1922MNRAS..82..149G jacobians takearelativelysimpleform.Inthispaperonlytwoofthe Jan. 1922.Long-PeriodInequalitiesinMovementsofAsteroids.149 appearance ofMoulton’sPeriodicOrbits.Thedetailstheexist- families ofperiodicorbitstreatedbyBuckarediscussed.Afull in 1912orearlier,theequationsofmotionaretransformedand with periodnearlyequalto—mayalsoberegardedasperiodicsolution» may beofinterest.InBuck’spaper,whichapparentlywascompleted prove theexistenceofperiodicsolutionswithperiod^whicharenot with periodnearlyequalto—-. account oftheotherfamilies,andalsoactualdevelopmentinseries at thesametimeperiodicwithperiodnearlyequalto—. of theperiodicsolutions,isgiveninBack’spaper. ence proofsaredifferentfromthoseofBuck,anditishopedthatthey by LaplaceandLeVerrier,theequationsofmotionareintegrateda Tisserand reducesthedisturbingfunctionto the principalnon-periodic tedious, asitwouldbenecessarytoproceed toalargernumberof for alltime.Generallyspeaking,thefirsttwoapproximationswillbe method ofsuccessiveapproximationwithregardtothemasses.The double thatofJupiterhasattractedalarge amount ofattention.It approximations. concerned arenearlycommensurable,andinthis caseperiodictermswith solutions inseriesthusobtainedwillbeconvergentforsufficientlysmall is sketchedbyTisserandint.iv.ch.xv.of hisMécaniqueCéleste. succeeding approximations.Theordinaryprocess wouldthenbevery small denominators(andoflongperiod)will appearinthefirstand sufficient, butitmayhappenthatthemeanmotions ofsomethebodies values ofthetime,butthereisnoguaranteethattheywillremainvalid successive approximations withrespecttothemassofJupiter. and theprincipallong-period term,andendeavourstointegrate the On Long-PeriodInequalitiesintheMovementsofAsteroidsivhoseMean equations withthesimplified disturbingfunctionwithoutappealing to In thecase=aninteger~isamultipleofandsolutions But wehavenotbeenable(inthecasewhen^isaninteger)to Note.—The aboveworkwascompletedin1920November,beforethe In theordinarytheoryofmovementsplanetsasdeveloped The problempresentedbyasteroidswhosemean motionsarenearly Isaac NewtonStudentintheUniversityofCambridge. Motions arenearlyhalfthatofMars.ByWM.H.Greaves,B.A., t © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System X a 21 (Communicated byProfessorH.F.Baker.) • 2TT Ai A . A 2 A x A 2 ISO Mr. W. M. H. Greaves, On Long-Period lxxxii, 3,

In this paper we shall be concerned with asteroids whose mean motions are nearly half the mean motion of Mars. The existence of a group of such asteroids is shown by the following table :— Asteroid. Mean Daily Motion.

No. 44 Nysa 9417363 67 Asia 942-3560 142 Polana 943'5246 Three values are taken from the 182 Elsa 944*5132 Connaissance de Temps, 1915. 265 Anna 942-0467 622 Esther 944-890 Mean motion of Mars = 1886-5183 (Le Terrier) = 2X943"*2592. At first sight one is tempted to cavil at Tisserand’s procedure of integrating the equations of motion with the disturbing function reduced to a few terms. It is not my intention to dwell on this point, but it is perhaps worth while to point out that the method is formally equivalent to picking out various terms which would arise at different stages in the ordinary method with the complete disturbing function and grouping them together. For if we integrate by successive approximations with the reduced disturbing function, every term we obtain in the result wilJ appear (among others) when we integrate by successive approximations with the complete disturbing function. And if when using the reduced disturbing function we do not integrate by successive approximations, but by some other method, we shall have obtained a way of grouping all these terms together. The method of reducing the disturbing function to a few terms and then integrating was first used by Laplace in his discussion of the secular inequalities of the major planets. A sketch of this work, which was completed by Le Yerrier, is given by Tisserand {Mécanique Céleste, t. i. ch. xxvi.). The same method will be used in this paper, the object of which is to get a general idea of the nature and magnitude of the long-period inequalities due to the action of Mars in the movements of such asteroids .as those mentioned above, and to compare them with the long-period inequalities due to the effect of Jupiter (the so-called u secular inequali- ties ”). Precise results are not aimed at, and the problem is simplified at the outset by supposing the motion to be confined to one plane. For illustrative numerical purposes the four asteroids 44, 67, 142, and 182 will be taken. The inclinations of these are all comparatively small, and the results obtained should give a fairly good idea of the actual phenomena. § i. The Simplified Disturbing Function, ATe shall retain in the disturbing function the principal long-period terms due to the effect of Mars and the principal non-periodic terms due to the effect of Jupiter.

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 1922MNRAS..82..149G 2 2 usual notation,andputting_=a,t* be calculatedforthesemi-majoraxesaanda!(a*J sion forEtheyaretobecalculatedaanda'(amass ofthesunbeingunity.Weshalluseastronomicalunits.Letk 3 (i) Transforming theexpressionsforEjandRweobtain,using In theexpressionsforEjandRLeVerriercoefficientsAareto Furthermore, ontransformingtheexpressionfor E,t Let m'denotethemassofMarsandm"Jupiter, Let a",e",GT",e"denotetheelementsofJupitersupposedconstant. Let a,e',ïït',¿denotetheelementsofMarssupposedconstant. Let L=meanlongitudeofasteroidintheosculatingellipse. We thenhave 2 2s 3 Let a,e,ïïT,€denotetheosculatingellipticelementsofasteroidat 2,/e// © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System L' =meanlongitudeofMars. R =/} 12 2(1) KmY d&i\. / TT/ 2a 2 a (1) + ^{A-2AW}cos(CTor"). 2 CL M, =£A<°>1 N =iB<«[ 0 2 () (3) (5) (4) 1922MNRAS..82..149G these equationsbecome, inequalities oflongperiod,thefirstdependingoneccentricity where asteroid, theseconddependingoneccentricityofMars,and and integratetheequationsofmotionineachcase,thusobtainingthree where where nisthe“meanmotion”ofasteroid. third arisingoutofthesecularactionJupiter. We shallreducethedisturbingfunctiontoR,andsuccessively, 152 Mr.W.M.H.Greaves,OnLong-Periodlxxxti.3, 23 I =meananomalyinosculatingellipse. We alsohavetherelations Putting g=VS-rít—¿T3L',n'beingthemeanmotionofMars, The equationsofmotiontheasteroidare We reducethedisturbingfunctiontoportionR. We write Write x § 2.InequalitiesdependingontheEccentricityofAsteroid. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System E =F+«'G—+—eQcos(2i£i).(7) cZG_0F _0F' dt 0ÏÏT0G ¿L_aF Ja *•2 K^m‘ 2 a 2a F =—-+R r eQ 1 ia (7) (8) (6) 1922MNRAS..82..149G 2 functions ofLandG.Hencebyusing(9)itispossibletofindthemas functions ofLonly,andtheproblemisreducedtoinversion e canbeexpandedinpowersofxbyusing(9).Wefind where Jan. 1922.InequalitiesintheMovementsofAsteroids..153 A2_(C-B) =-^[m'{eQ+2eE^+. .} e, GT,Z,g,0,andQ. the integral(11). where Cisaconstant. We alsohavetheintegral Hence 1 Then from(6) Expanding inpowersofxweobtain, Write To dothis,letaeg0Qbetheinitialvaluesofa,n, A andBaregivenasfunctionsofaeare,therefore, We nowget vVx © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 23 C —B=[jft'ejQjCos-(Di*+Dxo(a;))] 2 Ej =-+QiVi 2i 4«i 4^i - A= e =ej+aVx-ë^WT^ë,i}o(x). „4 dL dt /dL\2 [U say * n Khri 2 a 2 a, 2dt = 0a Ji 11 «1 - 2Asin0 2 F =Acos0+BC. 2 (ejQj +E^o^)] =4Asin0 =4^A-CB ) denotesthe initial valueof

2 L—2G =aconstant. \/A-(GB) ' a =(i+^). 1 2 - {meQcos^(D^+ D^+. l1 2 ± dh d£x dt 26, — Asin0. 0Q 0a 1U1 V3«A IS 2 i •)}} (!3) (12) (n) (16) (s) (h) (10) (9> 1922MNRAS..82..149G 2 zn^dt ^Sm'D^Q,sin'"-D/jfSm-D^Q^cosH^+D^ s 2 shall neglecttermsoforderx,m'x,and giving D^. so that where Hence from(1),usingL=K'\/aK'\/^(i+oc),weobtain 154 2n^dt = Write We nowhaveapproximately For theasteroidswithwhichwearedealingDwillbesmall.We We canwrite We nowhavetwocasestoconsider. We thenfind Notice thatDispositive. 2 2 Writing ^=2/-^-,thisgives x Case A:—Suppose 2 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System U =m'{2eQcos|^+E^+.}—(Djir4-Dæ Ui =m'{2eiQisinJ6>i+.}(DrD^ 21i lt2 2 fyim'e-fy-L sincos^—Da;1 2 Mr. W.M'H.Greaves,OnLong-Periodlxxxii.3, cos / /2 2 0 meQcos—-D; 1)12 El =2meQsin^0-p+Do; U =2m'eQcos—D^x-Da: 12 213 2 XJn P2 2 2 P2 4D 2 A —C+B=2Asin-—U 2 2 A +C-B=2Acos-—U, ~4D 4D9 2 2 , Dj-Sm'D^jQjcos 2 2 2 _Dj-8m'D2eQsin 1 Dj >Sm'D^Qjsin 2ndt = Y ^'(eiQi +E^d-.•.) U =UjUg. V w(i+x) ±dy + dx ± dx ¿ - l)x. Q 2 2a-, 2 2a 1 4D 2 V (,9) >} • (18) • (*7) • (20) • (23) • (22) 1 (2 1922MNRAS..82..149G where kisthecompleteellipticintegral Hotice that0cantakeallvalues. and ifRistherangeofvariationaweget Jan. 1922.InequalitiesintheMovementsofAsteroids.155 c beingaconstant. so that,if that æ=oinitially.IfDjispositivewetakethe+sign,andifD a isthengivenby=(1+x). (21) becomes negative wetakethe-sign. given by where Hence a 1 1 The rangeofvariationxisdenotedbyandgiven The ambiguityofsignin(27)mustbesettledbytheconsideration æ isaperiodicfunctionoft.TheperiodT given by And theellipticfunctiondnuistobeformedwithmodulusk This givesonintegration Substituting forxintermsofyandsimplifying,weget With ourmethodsofapproximation(20)becomes © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System .’. 7T—0=±2$ cos 0= 2W2 l , d^ V(?/-p)(/y) 12 R* =p'(-VT^k*)=/-p, = amucos(77-—0)2<ï>. = i2am(2p7iD(+c)) 12 ±p'dn{2p'nfDf(t +c)}..(27) 2 2 ^ ~Jo'x/1—ksin ra'âfQi cos0-DflDæ 2, 12 cos 6—isriui, u =2p'nfD(t+c).•(25) a2 2 I^a~i(p ~~P)*•(^) y=±pdnu ..(24) _ K T rz d<í> ñ p'n^Dz m'ejQi 2 P /yyi 2Ó () 2 2 (S®) (9) 1922MNRAS..82..149G x =o)orfromthefactthatwhen¿o,Q0 156 p beinggivenby(23). also B=rangeofvariationa=zap either ofthecasesAorBbyputtingft=1.In thiscasewehave Put we have V In thiscase0cannottakeallvalues,butlibratesbetweenthevalues a 18o° ±x",where 2/ The constantccanbedeterminedfrgmthefactt^iat,whent=o The caseinwhichD=8?7ieQsin\0 can bededucedfrom x and6areperiodicfunctionsofperiodT,where In thiscasewefind Leading to Putting ft, themodulusofellipticfunction,isgivenby Case B:— y We get x21X © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System dnu =cnusechu J Mr. W.M.H.Greaves,OnLong-Periodlxxxii.3, u^2p'n-D{trc) 0=i 8o°+2^. x —±psechu [2 = amugduulogtan^—-f- e 2ndt= V2 ^vWWW^7) 4D 2 ,2 ,, _8mDeQsin\0-Df x= ±pcnu 2 21 D< sin\0 xV K /2 =* 2?ZD'\/ (^+^")(¿c T 2, 12 7 itdy cos 0—i2dnu. dnu —cosx 2 2//2 r-1 0 ft: WjDgV p'+p X =sinft. $ —180±2x 2D 2 2 p' +p" _ ü 2D 0 V '2 • (38) • (37) 1 (33) (3^) (3) (36) (3+) (35) 1922MNRAS..82..149G Jan. 1922.InequalitiesintheMovementsofAsteroids. so that K*m E! beinggivenbyequation(2). neglecting termsoforderm'x,weobtain through thequantitynoccurringinL.* where x and0intermsofthetime. where where, asusual, and 2a Here informingthederivativeweonlymakeavaryfactor We have Performing thedifferentiations,expandinginpowersofxand Case A.—Wehavefrom(30) L istobeobtainedfrom(39),(40),(43),-andtheexpressionsfor From 8wededuce We havetoseparatethetwocasesAandBasbefore. eQ ofE,andnotintheargument2L—L'67,whereaisinvolved 1 © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System * Tisserand,MécaniqueCéleste, t.i.ch.xi.pp.191to193. 2 cos zQdu=u-2snudu € 1 1 .A/JT72I-VI9^1 dt nada?ede = na cos Odt=~ de v *=Im'QjVi-e^——?i-coç6 dt 2 n =(i+x)-% E(w) =/dnudu. 1 = ^(jU,+v)cos0 Inequalities inMeanLongitude. =2 =ni(f-§% +^x.• n + dL de dt ~ 2PXD 2 L =^ndt-J- 0R, COS 2$>du e \ (39) IS? (4^) (41) (40) (43) 1922MNRAS..82..149G 1 1 function ofuwithperiod2K. that SL=0when¿o, J ^ where EandKarethecompleteellipticintegrals,Z{u)beingaperiodic 158 Mr.W.M.H.Greaves,OnLong-Periodlxxxii.3, We cannowobtainLfrom(39),using(45)and(46). X(u) isthenaperiodic function ofuwithperiod2K., where «ïqand^aretheinitialvaluesof<Ê> ¡ndt =nJ-£nA—- f v,,orEjí from (40) Again, We obtainfrom(43)afterintegration We thenobtain Writing L=w^+€SLwehave,onadjustingtheconstantsso We introducethefunctionZ(w)definedbyequation Introduce thefunctionX(u)definedby 1i © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System +ti ^[-^-Z(m)++{v'JrTTTg}]•(46) 12 2 fosM =(±- 8 2??DlK4D^J CK 8L,2<)z> 12 -È[&'-w]*’' 2 fdnudti —&andJdnudu=E(w). J i J-82 +lt+lp ^Td Kr 2 + (m4-v)m|piji. 1 , on^Di,.15f'2^lb\ ,,,< Í2K-E}, E(w) =Z(w)+^i, 2 K-E X(u) =<&- -IV-, ÏK‘ ITU + aconstant(45) Z() M (44) (47) (48) 1922MNRAS..82..149G are theirinitialvalues. where uandxaregivenbyequations(33) and(35)uXi procedure,* x the principalperiodicterm+_?_À(w)inSL, where 11isgivenby term andtheprincipalsecularterm,weget SL=+ 3_ Jan. 1922.InequalitiesintheMovementsofAsteroids. {a(m)A(Wi)} Cask B.—Inthiscaseweobtain,usingasimilarmethodof Then itiseasilyfoundthat(usingtherelation—amu—dnu) Let Adenotetheamplitude(i.e.halftotalrangeofvariation) If weapproximatestillfurtherandonlyretaintheprincipalperiodic We thenobtainfrom(47) zZ +í,Wlí+p («„)-M«,>)-w}<<*>K» + 37T,_IíD,7T/ Ííll5¡K^ T © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 4^2 D, asn * Notethat^cnudu—^where dnu=cosX)i(35). zi 2 + ^gÎV+(/0[-f]^j + L^-Jv^Secular , 3^]/Ï5'rIK-EI\ 2 V+2K r 2 K4D 2 P2 '2 ElJ +t 4*^2 A =A(w') dnu =^ Secular terms. 2K + •(49') du terms. Periodic terms. Periodic • (50 • (50) • (sO terms. • (49) I59 i6o Mr, W. M. H, Greaves, On Long-Period lxxxii. 3,

Note that the ambiguities of sign in (49) and (52) correspond to the ambiguities in (27) and (32), which are to be settled by the sign of Dp If D1 is positive we take the upper sign, and if D2 is negative we take the lower. Eetaining the principal terms in (52), we obtain

8L = + -4r(x - Xi) + fëîV (52') 4D D„ We also have -1 A = sin 1c . (53) 4D2 where as before A is the amplitude of the principal periodic term in 8L. Note that in the limiting case, when It— i,

A = -3- x 90° = 90° approx. 4D2

Inequalities in Eccentricity.

On page 153 we obtained the relation

= e^1 x sj\ - ej2{ J1 - - %} + o(x2).

From this e can be obtained from (27) for Case A, and from (3 2) for Case B. Let B*2 denote the range of variation of e2. It is easily seen that for Case A . . (54) For Case B R«2 = zp' Ji - n/I - e/ - |} . . . (55)

§ 3. Inequalities depending on the Eccentricity of Mars,

We reduce the disturbing function to the portion R2. The equations of motion are as before, R2 being substituted for Rr We write and the equations become

¿L^aF = dt dl dt dg dt 0L dt 8G’ where

F = — + \nfK Ja(i — e2) — e'S cos 2(1 + g), 2 a 2 a We write 1 +g = 0, F = — A cos 20 + B = C, where C is a constant.

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System 1922MNRAS..82..149G where risanyinteger. Notice that9cantakeallvalues. Dj isnegative. where theuppersignistohetakenifDj positive,andthelowerif and weget results. Themethodofprocedureisthesame. where 2ftD2 =le . _cos0+D^xDx l2 2dt —■ 2 I*! >Sra'D^Sj^cos0 k2I V 67 =aconstant. ± dy / ='-72 x= ±qdnu— w =2gftD(¿+C) 12 x =y- 0 =T7T±<î>, 2 q + d\é 5 7^fW 1 2P constant, 0 2 D 0 2 4P2 (59) (58) (57) 161 1922MNRAS..82..149G and weget the upperorlowersignbeingtakenaccordingasDispositive negative. Wealsohave where X(u)isdefinedasbeforeand/“*» _ 3 4D, 8L= 1a2 + 1 if5g' E15D15^/Pi 2 t +2+? q -A(w') wherednu= ^ _8?^'De'Scos0—D^ 2 //221l L 8K2Di6_, 4Ö? 32 2 k ir= ±gc?iu ,_ sin0+D^ (9 =r7T±^wherecItiu =cosy. 2l gWiD 2 2,/ 2 271^1)^ q'+g"(tc), g +q* 2 D<8m'D/S cos0 1V g '2 2 4^2 2D0 ÏK 7T (60') (61) (62) (60) 1922MNRAS..82..149G -1 from thiscause. settled bytherulealreadygiven. so thattherearenolong-periodinequalities inthemajoraxisarising Amplitude oflibration0=sink. Jan. 1922.InequalitiesintheMovementsofAsteroids. , SL=+.-^(x-)if[Z(m)ZK)]M^(XXi) Xl The ambiguitiesofsignintheexpressionsforxandSLaretobe The equationsofmotionbecomeapproximately * Put We reducethedisturbingfunctiontoportionR. We haveatonce Retaining onlytheprincipalterms,weget 8L=:p(xXi)+Wií 3 R =zg'Vi-iVi e2 R =2a^ -1 ¿¡-S • a1 T- A =amplitudeofprincipalperiodictermin-SL=+sink § 4.SecularInequalitiesarisingfromtheActionofJupiter. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 2 AW+g"’ iae2 + 3^1^+-^ “L ’ - [Z()-Z(] MWl) 4^2 'i ^2 +D 2K , ,YE1 * Tisserand,Mécanique Céleste^t.i.p.407. 1 1 dt nadl dh _i0R^ 3 dt nodh- Jl_ _J_9R 3 e cosTJ5—l e sin6T=h i ——- n= const. a =const. 2 k K i K-E 2 32 -^2 15 D! 2 1 16 Do Do n,t 41^2 (63') (6S) (64) (63) 163 1922MNRAS..82..149G 2 so thatweobtaintheequations also e=A+Z;andfrom(69)weeasilyfind where ÄandÀareconstants. giving where 104 Mr.Tf.M.H.Greaves,OnLong-Period e isnowknownfrom(69),sothat(72)canbe integrated. where Putting Z=oin(69)weobtain For cwehavetheapproximateequation* We easilyfind The resultis H = © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 2 na * Tisserand,MécaniqueCéleste, t.i.ch.xxvi.p.428. 2 2 B> =rangeofvariatione2^ e Ä =e,+^e"-^V'cos(67!67") h =Asin(ßt+À)Le" Z =Äcos(ßt+\)L"67'' € =€q+HzAsin(ßtÀ—^)• e 2 dt _z0Pe0R dt nadamade 3+3 1 2 d na dl ??a 0/?/ ± Jk=ßi-r^ y §.ß-y,' i 0R I" K /3 =i!¿N n 8 2 =-y/i' 2 K^'p na^ 2a- dm da ß 2 LXXXII. 3, 2e a t/n • (67) • (66) • (70) • (69) • (68) • (73) • (72) ■ (71) da J 1922MNRAS..82..149G inequalities. other threeasteroidsfallunderCaseAasregards eachofthefirsttwo long-period inequalitiesofthethreekindsdiscussed beinginvestigated. case oflibration)asregardsboththefirst twoinequalities.The des Temps,1915:— 182 wereselected. also, ifTbetheperiod, and (67), € beingaconstant. Jan. 1922.InequalitiesintheMovementsofAsteroids.165 and 0 142 Polana 182 Elsa In applyingthetheory oftheprevioussections,epochoscula- It isfoundthattheasteroid142Polanafalls within theCaseB(the The inclinationswillbeneglectedandthepreceding workapplied,the- Asteroid. i, a?,and&arereferredtomeaneclipticequinox for 1920'o. a =semi-axismajor.ßlongitudeofascending node. The followingdataaretakenfromthesupplementtoConnaissance 44 Njsa 67 Asia For illustrativenumericalpurposestheasteroids44,67,142,and ^'=: inclination.M=meananomalyatepoch ofosculation. 6 =sin<£>eccentricity.o>angulardistancefrom nodetoperihelion. We findonreduction,substitutingfory,Æ,P,andNfrom(5} Now ifLbethemeanlongitude=+cCj8Lsay. So thatAistheamplitudeofprincipalperiodictermin8L. So thatSL=€-e-L+{nebeingtheinitialvalueofe. 0 1 182 Elsa 142 Polana 67 Asia 44 Nysa 2/y ^ Km"Ae"./TP)/Epoch ofOsculation. 1891 Apr.i-o 1897 Mar.20*0 1897 Pec.5*o 1896 Dec.io’o 2 2 3 5 AAe+ 2Aea =lW>Ta(w>) § 5.NumericalApplications. (2) 41 57*2 H 3i*3 10 7-3 29 6*4 . ,T„B®7id/B\ T = 0-3838611 0-3831990 0*3835023. 0-3840515 Log a. 292 203 106 13t J 27T 53 6-8 ii 487 30 167 12 54*9 201 2125*2 2II 1322*8 102 5220*2 101 307-1 M. 289 308 340 103 56 33 18 19 10 5051*9 10 4754-5 7 44loó 8 4810*9 36*2 537 53'9 547 0. (75) (74> 1922MNRAS..82..149G Long-period InequalityI.(duetotheactionofMars,andarisingfromatermlong the tablesinConnaissancedesTemps,andadoptedforpurposes In thecomputationvaluesofLeVerriercoefficientswerederived of thecomputation. to beconstantinthetheory.Thefollowingvalueswerederivedfrom values oftheelementsa,e,Z3forJupiterandMarshavebeenassumed Le VerrierandGaillot,asquotedintheConnaissancedesTemps.The at theoriginoftimeforeachasteroidwerederivedfromtables tion wastakenastheoriginoftimeforeachasteroidandquantities T, A,R«,andRcomputedforeachofthethreeinequalitiesconsidered. by interpolationfromRunkle’stables.ThemeanlongitudesofMars i66 Mr.W.M.H.Greaves,OnLong-Periodlxxxii.3, e2 Long-period InequalityII.(duetotheactionofMars, andarisingfromatermof The followingvaluesofthemasseswereused:— The resultsareasfollows Asteroid. Asteroid. For Jupiter— For Mars— period inthedisturbingfunctioninvolvingeccentricityofasteroid). lon^ periodinthedisturbingfunctioninvolvingeccentricity ofMars). 142 142 182 182 44 67 44 67 , For Mars,m=Jupiter,m"r^-run;. log a"=0*716217. log a'=oi8293. © Royal Astronomical Society•Provided bythe NASAAstrophysics Data System 9 0 £T' =33429'51",beingthevaluefor1900Jan.i*oreduced anl0 £?"= 1259'22"*10,beingthevaluefor1895Jan.i*oreduced e" =0*04832654,beingthevaluefor1895J-' e'= 0*0933088,beingthevaluefor190*0Jan.i*o. T (inJulian T (inJulian 9030V Years). 2070-6 Years). 80657 1483*0 IÍ98-4 I95°’4 1368*1 1162*9 to meaneclipticandequinoxfor1920Jan.i*o. to meaneclipticandequinoxfor1920Jan.i*o. 39 57 61 580*003734 2 22 0 41 3 420*000380 I 130*000124 I 380*000254 1 25 o / A. A. R (inAstronomical E, (inAstronomical a a 0001960 0*000257 0*000201 0-000125 Units). Units). 0*00019722 0*00002488 0*00000273 0*00000057 0-0000394I 0*00005037 0*00000696 0*00000184 It2. e U. t2 Jan. 1922. Inequalities in the Movements of Asteroids. 167

Long-period Inequality III. (due to the action of Jupiter). T (in Julian Asteroid. Years). A. Res.

44 32938 35 30 o o 00871460 67 32985 31 29 o 0*00772540 182 33119 30 51 o 0*00920888 142 33057 37 34 o 0*00757000 In these tables— T = period of inequality. A = amplitude (i.e. half the total range of variation) of the principal periodic term in the expression for the perturbation in mean longitude.

Ra = range of variation of A. 2 Rea = range of variation of e .

Conclusion. It has been pointed out that the process of reducing the disturbing function to one or more terms and then integrating is equivalent to picking out certain terms which arise at various stages in the ordinary method of successive approximations with regard to the disturbing masses and grouping them together. Now the infinite series obtained by the ordinary methods are con- vergent for a certain range of values of the time ¿0<¿

only valid for ¿0<¿

© Royal Astronomical Society • Provided by the NASA Astrophysics Data System i68 Prof. G. E. Hale, LXXXII. 3,

of Mars. One of these terms involves the eccentricity of the asteroid as a factor, and the other involves the eccentricity of Mars. 4. Long-period inequalities in the major axes arise from the cause mentioned in 1. 5. There are no long-period inequalities in the major axes arising from the action of Jupiter. 6. Long-period inequalities in the eccentricities arise from each of the causes mentioned in 1 and 2. The ranges of variation of those due to the action of Mars are appreciably less than those due to the action of Jupiter. 7. The mean longitude may be expressed in the form

L = + €1 + + periodic terms, where is the initial value of the theoretical mean motion, this being deduced from the initial value of the major axis by Kepler s third law. This may be written L = + «q + periodic terms. N may conveniently be called the “real mean motion.” Of the periodic terms of long period in L the amplitudes of those arising from the action of Mars may be appreciably greater than the amplitudes of those arising from the action of Jupiter, and when the approach to exact commensurability of the mean motions is near enough, the phenomena of libration (the cases B in the theory) may manifest themselves, and the amplitude of the corresponding periodic terfn may be comparatively large. In fact, when &, the modulus of elliptic functions, tends to unity in any of the cases A and B, the amplitude A tends to x 90 , i.e. to 90 approx. In the example above we found 4-^2 amplitudes of approximately 40o and 62 o. It is a pleasant duty to acknowledge my indebtedness to Professor Baker for his continued interest in this and the previous investigation.

Invisible Sunspots. By Dr. G. E. Hale. {Extract from a letter to Professor Nexo all, 1921 Bee. 21.)

I have been able to give more time to my own solar work, and the measurement and reduction of the sunspot spectra, so long delayed by various causes, is now well advanced. Another piece of work which I had in mind when with you has also made progress recently. You bmay remember that I commented on the strong tendency of spots to appear in the bipolar form, which is so marked that some 60 per cent, of all spots are double, while almost all single spots are followed (sometimes preceded) by streams of calcium flocculi. It struck me that an incipient spot, not dark enough to be visible, might frequently lie in the flocculi, at a point corresponding with the position of the second member of a full-fledged bipolar group.

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