爱因斯坦论文(英语)

经典word整理文档，仅参考，双击此处可删除页眉页脚。本资料属于网络整理，如有侵权，请联系删除，谢谢！ONTHEELECTRODYNAMICSOFMOVINGBODIESItisknownthatMaxwell'selectrodynamics--asusuallyunderstoodatthepresenttime--whenappliedtomovingbodies,leadstoasymmetrieswhichdonotappeartobeinherentinthephenomena.Take,forexample,thereciprocalelectrodynamicactionofamagnetandaconductor.Theobservablephenomenonheredependsonlyontherelativemotionoftheconductorandthemagnet,whereasthecustomaryviewdrawsasharpdistinctionbetweenthetwocasesinwhicheithertheoneortheotherofthesebodiesisinmotion.Forifthemagnetisinmotionandtheconductoratrest,therearisesintheneighbourhoodofthemagnetanelectricfieldwithacertaindefiniteenergy,producingacurrentattheplaceswherepartsoftheconductoraresituated.Butifthemagnetisstationaryandtheconductorinmotion,noelectricfieldarisesintheneighbourhoodofthemagnet.Intheconductor,however,wefindanelectromotiveforce,towhichinitselfthereisnocorrespondingenergy,butwhichgivesrise--assumingequalityofrelativemotioninthetwocasesdiscussed--toelectriccurrentsofthesamepathandintensityasthoseproducedbytheelectricforcesintheformercase.Examplesofthissort,togetherwiththeunsuccessfulattemptstodiscoveranymotionoftheearthrelativelytothe``lightmedium,''suggestthatthephenomenaofelectrodynamicsaswellasofmechanicspossessnopropertiescorrespondingtotheideaofabsoluterest.Theysuggestratherthat,ashasalreadybeenshowntothefirstorderofsmallquantities,thesamelawsofelectrodynamicsandopticswillbevalidforallframesofreferenceforwhichtheequationsofmechanicsholdgood.1Wewillraisethisconjecture(thepurportofwhichwillhereafterbecalledthe``PrincipleofRelativity'')tothestatusofapostulate,andalsointroduceanotherpostulate,whichisonlyapparentlyirreconcilablewiththeformer,namely,thatlightisalwayspropagatedinemptyspacewithadefinitevelocitycwhichisindependentofthestateofmotionoftheemittingbody.ThesetwopostulatessufficefortheattainmentofasimpleandconsistenttheoryoftheelectrodynamicsofmovingbodiesbasedonMaxwell'stheoryforstationarybodies.Theintroductionofa``luminiferousether''willprovetobesuperfluousinasmuchastheviewheretobedevelopedwillnotrequirean``absolutelystationaryspace''providedwithspecialproperties,norassignavelocity-vectortoapointoftheemptyspaceinwhichelectromagneticprocessestakeplace.Thetheorytobedevelopedisbased--likeallelectrodynamics--onthekinematicsoftherigidbody,sincetheassertionsofanysuchtheoryhavetodowiththerelationshipsbetweenrigidbodies(systemsofco-ordinates),clocks,andelectromagneticprocesses.Insufficientconsiderationofthiscircumstanceliesattherootofthedifficultieswhichtheelectrodynamicsofmovingbodiesatpresentencounters.I.KINEMATICALPART§1.DefinitionofSimultaneity2aIfamaterialpointisatrestrelativelytothissystemofco-ordinates,itspositioncanbedefinedrelativelytheretobytheemploymentofrigidstandardsofmeasurementandthemethodsofEuclideangeometry,andcanbeexpressedinCartesianco-ordinates.Ifwewishtodescribethemotionofamaterialpoint,wegivethevaluesofitsco-ordinatesasfunctionsofthetime.Nowwemustbearcarefullyinmindthatamathematicaldescriptionofthiskindhasnophysicalmeaningunlesswearequiteclearastowhatweunderstandby``time.''Wehavetotakeintoaccountthatallourjudgmentsinwhichtimeplaysapartarealwaysjudgmentsofsimultaneousevents.If,forinstance,Isay,``Thattrainarriveshereat7o'clock,''Imeansomethinglikethis:``Thepointingofthesmallhandofmywatchto7andthearrivalofthetrainaresimultaneousevents.''3Itmightappearpossibletoovercomeallthedifficultiesattendingthedefinitionof``time''bysubstituting``thepositionofthesmallhandofmywatch''for``time.''Andinfactsuchadefinitionissatisfactorywhenweareconcernedwithdefiningatimeexclusivelyfortheplacewherethewatchislocated;butitisnolongersatisfactorywhenwehavetoconnectintimeseriesofeventsoccurringatdifferentplaces,or--whatcomestothesamething--toevaluatethetimesofeventsoccurringatplacesremotefromthewatch.Wemight,ofcourse,contentourselveswithtimevaluesdeterminedbyanobserverstationedtogetherwiththewatchattheoriginoftheco-ordinates,andco-ordinatingthecorrespondingpositionsofthehandswithlightsignals,givenoutbyeveryeventtobetimed,andreachinghimthroughemptyspace.Butthisco-ordinationhasthedisadvantagethatitisnotindependentofthestandpointoftheobserverwiththewatchorclock,asweknowfromexperience.Wearriveatamuchmorepracticaldeterminationalongthefollowinglineofthought.IfatthepointAofspacethereisaclock,anobserveratAcandeterminethetimevaluesofeventsintheimmediateproximityofAbyfindingthepositionsofthehandswhicharesimultaneouswiththeseevents.IfthereisatthepointBofspaceanotherclockinallrespectsresemblingtheoneatA,itispossibleforanobserveratBtodeterminethetimevaluesofeventsintheimmediateneighbourhoodofB.Butitisnotpossiblewithoutfurtherassumptiontocompare,inrespectoftime,aneventatAwithaneventatB.Wehavesofardefinedonlyan``Atime''anda``Btime.''Wehavenotdefinedacommon``time''forAandB,forthelattercannotbedefinedatallunlessweestablishbydefinitiatthe``time''requiredbylighttotravelfromAtoBequalsthe``time''itrequirestotravelfromBtoA.Letarayoflightstartatthe``Atime''fromAtowardsB,letitatthe``Btime''bereflectedatBinthedirectionofA,andarriveagainatAatthe``Atime''.InaccordancewithdefinitionthetwoclockssynchronizeifWeassumethatthisdefinitionofsynchronismisfreefromcontradictions,andpossibleforanynumberofpoints;andthatthefollowingrelationsareuniversallyvalid:--BAABBCThuswiththehelpofcertainimaginaryphysicalexperimentswehavesettledwhatistobeunderstoodbysynchronousstationaryclockslocatedatdifferentplaces,andhaveevidentlyobtainedadefinitionof``simultaneous,''or``synchronous,''andof``time.''The``time''ofaneventisthatwhichisgivensimultaneouslywiththeeventbyastationaryclocklocatedattheplaceoftheevent,thisclockbeingsynchronous,andindeedsynchronousforalltimedeterminations,withaspecifiedstationaryclock.Inagreementwithexperiencewefurtherassumethequantitytobeauniversalconstant--thevelocityoflightinemptyspace.Itisessentialtohavetimedefinedbymeansofstationaryclocksinthestationarysystem,andthetimenowdefinedbeingappropriatetothestationarysystemwecallit``thetimeofthestationarysystem.''§2.OntheRelativityofLengthsandTimes,aawheretimeintervalistobetakeninthesenseofthedefinitionin§1.Lettherebegivenastationaryrigidrod;andletitslengthbelasmeasuredbyameasuring-rodwhichisalsostationary.Wenowimaginetheaxisoftherodlyingalongtheaxisofxofthestationarysystemofco-ordinates,andthatauniformmotionofparalleltranslationwithvelocityvalongtheaxisofxinthedirectionofincreasingxisthenimpartedtotherod.Wenowinquireastothelengthofthemovingrod,andimagineitslengthtobeascertainedbythefollowingtwooperations:--))§1,aaInaccordancewiththeprincipleofrelativitythelengthtobediscoveredbytheoperation(a)--wewillcallit``thelengthoftherodinthemovingsystem''--mustbeequaltothelengthlofthestationaryrod.Thelengthtobediscoveredbytheoperation(b)wewillcall``thelengthofthe(moving)rodinthestationarysystem.''Thisweshalldetermineonthebasisofourtwoprinciples,andweshallfindthatitdiffersfroml.Currentkinematicstacitlyassumesthatthelengthsdeterminedbythesetwooperationsarepreciselyequal,orinotherwords,thatamovingrigidbodyattheepochtmayingeometricalrespectsbeperfectlyrepresentedbythesamebodyatrestinadefiniteposition.WeimaginefurtherthatatthetwoendsAandBoftherod,clocksareplacedwhichsynchronizewiththeclocksofthestationarysystem,thatistosaythattheirindicationscorrespondatanyinstanttothe``timeofthestationarysystem''attheplaceswheretheyhappentobe.Theseclocksaretherefore``synchronousinthestationarysystem.''Weimaginefurtherthatwitheachclockthereisamovingobserver,andthattheseobserversapplytobothclocksthecriterionestablishedin§1forthesynchronizationoftwoclocks.LetarayoflightdepartfromAatthetime4,letitbereflectedatBatthetime,andreachAagainatthetime.Takingintoconsiderationtheprincipleoftheconstancyofthevelocityoflightwefindthatwheredenotesthelengthofthemovingrod--measuredinthestationarysystem.Observersmovingwiththemovingrodwouldthusfindthatthetwoclockswerenotsynchronous,whileobserversinthestationarysystemwoulddeclaretheclockstobesynchronous.Soweseethatwecannotattachanyabsolutesignificationtotheconceptofsimultaneity,butthattwoeventswhich,viewedfromasystemofco-ordinates,aresimultaneous,cannolongerbelookeduponassimultaneouseventswhenenvisagedfromasystemwhichisinmotionrelativelytothatsystem.§3.TheoryoftheTransformationofCo-ordinatesandTimesfromaStationarySystemtoanotherSysteminUniformMotionofTranslationRelativelytotheFormeraXYZaaNowtotheoriginofoneofthetwosystems()letaconstantvelocityvbeimpartedinthedirectionoftheincreasingxoftheotherstationarysystem(K),andletthisvelocitybecommunicatedtotheaxesoftheco-ordinates,therelevantmeasuring-rod,andtheclocks.ToanytimeofthestationarysystemKtherethenwillcorrespondadefinitepositionoftheaxesofthemovingsystem,andfromreasonsofsymmetryweareentitledtoassumethatthemotionofkmaybesuchthattheaxesofthemovingsystemareatthetimet(thist''alwaysdenotesatimeofthestationarysystem)paralleltotheaxesofthestationarysystem.WenowimaginespacetobemeasuredfromthestationarysystemKbymeansofthestationarymeasuring-rod,andalsofromthemovingsystemkbymeansofthemeasuring-rodmovingwithit;andthatwethusobtaintheco-ordinates,,z,and,,respectively.Further,letthetimetofthestationarysystembedeterminedforallpointsthereofatwhichthereareclocksbymeansoflightsignalsinthemannerindicatedin§1;similarlyletthetimeofthemovingsystembedeterminedforallpointsofthemovingsystematwhichthereareclocksatrestrelativelytothatsystembyapplyingthemethod,givenin§1,oflightsignalsbetweenthepointsatwhichthelatterclocksarelocated.Toanysystemofvalues,,z,t,whichcompletelydefinestheplaceandtimeofaneventinthestationarysystem,therebelongsasystemofvalues,,,determiningthateventrelativelytothesystem,andourtaskisnowtofindthesystemofequationsconnectingthesequantities.Inthefirstplaceitisclearthattheequationsmustbelinearonaccountofthepropertiesofhomogeneitywhichweattributetospaceandtime.Ifweplacex'=-vt,itisclearthatapointatrestinthesystemkmusthaveasystemofvaluesx',,z,independentoftime.Wefirstdefineasafunctionofx',,z,andt.Todothiswehavetoexpressinequationsthatisnothingelsethanthesummaryofthedataofclocksatrestinsystem,whichhavebeensynchronizedaccordingtotherulegivenin§1.FromtheoriginofsystemkletaraybeemittedatthetimealongtheX-axisto',bereflectedthencetotheoriginoftheco-ordinates,arrivingthere;wethenmusthave,or,byinsertingtheargumentsandatthetimeatthetimeofthefunctionandapplyingtheprincipleoftheconstancyofthevelocityoflightinthestationarysystem:--Hence,ifx'bechoseninfinitesimallysmall,orItistobenotedthatinsteadoftheoriginoftheco-ordinateswemighthavechosenanyotherpointforthepointoforiginoftheray,andtheequationjustobtainedisthereforevalidforallvaluesofx',,z.Ananalogousconsideration--appliedtotheaxesofYandZ--itbeingborneinmindthatlightisalwayspropagatedalongtheseaxes,whenviewedfromthestationarysystem,withthevelocitygivesusSinceisalinearfunction,itfollowsfromtheseequationsthatwhereaisafunctionthatattheoriginof,atpresentunknown,andwhereforbrevityitisassumed,whent=0.Withthehelpofthisresultweeasilydeterminethequantities,,byexpressinginequationsthatlight(asrequiredbytheprincipleoftheconstancyofthevelocityoflight,incombinationwiththeprincipleofrelativity)isalsopropagatedwithvelocitycwhenmeasuredinthemovingsystem.ForarayoflightemittedatthetimeinthedirectionoftheincreasingButtheraymovesrelativelytotheinitialpointof,whenmeasuredinthestationarysystem,withthevelocity,sothatIfweinsertthisvalueoftintheequationfor,weobtainInananalogousmannerwefind,byconsideringraysmovingalongthetwootheraxes,thatwhenThusSubstitutingforx'itsvalue,weobtainwhereandisanasyetunknownfunctionof.Ifnoassumptionwhateverbemadeastotheinitialpositionofthemovingsystemandastothezeropointofconstantistobeplacedontherightsideofeachoftheseequations.,anadditiveWenowhavetoprovethatanyrayoflight,measuredinthemovingsystem,ispropagatedwiththevelocity,if,aswehaveassumed,thisisthecaseinthestationarysystem;forwehavenotasyetfurnishedtheproofthattheprincipleoftheconstancyofthevelocityoflightiscompatiblewiththeprincipleofrelativity.Atthetime,whentheoriginoftheco-ordinatesiscommontothetwosystems,letasphericalwavebeemittedtherefrom,andbepropagatedwiththevelocitycinsystemK.If(,,z)beapointjustattainedbythiswave,then++=.TransformingthisequationwiththeaidofourequationsoftransformationweobtainafterasimplecalculationThewaveunderconsiderationisthereforenolessasphericalwavewithvelocityofpropagationcwhenviewedinthemovingsystem.Thisshowsthatourtwofundamentalprinciplesarecompatible.5Intheequationsoftransformationwhichhavebeendevelopedthereentersanunknownfunctionof,whichwewillnowdetermine.Forthispurposeweintroduceathirdsystemofco-ordinates,whichrelativelytothesystemkisinastateofparalleltranslatorymotionparalleltotheaxisof,such*1thattheoriginofco-ordinatesofsystemAtthetimet=0letallthreeoriginscoincide,andwhent==yz=0letthetimet'ofthesystembezero.Wecalltheco-ordinates,measuredinthesystem,',y',z',andbyatwofoldapplicationofourequationsoftransformationweobtain,moveswithvelocity-vontheaxisof.Sincetherelationsbetweenx',y',z'and,,zdonotcontainthetimet,thesystemsKandareatrestwithrespecttooneanother,anditisclearthatthetransformationfromKtomustbetheidenticaltransformation.ThusWenowinquireintothesignificationoftheaxisofYofsystemkwhichliesbetween.ThispartoftheaxisofYisarodmovingperpendicularlytoits.WegiveourattentiontothatpartofandaxiswithvelocityvrelativelytosystemK.ItsendspossessinKtheco-ordinatesThelengthoftherodmeasuredinKistherefore;andthisgivesusthemeaningofthefunction.Fromreasonsofsymmetryitisnowevidentthatthelengthofagivenrodmovingperpendicularlytoitsaxis,measuredinthestationarysystem,mustdependonlyonthevelocityandnotonthedirectionandthesenseofthemotion.Thelengthofthemovingrodmeasuredinthestationarysystemdoesnotchange,therefore,ifvand-vareinterchanged.Hencefollowsthat,orItfollowsfromthisrelationandtheonepreviouslyfoundthattransformationequationswhichhavebeenfoundbecome,sothatthewhere§4.PhysicalMeaningoftheEquationsObtainedinRespecttoMovingRigidBodiesandMovingClocks6a,.KvTheequationofthissurfaceexpressedin,,zatthetimet=0isArigidbodywhich,measuredinastateofrest,hastheformofasphere,thereforehasinastateofmotion--viewedfromthestationarysystem--theformofanellipsoidofrevolutionwiththeaxesThus,whereastheYandZdimensionsofthesphere(andthereforeofeveryrigidbodyofnomatterwhatform)donotappearmodifiedbythemotion,theXdimensionappearsshortenedintheratio,i.e.thegreaterthevalueof,thegreatertheshortening.Forvcallmovingobjects--viewedfromthe``stationary''system--shrivelupintoplanefigures.*2Forvelocitiesgreaterthanthatoflightourdeliberationsbecomemeaningless;weshall,however,findinwhatfollows,thatthevelocityoflightinourtheoryplaysthepart,physically,ofaninfinitelygreatvelocity.Itisclearthatthesameresultsholdgoodofbodiesatrestinthe``stationary''system,viewedfromasysteminuniformmotion.Further,weimagineoneoftheclockswhicharequalifiedtomarkthetimetwhenatrestrelativelytothestationarysystem,andthetimewhenatrestrelativelytothemovingsystem,tobelocatedattheoriginoftheco-ordinatesof,andsoadjustedthatitmarksthetime.Whatistherateofthisclock,whenviewedfromthestationarysystem?Betweenthequantitiesx,t,and,whichrefertothepositionoftheclock,wehave,evidently,vtandTherefore,whenceitfollowsthatthetimemarkedbytheclock(viewedinthestationarysystem)isslowbysecondspersecond,or--neglectingmagnitudesoffourth.andhigherorder--byFromthisthereensuesthefollowingpeculiarconsequence.IfatthepointsAandBofKtherearestationaryclockswhich,viewedinthestationarysystem,aresynchronous;andiftheclockatAismovedwiththevelocityvalongthelineABtoB,thenonitsarrivalatBthetwoclocksnolongersynchronize,buttheclockmovedfromAtoBlagsbehindtheotherwhichhasremainedatBby(uptomagnitudesoffourthandhigherorder),tbeingthetimeoccupiedinthejourneyfromAtoB.ItisatonceapparentthatthisresultstillholdsgoodiftheclockmovesfromAtoBinanypolygonalline,andalsowhenthepointsAandBcoincide.Ifweassumethattheresultprovedforapolygonallineisalsovalidforacontinuouslycurvedline,wearriveatthisresult:IfoneoftwosynchronousclocksatAismovedinaclosedcurvewithconstantvelocityuntilitreturnstoA,thejourneylastingtseconds,thenbytheclockwhichhasremainedatrestthetravelledclockonitsarrivalatAwillbesecondslow.Thenceweconcludethatabalance-clockattheequatormustgomoreslowly,byaverysmallamount,thana7preciselysimilarclocksituatedatoneofthepolesunderotherwiseidenticalconditions.§5.TheCompositionofVelocitieskXK,awhereanddenoteconstants.Required:themotionofthepointrelativelytothesystemK.Ifwiththehelpoftheequationsoftransformationdevelopedin§3weintroducethequantities,,z,tintotheequationsofmotionofthepoint,weobtainThusthelawoftheparallelogramofvelocitiesisvalidaccordingtoourtheoryonlytoafirstapproximation.Weset*3aisthentobelookeduponastheanglebetweenthevelocitiesvandw.Afterasimplecalculationweobtain*4Itisworthyofremarkthatvandwenterintotheexpressionfortheresultantvelocityinasymmetricalmanner.IfwalsohasthedirectionoftheaxisofX,wegetItfollowsfromthisequationthatfromacompositionoftwovelocitieswhicharelessthan,therealwaysresultsavelocitylessthan.Forifwesetandbeingpositiveandlessthan,then,Itfollows,further,thatthevelocityoflightccannotbealteredbycompositionwithavelocitylessthanthatoflight.ForthiscaseweobtainWemightalsohaveobtainedtheformulaforV,forthecasewhenvandwhavethesamedirection,bycompoundingtwotransformationsinaccordancewith§3.IfinadditiontothesystemsKandkfiguringin§3weintroducestillanothersystemofco-ordinatesk'movingparallelto,itsinitialpointmovingontheaxisof*5withthevelocityw,weobtainequationsbetweenthequantities,,z,tandthecorrespondingquantitiesof',whichdifferfromtheequationsfoundin§3onlyinthattheplaceofv''istakenbythequantityfromwhichweseethatsuchparalleltransformations--necessarily--formagroup.Wehavenowdeducedtherequisitelawsofthetheoryofkinematicscorrespondingtoourtwoprinciples,andweproceedtoshowtheirapplicationtoelectrodynamics.II.ELECTRODYNAMICALPART§6.TransformationoftheMaxwell-HertzEquationsforEmptySpace.OntheNatureoftheElectromotiveForcesOccurringinaMagneticFieldDuringMotionwhere(X,Y,Z)denotesthevectoroftheelectricforce,and(L,M,N)thatmagneticforce.oftheIfweapplytotheseequationsthetransformationdevelopedin§3,byreferringtheelectromagneticprocessestothesystemofco-ordinatesthereintroduced,movingwiththevelocity,weobtaintheequationswhereNowtheprincipleofrelativityrequiresthatiftheMaxwell-HertzequationsforemptyspaceholdgoodinsystemK,theyalsoholdgoodinsystem;thatistosaythatthevectorsoftheelectricandthemagneticforce--(,,)and(,,)--ofthemovingsystem,whicharedefinedbytheirponderomotiveeffectsonelectricormagneticmassesrespectively,satisfythefollowingequations:--Evidentlythetwosystemsofequationsfoundforsystemkmustexpressexactlythesamething,sincebothsystemsofequationsareequivalenttotheMaxwell-HertzequationsforsystemK.Since,further,theequationsofthetwosystemsagree,withtheexceptionofthesymbolsforthevectors,itfollowsthatthefunctionsoccurringinthesystemsofequationsatcorrespondingplacesmustagree,withtheexceptionofafactor,whichiscommonforallfunctionsoftheonesystemofequations,andandbutdependsupon.ThuswehavetherelationsisindependentofIfwenowformthereciprocalofthissystemofequations,firstlybysolvingtheequationsjustobtained,andsecondlybyapplyingtheequationstotheinversetransformation(fromktoK),whichischaracterizedbythevelocity,itfollows,whenweconsiderthatthetwosystemsofequationsthusobtainedmustbeidentical,that.Further,fromreasonsofsymmetry8andthereforeandourequationsassumetheformAstotheinterpretationoftheseequationswemakethefollowingremarks:Letapointchargeofelectricityhavethemagnitude``one''whenmeasuredinthestationarysystemK,i.e.letitwhenatrestinthestationarysystemexertaforceofonedyneuponanequalquantityofelectricityatadistanceofonecm.Bytheprincipleofrelativitythiselectricchargeisalsoofthemagnitude``one''whenmeasuredinthemovingsystem.Ifthisquantityofelectricityisatrestrelativelytothestationarysystem,thenbydefinitionthevector(X,Y,Z)isequaltotheforceactinguponit.Ifthequantityofelectricityisatrestrelativelytothemovingsystem(atleastattherelevantinstant),thentheforceactinguponit,measuredinthemovingsystem,isequaltothevector(,,).Consequentlythefirstthreeequationsaboveallowthemselvestobeclothedinwordsinthetwofollowingways:--a,aaTheanalogyholdswith``magnetomotiveforces.''Weseethatelectromotiveforceplaysinthedevelopedtheorymerelythepartofanauxiliaryconcept,whichowesitsintroductiontothecircumstancethatelectricandmagneticforcesdonotexistindependentlyofthestateofmotionofthesystemofco-ordinates.Furthermoreitisclearthattheasymmetrymentionedintheintroductionasarisingwhenweconsiderthecurrentsproducedbytherelativemotionofamagnetandaconductor,nowdisappears.Moreover,questionsastothe``seat''ofelectrodynamicelectromotiveforces(unipolarmachines)nowhavenopoint.§7.TheoryofDoppler'sPrincipleandofAberrationaaawhereHere(,,)and(,,)arethevectorsdefiningtheamplitudeofthewave-train,andl,m,nthedirection-cosinesofthewave-normals.Wewishtoknowtheconstitutionofthesewaves,whentheyareexaminedbyanobserveratrestinthemovingsystem.Applyingtheequationsoftransformationfoundin§6forelectricandmagneticforces,andthosefoundin§3fortheco-ordinatesandthetime,weobtaindirectlywhereFromtheequationforitfollowsthatifanobserverismovingwithvelocityvrelativelytoaninfinitelydistantsourceoflightoffrequency,insuchawaythattheconnectingline``source-observer''makestheanglewiththevelocityoftheobserverreferredtoasystemofco-ordinateswhichisatrestrelativelytothesourceoflight,thefrequencyofthelightperceivedbytheobserverisgivenbytheequationThisisDoppler'sprincipleforanyvelocitieswhatever.WhenassumestheperspicuousformtheequationWeseethat,incontrastwiththecustomaryview,when.Ifwecalltheanglebetweenthewave-normal(directionoftheray)inthemovingsystemandtheconnectingline``source-observer''theform,theequationfor*6assumesThisequationexpressesthelawofaberrationinitsmostgeneralform.Iftheequationbecomessimply,Westillhavetofindtheamplitudeofthewaves,asitappearsinthemovingsystem.IfwecalltheamplitudeoftheelectricormagneticforceAorrespectively,accordinglyasitismeasuredinthestationarysystemorinthemovingsystem,weobtainwhichequation,if,simplifiesintoItfollowsfromtheseresultsthattoanobserverapproachingasourceoflightwiththevelocity,thissourceoflightmustappearofinfiniteintensity.§8.TransformationoftheEnergyofLightRays.TheoryofthePressureofRadiationExertedonPerfectReflectors,aaKin.,,naWemaythereforesaythatthissurfacepermanentlyenclosesthesamelightcomplex.Weinquireastothequantityofenergyenclosedbythissurface,viewedinsystem,thatis,astotheenergyofthelightcomplexrelativelytothesystem.Thesphericalsurface--viewedinthemovingsystem--isanellipsoidalsurface,theequationforwhich,atthetime,isIfSisthevolumeofthesphere,andthatofthisellipsoid,thenbyasimplecalculationThus,ifwecallthelightenergyenclos