ ,/ m m 5 provided forchangingtothefinalvalue. Minor PlanetsIris,Victoria,andSappho,”GapeAnnals, 6,partvi. occultations. BrownUadopts22639*5ashis finalvalue.Thevalue 22639*5o wasusedintheconstructionof the tables,butameansis resulting valueoftheprincipaltermis used byCowellisadoptedBrown§asHansen’svalue. principal periodictermintheMoon’seclipticlongitudeemployed Hansen’s Tables.NewcombJderivedthevalue2264o*i5.The The valuez/=2Ae"=-o"'66isadoptedasthecorrection. Moon’s eclipticlongitude,arerespectively:— theoretical uncertaintiestowhichNewcombreferred,andwere present discussionareaffectedbyerrorsofasystematicnature. of themotionMoon,asnegligible.Thereisthereforenoreason parallax, canberegarded,inviewofthecompletionBrown’stheory the parallacticinequalityandofsolarparallaxderivedfrom to believethatthevalueofMoon’ssemi-diameterandvalues observations orduringlunareclipses.Thepossible occultations underdiscussion,which,withfewexceptions,aredark-limb emphasised byGill*inhiscritiqueofthismethodderivingthesolar the systematicerrorstowhichmeridianobservationsareliablemay and thereappearancestoolate.Theseconditionsaresatisfiedby stars bright-limbdisappearanceswouldtendtobeobservedtooearly, reappearance atthedarklimbtostarsnotfainterthan2.For the brightlimbarerestrictedtostarsnotfainterthan3,andof inequality.’ Theobservationofoccultationsprovidesameansbywhich parallax. Inthisdeterminationthenumericaldivisorofinequality be avoided,subjecttotheprovisothatobservationsofdisappearanceat of theparallaxwillhelittlemorethanone-fifteenththat is nearly15,sothat,possibletheoreticaluncertaintyaside,theerror Nov. 1924.TheMoon'sMeanLongitude.27 ' t64,163. affords oneofthemostprecisemethodsdeterminingsolar , § M.N.R.A.S.,74,421. f t TransformationofHansen'sLunarTheory,p.60. II ResearchesontheMotionof theMoon,partii.,p.224. * ‘DeterminationoftheSolarParallaxfromHeliometer Observationsofthe // Newcomb IIobtainedthevalue22639"5ofrom hisdiscussionof Applying thecorrectionof-d'66foundabovetothisvalue, Cowell tadoptedthevalue22640*i68ascoefficientof The valuesderivedfory,thecorrectiontoprincipaltermin Tables oftheMoon,sect,i., chap,i.,p.6. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System , -o*68±oo37 'SolutionI. - o'6$±-037„III. -o-6s± '037„II. // 22639"*5i ±o"'04. The Eccentricity. 192 4MNRAS. .85. . .IIS / , zT for epoch1900*0gives for 1900*0,theresultingcorrectiontoBrown's longitudeofperigeefor perigee derivedbyNewcombinResearchesof theMoon,partii.,p.224, probable error.SincethedifferenceBrown minus Hansenis+8"*io a valuewhichisfivetimesitsprobableerror. Thelongitudeofthe this epochis for theepoch1900*0is,however,derivedwith acomparativelysmall rection tobederivedwithmuchweight,andthedifferencebetween present investigationisonlyabouttwicetheprobableerroroflatter. occultations underdiscussionisnotsufficientlylongtoenablethiscor- Brown’s correctiontoHansen’smeanmotionandthatderivedinthe of +i2*44percentury.InSolutionII.acorrectionwasderivedto Hansen’s meanmotionof+5"*82±Theperiodcoveredbythe days, from1900*0.Theoriginofreckoningisthemeanequinox Hansen’s *andBrown’sfvaluesforthelongitudeofperigeeepoch The value2=-leATl—o"yi±"‘036isadopted. assigned probableerror. resulting fromthecoefficientfoundinpresentinvestigationis This valueisinagreementwithBrown'swithinthelimitsof 1900 arerespectively:—■ of‘054900489, usingthecoefficient2e~\e+e.Thevalue 28 23 2 Brown: H=3341946*40+1110922*52¿-37*i7¿•0456.. Hansen: H=3341938*30+1110915o*o8¿-3Ó’25¿-‘037V. c cC Brown derivesacorrectiontoHansen'smeanmotionoftheperigee In theseexpressionstdenotesthenumberofcenturies,each36,525 The valueofthecorrectiontoHansen’smean longitudeofperigee The resultingcorrectiontoHansen'slongitudeofperigeeis The valuesderivedforzfromthethreesolutionsare:— c The coefficient22639"*55correspondstoavalueoftheeccentricity f M.N.R.A.S.,75,510;Tables oftheMoon,sect,i.,chap.L,p.28. * Newcomb,Researcheson the MotionofMoon,partii.,p.224. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System AH= +ó'^ó±o"‘33(epoch1900*0). o ///r ,/ Longitude oftheMoon'sPerigee. -073 ±0*037SolutionI. -o‘7i± *036,,III. - 0*71±*036„IL Brown -Newcomb=+5*53 e =*054900392±*000000097. un / Mr. H.SpencerJones, AH=-i*64±o"*33, 96 4608 LXXXV. I, 192 4MNRAS. .85. . .IIS but inoppositedirections. indicated correctiontoBrownis tions toBrown’sadoptedvaluefor1900ofapproximately equalamounts correction sin¿AO. for epoch1900are:— mean motionhasverysmallweight.Itiscertainlyconsiderablytoo approximately midwaybetweenBrown’svalueandNewcomb’sas with Brown’sfinallyadoptedvaluegivesadifference corrected byBrown||foralong-periodplanetary term,whencompared the nodeis and areidentical.TheresultingcorrectiontoHansen’slongitudeof large, duetoseveraloftheearlyyearsgivingsmallvaluesfor corrected byBrown. value ofthesecularaccelerationatepoch1850andtoincludea Nov. 1924.TheMoon'sMeanLongitude.29 Since thedifferenceforthisepochBrownminusHansenis+9"*4i, corrections, thedifferencebecomesforepoch1900*0, planetary termoflongperiodneglectedbyNewcomb.Afterthese Newcomb’s valuewascorrectedbyBrown*soastogivethetheoretical 10x3 23 Brown: 0=2595979“53483123¿+*oo8¿. Hansen: 0=259°1050*38-5134843*21^+8*2i^+-ooy^. c c Newcomb’s andthepresentinvestigationstherefore indicatecorrec- The correctiontoHansen’slongitudefor1900*0hasagoodweight. The longitudeofperigeederivedabovefortheepoch1900thusfalls The valuederivedinSolutionII.forthecorrectiontoHansen’s Newcomb’s §finalvalue,derivedfromhisoccultation discussion,but Hansen’s fandBrown’sJexpressionsforthelongitudeofnode The valuesderivedforsin¿AQfromthethreesolutionsare:— t Newcomb,ResearchesontheMotionofMoon,part ii.,p.224. % M.N.R.A.S.,75,510;Tables oftheMoon,sect,i.,ehap,p.28. § Ibid.,p.224. * M.N.R.A.S.\74,420. II 74,563. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System / Brown —Newcomb(corrected)=+2*86. Brown -Newcomb=+2"*41(1900*0). , The LongitudeoftheMoon'sNode. + n'^zS±o'59(epoch1900*0). + 1*01±0*054SolutionI. + i*oi±*053„III. + roi±*053„II. // Ail =+I"*87±0*59. 192 4MNRAS. .85. . .IIS aso The derivationofthesemi-diameterfromanequation ofthisformisnot where A?*isthecorrectiontoassumedvalue ofthesemi-diameter. the assumedsemi-diameterarederivedfromequationsofcondition semi-diameter arecloselyboundupwithone another, sothatunlessthe strictly legitimate.Thevaluesoftheparallactic inequalityandofthe bis 1876beobachtetenBedeckungenderPlejaden,”Ast. Nachr.,138,147. correct valueoftheprincipalcoefficient parallacticinequalityis the form bachtungen vonSternbedeckungen; VereinigungderResultatedreiBerliner of occultationsthePleiades.Inbothinstancescorrectionsto occultations duringtwolunareclipses,andtheseconduponobservations Reihen,” Beob.-Ergeb.derKön.-Sternwarte zuBerlin,^0.13. 135, 175. 1888 January28beobachtetenSternbedeckungen,”Dorpat, 18935lNachr., The correctiontothesemi-diameterresultingfrompresentdiscussion in theconstructionofhistables.Headopts the valueofmeanequatorialhorizontalparallaxasusedbyHansen mann *hasdiscussedthequestionastocorrectvalueadoptfor occultation observations:—. This maybecomparedwiththefollowingotherdeterminationsfrom where ttisHansen’stabularequatorialhorizontalparallax.Batter- occultations is must beappliedtothatvalueofthesemi-diameterwhichcorresponds to thisparallax,viz.933"*o6. 30 Mr.H.SpencerJones,lxxxv.i, f “BearbeitungderwährendtotalenMondfinsternisse 1884October4und * Beobachtangs-ErgebnissederKöniglichenSternwarte zuBerlin^No.13,p.11. X ‘BerechnungderCoordinatenunddesHalbmessers aus achtindenJahren1840 //,3 § “BeitragzurBestimmung derMondbahnunddesMondhalbmessersausBeo- The firstofthesedeterminationsisbaseduponobservations The threesolutionsgivethefollowingvaluefors:— The valueofthesemi-diameterusedinCapereductions The resultingvalueofthesemi-diameter933o6(i-0*39/1)is © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System H. Battermann§.932*72±0*15. J. PetersÍ.932*49±0*28(m.e.). L. Struvet.932*65±0*044. (a)Aa +(S)A8(7r)A7r±(?’)A?*=const., /, sin 7T=3422"*i2.342228. -o'39± -044„III. -o'39± -044„II. -0*39 ±0*045SolutionI. // The Semi-Diameter. ,/ 932"*7o ±o4. -Xlr [943559]> // 192 4MNRAS. .85. . .IIS , compiling whichbright-limbobservationsandofocculta- tions duringlunareclipseshavebeenexcluded. disappearance andreappearanceseparately,inordertotestwhetherthere just referredto. are anysystematicdifferences. according tothemagnitudeofstarsandforobservations gave avalue932"8i,werenotavailabletoNewcomb.Itwouldappear that Newcomb’sadoptedvaluerequirestobeincreasedbyabout upon theresultsofStruveandPeters,referredtoabove, into theequationsofcondition;sothatthisvalueisfreefromerror second ofarc. first twoofBatterman’sseries.Theresultsthethirdseries,which agreement withBattermann’smeanvalue.Newcombadoptedforhis involving thecorrectiontoparallacticinequalitywasintroduced occultation investigationasemi-diameterof932"*58.Thiswasbased regarded asrealunlessacorrectionisappliedfortheincorrectvalueof diameter obtainedfromanydiscussionofoccultationsinwhichthe Battermann fromthreeseriesofoccultations:inthereductionsaterm the parallacticinequalityemployedbyHansen. tabular placesoftheMoonarederivedfromHansen’stheorycannotbe about -o"‘11.Itisapparent,therefore,thatthecorrectiontosemi- to theassumedvalueofsemi-diameterischangedfrom-{-o"'i2 to correctBrown’svalueofthemeanequatorialhorizontalparallax the principalcoefficientofparallacticinequality(AP=+i"‘3o)and (A7t= -o"*3i)havebeenroughlyestimated,andtheresultingcorrection by Struve,thecorrectionstoreducefromHansen’sBrown’svalueof minimised, asthensinD=o,i.e.theprincipalterminparallacticin- semi-diameter. Duringalunareclipse,anerrorofthisnaturewillbe equality iszero.Theremainingtermsin(FP)donot,however,vanish. cient willbetoalargeextentthrownintothededucedcorrectionfor Nov. 1924.TheMoon’sMeanLongitude.31 employed inderivingtheMoon’stabularplace,anyerrorcoeffi- The resultingmeanresidualsaregiveninthefollowingtable, The residualsoftheequationsconditionhavebeencollected The thirdvalue,givenabove,isthemeanresultderivedby The valuederivedinthepresentinvestigationissatisfactory In thecaseofobservationsduringtwolunareclipsesdiscussed © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 9‘0-9 *9 4*0-4 *9 7 *o-7-9 6 *o-69 8 *0-8*9 5 *0-5*9 Mag. >4'0 Residuals arrangedaccordingtoSteer'sMagnitude. m 163 223 210 No. 165 64 87 35 Disappearances. Table IV. + 0*03 + 0*07 + 0-10 -0*17 -0*25 -O’U - 019 Mean. // 114 149 No. 29 53 16 12 10 Reappearances. + 077 + o*i8 + O'OI + 0*04 - 0*29 -o'o6 Mean. 0*00 // 32 Mr. H. Spencer Jones, lxxxv. i,

The residuals are taken in the sense computed minus observed. The coefficient of the semi-diameter correction in the equations of condition is in all cases negative. It follows that if the residuals are interpreted as due to magnitude equation in the semi-diameter, the mean values for each group should represent, both for disappearance and reappearance, the excess of the group value over the mean value. The residuals for each maguitude group should then, accidental errors apart, agree in magnitude and sign for disappearances and reappearances. °The residuals may also be interpreted as due to magnitude equation in the star-places. Since for disappearances the coefficient of Aa' (the correction to the star’s E.A.) in the equations of condition is positive for disappearances and negative for reappearances, the mean residuals should be approximately equal in magnitude but opposite in sign, for the two croups dealing with stars of the same magnitude. . ° To separate the two effects, the sum and difference of the mean residual for each magnitude group separately should be taken: the sum will give the magnitude effect in the semi-diameter and will eliminate magnitude equation in the star positions. The difference will give the latter and eliminate the former. Unfortunately the number of observed reappearances for stars fainter than 7m*o are so few that it is difficult to draw conclusions. Table Y. gives the weighted mean residual and the mean of the mean residuals for disappearances respectively, derived on the hypotheses that the residuals are due to magnitude equation in the semi-diameter and in the star-places respectively.

Table Y. Semi-diameter. Star-places. Magnitude. No. Weighted Mean of Weighted Mean of Mean. R. and D. Mean. R. and D. m >4*0 64 -0‘24 -o'24 + 0-02' + 0-05 -O'll -O’lO* 4*0-4‘9 117 - 0*07 -o'o6' + 0*06 + 0*04* 5*0-5 9 359 + 0‘06 + o*o5* + o-o3* 6*o-6 *9 337 + 0’05 + o-o3* + 0*04 -fO’OO* + 0*05 7-0-7 *9 99 + 0-04* + 0-IO* -0*10 -o*o4’ 8 *0-8 9 175 -0*11 - o’oS* -0*51 9 *0-9 *9 179 - o*i6 + 0*26 -0*30

The two columns headed u weighted mean ” in this table should, give the magnitude effects in the semi-diameter and star-places respectively, provided that the mean residuals arise from one cause only. . If both are operating, the columns giving the mean of the K. and D. residuals separ- ately should represent the two effects, but the probable errors of the values for the three last groups are large, on account of the small number of reappearances in these groups. • v 4.1, i. * There is some evidence of a progressive change in both sets ol residuals In the columns headed “semi-diameter” a positive residual implies that the semi-diameter for that group is larger than the mean

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 192 4MNRAS. .85. . .IIS , comparison. TheresultingcorrectionstoHansen’s tabularpositionsare tabulated byBrown,tthecomparisonincluding allthetermsoffairly represents theapproximatecorrectionnecessary toreducetheerrors eorrection tothevalueofmeanlongitudeemployedinconstruc- the quantityST-o'^ió.ThecalledbyBrownST*is long period.Thequantitiesincolumn3are derivedfromBrown’s tion ofhistables,resultingfromthecorrection tothesecularacceleration dition. Thesevaluessufficientlywellindicatetheprobableaccuracyof found byFotheringham,withconsequentialchanges inthemeanmotion mean differencesbetweenBrownandHansen-Newcomb havebeen given incolumn2tocorrectionsHansen’s tabularpositions.The of BrownandHansen-Newcombthe quantityST-o'^ióand and longinequality. not distributedasuniformlycouldbedesired,severaloftheyears mean longitudeofBrown’sTables,'afterthelatterhasbeencorrectedby near thebeginningandendofseriesbeingparticularlyweak. (term No.44)hasbeentakenintoaccount.Theprobableerrorsassigned the errorsofmeanlongitudeforvariousyears.Theobservationsare have beenderivedfromtheweightsinpartialsolutionssmallgroups, free fromanyseriousmagnitudeeffects. assigning avalueof±o'^Siastheprobableerroroneequationcon- substituting y=-o"‘6$,z=AP=+i"3o,sin¿AO=-fi*oi. Az= -i'^só,A(sin¿AO)=+havebeenused.Theinequalityinz The theoreticalcorrectionstothemeanmotionsofperigeeandnode, have beenderivedfromthenormalequationsforlongitudeerrors, effect hasbeendetectedforNewcomb’scataloguebyBattermann.This group issmallerorlargerthanthemean.Thereaslightindication that thebrightestandfainteststarsareobservedonmeridiansome- semi-diameter, ifpresent,issufficientlysmalltobenegligibleinthemean. magnitude equation,ifreal,issmallinamountandnegligiblethemean. semi-diameter. Itmaybeconcludedthatthemagnitudeeffectin what lateascomparedwithstarsofintermediatemagnitude.Asimilar residuals showaslighttendencyinthisdirection,butwiththeexception value; inotherwords,accordingasthemeanR.A.ofstarsfor negative accordingasthegroupvalueissmallerorlargerthanmean not muchlargerthantheprobableerrorofdetermination of thefirstgroup,whichisacomparativelysmallone,residualsare value :itmightbeanticipatedthatforthefaintstarsdisappearances the meanresidualswouldtendtobepositiveforfaintstars.The would beobservedearlyandthereappearanceslate,thattherefore Nov. 1924.TheMoon’sMeanLongitude.33 The thirdcolumninTableVI.containsthe sum ofthedifference The errorsasgiveninthesecondcolumnrepresentcorrectionsto The errorsofmeanlongitudearegiveninTableVI.Thesevalues From thisdiscussionitmaybeconcludedthattheobservationsare In thecolumnsheaded“star-places”residualswillbepositiveor © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System * Astron,Journ.,No.799. t M.N.R.A.S.,73,692;and Astron.Journ,,No.799. The MeanLongitude. 3 192 4MNRAS. .85. . .IIS given incolumn4.ThenextcontainsthecorrectionstoHansen and Greenwichcorrections.Inviewofthefactthat,apartfrom derived fromthemeridianobservationsatGreenwich,*eachvaluebeing from acomparativelysmallnumberofoccultations. reduce totheHansen-Newcombbasiserrorofmeanlongitudederived based uponobservationsonabout100days. 34 period 1896to1900,theCapevaluesdependuponanaverageofonly Royal Observatory,Greenwich, from1851Januaryto1922December,”Greenwich terms, fromthetruevalueofreductionwhichshouldbeappliedto may appreciablydiffer,onaccountoftheexclusionshort-period mean reductionsfromBrowntoHansen-bTewcomb,asgivenbyBrown, the differencesmaypossiblybeaccountedforongroundthat 24 occultationsperyear,theagreementissatisfactory.Aportionof Observations, 1920,sect.G. * “ComparisonofHansen’s TableswithObservationsoftheMoonmadeat The lastcolumnofthetablegivesdifferencesbetweenCape Year. 1890 1880 1900 92 87 85 86 94 91 89 84 93 88 83 82 81 02 01 96 95 97 98 99 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System , -I -SAI -2-49 - 2‘01 -279 -3*09 -2*03 - i’40 - 2’óI -3*88 - 3‘°4 - 2’24 ~ 3*39 -3*40 - 2‘44±oi8 - I‘29± -2*46 -3*5o “ 3‘45 3*3 -2*48 Occultations. - 206 -3*33 - 2*90 Cape *26 '29 •19 '22 •16 '22 •28 •41 *23 •30 ■23 •19 *33 •IS •23 '09 '20 *22 '12 •l6 'H 17 Correction toMeanLongitude. Mr. H.SpencerJones, with Newcomb’s Hansen’s Tables Eeduction to Corrections. I + 3’4 2'l6 4*22 2*52 2-87 2 ’12 2*19 229 2'I7 2‘53 4-46 2*55 2'50 3'20 4-28 4'39 3'69 4*47 5'i8 5*i4 4*45 574 1 *92 Table VL 2 + 0-15 + 077 - 0*26 + 0'26 + 0-59 - o'i9 - 0-52 + 2-99 + I'OI - o*97 + 0-52 + i'57 + i*68 - o*54 - 070 + 070 + 2*33 + 0-34 + 0-38 + i'99 + i‘81 - I*48 + ‘35 Cape. Correction toHansen- Newcomb Tabular // Positions. Greenwich. + 0*29 + 0*21 - 0*13 + 0*16 - 0*47 + 0*06 + i'95 + I'33 + o'oS + 0-99 + ITS + 0*92 - o*95 + 2*07 - O'lO - o71 + 2-49 + 1*67 + 2*58 + 3'36 + 2*54 + 2*46 (T.C.) O'OO // LXXXV. I, Greenwich. + 0*28 + 0-28 -0*02 + 0*62 + 071 + 0-I7 -073 -1-05 -°'55 -073 - I'36 + 0*38 - 0*46 -I'56 -0'58 -077 + 0'or -0’32 -°'37 -o'i6 -O'l I -o*55 -o'5o Cape- 192 4MNRAS. .85. . .IIS / ,/ is foundtobei25*i.Theresultingvalue ofthesolarparallaxis 8*o5 ±o"*oo5. of theMoon’snodeis+±o"*59. *000000097. of theMoon’sperigeeis—i"*64±o"*33. and comparedwithcorrectionsindicatedby the Greenwichmeridian observations. corrections requiredbytheHansen-Newcomb longitudes arealsogiven, by BrowninAstron.Journ.,No.799,aregivenTableYT.The period 1880to1922havebeendiscussed. sequences ofpositiveandnegative,butnoperiodicityisapparent. Nov. 1924. 7. Thederivedsemi-diameter oftheMoonis932"*7o±o'^oq. 6. Thecoefficientoftheprincipalterm parallacticinequality 4. Thederivedcorrectionfortheepoch1900*0 toBrown’slongitude 5. TheeccentricityoftheMoon’sorbitisfound tobe*054900392± 3. Thederivedcorrectionfortheepochiqoo’o toBrown’slongitude 2. ThecorrectionstothemeanlongitudeofBrown’sTablesasrevised The differencesCapeminusGreenwichshowseveralalternating 1. TheoccultationsobservedattheCapeObservatoryduring 1920 Year. 1910 1903 22 21 19 18 I? 16 15 IS 09 07 08 H 12 11 05 04 06 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System , + 3'58±'23 + 3'65 + 4'49 + 370 +4-53 + 3'27 + 3'48 + 373 + 3-96 + 3'67 + 2-95 + 2-93 +2-23 + 1*03 +2-09 + 1-36 + *020 -o57±o*2 + 0-07‘27 -0*01 Occultations. Cape •3O •26 •23 •32 •28 '23 *22 •23 •23 •22 •27 •30 •23 *21 •21 •20 •27 The Moon’sMeanLongitude, with Newcomb’s Hansen’s Tables Table VI.—continued. Corrections. Reduction to + 4*36 872 876 8*86 9*28 876 8-98 8*63 Summary. 9*17 8-8o 7*61 8-84 6*53 578 6-II 4'84 5*20 5*05 5’4I 5*11 + 12*30 + 12*41 + 12*56 + I3*Sl + 13*25 + 12*25 + 12’28 + 12-36 + 13*13 + 12-51 + 10-56 + 7'87 + 9'46 + 8-34 + 6-56 + 5-87 + S'I2 + 5'3i + S'4° + 379 Cape. Correction toHansen- Newcomb Tabular Positions. Greenwich. 1 + 13*13 + 13*08 + 13*01 +13-21 + 12*52 + 14*02 + 12*66 + 9*89 + ii*71 + 13*64 + 12*55 + 6-53 + .8*40 + 7'97 + 5'98 + 5'56 + 5'55 + 4'07 + 3*! + 5-42 (T.C.) u Greenwich. -0*83 + 073 -0*67 ~o*45 - 0*21 - 0*96 + 0*65 -0*38 + i*o6 -0*51 + 0*67 + 0-37 + i'34 -0*04 + 0*31 +0-58 + 1*05 - 0*24 -0*02 + 0*68 Cape- n 35 192 4MNRAS. .85. . .IIS /, // // //,// , ,/ //, tional to(e-whereedenotestheellipticity oftheEarth’sfigure observations oftheMoon,andmayberegardedassubstantiallycorrect coefficient ofanytermdependinguponthefigure oftheEarth fifty yearsoferrorsintheadoptedmeanmotionsperigeeand century inthemeanmotionofperigeeand21"per epoch 1900maythereforebeattributedtothecumulativeeffectduring and <$>istheratioofcentrifugalforceat theequatortogravityat the equator.hasvalue*003468.Writing i/e=7c,ifCisthe longitude ofperigeethevalue for theweightedmeanepochof1850.Thecorrectionsfound derived fromthediscussionoflongseriesGreenwichmeridian their probableerrors,andcanthereforeberegardedaswellestablished. longitude oftheperigee,forepoch1900,+±o*3andto and forthecorrectiontolongitudeofnode the epoch1900*0.Thecorrectionsarerespectivelyfiveandthreetimes where tdenotescenturiesfrom1900.Theseexpressionsvanishfor the Earth.AchangeinvalueofellipticityfromFaye’s to whichgreatestuncertaintyattachesisthatarisingfromthefigureof node. Ofthevarioustermswhichenterintomeanmotionsone during theyears1880-1922*acorrectionwasderivedtoHansen’s mean motionofthenode. Brown minusHansen,is+8"*ioandofthenode+9*4i. Hansen’s longitudeoftheascendingnode,forsameepoch, 36 Mr.H.SpencerJones,TheMeanMotionslxxxv.i, The MeanMotionsoftheMoovUsPerigeeandNodeEllipticity 1/292*9 toHelmert’svalue1/298*3resultsinachangeof24per 0 1850 andhavevalues-i"*64,+i"*87respectively for1900. — i64±o*3,andtothelongitudeofnodeis+ff'-Syo"'59,for — 3'*28andtothemeanmotionofnode+ 3*74. + ii-28±o*59 < The derivedcorrectiontoBrown’slongitudeofperigeeistherefore The correctiontothemeanmotionof perigeeistherefore We maythereforeassumeforthecorrectionrequiredbyBrown’s 2. Brown’sexpressionsforthelongitudeofperigeeandnodewere For theepoch1900*0differenceoflongitudesperigee, 3. Thetermsdependinguponthefigureof the Eartharepropor- 1. FromthediscussionofCapeobservationsoccultations H.M. Astronomer. of theEarth’sFigure.ByH.SpencerJones,M.A.,B.Sc., © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System * 85,ii. i,,i , + ‘87(2f), - i'*Ó4(i+2f), c c