1958Mnras.118..551L a Short Method for The

1958MNRAS.118..551L / f r less laboriousmethodthatwouldhaveachievedthepredictionwithanaccuracy discovery ofNeptunewhetherinfacttherecouldhavebeendevisedanymuch of itsnumericalapplicationhavebeengiveninfullelsewhere*.Thewhole comparable withthatattainedbyAdams(2|°)andLeVerrier(i°).Sucha method isdescribedinthepresentpaperastoitstheoreticalbasis;details time ofheliocentricconjunction,whichisthefirststeppresentmethod, J. E.Littlewoodfollowingthe1946centenarycelebrationsoforiginal of thepresentinvestigationwasinspiredbydiscussionsbetweenauthorand the orbitofUranus,ha(orSn),Se,andhm;fourelements(coplanar) known planetshavingbeendealtwith,theunknownsthathavetobesolvedfor is dueprincipallytohimf. h' =e'coste7,£'sinm',asisusual,andh! andkthenoccuronlylinearly consist ofthefourcorrectionsnecessarytoexistingelementsassigned and forthistheoneusedbyLeVerrierwillbesufficientJ.Allperturbations discovery, andtheprocedureheredescribedinSection2forestimating in theperturbationslongitudeofUranus§. But a'ande'donotoccurlinearly, pp. 117-134,Methuen,1953. but e'occurstrigonometricallyas.e. 2€, etc.Todetermineafirst while mmultiplieseverythingthroughout.The meandistancea'wasassigned making apparentlynineinall.Thequantitiese'andcanbereplacedby orbit ofNeptune,a',e\e',andm';andalsothemassm'unknownplanet; I, Ch.23,Paris1889;W.M. Smart,CelestialMechanics,p.259,Longmans,1953. (with considerableinaccuracy,itwillberecalled) bymeansofBode’slaw, © Royal Astronomical Society • Provided by theNASA Astrophysics Data System f ForProf.Littlewood’sownaccountoftheproblem see hisAMathematician'sMiscellany, X ForamoredetailedaccountofLeVerrier’smethod see: F.Tisserand,MécaniqueCéleste, * VistasinAstronomy,Vol.Ill,1959,PergamonPress,London. § Tisserand,MécaniqueCéleste, Vol.I,p.378. I. Introduction.—Itisanoutstandingquestionofinterestinrelationtothe It ishelpfulfirsttoexplainbrieflywhytheoriginalmethodsweresoextensive, A SHORTMETHODFORTHEDISCOVERYOFNEPTUNE 0 finding thedistanceinacircularorbitappropriatetobestfitof Neptune thatinvolvesfarlessnumericalcalculation.Thetimeofhelio- would otherwiseenterfororbitsnearthe2:1resonance. equations ofconditionthenumberunknownscanbereducedtothree prediction possiblewithinlessthan15onthebasisofBode’slaw.By centric conjunctioncanbefoundsolelyfromconsiderationsofthe achieved byLeVerrier.Itisshownhowsuitablecombinationofthe observations apredictioncanbemadecomparableinaccuracywiththat discrepancy inthelongitudeofUranus.Thisinformationalonemakes distance, andthesameprocessremovescertainawkwardfeaturesthat (as comparedwitheightintheoriginalmethods)foranyassumedmean The mathematicalbasisisdescribedofamethodforthediscovery 0 J sm (Received 1958September25) R. A.Lyttleton Summary cos 1958MNRAS.118..551L 0 o 16 2 r r took intum40valuesfore'spacedat9intervalscoveringthewholepossible approximation toe',withavieweventuallyhavinglinearequations,LeVerrier equations ofconditionrelatingto1690-1845,nowinsevenvariables.Itwas 552 R.A.Lyttleton finally atasetof“best”elementsforNeptune. then possibletoselectthatvalueofe'givingthebestfitwholeseries range fromo°to360,andforeachvaluesolvedbyleastsquares18consolidated observations. Usingthis,hethenproceededtoimprovethesolutionarrive and thefirststepofmethodtobedescribedinthispaperleadsanequivalent to thefirstorderineccentricity.WenotethatforUranus£=0*047.^tf longitude, andisonethatinvolveslittlecalculationinapplication. of angularmomentum,dh/dt,mustvanishatconjunctionwouldenablethe alternative procedureispossiblethatdependsonlyonconsiderationsofthe determined becauseofitsstrongdependenceontheradiusvectorr.Butan instant tobefoundwereitnotthath=rdv/dtisitselfsufficientlywell but evenmorespecificpieceofinformationconcerningtheunknownplanet, the firstfourtermsareoforderm\whilelasttermisemandso elements areinerrorasaresultofsmallperturbationsdependingonthemassm' namely thetimeofheliocentricconjunctionwithUranus. and ontheright-handside,fromconsiderationofequationsmotion, is muchsmallerthantheothers.Henceifweagreetoneglectthislastterm of theunknownplanet(Sun=1),resultingerrorinvwillbegivenby we canwrite,withanobviousnotation, standard notation, be properlyrepresentablebyEbecausethey containeffectsoftheunknown for anydiscrepancyinlongitudeduetoperturbations,correctthepresentorder. *;(£) =actualheliocentriclongitudedetermined fromobservation,thenthe body m.Writingv(t)=calculatedheliocentric longitudeofUranusinellipseE, (after allowingForallknownperturbations). The observationswillnotinfact well-known discrepanciesinlongitude,which constitutetheobservational have beenusedinapplyingthepresentmethod. LeVerrierhadobservational The valuesofS(t)from1690-1840werethe quantities utilizedbyAdams*and material tobeexplained,aregivenby data from1690-1845. E find, letEdenotethe instantaneous,orosculating,ellipticorbit of Uranus 0 © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 2. Determinationofthetimeconjunction,—Thefactthatratechange The quantitye'iscloselyassociatedwiththelongitudeofunknownplanet, In undisturbedellipticmotionwehavefortheheliocentriclongitude,in Consider nowtheellipticorbitEbestfittingobservationsofUranus Next, denotingbytthe instantofheliocentricconjunctionthatwe wishto 0 àv =tkn+Ac2A£sin(nie—m)2£(AcAm)cos(nt * Coll,Works,I,p.ii,Cambridge, 1896. v =nt+€2esm(nt—'m),*(1) = w'(a+£cosnt-\-dsinnt)(3) 0 + zetkncos{nte—m),(2) 8(t)=v(t)-v{t). (4) E / Vol. ii8 1958MNRAS.118..551L w(t) =v(t)-v(t). given by hy (11)butisnotrequiredforthepresentpurposeoffinding¿. 0 instant ¿,calculatetheseconddifference S(¿o +t)—2S(¿o)S(£—t)foranumberof shown inFig.1.Thequantityp(r)turns conjunction. and theresultingquantitywillbe{approxi- simple rulefordeterminingtheapproxi- out tobemostnearly constant foravalue observational data8(i)(Adams,loc.dt.)is mately) constantifthasbeenselectedat will beconstant. actual value1821-74. by onlyjustoversix monthsfromthe neglecting termsoforderem'thefunction which maybewritten,introducingnewconstants,intheform and so Hence, since8(t)=A,itfollowsthat so that heliocentric conjunction inlongitudediffers of £about1822-3. This timeforthe values ofr,divideeachby(1—cosht), mate timeofconjunction:Selectan #>(*© —r)=w{t+r),orwriting separated onthetwosidesofconjunction.Soifrmeasurestimefromconjunction terms oforderm',thenincalculatingw(f)bothorbitsEandi?canberegarded and inthisbothtermsareoforderm'.Sinceweproposingtoneglectall Then wehavefrom(4) corresponding toi,andVq(í)thelongitudeofUranusinthisinstantaneous as circular,andtherefore«;(£)willtakeequaloppositevaluesattimesequally then W(t)isanoddfunctionofr.TheanalyticalformWgivenbelow 0 0 orbit. Alsolettheperturbationsproducedbym'sincetime£bedenotedw(t\ 0 Since W(t)isanoddfunction,wealsohave 0 0 0 No. 6,1958AshortmethodforthediscoveryofNeptune Q 0 0 0 T © Royal Astronomical Society • Provided by theNASA Astrophysics Data System p(r)= +)-2¿>(*o)S(*ot) The resultofapplyingthisruletothe This resultprovidesthefollowing Now tothefirsttermonrightin(5),form(3)willclearlybeapplicable, 8(i) =m'{a+btbrccos(ntnr)dsinnr)}JT(t), 0 Tic rI S(io +)=^45(—osnT{CösinnT+W(t)}.(7) S(i —)=^4+ß(coswr)—{CrDsinwT+W(t)}. 0 i —COSWT {T) =w{t)w{t+T)(6) WQ (8) 2 at about1822-3. weightings. Foreachtheminimum occurs constancy ofp(r).Theordinaterepresents the rootmeanvalueof(p—~p)plottedatt The differentcurvescorrespond todifferent O weightsincreasingandthen decreasing. = 1813,1816,1819,1822,1825,and1828. + equalweightingsofp(r); Q X weightingsincreasingwith r; Fig. i.—Graphshowingdegreeof 553 (5) 1958MNRAS.118..551L 0 unknown planetbeinginacircularorbit,itissimplematteroncethetime of conjunctionisavailabletocalculateitslongitudeforanyassumedsize triflingly easybythepresentmethod,whetheranybetterestimateofa'/acan orbit. EvenonthecrudeassumptionofBode’slaw,viz.a'la=2,longitude 554 The actualplanetwouldhavebeenwellinsideazodiacalbelt30longbyio° so predictedforthedateofdiscoveryisonlyabout13°behindactualposition. failure ofBode’slaw,whichhaditheldwouldhavemadediscoveryalmost be searchedbyChallis. labour, takingatmostafewhours,butthequestionarises,inviewofserious be reachedwithoutundulyextensivecalculation.Forthispurpose,Bode’s wide centredonthisplace,whichwasthekindofregionAirysuggestedshould law isdroppedandinsteadweassumealwaysacircularorbitfortheunknown body, whichisatleastanequallyvalidassumptiontomakeforanyplanetary ^ ={(1—n'/n),andA®isacertaininfiniteseriesina/a']whilesimilarform orbit. perturbations inlongitudeofUranusaregivenby* wherein therefore nopossibilityofarrivingatlinearizedequations,andthisplainlymust have beenthegreatattractionofBode’slaw. gives G¿.Unlessadefinitenumericalvalueisadoptedfora/a'thereclearly known tcanbemeasuredfromit.Alsosinceeissmall,thetermsinm'e(9) namely m'anda',alsoremovealtogethertheawkwardfeatureotherwise produced bytheappearanceofe'trigonometrically,foronceconjunctionis immediately reducethenumberofunknownsassociatedwithNeptunetotwo, wherein now in m'ecanhoweverreadilybeincludedfor higher accuracy.Accordingly,if say. If,foranyassumed valueofa/a',directcomparison(10) withthe calculated asifUranusitselfmovedinacircular orbit.Theprincipalterms are smallcomparedwiththosesolelyinm', andtheperturbationscanbe and m',alreadyaconsiderable reductiononthenumberusedby Adams and observations ismade,the variablesarenowonlyfiveinnumber,Se, 8n,Sa,8ß, from nowonthetimeismeasuredconjunction, thewholeexpressionfor the discrepanciesinheliocentriclongitudewill beoftheform © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 3. Predictionpurelyfromaknowledgeofconjunction.—Onthebasis The foregoingprocedurecanbecarriedoutwithaminimumofarithmeticál 4. Equationsforfindinganimprovedvalueofa'.—Onthisbasisthen,the The assumptionofacircularorbitandknowledgethetimeconjunction P{t)= —sin+e'<:} P(t)= —m'^Fsini(n—n')t=—m'^FjSiniD, (11) i 2+3({) F =^»)aA+ 8v =?>€-\-t8n+Sasinnt8ßcos4-P(t) (10) 2 ' ^(i-sV) 1 + 2Qsin[z{(w'—w)£e—e}w£€—'nr] 1 1 * Tisserand,Vol.I,p.365. — 00 + GO R. A.Lyttleton 12 Vi -*i) 22V a‘ dA® Vol. ii8 (9) 1958MNRAS.118..551L f /(t+14) +f(t-14) to findwhatvalueofthe ratioofaxesgivesthebestfittoobservations. condition foreachofa shortseriesofvaluesa/a',andsolvebyleast squares shows thevaluesofF^ F,foranappropriatesetofvalues ofa/a\ and therevisedequationsofconditiontake form factors applied,wecanwrite,denotingthenew coefficientsby/, Sft, andm. and these,foreachselecteda/a',involvenow onlythreeunknowns,namelySe, coefficient Fhastobemultipliedby2cos{(i—v)6o°}—1,the to bemultipliedby2Cos{(i—v)i20°}—1,andsoon.Withtheseconversion instance, in P(£)havetobeadjustedastheircoefficients.Forthetermsin2),for later itsimultaneouslyhastheimportanteffectofmakingmethodsafely condition, incidentallyreducing.theirnumbersince,tosomeextent,14years where infactthecoefficientFcanbecomeverylarge. applicable neartheresonancen¡rí=2(forwhicha/a'=0*63approximately) are lostateachendoftherangeavailableobservations.Butaswillbeshown in which,if0=6o°,theright-handsidesvanish. described bytheplanetinalmostexactly14years.Henceifinsteadofaparticular the simpleidentities where tisnowmeasuredinyears,thenthetermsSaandSj8willdisappear. 234 observation (discrepancyinlongitude),f(t)say,weadopt Sa andSjS,aswellacertainotherimportantconsequence.Forthis,wenote ?: So, writingv—n'/n,wehave0=(i—v)6o°,andhencetotransformP(t)the of asimpleregroupingtheequationsconditionthathaseffectremoving feature thataseriesofvaluesaja'mustbetaken,itisconvenienttotakeadvantage 12 2 shall nextexplain. of valuesaja'.ButitispossibletoeliminatethevariablesSaandSjS,aswe Le Verrier,thoughpracticalapplicationwouldrequiresolutionforanumber No. 6,1958AshortmethodforthediscoveryofNeptune555 © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 6. Effectofthetransformation onthecoefficientF.—Thefollowing table The procedureadoptedwasthereforetosetup thesemodifiedequationsof Under thisregrouping,thetermsSe+tbncomeoutsame,but This deviceisofcoursenomorethanaregroupingtheequations Now forUranus277/^=84*015years,andaccordingly6o°oflongitudeis 2 5. Furtherreductionofthenumberunknowns.—Inviewunavoidable sinsin cos sin/^ 0sin/^_m*=:(2cos0-i)(12) Q(t)= P(t+14)+P(t-14.)-P(t)=m'^fisiniD, (13) +) he +thn-fQ{t)=hv(t14)S^(i—hv{t), (14) Z) +0=(ft—tt')(£+14)whereft.14=60° = Z>+(i-ft7ft)6o°. 1958MNRAS.118..551L 556 excessive calculationofthevariousFfordifferenta/a',itwouldobviouslybe n'ln =2,or«/a'=0-63,isapproachedfromeitherside.Theexpression(9), of theresonance.(Allthesecoefficientsarenowadaysavailableintabularform*, or (11),fortheperturbationsinlongitudeisnotofcourseapplicableexact It willbenoticedthatFbecomesverylargeinmagnitudeastheresonance convenient ifwecouldinterpolatewithintheabovetable,butincaseofF resonance, atwhichthecoefficientFwouldbecomeinfinite.Nowtoavoid must supposetheFtobecalculatedaspartofwork.) but astheywerenotsotoAdamsandLeVerrieranycomparisonofmethods coefficient Fhasitsinfinitythroughthefactor1—#¿=2vindenominator. has limittt/vjasv->J,andthecoefficientfpassessmoothlythrough for eachsolutionisshown plottedagainsta'¡ainFig.2,fromwhich itisseen for otherintermediatevaluesofaja'easilypossible, foritismainlythegeneral coefficients shows: resonance valuewithoutsingularity,asthefollowingtableoftransformed that thereisapronounced minimumatjustabouta'¡a=i-6,andthere is obviously •0*50 (Bode’slaw)to0*65. Themeansquarevalueoftheresultingsets ofresiduals been setupandsolved foraseriesofsevendifferentvaluesaja'ranging from form ofthecurverepresentedbyQ{t)asaja' changesthatmatters,andour The smoothtrendofthesecoefficientsmakes sufficientlyaccurateinterpolation aim istofindthatvalueofaja'givingtheclosest fittothe(adjusted)observations. interpolation fromonlyafewvaluesisnotpossiblewithmuchaccuracybecause the generalrunofearliercoefficients. ( But itiseasilyseenthattheexpression 2 2 2 { Another featureofthistableisthatthecoefficients correspondingto 2 and sin6Z)canalsoberoughlyestimatedfor any givenaja'frominspectionof 2 © Royal Astronomical Society • Provided by theNASA Astrophysics Data System But theconversionfactor2cos8—1vanishesatresonancewhen 7. Thesolutiongivingthebestfit.—Theequations ofcondition(14)have 0-70 o*6o 0*65 o*55 o*5o aja 0-70 0*65 0-50 aja 0*60 0-55 * YaleTrans.,VIandVII, Cambridge, 1932. - 9-911 -19-944 -13-718 - 7-432 - 5738 Coefficients oftermsinP(t)=m'^FjsiniD -16-165 —10-382 - 3-208 - 6-877 - 4-664 Coefficients oftermsinQ(t)=rn'TifysiniD sin D sin D 2 COs{(l—v)l20°}I R. A.Lyttleton -18-655 - 38-422 Table II Table I 2V— I 15-712 sin zD sin 2jD 3-380 2- 0 52 0-814 5-471 3- 2 99 1-429 1*290 sin 3Z) 5-461 0- 672 sin 3Z) 0-299 1- 692 0-141 0- 8 21 0-266 0-469 2- 558 1- 4 36 -o*345 — 1-284 — 0-087 -0-666 — 0-176 sin 4Z) 0-967 0-366 sin 4Z) 0-154 o-o68 0-031 Vol. n8 1958MNRAS.118..551L 0 0 just underi°behind. the dateofdiscoveryisreadilyfoundto compared withthetruevalueof30-07. little pointinendeavouringtoimproveontheroundfigureofi-6forratio No. 6,1958AshortmethodforthediscoveryofNeptune557 gives acloservalueofthemass is m'/O=1/25000inroundfigures. axes. Thiscorrespondstoameandistancefortheunknownplanetof30-71a.u. obtained by.thevariousmethods,adoptingbetteroftwosolutionsgiven is thereforeabouti°aheadofthetrue of 328‘4.Theerrorthepresentmethod be 329*4comparedwiththeactualvalue discovery (equinox1950) the truevalue,presentmethodalso position, whereasthatofLeVerrierwas that someelementofgoodfortuneenteredintothepredictionstomakethem by Adams,areasfollows. unknown bodythanwasobtainedbyeither Longitude at330°*93^7°'4 have beenso)sincetheerrorchangessignat about 1842.Bothsolutionsplace as goodtheywere.LeVerrier’ssolutionappearssuperiortothatofAdams, above valuesandtheactuallongitudeofNeptune.Itisevidentfromcurves at anyothertimethan1846-73,theofdiscovery.Thethreecurves Le VerrierorAdams.Thevaluefound their orbitswereintendedtofittheobservations priorto1840inAdams’case, more slowlythantheactualoneandtherefore musteventuallycoincidewithit so thatwitha'¡aassumednearlyequalto2, the hypotheticalplanetwillmove earlier thandiscovery. This showsthattheerrorsimpliedbyAdams’ solution the unknownplanetaheadoftrueposition intheearlierpartofrange, and hispredictionwouldhavebeenbetter still atasomewhatearlierdate Fig. 3showthedifferencebetweenlongitudecalculatedinturnfrom arrived atbythepresent methodshowsanaccuracycomparablewith thatof are onthewholeroughly twiceasgreatthoseofLeVerrier’s. (A possible in longitude.ForAdams’solutionthisdoesnot occurtillaslate1856.But reason forthisisdiscussed below.)Ontheotherhandcircular solution and 1845inLeVerrier’s,socomparisoncan onlyproperlybemadefortimes Le Verrier’sorbit,but persisting overamuchlongerrangeintime. The three (supposing thesamematerialtohavebeenavailable, whichcouldnotinfact © Royal Astronomical Society • Provided by theNASA Astrophysics Data System The longitudetherebypredictedfor Consistently withtheclosenessofa'to 8. Predictionatadateotherthan1846.—Theorbitalelements(andmass) It isasimplemattertofindthelongitudethatthesesolutionswouldpredict Qlm 66709350 o TOT 299285° a 37-2536-15 e 0*12060-1076 Adams LeVerrier Table III computed foranumberofvaluesa'ja. The bestfitisseentooccuratabouta'la=i-6. Fig. 2.—Curveofmeansquareresiduals Present method 2 25500 39°*4 30-71 o Neptune o 328-4 30-07 44° 19300 0-0086 1958MNRAS.118..551L 2 time seemssomethingofapuzzlewhentheobservationsutilizedextendedback feature thattheirdifferenceinlongitudefromthetruepositionisatitsleast strong influence,butinanyeventitwouldseemthatsomeelementofgood fairly nearthetimeofprediction—sixyearsearlierforLeVerrierandnine some 150years.Itmaywellbethatthelatestobservationswereofparticularly later forAdams.Whythisplaceof“bestfit”shouldhavecomenearthecrucial curves showthat±i°accuracywouldbeobtainedbyAdams’solutiononly 558 R,A.Lyttleton fortune attendedthedegreeofclosenessactualprediction. for about5years,byLeVerrier’ssolution12whilethepresent solution itwouldobtainforabout24years. the correctionsappliedtoelementsof orbit,ismoreconvergentthan that whichgivesthecorrectionoftruelongitude, andthesamethingistrue him attheveryoutsetofhiswork.Adamssays(loe.dt.,I,p.11).“Itiseasily seems veryprobablytoarisefromwhatappearsbeaninvalidstepmadeby the meanlongitudeinanundisturbedorbit virtueoftheequation for theperturbationsofmeanlongitude,as comparedwiththoseofthetrue. seen thattheseriesexpressingcorrections ofmeanlongitudeinterms such arelationasthathereclaimedsubsists betweenthetruelongitudeand The correctionsfoundabovewereaccordingly convertedintocorrectionsof mean longitudebymultiplyingeachofthem thefactorr¡ab”Nowwhereas there wouldseemno validityinsupposingthatperturbed motion the which maybewritten variation intruelongitude AvandthatinmeanlongitudeAZaresimilarly related. © Royal Astronomical Society • Provided by theNASA Astrophysics Data System The orbitsarrivedatbyAdamsandLeVerrierbothexhibitthecurious 9. Otherpoints.—TheweakeragreementofAdams’orbitwiththeactualone Fig. 3.—ShowingtheamountsbywhichorbitsofAdamsLeVerrierandcircular y orbit solutiondifferfromthetruepositionofNeptune. 2 (rlab) dv—ndt=dl y r2 “~T —h=nab, dt Vol. ii8 (iS) 1958MNRAS.118..551L 0 0 impression thatthiscouldbeaconsiderabledefectwhenitcomestoapphcation. and thattheratioofthesecannotreducetoaformindependentvariations and soforavariationA,inwhichalltheelementsarechanged, of theelements,suchas(15),seemsevident.Butactualcaseismorecomplex No. 6,1958AshortmethodforthediscoveryofNeptune559 three termsofthisfunctionareadequate.Independentpreliminarydetermin- No suchrelationseemsknownincelestialmechanics.Wehaveanycasethat ation oftheinstantconjunctionisnotincludedinhismethod,andonegets would leadtoalongitudeatthetimeofdiscoverysome27aheadactual numerical valueshegivesforseveralofthevariouscoefficients.Atallevents, his paper,sincethepresentauthorhasnotfounditpossibletorecover of theunknownplanet,thoughheappliesmethodonbasisthatfirst some simplecriterionforthepredictionofanunknownplanet,andmethod The validityofthisstephasalsobeenqueriedbyE.W.Brown*. discussion showshowthegeneraltrendofthisfunction(withchanginga/a'), still becausethediscrepanciesinvolvealsocorrectionstoorbitofUranus. occurring closeto1840,whichinfactisalmost90fromthetrueposition,and There isdifficultyinbeingcertain,however,fromtheactualworkgiven the principalperturbationsinlongitude,naturallyenough,ashere.Brown’s he proposedthereinrestsonconsiderationoftheseries—m'^F^iniDfor place ofNeptune,evenassumingacircularorbitatexactlytherightdistance. rather thanitsprecisevalueatallstages,issufficienttoestimatetheposition those ofAdamsandLeVerriermightverywellrequireconsiderablenumerical Brown himselfreachestheconclusionthathismethodpredictsconjunctionas knowledge ofconjunction,toreachapredictionaccuracycomparablewith Nevertheless, thegeneraltheorybehindBrown’smethodwouldappeartobe labour. sound, though,handicappedasitwouldbeindetailedapplicationbylackof © Royal Astronomical Society • Provided by theNASA Astrophysics Data System 1958 September19. St. John'sCollege, In hispaperE.W.Brownalsowasconcernedwiththepossibilityoffinding Cambridge : J (r-z&h—zhr-sArfdt,A/=¿Aw+A€, * M.N.,92,93,1931.SeeSection14. 2 v— Jhr~dt,l=nt+€ y