u x, u ) is not necessarily

Transcription

1 CHAPTER III Dffeetl Eqtos of Hghe Ode Ptl dffeetl eqtos of hghe th fst ode peset so m dvese spets tht fed geel theo s Chpte II s ot possle Thee s desve dstto etwee sevel tpes of dffeetl eqtos lled ellpt hpeol d pol eh of whh shows etel dffeet ehvo egdg popetes d ostto of soltos I ths hpte we shll tode ths lssfto gded emples of phsl teest Moeove we shll dsss pelm w methods of ppoh towd the solto of elevt polems The sseqet hptes wll e pml oeed wth sstemt theo of ellpt d hpeol polems Some lssl dffeetl eqtos of seod ode fo fto z e epesettve emples: The Lple eqto ellpt tpe: zz The wve eqto hpeol tpe: The het eqto pol tpe: z zz Noml Foms fo Le d Qs-Le Dffeetl Opetos of Seod Ode Two Idepedet Vles Fo le d lso fo qs-le dffeetl eqtos of seod ode o oespodg sstems of two fst ode eqtos two depedet vles the lssfto e ed ot eplt elemet steps wthot efeee to geel theo It ogtes fom the ttempt to fd smple oml foms Ellpt Hpeol d Pol Noml Foms Med Tpes A le dffeetl opeto of seod ode fo the fto s gve L [ ] ; the oeffets e ssmed to e otosl dffeetle d ot smlteosl vshg ftos of d dom G We osde L[ ] g L[ ] L whee the dffeetl epesso g s ot eessl le d ots o seod devtves O ojet s to tsfom the dffeetl opeto o the oespodg dffeetl eqto L [ ] L to smple oml fom todg ew depedet vles

2 ξ η ψ Deotg ξ η the fto to whh s tsfomed we hve the eltos ψ ξ η ψ ξ η ξξ ξηψ ηηψ L ψ ψ ψ ψ L ξξ ξη ξξ ψ ψ ξη ηη ηη L Hee g the dots me tems whh o seod ode devtves of ppe Ths the dffeetl opeto ssmes the fom 5 Λ[ ] αξξ βξη γηη wth 6 Moeove d β α ψ ψ ψ ψ γ ψ ψ ψ ψ α β γ e elted 7 ψ ψ αγ β d the dett fo the "htest qdt fom" Q l m l lm m αλ βλµ γµ whee the vles l m d λ µ e oeted t fed pot the le tsfomto oeted the msptg l λ µψ m λ µψ I the tsfomto two ftos ψ e t o dsposl so tht we m mpose two odtos o the tsfomed oeffets α β γ mg t smple oml foms of the tsfomed dffeetl eqto 5 We osde the followg sets of odtos: I α γ β II α γ β o α γ III β γ Whh of these odtos e stsfed the tsfomtos of ose lws el tsfomtos e ssmed -- depeds o the lge hte of the fom Ql m o geometll spekg o the hte of the qdt ve the l m-ple fo fed αγ β ψ ψ ψ ψ L ψ ψ Q l m λ µψ λ µψ αλ βλµ γµ λ ψ ψ ψ ψ µ ψ ψ ψ ψ λµ λ µψ λ µψ

3 : Q l m ; ths ve m e ellpse hpeol o pol Aodgl t pot we ll the opeto L[] I ellpt f > II hpeol f < III pol f The oespodg oml foms of the dffeetl opeto e I Λ ] α L II III [ ξξ ηη Λ[ ] α ξξ o Λ[ ] β Λ[ ] αξξ ξη L ηη L L d the oml foms of the dffeetl eqto e I L II ξξ ηη ξξ ξη ηη o L L III L ξξ Fo fed sh oml fom lws e oted smpl the le tsfomto whh tkes Q to the oespodg oml fom Howeve ssmg tht the opeto L s of the sme tpe eve pot of dom G we wt to fd ftos d ψ whh wll tsfom L[] to oml fom t eve pot of G Sess depeds o whethe et fst ode sstems of le ptl dffeetl eqtos e solved Wthot loss of geelt we m ssme evewhee the dom G; othewse ethe the eqvlet ssmpto wold hold o o epesso wold led e the oml fom II To deteme tsfomto ftos d ψ fo the whole dom G we fst ssme tht L[] s hpeol G d tht the ew oeffets mst stsf the odto α γ Eqtos 6 the led to the qdt eqto 8 Q λ λµ µ fo the to λ / µ of the devtves / d ψ / ψ Hpeol If L[] s hpeol G the < d ths eqto 8 hs two dstt el soltos λ /µ d λ /µ Se m ssme tht we

4 µ µ ; the 8 defes the qttes λ d λ G s otosl dffeetle ftos of d Ths the hpeol se we ot the oml fom 9 β L ξη detemg the tsfomto ftos d ψ fom the dffeetl eqtos λ ψ λψ These two fst ode le homogeeos ptl dffeetl eqtos eld ft two fmles of ves ost d ψ ost whh lso m e defed s the fmles of soltos of the od dffeetl eqtos o ' λ ' λ ' ' whee s osdeed s fto of log the ves of the fml The elto λ λ shows tht ves of the two fmles ot e tget t pot of G d tht ψ ψ If α γ eqto 7 mples β The ves ξ ost d η ψ ost e lled the htest ves of the le hpeol dffeetl opeto L[] Se we m dvde 9 β we stte: If L[] s hpeol e < the seod ode dffeetl eqto m e tsfomed to the oml fom L ξη todg the two fmles of htest ves ξ ost d η ost s oodte ves Ellpt If > holds the the opeto s ellpt G I ths se the qdt eqto 8 hs o el soltos t t hs two ojgte omple soltos λ d λ whh e otos omple-vled ftos of the el vles The eqtos α γ e stsfed o fml of el ves ost ; e

5 5 thee e o htest ves If howeve e lt ftos of d f we ssme tht d ψ e lt the we m osde the dffeetl eqtos fo omple d d s efoe tsfom them to the ew vles ξ d η whh eome omple ojgte Itodg el depedet vles d the eqtos η ξ η ξ we ot ξη Ths we ve t the oml fom L L the ellpt se Fo the peedg tsfomto volvg omple qttes we hd to eqe odto of ltt fo these oeffets odto sve t essetll le to the polem To vod ths estto we m se the followg poede ot volvg omple qttes of tsfomg ellpt epesso to oml fom: Wtg d sted of ξ d η eqtos we stplte the odtos γ α β o epltl These dffeetl eqtos e eded l elemet lge mplto to the followg fst ode sstem of le ptl dffeetl eqtos: W W whee W d ethe sg s pemssle fo W Fom these so-lled Beltm dffeetl eqtos we ot mmedtel elmtg oe of the kows eg the followg seod ode dffeetl eqto fo the othe qtt: 5 W W The tsfomto of the dffeetl eqto to oml fom eghohood of pot s gve p of ftos ψ η ξ ] [ ] [ α ] [ ] [ ψ ψ ψ ψ γ ψ ψ ψ ψ β

6 stsfg d hvg ovshg Jo W Sh ftos e detemed oe we hve solto of 5 wth ovshg gdet We shll see Ch IV 7 tht de et smoothess ssmptos o the oeffets eg estee of otos devtves p to seod ode of e sh solto lws ests -- t lest loll -- d hee omlzg pmete sstem m e toded eghohood of pot Pol The thd se s the pol se: The qdt eqto 8 the hs oe el oot d we odgl tode oe fml of ves ξ sh w tht α holds; the o ot of elto 7 we mst lso hve β whle fo ste fo the oml fom ψ G γ I the pol se we ot ηη L The theoem stted t the egg hs ths ee poved Note tht the tsfomto to the oml fom s o mes qel detemed Fo emple the ellpt se the oml fom ems hged f we sjet the mppg 6 -dom to ofoml Emples Sevel emples of the dffeet tpes of dffeetl eqtos hve led ee dsssed Ch I The smplest hpeol eqto tht of the vtg stg ws tt ompletel solved The pototpe of the ellpt dffeetl eqto s the potetl eqto see eg Ch I The pol eqto of het odto ws dsssed Ch I t Fom the tpe of the eqto we shll dede mpott popetes whh sggest ot ol methods of solto t lso te to deteme whethe o ot polems e esol posed Idetll gve dffeetl eqto m e of dffeet tpe dffeet doms med tpes; fo emple the eqto

7 6 s ellpt fo > d hpeol fo < se I the dom < eqto 8 e the eqto λ µ possesses the two el oots λ / µ ± ; ths the two dffeetl eqtos 7 ψ ψ hold fo d ψ The hve the soltos B the tsfomto 8 ξ η ψ 6 ssmes the hpeol oml fom 7 9 ξη ξ η ξη fo < The htest ves e gve the pols ptl the ves ost e the hes of the pols hvg postve slope the ψ ost those hvg egtve slope f Fge Fo > we wte ξ ; η ; ths tsfomto eqto 6 ssmes the ellpt oml fom ξξ ηη η η Smll the dffeetl eqto kow s Tom s eqto s ellpt fo > d hpeol fo

8 < ese I the: hlf-ple < the eqtos ξ η ψ tsfom to the oml fom 9 ξη ξ η ξ > η 6 ξη The htest ves e the pols ± ; the hes wth dowwd deto eld the ves ost those deted pwd eld the ves ψ ost f Fge d set 5 Fo > we wte ξ η ξ η ξη we ot ths tsfomto the oml fom 6 9 The ftos 5 stsf the Beltm dffeetl eqtos ; 7 Noml Foms fo Qs-Le Seod Ode Dffeetl Eqtos Two Vles Emple Mml Sfes Let s osde the dffeetl eqto of mml sfes 5 q pqs p t ; se p q > ths dffeetl eqto s evewhee ellpt d e tsfomed to oml sstem of 8

9 the fom I ft shot llto elds the followg eqtos: 6 o 7 wth 9 Ths sstem e pt to smple fom: Dffeettg eqtos 6 we hve hee ; d d O the othe hd eqto 7 mples tht omto s le α β of the vetos d Hee α β d ths Ths mml sfe pmet epesetto wth stle pmetes d m e htezed the followg odtos: Eh of the thee oodtes stsfes the potetl eqto e 8 Moeove the stsf the odtos 9 A B Wth the sl otto of dffeetl geomet 9 mples the odtos E F G E G F fo the fst fdmetl fom of the sfe These ddtol odtos ppetl dd two ew dffeetl eqtos to the thee eqtos 8; howeve the meel epeset od odto We eed ot mpose the ddtol esttos 9 whole two-dmesol -dom t ol log some losed ve the eltos -dom Fom eqtos 8 the two A B A B follow mmedtel The hteze A B s lt fto of the omple vles ; theefoe A B vshes detll f the el pt A vshes o losed ve eg the od d f B s zeo t some pot Two mmedte olsos e sgft fo the theo of

10 mml sfes: The mppg of the -ple o the mml sfe s ofoml The epesetto of the mml sfe hmo ftos s eqvlet to the lssl Weestss epesetto mes of lt ftos of the omple vle ω To ot the fomls of Weestss we osde the potetl ftos of s the el pts of lt ftos f f f If ~ ~ ~ e the ojgte potetl ω ω ω ftos we hve ~ f ω ~ f ω ~ f ω Se the Ch-Rem dffeetl eqtos ~ ~ ~ we hve f ' ω f ' ω f ' ω so tht odtos 9 eome ω E G F f ' ν ω ν Ths ll mml sfes m e epeseted Re f ω Re f ω Re f ω whee the othewse t lt ftos f ω e sjet to the odto ν ν ω f ' Isted of ω we m tode oe of the ftos ν f ν eg f ω s the depedet vle Theefoe the totlt of mml sfes depeds essetll o ol oe t lt fto of omple vle 5 Sstems of Two Dffeetl Eqtos of Fst Ode Clssfto Geel d Chtests 6 Emples Mwell s d D s Eqtos It wll e es fo the ede to detf the wve eqto s hpeol Lple s eqto s ellpt Ch-Rem s eqtos v v s ellpt sstem v v s hpeol sstem d v v s pol sstem Fo the ellpt se we gve the followg ddtol emples:

11 Fst o k k wth the htest fom m Q d seod the dffeetl eqto wth the htest fom Q A emple of "pol" dffeetl eqto s tt fo depedet vles wth the tme vle t sgled ot Hee the htest fom s degeete se t does ot ot the vle The opeto tt tttt t t s hpeol ese ts htest fom the vles L τ Q τ τ lel hs the eqed popet O the othe hd the opeto t t t epesets temedte tpe; t s ethe ellpt o pol o hpeol se the fom τ Q hs ol two ot fo el oots τ f the vles of the vles L e fed

12 Beltm eqto Fthe emples of sstems of fst ode e the Beltm dffeetl eqtos v v W v v W whee the mt s ssmed to e postve defte Hee the oespodg htest fom s W W W Q I the spel se whee W Ch-Rem t s Q Mwell eqto D eqto The htest eqto elogg to D s dffeetl eqtos s sml to tht of the Mwell eqtos The D eqtos volve sstem of fo omple-vled ftos fo fo vles d whee t To fomlte them smpl we tode the followg mtes: α α α α β The eqtos the e k k k k β α

13 Hee the veto s popotol to the mget potetl to the elet potetl d to the est-mss Aodg to o les the htest detemt eomes k Q α kk fom of the foth degee the vles Ths the htest mfolds e g the sme s those fo the wve eqto Le Dffeetl Eqtos wth Costt Coeffets Fdmetl Soltos fo Eqtos of Seod Ode Iespetve of ode o vlt of oeffets fo ll le dffeetl eqtos ellpt o hpeol fdmetl soltos defed et sgltes pl mpott ole s wll eome ppet the et hptes Hee we shll meel set ef pelm dssso fo ellpt seod ode eqtos wth ostt oeffets We osde the eqto L[ ] d stt skg fo fdmetl soltos whh deped ol o the dste Σ ξ fom the pot to pmete pot ξ B tsfomg the Lpl opeto to pol oodtes we ot see Vol I p 5 As s esl vefed the fto w stsfes the sme eqto s wth - epled : w w w Ths g wtg sted of w we ot the fdmetl soltos fo eso s soo s we kow fo d fom the od dffeetl eqtos ' ' ' w w w w w { } { }

14 d ' ' ' espetvel Fo e fo the Lple eqto the soltos eept fo ostt fto t o dsposl e log d Ths oe ots fo the fdmetl soltos Wth s Hee fo ost ω we fd omple otto fo ω e d fo 5 ω e ω ω ω et Ths ll soltos fo odd e epessed tems of e ω tgoomet ftos o hpeol ftos f ω < Wth eve we hve fo whee J d α J ω βn ω egl _ fto N / π J log L e the Bessel fto ω d the Nem fto of ode zeo espetvel d α β e ostts If α s hose s zeo we fd fo the sgl solto J ω ω J ' ω log L J ' ω / s egl fo Ths solto we ll the fdmetl solto Oe esl sets geel: Fo odd > we hve the sgl "fdmetl" soltos d fo eve U L U W log L whee the dots deote egl tems d whee U d W e egl soltos of L[U] L[W] Fo < e fo mg ω oespodg eltos lso

15 pevl I the se of the hpeol dffeetl eqto L[ ] tt L qte pllel esog leds to the followg eslt: We seek sgl fdmetl solto of whh deped ol o the hpeol dste tτ ν ν ξ ν fom the pot t to the pmete pot 5 τ ξ the spe of m dmesos Fo we ot the od dffeetl eqto ' ' ' As efoe fdmetl soltos whh e sgl o the oe e fod to e of the fom desed ove fo the ellpt se The m dffeee s tht ow the sglt s sped ove whole oe d tht otsde the oe the fto s ot defed o m e defed s detll zeo whle the ellpt se ol the pot ξ s sgl The sgfe of sh fdmetl soltos whh m e modfed mltplto wth ostt d ddto of egl solto of L[] wll eome le Chpte VI I Volme I we hve led met sh soltos the fom of Gee s fto see Vol I Ch V It m e stted hee tht these fdmetl soltos ; ξ s ftos of the pot d the pmete pot ξ hve the followg s popet: v I the ellpt se the tegl G f ξ L ξ ξ dξ Ldξ eteded ove dom G ldg the pot stsfes wth stle ostt the Posso eqto I ptl fo the tegl v G L [ v] f stsfes the eded wve eqto v ω v ω e f ξ d Ld ω I the hpeol se t e poved tht v lso stsfes the

16 dffeetl eqto f the dom of tegto G flls the htest oe ssg fom the pot to the ξ -spe See Ch VI 5 Ple Wves Tg to eqtos of t ode k we wte the dffeetl eqto depedet vles the smol fom Pk D Pk D L P f whee L g P k s homogeeos poloml wth ostt oeffets of degee k the smols D / d f deotes gve fto of the depedet vles We eed osde ol the homogeeos eqto; e we ssme f The ohomogeeos eqto the e esl teted see eg The s ft s: Fo me of depedet vles the homogeeos eqto possesses soltos the fom of epoetl ftos wth ostts ν e whee L O oso we shll lso wte o The eess d sffet odto fo to e solto s tht L stsf the lge eqto of degee k Q * Pk Pk L P whh defes lge sfe Q * of degee k the spe of the oodtes L The lssfto to tpes howeve efes moe smpl to the homogeeos eqto Q Pk ; ths htest eqto depeds ol o the ppl pt of the dffeetl eqto; t detemes the omls of htest sfe elemets geemet wth the deftos of Fo emple thee dmesos fo the Lple eqto we ot the elto zz Hee t lest oe of the epoets oespodg soltos mght e wtte eg the fom ν s mg; the e z e Fo the wve eqto we hve soltos ω we hve the elto 6 e t wth d fo the "eded" wve eqto ω Fo the het

17 eqto t we hve the elto If the eqto Q ot e stsfed el vles of L the the dffeetl eqto s lled ellpt Ple Wves Coted Pogessg Wves Dspeso I the followg setos we shll e pml oeed wth soltos whh epeset popgto pheome ptl wth ple wves sg the hpeol se I ddto to the depedet spe-vles we shll osde fthe vle t ; we fom the e podt L A wth the -dmesol veto : L d defe the phse wth ostt B t A t Let s ssme fst tht the dffeetl eqto ots ol the ppl pt e the tems of ode k o othe wods tht P fo κ < k The the mpott ft holds: Not ol the epoetl ftos s ove e soltos t qte geell ll ftos of the fom 5 f B e soltos whee the wve fom f B s t fto of the phse B A - t d the oeffets htest eqto Q - Compe ν e sjet to the Povded tht ths eqto e stsfed el vles of L d the ftos f B epeset dstoted pogessg wves B the tem pogessg ple wve fo homogeeos le dffeetl eqto L[] we me solto of the fom 5 Ple wves of ths tpe hve ostt vles o eve phse ple of the fml B t ost the -dmesol t-spe To motvte the tem pogessg wve we osde the -dmesol spe R of the spe vles κ L whh the feld ves wth the tme t A solto of the fom 5 s ostt o whole ple of ostt phse B of fml of pllel ples A ple wth ostt phse moves wth ostt speed pllel to tself thogh the spe If we set R 7

18 l αl α l l γ l l d wte B A t α γt α γt E f B E we ot epesetto whh the mes α l e the deto oses of the omls to the ple wves d γ deotes the speed of popgto of the wves E s g lled the phse of the wve d the fto o f s lled the wve fom Fo emple the od wve eqto spe vles tt dmts ple wves of the fom hee the oeffets ; α t α l m fom t t veto α wth α d the wve fom m e t fto I othe wods the wve eqto s solved ple tt wves of t deto d tl gve fom; ll these wves pogess wth the speed γ The wves f B e lled dstoted o fee of dspeso ese the epeset fo t fom of the wve o sgl f B dstoted tslto wth speed γ the deto α of the oml of the phse ples If fo t deto α the htest eqto Q possesses k el d dffeet oots γ e f thee e k dffeet speeds possle fo dstoted wves eve deto the speeds depedg geel o the deto α the the dffeetl eqto s lled hpeol We shll lte geelze ths defto dmttg mltple oots et ses Ths defto of hpeolt efeg to the htest eqto Q s lso eted f the dffeetl eqto ots tems of lowe ode 5 Emples Telegph Eqto Udstoted Wves Cles Fo the wve eqto wves wth the speed d the t fom / tt pogessg dstoted ple αl l t α l l l 8

19 e possle eve deto A moe geel emple s gve the telegph eqto 9 α β αβ tt stsfed the voltge o the et s fto of the tme t d the posto log le; hee meses the legth of the le fom tl pot Uless α β ths eqto epesets t dspeso If we tode eqto / α β t v e we ot the smple v v tt αβ 9 v fo the fto v Ths ew eqto epesets the dspesoless se f d ol f α β I ths se the ogl telegph eqto of ose possesses o soltel dstoted wve soltos of tl pesed fom Howeve o eslt m e stted the followg w: If odto holds the telegph eqto possesses dmped et eltvel dstoted pogessg wve soltos of the fom / α β t e f ± t wth t f pogessg oth detos of the le Ths eslt s mpott fo telegph; t shows tht gve ppopte vles fo the pt d dte of le sgls e tsmtted - dmped tme - eltvel dstoted fom f Ch V App 6 Cldl d Sphel Wves The pple of speposto leds to othe mpott foms of soltos fo o dffeetl eqtos ptl to ldl d sphel wves Cldl Wves The wve eqto two dmesos tt s solved fo t θ { osθ sθ } ep{ t} ep whee s me whh e tl hose Itegto of ths "ple wve" wth espet to the deto gle θ elds the ew solto t π t { os θ } dθ πe J t e ep whee the pol oodte s toded Ths solto epesets stdg wve os s Ths ottoll smmet solto of the wve eqto

20 so-lled ldl wve s gve the Bessel fto Jo Ths solto s egl t the og B the speposto of ple wves we lso ostt solto whh s sgl t the og oespodg to dto poess f wth soe t the og Fo ths ostto we se mpope wves We osde the omple pth L of tegto the θ -ple llstted Fge 5 f Vol I Ch VII d fom the omple tegl whee t osθ t e e dθ πe H L H deotes the Hkel fto The s solto of the wve eqto Both ldl wves e peod t of ose d oslltg t ot peod the spe vle Sphel Wves I thee-dmesol spe the stto s slghtl dffeet Fom the solto { t} ep{ G α β γz } ep{ t}w ep we ot tegtg w ove the t sphee ew fto whee v α β γz e Ω dω α β γ -spe the d Ω s the sfe elemet of the t sphee Se ths fto s evdetl vt de otto of the oodte sstem we m fo pposes of llto set z Itodg pol oodtes θ α β γ -spe we ot o π π v d e osθ sθdθ Ths ep{ t} s / v π s s stdg sphel wve ottoll smmet egl t the og d deved the speposto of egl pogessg ple wves Wves wth sglt t the og whh oespod to dto pheome mst g e fomed mes of mpope ple wves The pth of tegto L of Fge 6 leds to

21 e v osθ e sθdθ π L I tems of el qttes we hve smlteosl ostted the two sphel wve foms os / d s / the seod of whh s the egl oe jst ompted We oseve: The sphel wve fom m e deved speposto of the ple wves ep{ α β γz } fo t posto of the pot z wth z > Idepedetl of the posto of ths pot e π π d e L holds wth omtted α βγz sθdθ z The es vefto of m e Se the wve eqto does ot ot dspeso tems we ostt the ottoll smmet wve π π π f tαβγzsθdθd wth t fto f λ Ths epesso s vt de otto; ths we m evlte the tegl ssmg I pol oodtes we ot π π f t osθ sθdθ π [ F t F t ] whee F the defte tegl of f s t Ths fo t twe dffeetle F the fto F t F t e s solto Lkewse eh of the ftos F t d F t tself s lso solto Ths e esl see mkg ppopte hges the fto f o F s well s det vefto These soltos whh ovosl e sgl t the og epeset pogessg sphel wves tteted spe Moeove these e the ol soltos of the wve eqto thee-dmesol spe whh deped sptll o loe ese

22 fo fto t the epesso eomes zz f Vol I p 5 Hee the wve eqto goes ove to the eqto [ ] whose geel solto odg to Ch I p 6 s F t G t wth t F d G tt tt 6 Tpl Polems Dffeetl Eqtos of Mthemtl Phss Itodto Remks To fd the geel solto e the totlt of ll soltos of ptl dffeetl eqto s polem whh hdl eve os The gol sll s to sgle ot spef dvdl soltos ddg fthe odtos to the dffeetl eqtos Fo depedet vles these ddtol esttos sll efe to -dmesol mfolds whh sometmes ppe s odes sometmes s tl mfolds d sometmes s dsott sfes of doms wth whh the soltos e to e fod od tl d jmp odtos I ptl tl vle polems o Ch polems wee dsssed : Alog the ple t vles fo the solto d s the se eqes fo ts devtves wth espet to t wee pesed s ftos of the oodtes L We seek solto fo t whh t t epesets the pesed tl stte Sh soltos of tl vle polems m e oted fo t < so tht the mfold t les wth the dom of defto of the solto I othe wods the stte t t m e tepeted s the eslt of pevos stte whose otto to the fte s goveed the sme lws I the se of dffeetl eqtos of fst ode fo whh the solto of the tl vle polem ws ostted Chpte II otto of ths kd s tomt Fo polems of hghe ode sml eslt s oted Ch I 7 whee lt dffeetl eqtos d lt tl odtos e teted Howeve to ssme tht ethe the dffeetl eqtos o the tl

23 odtos e lt s geel tfl estto ; moeove eve fo lt dffeetl eqtos the lt hte of ll the soltos s ot evdet po It s theefoe esole to osde tl o od vle polems wthot egd to otto of soltos eod these odes A tpl od vle polem