Undulating Relativity ABSTRACT

Unling Relii h: lfe is Mei Gi -il: li@sj.sp.g.b STRCT The Speil The f Relii es s esls h pesenl e nsiee ineplible n enne sieniss, n: -The ilin f ie, n -The nin f he enz engh. The slin hese he ien he h he eelpen f he Unling Relii (UR) he, hee he Tepl iin is e he iffeenes n he e f he ligh ppgin n he lenghs e nsns beeen lns in nif elie een. The Unling Relii pies nsfins beeen he lns h iffes f he nsfins f enz f: Spe (,,z), Tie (), Spee ( ), elein ( ), neg (), Men ( p ), e ( ), leil iel ( ), Mgnei iel ( ), igh eqen ( ), leil Cen ( J ) n leil Chge ( ρ ). he nlsis f he eelpen f he Unling Relii, he flling n be snhesize: - I is he ih piniples pleel n phsis; - The nsfins e line; - Keeps nhe he liin piniples; - Cnsies he Glile s nsfin isin n eh efeenil; - Ties he Spee f igh n Tie niqe phenenn; - The enz fe n be ine b isin pes f ile es, n - Wih he bsene f he spil nin f enz, eh he se lssil esls f he speil elii ning is n neess s nle n he pple effe. h, he Unling Relii n he Speil Relii f lbe insein eplin he epeiene f Mihel- Mle, he lngiinl n nsesl pple effe, n spplies el ienil flin : bein f zenih gα / esnel s fl ( ) n n.. Mss ( ) ih eli ( ) = [esing ss ( )]/... Men p.. Relin beeen en (p) n neg ().. p. Relin beeen he elei fiel ( ) n he gnei fiel ( V ). μο.i i-sn s fl μ..π.r is e glie s e eqin ψ(,).si nπγ ; /55

lng ih he eqins f nsfins beeen efeenes f he UR, e ge he inine f shpe Mell s eqins, sh s: ρ i ; i. i. R. εο R μο. j εο.μο. ; R εο.μο.. We ls ge he inine f shpe he eqin f e n eqin f nini ne iffeenil shpe: z ρ.j. Ohe Ws: 9 plining he Sgn ffe ih he Unling Relii. plining he epeiene f Ies-Silell ih he Unling Relii. Tnsfin f he pe f lins beeen efeenils in he Speil The f Relii. inei. Rih C. Tln. Veliies psiin. 5 Inine. 6 Tie n eqen. 7 Tnsfin f. enz. 8 The Mihelsn & Mle epeiene. 9 Regessin f he peihelin f Me f 7,. 9 ne f Me s peihelin f.79. Inei. Inei (lifiins) ne f Me s peihelin f.79 lle ih he Unling Relii. Spil efin. Spe n Tie en. Viinl Piniple. Viinl Piniple ninin. 5 gihi Spil. 6 Me Peihelin ne f.99". 7 neen f Peihelin f Me f.99 "n Cniins" 8 Siplifie Peielli ne /55

h: lfe is Mei Gi. e-il: li@sj.sp.g.b Unling Relii Tnsl: Rlf Ms Venâni e-il: se@e..b Tnsfin spe n ie The Unling Relii (UR) eep he piniple f he elii n he piniple f Cnsn f ligh spee, el lie lbe insein s Speil Relii The efine: ) The ls, ne hih he se f phsis sses e hnge e he se, eihe hen efee eeine sse f ines n he h hs nif nslin een in elin he fis. b) n f ligh es in he esing ines sse ih eeine eli, h is he se, hee his is eie b esing b b b in een (hih eplins he epeiene f Mihel-Mle). e s igine fis h bsees O n O (in ), ing in nif nslin een in elin eh he, h is, he bsee n e eliel eh he. In his, he bsee O gehe ih he is,, n z f sse f engle Cesin ines, sees he bsee O e ih eli, n he psiie is, ih he espeie pllel is n sliing lng ih he is hile he O, gehe ih he, n z is f sse f engle Cesin ines sees O ing ih eli, in negie iein s he is ih he espeie pllel is n sliing lng ih he is. The bsee O eses he ie n he O bsee eses he ie ( ). e s i h bh bsees se hei ls in sh h, hen he iniene f he igin f he ine sse hppens = = ze. In he insn h = =, f ligh is pjee f he n igin bh bsees. fe he ie inel he bsee O ill nie h his f ligh h silnesl hi he ines pin (,, z) ih he f he O bsee ih eli n h he igin f he sse f he O bsee hs n he isne lng he psiie f he is, nling h: + + z =. =.. The se fe he ie inel he O bsee ill nie h his f ligh silnesl hi ih he bsee O he ine pin (,, z ) ih eli n h he igin f he sse f he bsee O hs n he isne n he negie f he is, nling h: + + z =. = +.. Ming. eql. e he + + z = + + z..5 ese f he se = en z = z, h siplif.5 in =..6 T he bsee O = (.) h pplie in.6 spplies = ( ) f hee..7 T he bsee O = + (.) h pplie in.6 spplies ( + ) = f hee /55

..8 Tble I, nsfins he spe n ie =. = +. =.. =.. z = z.. z = z...7 he eqin sse fe b. n. e fin.8 = (nsieing > e >).9 h enses he inine f he spe in he Unl Relii. he eqin sse fe b.7 n.8 e fin. =.. If in. = hen =, h pplie in. spplies,. =.. If in. = n = hen... T he bsee O he piniple f ligh spee nsn gnees h he pnens, n z f he ligh spee e ls nsn lng is is, hs z z,, z. n hen e n ie =.. Wih he se f.7 n.9 n. e n ie = =..5 iffeeniing.9 ih nsn n, else, nl he ie ing e he,.6 b f.5 hen eing n nsns, he ezns iffeenil f he se s....7 n /55 in.5 s ls be nsn bese f his he s be eql ze f hee e nle, h is el

T he bsee O he piniple f Cnsn f eli f ligh gnees h he pnens,, n z f eli f ligh e ls nsn lngsie is is, hs z z,, z,.8 n ih his e n ie, =..9 Wih he se f.8,.9, n.9 e n ie = =.. iffeeniing.9 ih n nsn, h is, nl he ie ing e he,. b f. hen eing n nsn he iisins iffeenil f el lie.8... n in. ls he be nsn bese f his he s be eql ze f hee e nle Repling. n.9 in. e he, h is. =.. T he bsee O he e psiin f he pin f ines (,,z) is R i j z,. n he e psiin f he igin f he sse f he bsee O is R i j R i..5 T he bsee O, he e psiin f he pin f ines (,,z ) is R i j z,.6 n he e psiin f he igin f he sse f he bsee O is R i j R i..7 e.9,.5, n.7 e he, R s. is eql.5 pls.6 e he R R R R R R R..8..9 ppling.8 in.9 e he, R R R.. 5/55

6/55 T he bsee O he e eli f he igin f he sse f he bsee O is i j i R.. T he bsee O he e eli f he igin f he sse f he bsee O is i j i R...5,.,., n. e fin he flling elins beeen n... Obsein: in he ble I he fls.,.., n.. e he pnens f he e.9 n he fls.,.., n.. e he pnens f he e.. f eli nsfins n iffeeniing.9 n iiing i b.7 e he K R R R.. iffeeniing. n iiing i b. e he K R R R.. Tble, f eli nsfins n K. K. K. K. K.. K.. K z z.. K z z.. K.5 K. K.5 K.6

7/55 Mlipling. b iself e he..7 If in.7 e e = hen = s i is eqie b he piniple f nsn f eli f ligh. Mlipling. b iself e he..8 If in.8 e e = hen = s i is eqie b he piniple f nsn f eli f ligh. If in. e e = hen s i is eqie b he piniple f nsn f eli f ligh. If in. e e = hen s i is eqie b he piniple f nsn f eli f ligh. Reeling.7 n.8 e he =..9 =.. The ie elins beeen he ies n eliies f pins in spe n be bine ih he eqliies ing f., h pplie in.7,.,., n.5 sppl,.,.,...

bein f he zenih T he bsee O lng ih he s =, = n z =, n he bsee O lng ih he h e he he njn.,, z =, z el s feseen b he piniple f elii. T he bsee O he ligh ppges in iein h es n ngle ih he eil is gien b / ngα.5. h is he bein fl f he zenih in he speil elii. If e inee he bsees e l he he njn.,, z =, z / ngα.6. h is eql.5, ih he negie sign iniing he n iein f he ngles. Cnsieing in., elin he pps hen esnel s fl / n he eli f ligh eliil he e, ill be he eli f ligh eliel he lb / n / n / n n n n n Igning he e / e he n n n n n n igning he e / n e he he esnel s fl n n n n he eli f e in n..7 8/55

Ming z n s pple effe epling hen, hen e ill efine z in.5 e he n.7 e fin hee en he piniple f elii.8 Resling in he epessin sei n inible beeen he bsees. T he bsee O n epessin in he fl f ψ, f.9 epesens e h ppges in he iein f R. T he bsee O n epessin in he fl f ψ, f. epesens e h ppes in he iein f ppling in.8 λ λ e K h pplie in R. π π,,.,.9,.,.5, n.6 e he λ λ λ λ, K. λ λ sppl, K n K.. Cnsieing he elin f Pln-insein beeen eneg ( ) n feqen ( ), e he he bsee O h n he bsee O h h eple in. sppl K n K.. If he bsee O h sees he bsee O ing ih eli in psiie he is, eis es f feqen n eli in psiie he is hen, ing. n he bsee O ill ese he es ih eli n feqen,. h is el he lssi fl f he lngiinl pple effe. If he bsee O h sees he bsee O ing ih eli in he negie f he is, eis es f feqen n eli, hen he bsee O ing. n ill ese es f feqen n eli in pepenil plne he een f O gien b γ γ,.5 h is el he fl f he nsesl pple effe in he Speil Relii. iffeeniing. n iiing i b.7 e he Tnsfins f he eleins n / K / K K.. K K K K iffeeniing. n iiing i b. e he / K K / K K K K K.. 9/55

Tble, nsfins f he eleins n.. K K K K.. K K K K K K.. K K z z z z K K.. z z K K.8.9 K K K.5 K.6.... he bles n e n nle h if he bsee O hen i is ls he bsee O n is pepenil. ze n s he es he eqies.. ze n z, hs is pepenil z, iffeeniing.9 ih he eliies n he ies hnging e he,, b nsieing.6 e he,.7 Whee epling.5 n iiing i b.7 e he, We n ls eple. in.7 n iie i b. eing K K..8..9 K K The ie elins beeen he les f he eleins n f pins in spe n be bine ih he ing f., h pplie in.8 n.9 sppl n.. Th n ls be ee f. n. if e se he se eqliies ing f.. efine s hee Tnsfins f he Mens p n p p n p n n,. sblizes he fnin sses f he les f eliies n. We ill he he elins beeen n he esing ss, nlzing he elsi llisin in plne beeen he sphee s h f he bsee es lngsie he is ih eli = n he sphee s h f he bsee O es lngsie he is ih eli = -. The sphees hile bsee in elie esing e ienil n he he ss. The nsiee llisin is sei in elin pllel line he is n pssing b he ene f he sphees in he en f. Cllisin. efe n fe he llisin he sphees he eliies bsee b O n O ing he flling ble gen f ble /55

Sphee Obsee O Obsee O efe s s ze, s s, Cllisin s s, s fe s s ze, s Cllisin s s, s s s ze, s s, s s ze, s T he bsee O, he piniple f nsein f ens esblishes h he ens p p n, f he sphees s n s in elin he is n, ein nsn befe n fe he llisin hs f he is e he s s s s s s s s s s s s hee epling he les f he ble e he f hee e nle h, n f he is s s s s s s s s s s s s hee epling he les f he ble e he,, siplifing e he,, hee hen bees b is eql he esing ss hs, ih elie eli. h pplie in. spplies p Wih he se pees e l he f he O bsee.., /55

. n p.. Siplifing he siblg e ill p. n. h siplif he ens in p n p.. ppling. n. in.9 n. e he K n K efining fe s Nen e he inei eneg n, s p n.r.r... p.r.r. Reeling. n. n iffeeniing e he,., ih his e n efine hen n, h pplie in he fls f inei eneg spplies n hee n,.5.6 e he l enegies s in he speil elii n he esing eneg..7 ppling.6 in. e he el...6,.,., n. e fin p n p.8 ienil elins he Speil Relii. Mlipling. n. b e ge /55

n p p p p.. Tble, nsfins f ens p n p p p.9 p p. p p. p p. p p.. p p.. p z pz.. pz p z.. K. K... K. K..5.5.6.7 p.6.7.8 p.8 We eqin f is e glie The bsee O ssies esing pile in is igin he flling ppeies:.9 -Resing ss -Tie -Resing neg -eqen -We fnin h h ψ senπ ih = nsn. The bsee O ssies pile ih eli he flling: -Mss -Tie -neg (f. hee ) (f.7 ih n ) (f. ih n ) /55

-eqen /h (f. ih n ) -isne = (f. ih = ) -We fnin -We lengh ih ψ sen senπ senπ h h λ λ (f.9 ih p p p p p ) T g b he O bsee efeenil hee, e ill nsie he flling ibles: -isne = (f. ih = ) -Tie -eqen (f.8 ih ) (f. ih ) -Veli (e.) h pplie he e fnin spplies ψ senπ senπ sen b s n hen ψ ψ. 5 Tnsfins f he es n iffeeniing.9 n iiing b.7 e he p p K K K K iffeeniing. n iiing b. e he,. p p K K K K he sse fe b 5. n 5. e he.... 5.. 5., 5. h is n inin beeen he bsees in he Unling.Relii. /55

Tble 5, nsfins f he es n. K 5.. K. K. K 5. / K 5.. / K 5.5. z z/ K 5.. z z / K 5.5. 5... 5. 6 Tnsfins f he ensi f hge ρ, ρ n ensi f en J n J q Mlipling. n. b he ensi f he esing elei hge efine s ρ e he ρ ρ ρ ρ ρ ρ J J ρ ρ ρ ρ n ρ ρ ρ J J ρ Tble 6, nsfins f he ensi f hges ρ, ρ n ensi f en J n J J J ρ 6. J J ρ 6. J J ρ 6. J J ρ 6. J J 6.. J J 6.. J z Jz 6.. Jz J z 6.. J ρ 6.5 J ρ 6.6 ρ ρ ρ ρ 6.7 6.8 ρ ρ K 6.9 ρ ρ Κ 6. he sse fe b 6. n 6. e h 6.9 n 6.. 7 Tnsfin f he elei fiels, n gnei fiels, ppling he fes f enz q n q q q q. K K n q q K. hee e ge he inine f flling pnens f eh is. q. 5. 5.5 6.. 6. in 5. n 5. e he, h siplifie bee. K n f beeen he bsees s nseqene f 5. n he 5/55

z z zz z z K z z z z K z z K zz z z z z K z z z z K z z K 7. 7.. 7.. 7. 7.. 7.. T he njn 7. n 7. e he slins esibe in he bles 7 n 8. Tble 7, nsfins f he elei fiels, n gnei fiels e 7. 7. K K z K K 7.. z K K 7.. z z z K 7.. z K K K 7.5 7.6 z 7.5. z z z 7.5. z z K 7.7 K 7.8 7.7. K z z K z z z 7.9 z z 7.9. z 7.. 7.6. 7.6. 7.8. 7. 7.. Tble 8, nsfins f he elei fiels, n gnei fiels e. K. K 7. z 7.. z K K 7.. z z 7.. z z K K 7.. 7. 7. 7.. 7.. z z 7.. z z 7.. 6/55

Relin beeen he elei fiel n gnei fiel ze n he. T he bsee O his fiel is epesene ih bh pnens, being he If n elei-gnei fiel hs he bsee O he ngh gnei pnen elei pnen gnei fiel esibe b he njn 7.5 n hás s pnens z ze, h e eqilen, z, 7.5. 7.6 l f i-s The bsee O ssies esing elei hge, nifl isibe lngsie is is he flling elei-gnei ppeies: -ine ensi f esing elei hge ρ -Ngh elei en I ze -Ngh gnei fiel ze ze ρ -Ril eleil fiel f le z n pin f is R πεr he pnen ze. q z ih T he bsee O i eles n elei hge nifl isibe lngsie is is ih eli hih i ssies he flling elei-gnei ppeies: -ine ensi f he elei hge -lei en I ρ ρ -Ril eleil fiel f le ze ze n ) ρ ρ (f 6.7 ih = ) -Mgnei fiel f pnens ze (ing he njns 7. n 7.5 ih z, ρ I πε R R hee I R, z n le, being in he eil f 7.7 hee is ni e pepenil he eleil fiel n ngen he ifeene h psses b he pin f is R z bese f he njn 7. n 7.6. ze. 7/55

8 Tnsfins f he iffeenil pes Tble 9, iffeenil pes z z K K 8. 8.. 8.. 8. z z K K 8. 8.. 8.. 89. he sse fe b 8., 8., 8., n 8. n ih.5 n. e nl fin he slins / n /. 8.5 hee e nle h nl he fnins ψ (.9) n ψ (.) h sppl he niins ψ / ψ n ψ / ψ, 8.6 n epesen he ppgin ih eli in he Unling Relii iniing h he fiel ppges ih efinie eli n ih isin being pplie. n.8. ese f se e n ls ie he he is ψ / ψ ψ / ψ, n ψ z / ψ z ψ z z / ψ,. 8.7 he nsfins f spe n ie f he Unl Relii e ge Jb s hee J,,z,,,z, K n J,,z,,,z, K, 8.8 ibles ih n s nseqene f he piniple f nn f he ligh eli b e eql is J J n ill be eql ne J J hen. The e eqin he bsee O is Inine f he e eqin z ze hee ppling he fls f bles 9 n. e ge f hee e fin z K K ze 8/55

9/55 ze z K K K 6 6 6 6 6 h siplifing spplies ze z K K K 6 hee eeing he es e fin ze z K K K 8.9 b f 8.5 n. e he ze / h pplie in 8.9 spplies he e eqin he bsee O ze z. 8. T en he efeenil f he bsee O e ill ppl 8. he fls f bles 9 n.8, geing ze K K z f hee e fin ze z K K K 6 6 6 6 6 h siplifing spplies ze z K K K 6 hee eeing he es e fin ze z K K K b f 8.5 n.8 e he ze / h eple in he eee eqin spplies he e eqin he bsee O. Inine f he Cnini eqin The nini eqin in he iffeenil f he bsee O is ze z Jz J J ρ ze.j ρ 8. hee epling he fls f bles 6, 9, n. e ge ze z Jz J ρ J K ρ K K

ing he peins e fin ρ ρ ρ ρ J J ρ ρ J Jz z ze h siplifing spplies ρ ρ J J J Jz z ze hee ppling ρ ρ J ρ ih nsn e ge J J Jz z ρ J J Jz ze z ze 8. h is he nini eqin in he iffeenil f he bsee O. T ge gin he nini eqin in he iffeenil f he bsee O e ill eple he fls f bles 6, 9, n.8 in 8. geing K K ρ K J J z z J ρ ze ing he peins e fin ρ ρ ρ ρ J J ρ ρ J J z z ze h siplifing spplies ρ ρ J J J J z z hee ppling J ρ ih nsn e ge ρ ρ J J J z z ze ρ J J J z ze z h is he nini eqin in he iffeenil f he bsee O. Th in he iffeenil f e ien his Inine f Mell s eqins ze Wih eleil hge T he bsee O T he bsee O z z z z z z z z z Jz z 8. z z 8.5 8.7 8.9 8. 8. z z z z z z z J z z 8. 8.6 8.8 8. 8. 8. /55

z J z z J z 8.5 8.7 z z z z J J Wih eleil hge ze n J J ze T he bsee O T he bsee O z z 8.9 z z z z 8. z z z z 8. z z 8.5 z z z z 8.7 z z z z 8.9 z z 8. z z z z 8. z z 8.5 8.6 8.8 8. 8. 8. 8.6 8.8 8. 8. 8. We ense he inine f he f Gss in he iffeenil f h f he bsee O is z z hee epling he fls f he bles 6, 7, 9, n.8, n nsieing nsn, e ge z z K K K K ρ Κ ε ing he ps, sing n sbing he e z z z z z z z z K ρk, e fin z 8. h eeing esls z z z z ρk /55

hee he fis penheses is 8.5 n bese f his eql ze, he sen bln is eql J gen f 8.5 n 8.5 esling in z ρ z z f hee e ge z h is he f Gss in he iffeenil f he bsee O. ρ ρ T e he inese e ill eple in 8. he fls f he bles 6, 7, 9, n., n nsieing nsn, e ge z z K K K K ρ Κ ε ing he ps, ing n sbing he e z z z h eeing esls in hee he fis bln is z z z z J z K z z z z z ρ K, e ge ρ K 8. 8.5 n bese f his eqls ze, he sen bln is eql gen f 8.6 n 8.5 esling in f hee e ge bsee. z ρ z z z ρ ρ h is he f Gss in he iffeenil f he O Peeing his e n pe he inine f f f ll he he eqins f Mell. 9 plining he Sgn ffe ih he Unling Relii We s nsf he sigh een f he bsees O n O se in he ein f he Unling Relii in plin il een ih nsn is. e s igine h he bsee O sees he bsee O ning n ih ngenil spee in lise (C) eqls he psiie se f he is f UR n h he bsee O sees he bsee O ning n ih ngeil spee in nlise (U) eqls he negie se f he is f he UR. In he en = = ze, he bsee O eis s f ligh f he n igin bh bsees, ne in nlise f U n nhe in lise f C, heefe U = C n U = C, bese is he spee f he nsn ligh, n U n C he ie. /55

In he en = = ze he bsee O ls eis s f ligh f he n igin bh bsees, ne in nlise (seless) f U n nhe ne in lise f C, hs U = C n U = C bese is he spee f he nsn ligh, n U n C he ie. Reiing he eqins.5 n. f he Unling Relii (UR): =..5 =.. Ming = = ( f ligh pjee lngsie he psiie is ) n spliing he eqins e he: 9. 9. 9. 9. When he igin f he bsee O ees he nlise f he bsee O, ill be he isne f he bsee O n silnesl ill ee is lise f ligh he se pin f C U he bsee O, in sei psiin he iee h ges hgh he bsee O bese U C U C n U C U C, flling he f eqins be e he: U R C R C 9.5 C R U R C 9.6 When he igin f he bsee O ees he lise f he bsee O, silnesl ill ee is n lise n ill be he isne,, n be e he: f he bsee O, hen flling he eqins C U C R R C C 9.7 C R R C 9.8 The ie iffeene he bsee O is: R R R C C 9.9 The ie iffeene he bsee O is: R R R C C 9. Repling he eqins 5 in e pe h he nfi he nsfins f he Unling Relii. /55

plining he epeiene f Ies-Silell ih he Unling Relii We shl eie he eqins (.) he e lengh in he Unling Relii: n,. Ming = = ( R f ligh pjee lngsie he psiie is ), e he he eqins: n,. If he bsee O, h sees he bsee O ging ih he eli in he psiie f he is, eis es, penien f esing se in is igin ih eli n e lengh in he psiie f he is, hen ing he eqin. he bsee O ill ese he es ih eli n he e lengh ing he fls: n,. If he bsee O, h sees he bsese O ging ih eli in he negie f he is, eis es, penien f esing se in is igin ih eli n he e lengh in he psiie f he is, hen ing he eqin. he bsee O ill ese es ih eli n e lengh ing he fls: n,. The esing ses in he igin f he bsees O n O e ienil hs. We lle he ege e lengh f he ese es, sing he eqins. n., he lef sie in eh eqin: We lle he iffene beeen he ege e lengh n he eie e lengh b he ses : /55

. Refeene hp://.bbin.ne/phsis/fj7.h Ies-Silell (ninin) The pple s effe nsesl he Unling Relii s bine in he s flls: If he bsee O, h sees he bsee O, es ih he spee in negie he is, eis es ih he feqen n he spee hen he bsee O ing. n ill ese es f feqen n spee in pepenil plne he een f O gien b.5 e ill he ze n n he se feqen lie his he nsesl feqen ih his e n ie he elin beeen.5 Wih e he he elin beeen he lengh f he nsesl e he se e n he lengh f.6 The iin f he lengh f he nsesl e in he elin he lengh f he se e is:.7 h is he se le gen in he The f Speil Relii. ppling.7 in. e he.8 Wih he eqins. n. e n ge he elins.9,., n. esibe s flls.9 n f his e he he fl f spee.. ppling. n. in.6 e he. 5/55

.8 n. e nle h.. S h e he les f n bine f he Ies-Siell epeiene e n ele, n nle hehe hee is n he spe efin peie in he The f Speil Relii. Tnsfin f he pe f lins beeen efeenils in he Speil The f Relii The elinship ihin he pe eelpe b he fes beeen efeenils is ien in he Speil The f he Relii in he flling :... The efiniin f he pnen f he fe lng he is is: p. lins, he piniple f ligh spee nsn gnees h he pnen f he ligh spee is ls nsn lng is is, hs The fl f eneg is ze. f hee e he. nsn, ensing h in he efiniin f eneg e he ppling 5 in e hee:. h ppling in n e he..5,... hee e fin h...6 esl eql 5. f he Unling Relii h n be epeienll pen, nsieing he Sn s he se. inei The The f Unling Relii hs s is fnenl i he neessi h ineil efeenils be ne elsiel s hse nes in hih f ligh eie in n iein f is igin spes in sigh line, h is heill esibe b he fle (.,.8, 8.6 e 8.7) f he Unling Relii: z z, z,. z z,, z.8 Wle Vig e in.887 he line nsfin beeen he efeenils s he bsees O e O in he flling : 6/55

.. Wih he espeie inee eqins:.. Whee,, n e nsns n bese f he se e n nsie he es ih, z n, z. We n h n e pjeins f he s f lighs n h spe ih Cnsn spee (e he nsn piniple f he R f ligh), eie in n iein f he igin f he espeie ineils efeenil he en in hih he igins e inien n he en hee: = = ze.5 bese f his in he eqin. he en hee = ze e s he = ze s h e ls he = ze, e n sse h hen = ze, ls be eql ze, bese if he speing hppens in he plne z e ill he = ze pls ze. We shl eie he ee eqins ( = ze):.6.7 Wih he espeie ee inee eqins:.8.9 If he speing hppens in he plne z e he = ze n iiing.6 b.7 e he:. hee is he le f he spee in hih he bsee O sees he efeenil f he bsee O ing lngsie he is in he psiie bese he sign f he eqin is psiie. If he speing hppens in he plne z e he = ze n iiing.8 b.9 e he:. hee is he le f he spee in hih he bsee O sees he efeenil f he bsee O ing lngsie he is in he negie bese he signl f he eqin is negie. The eqin.6 esibes he nsn piniple f he spee f ligh h s be sse b he eqins.6.9:.6 ppling.6 n.7 in.6 e he: hee e he: 7/55

hee ing = in he bes in n in he sigh bes e he he eqli beeen bh sies f he eql signl f he eqin. pplling = in e he. pplling = in. e he. Th pplie in. splies:,. s (, ) is eql he fnin epening f he ibles n. ppling.8 n.9 in.6 e he: hee e he: hee ing = in he be in n in he sigh be e he he eqli beeen bh sies f he eql signl f he eqin. ppling = n. in e he:,. s (, ) is eql he fnin epening n he ibles n. We s e he flling ning ing.5 n.6: K. K.5 K K s he eqin (, ) f. n (, ) f. s be eql, e he: 8/55

.6 Ths: K K l eql...7 Reiing he eqins.6,.7,.8 n.9 ing he fnin f, n e he:.6.7 Wih he espeie inee ee eqins:.8.9 We he he eqins.6,.7,.8 n.9 finls epling b he espning fle:.6.7 Wih he espeie inee finl eqins:.8.9 Th e el he eqins f he ble I s n hen he elins beeen n e..8 We ill nsf (.) fnin f he eleens,, n f (.) fnin f he eleens, n, epling in. he eqins.8,.9 n.8: 9/55

Th is el he eqin.. We ill nsf (.) fnin f he eleens,, n f (.) fnin f he eleens, n, epling in. he eqins.6,.7 n.8: Th is el he eqin.. We he lle he l ifeenil f (, ) (.): s: K n K.9 e he: K K hee ppling.8 e fin:. K K hee e nle h fnin f n is nsn. We he lle he l ifeenil f (, ) (.): s: K n K. e he: K K. /55

hee ppling. e fin: K K hee e nle h fnin f n is nsn. The eqins. n.8 epesen he bsees O n O he piniple f nsn f he ligh spee li f infiniel sll he infiniel big n en h in he Unling Relii he spe n ie e ese silnesl. The shln be inepee ih epenen beeen spe n ie. The ie hs is n inepein h n be nes if e nlze eeine bsee he eissin f s f ligh f he insn =ze. If e he ies e ge, f eh f ligh, e ill ge esl ih n se f he phsis. If in he insn = = ze, he bsee O eis s f ligh, ne lngsie he is n he he lngsie he is, fe he inel f ie, he s hi f he bsee O, silnesl, he pins n he isne f he igin, lhgh f he bsee O, he pins n be hi silnesl. bh s f lighs be silnes bh bsees, he s hi he pins h he he se is in elin he is n h pie he se ie f bh bsees ( = n = ), hih ens h nl ne f ligh is neess he he ie beeen he efeenils. ing, bh efeenils f he bsees O n O e ineil, hs he ligh spes in sigh line ing h is ene b he fnenl i f he Unling Relii, bese f his, he iffeene in eliies n is e nl iffeene in ie beeen he efeenils... We n ls ele n ineil efeenil f hih he ligh spe in sigh line ing h is ene b he fnenl i f he Unling Relii, ih n elee ing efeenil f hih he ligh spe in e line, nsieing h in his se he iffeene n isn e nl he iffeene f ie beeen he efeenils. ing, if he bsee O he insn = = ze, eis f ligh f he igin f is efeenil, fe n inel f ie, he f ligh his he pin ih ines (,, z, ) he isne f he igin f he bsee O, hen e he: fe hiing he pin he f ligh sill spe in he se iein n in he se, fe n inel f ie, he f ligh his he pin ih ines ( +, +, z + z, + ) he isne he pin, hen e he: n ih his e ge: The gee f spe n ie in he Unling Relii is size in he fige bel h n be epne n pins n seel bsees. /55

In he fige he ngles he elin n e eql he flling segens: O O O is eql O O O O O O O is eql O O O O n e pllel he flling segens: O O O O O is pllel O O is pllel O X X is pllel X X The sine f he ngles f inlinin n he s f he bsees O n O ing. n. e: s / s s s / s K. sen n ih his e he: sen K. s / s s s / s K.5 /55

n ih his e he sen sen K.6 The sine f he ngle ih inesein f s eql : s s s K K K K.7 n ih his e he: The inine f he Unling Relii. sen sen sen K K.8 s shs he hn f ll pe hpheses f spe n ie in he The s is eql he Jbins f he nsfins f he spe n ie f he pie I, hee he ils s s n K K J i j e nsiee ibles n e eie.,,z,,,z, K K / K 8.8 J l K,,z,,,z, K K / K K 8.8 Rih C. Tln The Tnsfins f he Men f Unling Relii s eelpe bse n he epeiene ne b eis n Tln, ing he efeene []. Whee he llisin f sphees peseing he piniple f nsein f eneg n he piniple f nsein f en, shs h he ss is fnin f he eli ing : hee is he ss f he sphee hen in esing psiin n /55 he le f is spee. nlzing he llisin beeen ienil sphees hen in elie esing psiin, h f he bsee O e ne S n S e ing lng he is in he n ih he flling eliies befe he llisin: Tble sphee S sphee S ze ze z ze z ze he bsee O he se sphees e ne S n S n he he eliies,, i zi ze befe he llisin lle ing he ble s flls:

The eli f he sphee S is eqls :. The nsfin f ing. f Tble is:. Th pplie in spplies: The eli f he sphee S is eql : Tble Sphee S Sphee S ze ze ze z ze z ze ze he bsees O n O he sphees he he se ss hen in elie esing psiin. n f he bsee O he sphees llie ih eliies f eql le n ppsie iein bese f his he en p p nll heseles ing he llisin, fing f bief ie nl ne b f ss. ing he piniple f nsein f en f he bsee O e ill he ipse h he en befe he llisin e eql he en fe he llisin, hs: Whee f he bsee O, is he bi eli h sppsel f bief ie ill ls see he sses nie ing. s he sses i he iffeen eliies n he sses ing hei n eliies, his eqin nn be siplifie lgebill, hing his iin f sses: T he lef sie f he eql sign in he eqin e he: /55

5/55 ze ze T he igh sie f he eql sign in he eqin e he: ppling in he eqin f nsein f en e he:. hee e he: s f he bsee O he sses nie ln e enil lngsie he bsee O hih is neible if e nsie h he insns e iffeen hee sppsel he sses l be in esing psiin f he pin f ie f eh bsee n h he ss ing ih eli is bigge hn he ss in esing psiin. If e pee ih hese ibles in line e l he:.

hee e nle h hih s be eql he peis le f, h is: elin beeen n h is bine f Tble hen he eli ing e he sphee in esing psiin. Refeene Millenni Relii Veliies psiin UR: hp://.elii.ne/miefs/vcp_si_sb_w.h h espns f he bsee O e s ie he nsfins f eni. enz f spe n ie in he Speil The f Relii:...b.b z z. z z... he e bin he eqins f eli nsfin: z.5.5b z.5 z.6.6b z.6 e s nsie h in elin he bsee O n bje es ih eli: 5,5. / s,5. 6/55

n h he eli f he bsee O in elin he bsee O is: 5,5. / s,5. The eli f he bje in elin he bsee O s be lle b he fl.6: 5 5,5.,5. 5,5..,5. 5,. 5 5,. / s,8. 5 Whee e se,. / s,. Cnsieing h he bje hs e ing ne sen in elin he bsee O hen ih. lle he ie psse he bsee O : 5,5..,.,,. 5,5. 5,. 5 5,6, 75,69s T he bsee O he bsee O is he isne gien b he fl: 5 5,5..,,5.. T he bsee O he bsee O is he isne gien b he fl: 5.,6 5,5.,9.., 75.,s e n T he isne f he bje, in elin he bsees O n O is gien b he fle: O O O O 5 5,..,,.. 5,6 5,5..,9.., 75 T he bsee O he isne beeen he bje n he bsee O is gien b he fl: 5 5 5,.,5.,9.. O T he bsee O he eli f he bje in elin he bsee O is gien b: 5,9.,9. / s,s 5 (=,) Reling he ies n sing he fl is nl pssible n elsiel hen n ze h isn he se be, e i pssible nesn his e ie he eqins. n. in he fl bel: s. s. 7/55

Whee s n s. The eqins be n be ien s: f, e f,.7 In eh efeenil f he bsees O n O he ligh ppgin ees sphee ih is n h ineep eh he fing ifeene h ppges ih eli. The is n n he psiie f he is n f he ngles n nsn beeen he efeenils. If f he se pi f efeenils e ngles ee ible he ie l be lle n l bee seless f he Phsis. In he eqin f, e he ienil fnin f n, if e he in i nsn n ies ing e ge he n elin beeen he ies n beeen efeenils, hee if e he nsn n ies ing e ill he f eh le f f,. ne le f n beeen iffeen efeenils, n his nlsis is ls li f iiing.5 b e he: s s s Whee s n s Isling he eli e he:..8. ss ss.9 hee e nle h e s he ngles n beeen he efeenils. nsn s h e he he se eli This en f nsn ngles beeen he efeenils s sle he nesies f ebe ingle. 5 Inine The nsfins he spe n ie f ble I, gp. pls.7, in he i f is ien lie his: z z K 5. Th ien in he f bel epesens he se ine nsfins: / z z K 5. We ll s: i z /, ij, K j z 8/55 5.

Th e he fnins i i j i i,,,,, z, 5. Th in he sbli f is ien:. in he inee f i j ij j i ij j 5.5 Whee e se insein s s nenin. The nsfins he spe n ie f ble I, gp. pls.8, in he i f is ien: z z K 5.6 Th ien in he f bel epesens he se ine nsfins: / z z K 5.7 Th e ll s: / z, l, K l z l Th e he fnins,,,,, z, 5.8 5.9 Th in he sbli f is ien:. in he inee f l l l eing K (.7), K (.8) n K. K (.). l l 5. The nsfin ies ij n l he he ppeies: / / i. ij l ij jl I l j K K ji l ji i I i / K / K j 5. 5. Whee ji is he nspse i f ij is he Knee s el. n l is he nspse i f l n / /. l ij l lj I j l K K 5. 9/55

l ji l i I / K / K l i 5. Whee l is he nspse i f l n is he Knee s el. Obsein: he ies n. ij l The pil eiies ij n i i j ele ing nsiee nsn n eql : l n ji is he nspse i f ij e inese f ne nhe b e n hgnl, h is: ji l i i i j f he l iffeenil f he ine pnens h j j, hee in he nsfin i ij he il K is Tble, pil eiies f he ine pnens: i j j i j j i j j i j j K The l iffeenil f he ines in he i f is eql : / K 5.5 Th e ll s: i, / i i j j, K j 5.6 Then e he i j i j j i i j j 5.7 The pil eiies ele ing nsiee nsn n eql : l f he l iffeenil f he ine pnens h l l l, hee in he nsfin i l he il K is /55

Tble pil eiies f he ine pnens: l l l l l l l l K The l iffeenil f he ines in he i f is eql : / K 5.8 Th e ll s:, / l l, K l 5.9 Then e he: l l l l l 5. The Jbins f he nsfins 5.5 n 5.8 e: J J l i j,,,,,,,,,,,, / K / K K K 5. 5. Whee K (.5), K (.6) n K. K (.). The ies f he nsfin n ies n. ls he he ppeies 5., 5., 5. n 5. f he l l he fnin hee he ines ele in he f he esibe s: l l e /55

/55 Th in he i f n ih pesening he fnin bees: l K K Whee epling he ies bel: K K K K Obsein: his ls elin shs h he ie ies in n eql f beeen he efeenils. We ge: l K K Th is he gp 8. pls 8. f he ble 9, iffeenil pes, in he i f. he fnin j i i hee he ines ele in he f j i i e he j i i j esibe s:

/55 i i i i i i i i Th in he i f n ih pesening he fnin bees: j K K Whee epling he ies bel: K K K K Obsein: his ls elin shs h he ie ies in n eql f beeen he efeenils. We ge: j K K Th is he gp 8. pls 8. f he ble 9, iffeenil pes in he i f. ppling 8.5 in 8. n in 8. e siplif hese eqins in he flling :

Tble 9, iffeenil pes ih he eqins 8. n 8. siplifie: K ze 8. 8.. 8.. K 8. 8.5 ze 8. 8.. 8.. 8. 8.5 The ble 9, in he i f bees: 5. / K 5. / K The sqe ies f he nsfins be e nspse f he ies n. Inine f he Tl iffeenil In he bsee O efeenil he l iffeenil f fnin is eql : 5.5 l Whee he ines ele ih he nes f he bsee O ing nsfins 5. n 5.8 n ih pesening he fnin e he:, epling he / 5.6 / K K The lipliin f he ile ies spplies: / / K K / / 5.7 Resl h n be iie in ies: / / / / 5.8 /55

Th pplie he l iffeenil spplies: / 5.9 / eing he peins f he sen e e he: / Whee ppling 8.5 e he: / ze Then e he: / / ze 5. Wih his esl e he in 5.9 he inine f he l iffeenil: l l In he bsee O efeenil he l iffeenil f fnin i is eql : 5. i i 5. i i i j Whee he ines ele ih he nes f he bsee O efeenil ing, epling he nsfins 5. n 5.5 n ih pesening he fnin e he: / i 5. i / K K The lipliin f he ile ies spplies: / / / K K / 5. Resl h n be iie in ies: 5/55

6/55 / / / / 5.5 Th pplie he l iffeenil spplies: / / i i 5.6 eing he peins f he sen e e he: / / Whee ppling 8.5 e he: ze Then e he: ze / / 5.7 Wih his esl e he in 5.6 he inine f he l iffeenil: j j i i 5.8 Inine f he We qin The e eqin he bsee O is eql : 5.9

Whee ppling 5. n he nspse f 5. e he: 5. K K The lipliin f he hee ile ies spplies: K K 5. Resl h n be iie in ies: 5. Th pplie in he e eqin spplies: 5. eing he peins f he sen e e he: eing he peins e he: Whee ppling 8.5 e he: ze 7/55

8/55 Then e he: ze 5. Wih his esl e he in 5. he inine f he e eqin: 5.5 The e eqin he bsee O is eql : 5.6 Whee ppling 5. n he nspse f 5. e he: K K 5.7 The lipliin f he hee ile ies spplies: K K 5.8 Resl h n be iie in ies: 5.9

9/55 Th pplie in he e eqin spplies: 5.5 eing he peins f he sen e e he: eing he peins e he: Whee ppling 8.5 e he: ze Then e he: ze 5.5 Then in 5.5 e he he inine f he e eqin: 5.5 Inine f he eqins 8.5 f line ppgin Repling., 8., 8. in 8.5 e he: ze K K

eing he peins e he: ze Th siplifie spplies he inine f he eqin 8.5: ze Repling., 8., 8. in 8.5 e he: K K ze eing he peins e he: ze Th siplifie spplies he inine f he eqin 8.5: ze The ble in i f bees: p / p p p p p / K / p / p p p p p / K / 5.5 5.5 The ble 6 in i f bees: J / J J J J J 5.55 K J / J J J J J K 5.56 Inine f he Cnini qin The nini eqin he bsee O is eql : J ρ J J J ρ. J J ze J 5.57 5/55

Whee epling 5. n 5.56 e he: / J ρ. J J J ze / K K 5.58 The p f he nsfin ies is gien in 5.7 n 5.8 ih his: ρ.j / / J J J 5.59 eing he peins f he sen e e he: / / J J J J Whee epling J n 8.5 e he: ze Then e he: / / J J ze J 5.6 Wih his esl e he in 5.59 he inine f he nini eqin: ρ.j J J ρ.j J 5.6 The nini eqin he bsee O is eql : J ρ J J J ρ. J J J ze Whee epling 5. n 5.55 e he: / J ρ. J J ze J / K K 5.6 5.6 5/55

The p f he nsfin ies is gien in 5. n 5.5 hen e he: ρ J / / J J J. 5.6 eing he peins f he sen e e he: / J J J J / Whee epling J n 8.5 e he: ze Then e he: / J J ze J / 5.65 Wih his esl e he in 5.6 he inine f he nini eqin: ρ.j J J ρ.j J 5.66 Inine f he line iffeenil eleen: Th he bsee O is ien his : s Whee epling 5.8 n he nspse f 5.8 e he: 5.67 s 5.68 K K The lipliin f he hee enl ies spplies: K K 5.69 5/55

Resl h n be iie in ies: 5.7 Th pplie in he line iffeenil eleen spplies: s 5.7 eing he peins f he sen e e he: ze Then e he: ze 5.7 Wih his esl e he in 5.7 he inine f he line iffeenil eleen: 5.7 s s T he bsee O he line iffeenil eleen is ien his : s Whee epling 5.5 n he nspse f 5.5 e he: 5.7 s 5.75 K K 5/55

The lipliin f he hee enl ies spplies: K K 5.76 Resl h n be iie in ies: 5.77 Th pplie in he line iffeenil eleen spplies: s 5.78 eing he peins f he sen e e he: ze Then e he: ze 5.79 Wih his esl e he in 5.78 he inine f he line iffeenil eleen: 5.8 s s In 7 s nseqene f 5. e h he inine f.. hee n ppling 7.., 7.., 7.., 7.. n he eli nsfin fle f ble e he ne elins beeen n isin f 7. n 7. n ih he e eie he ble 7 in he f bel: 5/55

Tble 7 K K K 7. 7.. z z K 7.. z z K 7.. 7.5 7.6 z z 7.5. 7.6. z z z z 7.5. 7.6. z z 7.9 7. z z 7.9. Wih he bles 7 n 9 e n he he inine f ll Mell s eqins. Inine f he Gss f he eleil fiel: K 7. 7.. 7.. z z 8. Whee ppling he bles 6, 7 n 9 e he: K K z K K / z Whee siplifing n epling 8.5 e he: Th eee spplies: z / z z / z Th siplifie spplies he inine f he Gss f he eleil fiel. Inine f he Gss f he gnei fiel:. z ze z 8.6 Whee ppling he bles 7 n 9 e he: z z z 55/55

Th eee spplies: z z z z Whee he e in penhesis is he -en s (8.9) h is eql ze f hee e he he inine f he Gss f he gnei fiel. Inine f he -en s : z 8.8 Whee ppling he bles 7 n 9 e he: K K / K z Th siplifie n liplie b / e he: z Whee eeing he ps n epling 7.9. e he: z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he -en s. Inine f he -en s : z z 8. Whee ppling he bles 7 n 9 e he: z K z K K Th siplifie spplies he inine f he -en s. Inine f he -en s : z z 8. Whee ppling he bles 7 n 9 e he: K / z K K z Th siplifie n liplie b / e he: z z z z z 56/55

Th siplifing n ing he peins e he: z z Whee ppling 7.9 e he: z z z z z. s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he -en s. Inine f he pee-mell s : z J z 8. Whee ppling he bles 6, 7 n 9 e he: z Jz K z K Th siplifing n ing he peins e he: z Jz z Whee siplifing n ppling 7.9 e he: z Jz Th egnize spplies z z z z Jz z z z z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s : Inine f he pee-mell s : z z J z 8.6 Whee ppling he bles 6, 7 n 9 e he: z z J z K K / Ming he peins e he: z z J z z / 57/55

58/55 Repling in he fis penhesis he Gss n lipling b e he: J z z J z z Whee epling J, 7.9., 7.9 n 8.5 e he: z z J z z Th siplifie spplies: z z J z z Repling in he fis penhesis he Gss e he: J z z Th egnize es: J z z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s : Inine f he pee-mell s : J z z 8.8 Whee ppling he bles 6, 7 n 9 e he: K K J z z Ming he peins e he: z J z z Whee siplifing n ppling 7.9. e he: J z z Th egnize es: J z z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s :

Inine f he Gss f he eleil fiel ih eleil hge: z ze z 8. Whee ppling he bles 7 n 9 e he: K K z K / z ze Whee siplifing n epling 8.5 e he: z ze / z Th egnize es: z ze / z. Th siplifie spplies he Gss f he eleil fiel ih eleil hge. Inine f he pee-mell s ih eleil hge: z 8. Whee ppling he bles 7 n 9 e he: z Ming he peins e he: K z K z z Whee siplifing n ppling 7.9 e he: z Th egnize es: z z z z z z z z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s ih eleil hge: Inine f he pee-mell s ih eleil hge: z z z 8. Whee ppling he bles 7 n 9 e he: 59/55

6/55 / K K z z z Ming he peins e he: / z z z z Repling in he fis penhesis he Gss ih eleil hge n lipling b / e he: z z z z Whee epling 7.9, 7.9. n 8.5 e he: z z z z Th siplifie spplies: z z z z Repling in he fis penhesis he Gss ih eleil hge e he: z z Th egnize es: J z z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s ih eleil hge: Inine f he pee-mell s ih eleil hge: z z 8. Whee ppling he bles 6, 7 n 9 e he: K K z z Ming he peins e he: z z z Whee siplifing n ppling 7.9. e he: z z

Th egnize es: z z s he e in penhesis is he eqin 8.5 h is eql ze hen e he he inine f he pee-mell s ih eleil hge: fnin f f 5 Inine (ninin).9 Whee he phse is eql 5.8 In e epesen n nling een h ges n in ne bi iein s pl ih he e eqin n bese f his e he: z f f f z ze 5.8 Th esn ee ih he e eqin bese he ls eleens ge nle b he fis ne esn. In e ee his pble e efle he phse f he fnin in he flling. ni e sh s n si sj s 5.8 hee s, s, s z z 5.8 hs he le eql n n n.n s s s. 5.85 Ming he p n.r e he z si sj s. i j z s s sz n.r s s s z n.r s s s z ih he se ening f he peis phse h pplie he phse spplies ne phse. 5.86 5.87 Repling phse in he f n.r s s s z e in he phse liplie b e ls ge nhe s s s z 5.88 ih he se ening f he peis phse Ths e n ie ne fnin s: f s s sz. f 5.89 6/55

Th eple in he e eqin ih he ie sine nsiee nsn spplies: f f f f s s s ze 5.9 h siplifie ees he e eqin. The psiie esl f he phse in he e eqin is n elsie nseqene f he ie sines being nsn in he pil eiies shing h he e eqin ens he ppgin he ne se iein in he spe (plne e). he bsee O se le in he igin f is efeenil pes in n pin le he isne z f he igin, n eleil fiel esibe b: i j z 5.9 Whee he pnens e esibe s: z z.f.f.f 5.9 Th pplie in spplies: f i f j f i j f f z ih le eql.f. f eing z z. 5.9 5.9 i j 5.95 z The i plie e Cnsn ih he pnens,, z 5.96 n le z 5.97 eing f fnin ih he phse eql 5.87 5.88. eiing he pnen in elin n e he: f f f f f f f 5.98 5.99 h pplie in 8.5 spplies / ze f / / f f f / ze ze ensing h i is he phse h s pl ih 8.5. 6/55 ze / ze 5.

s / ze / hen plies ih 8.5. ze / ze ze s he phse is he se f he pnens n z hen he ls pl ih 8.5. s he phses f he bsees O n O e eql bsee O ls pl ih 8.5. / / hen he pnens f he ze 5. The pnens eliel he bsee O f he eleil fiel e nsfe f he efeenil f he bsee O ing he bles 7, 7 n 8. ppling in 8.5 e fnin ien in he f: e i e s isin s isin i 5. hee i. eiing e he: sen is en sen i s 5. e i n e i 5. Th pplie in 8.5 spplies: / ze / sen i s sen i s ze h is eql : i sin i s ze / i / ze e e ze hee e s he he effiiens eql ze s h e ge n ieni, hen: ze i i ze i / i e e Whee ppling ze e he: i 6/55

Then ee ih he eqin 8.5 e s he e ppgin lng he is ih he spee. If e ppl n e he:. esl ls gen f he is e glie s e eqin. Cnsieing he pple effe s l f phsis. 6 Tie n eqen We n efine l s n eie h pes feqen f ienil eens in seies pssible be enlise n e in sh h n een n f eie ill be ienil n een in he seies f eens pe b epli f his eie hen he eens e pe in elie esing psiin. The lil een f l in esing psiin ing he bsee O efeenil ses he ie in his efeenil n he lil een f he s f l in esing psiin ing he bsee O ses he ie in his efeenil. The fls f ie nsfin.7 n.8 ele he ies beeen he efeenils in elie een hs, ele eens in elie een. The elie een beeen he ineil efeenils pes he pple effe h pes h he feqen ies ih eli n s he feqen n be inepee s being he feqen f he lil een f he s f l hen he ie ies in he se ppin h ies he feqen ih he elie een h is, i is engh eple he ie n in he fls.7 n.8 b he feqenies n ge he fls f feqen nsfin, hen: K K.7 bees. K K.8 bees. The Glile s nsfin f eliies beeen ineil efeenils pesens ininsill hee efes h n be esibe his : ) The Glile s nsfin f eli he is is. In h ne if e he hen n if e he hen. s bh esls e n silnesl pssible else e he hen he nsfin esn ll h f ligh be silnesl bsee b he bsees O n O h shs he piilege f n bsee in elin he he bese eh bsee n nl see he f ligh nning in is n efeenil (ininsi efe he lssi nlsis f he Sgn s effe). b) I nn ls pl Nen s fis l f inei bese f ligh eie pllel he is f he igin f he espeie ineil efeenils he en h he igins e inien n he en in hih = = ze ill he b he Glile s nsfin he eli f ligh lee b he efeenils, n he n f he ineil l h ln ll he eisene f iin in eli bese hee is n eenl in ing n he f ligh n bese f his bh bsees shl see he f ligh ih eli. ) s i nsies he ie s nsn beeen he efeenils i esn pe he epl iin beeen he efeenils in een s i is eqie b he pple effe. The piniple f nsn f ligh eli is nhing b eqieen f he Nen s fis l, he inei l. Nen s fis l, he inei l, is ine in Glile s nsfin hen he piniple f nsn f ligh eli is pplie in Glile s nsfin piing he eqin f bles n f he Unling Relii h esn he he hee efes esibe. 6/55

The ie n eli eqins f bles n n be ien s: s.7.5 s s.8. s The isne beeen he efeenils is eql he p f eli b ie his :.9 I esn epen n he ppgin ngle f he f ligh, being elsiel fnin f eli n ie, h is, he ppgin ngle f he f ligh, nl les beeen he ineil efeenil he ppin beeen ie n eli, eeping he isne nsn in eh en, n ppgin ngle. The eqins be in fnin f e ien s:, e, e.9 f,,.7 g,.5 f,,.8 g,. Then e he h he isne is fnin f ibles, he ie fnin f hee ibles n he eli fnin f ibles. he efiniin f en. n eneg.6 e he: p 6. The elee he pe f spplies: p 6. leing he pe f he eneg fl e he: Whee ppling 6. e he: 65/55

p p.8 hee e nle h if he ss in esing psiin f pile is nll ze he pile eneg is eql p. 6. Th pplie in 6. spplies: p p 6. p hee e nle h he een f pile ih nll ss in esing psiin ls be he eli f ligh. ze ill ppling in p he elins h n e he: p p h h n in he se p h 6.5 qin h eles he en f pile ih nll ss in esing psiin ih is n lengh. leing he pe f he fl f en nsfin (.9) e he: p p p p p Whee ppling p n p ps p e fin: p p p p p p p p p K 6.6 Whee ppling 6.5 esls in: h h p p K K inee. K K Whee ppling n e he: K inee K. In e he he eqins. n. ppling he piniple f elii he e phse. 7 Tnsfin f. enz bsees in elie een, he eqin h epesens he piniple f nsn f ligh spee f n pin is: z z 7. In his eqin neling he sei es e he: Nes neln s es siéis bes: 7. 66/55

67/55 Th e n ie s: 7. If in his eqin e efine he ppin fs n s: 7. hee e s he. pl 7.. The eqins 7. hee fis gen b lbe insein. When f ligh es in he plne z he bsee O e he = ze n = n sh niins pplie he eqin 7. spplies: 7.5 This esl ill ls be spplie b he eqins n f he gp 7. ne he se niins: 7.6 hse e he: n 7.7 Whee e he pen h.. he gp 7. e he he Tnsfins f. enz: 7.8 7.9 7. 7. Inees eqins, n : 7. 7.

7. Sgn effe When bh bsees igins e eql he ie is zee ( = = ze) in bh efeenils n s f ligh e eie f he n igin, ne in he psiie iein (lise ine ) f he is n ih e fn n nhe in he negie iein (ne-lise ine ) f he is n ih e fn. The ppgin niins be pplie he enz eqins sppl he bles n bel: Tble qin Clise () qin Cne-lise () S f s Resl Resl Cniin Cniin 7.8 7.8 7.9 Tble 7.9 qin Clise () qin Cne-lise () S f s Resl Resl Cniin Cniin 7. 7. 7. 7. We bsee h he bles n e inese ne nhe. When e f he gp f he s eqins f he s f bles n : 7.5 Whee he bsee O is he isne beeen he fn es n n hee he bsee O is he isne beeen he fn es n. f he In he eqins 7.5 be, e he isp f spe n ie n he fn es s f ligh being he se f bh bsees, he s f s f ligh e ies s be inible beeen he bsees, hih e n epess b: 7.6 This esl h genees n eqin f isp f spe n ie n be lle s he nsein f spe n ie piniple. 68/55

The hee hphesis f ppgin efine s flls ill be pplie in 7.5 n ese pe he nsein f spe n ie piniple gien b 7.6: phesis : If he spe n ie e ispi n hee is n een ih n piilege f ne bsee nsiee e he he in n ep spe hen he ppgin gee f s f ligh n be gien b: n 7.7 This hphesis pplie he eqin f he gp 7.5 plies he spe n ie nsein piniple gien b 7.6. The hphesis 7.7 pplie he bles n esls in: phesis : C 7.8 If he spe n ie e ispi b he bsee O is in n bsle esing psiin in n ep spe hen he gee f ppgin f he s f ligh is gien b: 7.9 Th pplie he ble n esls in: Sing n in 7. e he: C 7. 7. 7. i This esl esn pl ih he nsein f spe n ie piniple gien b 7.6 n s esls in siin f f s f ligh, eh bsee, n eh f ligh ih is espeie inepenen fn e f he hes. phesis C: If he spe n ie e ispi b he bsee O is in n bsle esing psiin in n ep spe hen he ppgin gee f he s f ligh is gien: 7. Th pplie he bles n esls in: C 7. 69/55

Sing C n in 7. e he: 7.5 7.6 This esl esn pl ih he nsein f spe n ie piniple el he se s hphesis gien b 7.6 n s i esls in siin f f s f ligh, eh bsee n eh f ligh ih is espeie inepenen fn e f he hes. Cnlsin The hphesis, n C e pleel pible ih he en f isp f spe n ie s e n nle ih he gee f ppgins. The esl f hphesis is n he esl f hphesis n C espie f he elie een f he bsees n hnging he fn e eliel he fn e bese he fn es he inepenen een ne f he he n f he bsees. The hphesis pplie in he nsfins f. enz plies ih he nsein f spe n ie piniple gien b 7.6 shing he pibili ih he nsfins f. enz ih he hphesis. The ppliin f hphesis n C in he nsfins f. enz spplies he spe n ie efins gien b 7. n 7.6 bese he nsfins f. enz e n pible ih he hphesis n C. s bin he Sgn effe e s nsie h he bsee O is in n bsle esing psiin, hphesis C be n h he ph f he s f ligh be f R : R 7.7 he bsee O he Sgn effe is gien b he ie iffeene beeen he lise f ligh n he ne-l f ligh h n be bine sing 7. (C-), 7.7 n 7.: R 7.8 R 9 The Sgn ffe (ninin) The en he igins e he se he ie is zee ( = = ze) bh sies f he efeenil n he s f ligh e eie f he n igin, ne in he psiie (lise ine ) f he is n ih e fn n he he ne in he negie (ne lise ine ) f he is n ih e fn. The pjee f ligh in he psiie (lise ine ) f he is n is eqine b n h pplie he Tble I spplies: (.8) 9. K (.7) K K (.5) K (.) 9. 7/55

hse e e h he isne beeen he bsees is gien b: 9. Whee e he: KK 9. The f ligh pje in he negie (ne lise ine ) f he is n is eqine b n : h pplie he Tble I gies: (.8) 9.5 K (.7) K K (.5) K (.) 9.6 hse e e h he isne beeen he bsees is gien b: 9.7 Whee e he: KK 9.8 We s bsee h fis hee is n elinship beeen he eqins 9. 9. ih he eqins 9.5 9.8. Wih he ppgin niins esibe e f he flling Tbles n : Tble qin Cniin Clise f Cne lise f qin S f he s f ligh ligh () ligh () Resl Resl Cniin K. K K K K K K.7 K K K. Tble.7 qin Cniin Clise f Cne lise f qin S f he s f ligh ligh () ligh () Resl Resl Cniin K. K K K K K K.8 K K K..8 7/55

We bsee h f he s f ligh ih he se iein he Tbles n e inese f eh he. ing he eqins gp f he s f he s f ligh f he Tbles n : K K K K 9.9 Whee f he bsee O is he isne beeen he e fns n n hee f he bsee O is he isne beeen he e fns n. f he In he eqins be 9.9 e he isp f he spe n ie n he e fns s f ligh being he se f bh bsees, he s f he s f ligh n f ies s be inible beeen he bsees, hih is epesse b: 9. This esl h eqins he isp f spe n ie n be lle s he spe n ie nsein piniple. The hee hphesis f ppgins efine ne ill be pplie in 9.9 n ese pe he pline f he nsein f spe n ie piniple gien b 9.. Wih hese hpheses e ee bn beeen he eqins 9. 9. ih he eqins 9.5 9.8. phesis : If he spe n ie e ispi n hee is een ih n piilege f n bsee e eh he in he ep spe hen he ppgin gee f he s f ligh is eqine b: K K K K 9. Wih hse e e h he isne beeen he bsees is gien b: 9. Resls h pplie in he eqins f he gp 9.9 plies ih he nsein f spe n ie piniple gien b 9., shing h he pple effe in he lise n ne lise s f ligh e pense in he efeenils. phesis : If he spe n ie e ispi b he bsee O is in n bsle esing psiin in he ep spe hen he ppgin gee f he s f ligh is eqine b: C 9. Wih hse e e h he isne beeen he bsees is gien b: 9. Resls h pplie in he eqins f he gp 9.9 plies ih he nsein f spe n ie piniple gien b 9., shing h he pple effe in he lise n ne lise s f ligh e pense in he efeenils. 7/55

phesis C: If he spe n ie e ispi b he bsee O is in n bsle esing psiin in he ep spe hen he ppgin gee f he s f ligh is eqine b: C 9.5 Wih hse e e h he isne beeen he bsees is gien b: 9.6 Resls h pplie in he eqins f he gp 9.9 plies ih he nsein f spe n ie piniple gien b 9., shing h he pple effe in he lise n ne lise s f ligh e pense in he efeenils. In e bin he Sgn effe e nsie h he bsee O is in n bsle esing psiin, hphesis C be n h he s f ligh se s be f R : R 9.7 ppling he hphesis C in 9. n 9.5 e he: K 9.8 K 9.9 he bsee O he Sgn effe is gien b he ie iffeene beeen se f he lise f ligh n he ne l f h n be bine ing (9.8 9.9) n ppling 9.7 ing: R 9. The eqin is el he esl bine f he gee nlsis f he ppgin f he lise n ne lise s f ligh in ifeene shing he heene f he hphesis pe b he Unling Relii. In 9. ppling 9. n 9.6 e he he finl esl e n : R R R 9. The lssi fl f he Sgn effe is gien s: R 9. he ppgin gee e he: 9. 7/55

The lssi ies l be gien b: R 9. R 9.5 R 9.6 ppling 9., 9.5 n 9.6 in 9. e he: R R 9.7 R R 9.8 R R 9.9 The esls 9.7, 9.8 n 9.9 e pleel iffeen f 9.. 8 The Mihelsn & Mle epeiene The iinl nlsis h spplies he slin f he nll esl f his epeiene nsies eie in esing psiin he efeenil f he bsee O h eis s f ligh, ne hiznl in he iein (lise ine ) n nhe eil in he iein. The hiznl f ligh (lise ine ) ns nil i ple in = his pin he f ligh efles (ne lise ine ) n ens he igin f he efeenil hee = ze. The eil f ligh ns nil i ple in = efles n ens he igin f he efeenil hee = ze. In he iinl nlsis ing he spee f ligh nsn piniple f he bsee O he s f ligh is gien b: 8. he bsee O he s f ies f he f bh s f ligh lng he is is: 8. In he iinl nlsis f he bsee O he s f ies f he f bh s f ligh lng he is is: s e he 8. hee is n inefeene finge n i is pplie he nll esl f he Mihelsn & Mle epeiene. In his iinl nlsis he ienil f he lise n ne lise s f ligh in he eqin 8. h igines he nll esl f he Mihelsn & Mle epeiene nis he Sgn effe h is el he ie iffeene eising beeen he f he lise n ne lise s f ligh. se n he Unling Relii e e eepe nlsis f he Mihelsn & Mle epeiene bining esl h plies pleel ih he Sgn effe. Obseing h he eqin 8. espns he hphesis C f he pgph 9. 7/55

ppling 8. in 9.9 e he: K K K K K K K K K 8. e he: K 8. 8.5 Whee ppling 9.6 e he: 8.6 In 8. e he: Whee ppling 9.5 e he: 8.7 8.8 The eqins 8.6 n 8.8 ense h he pple effe in he lise n ne lise s f ligh pense iself in he efeenil f he bsee O esling in: 8.9 ese f his, ing he Unling Relii in he Mihelsn & Mle epeiene e n pei h he lise f ligh hs iffeen f he ne lise f ligh ing he fl 8.8 bining ls he nll esl f he epeiene n hing hen ih he Sgn effe. This sppsiin nn be e bse n he insein s Speil Relii bese ing 7.6 e he: 8. 75/55

76/55 9 Regessin f he peihelin f Me f 7, e s igine he Sn le in he fs f n ellipse h inies ih he igin f sse f ines (,,z) ih n een in elin enine fie ss n h he plne Me is in een gene b he fe f giinl in ih he Sn esibing n ellipi bi in he pln (,) ing he ls f Keple n he fl f he Nens giinl in l:,8.,98. 67. 6, 9. The sb ine "" iniing ss in elie es he bsee. T esibe he een e ill se he nn fls: 9. 9.. 9. 9.5 The fl f he elii fe is gien b: / 9.6 In his he fis e espns he iin f he ss ih he spee n he sen s e ill see le in 9. espns he iin f he eneg ih he ie. Wih his n he peis fls e bin: / 9.7 / / 9.8

77/55 In his e he he nsese n il pnen gien b: / / 9.9 / / 9. s he giinl fe is enl e shl he nll he ese pnen ze s e he: ze / / 9. hee e he: 9. he il pnen e he: / / 9. Th ppling 9. e he: / / 9. Th siplifing esls in: 9.5 This eqle Nens giinl fe esls in he eliisi giinl fe: 9.6

s he giinl fe is enl i shl ssis he he f nsein f he eneg () h is ien s: = nsn. 9.7 p Whee he inei eneg ( ) is gien b: 9.8 n he penil eneg ( p ) giinl b: p 9.9 Resling in: s he l eneg () i is nsn e shl he: Cnsn. 9. p ze. 9. Then e he: 9. p 9. Resling in: p ze ze 9. This pplie in he eliisi fe 9.6 n eqle he giinl fe 9. esls in: 9.5 In his sbsiing he peis ibles e ge: 78/55

9.6 his e bin he il pnen eqls : 9.7 Th esil bees he eliisi giinl fe 9.6. 9.6 e bin he ese pnen eqls : ze 9.8 his ls ne e he: 9.9 s he giinl fe is enl i shl ls ssis he he f nsein f he ngl en h is ien s: p p nsn. 9. 9. = nsn. 9. Resling in h is nsn. In 9. e h ze ze ze bese he een is in he plne (,). 9. 79/55

eiing e fin: h e he: ze 9. 9.5 qling 9. igining f he he f he enl fe ih 9.9 igining f he he f nsein f he eneg n 9.5 igining f he he f nsein f he ngl en e he: 9.6 he ls eqli e bin 9. n f he f he ile e bin 9.6. slin f he iffeenil eqins e ill se he se eh se in he Nens he. e s sse 9.7 The iffeenil l f his is 9.8 hee e he e 9.9 he le f he ngl en e he hee e he Whee ppling 9.9 e he 9. 9. 9. Th eie spplies 9. 8/55

8/55 Whee ppling 9. n eiing e he: 9. In his ih 9.6 he il eie is bine his : / 9.5 / 9.6 Th pplie in 9. spplies: 9.7 Siplifie esls: 9.8 e s fin he sen eie f he ngle eiing 9.: 9.9 In his ppling 9. n 9.5 n siplifing e he: 9.5 ppling in 9. he eqins 9. n 9. n siplifing e he: 9.5 The eqin f he eliisi giinl fe 9.6 eele is: 9.5 In his ppling he fls be e he:

8/55 8 8 8 8 8

8/55 8 8 8 6 8 8 6 8 8 6 8 In his e ill nsie nsn he Nens ngl en in he f: 9.5 Th i is ell he nn heeil ngl en. ze 9.5 Whee e he: 9.55 9.56

8/55 The eqin 9.5 hs s slin: s s 9.57 Whee e nsie ze. I is enine in 9.57. 9.58 The eqin 9.58 is fnin nl f ensing he ininsi nin beeen he iin f he ss ih he iin f he eneg in he ie, bese bh s le esibe, piipe in he eliisi fe 9.6 in his elies he essenil iffeene beeen he ss n he elei hge h is inible n iniisible in he elegnei he. 9.57 e bin he f nil: s s 9.59 Whee is he eenii n he ie isne f he fs. eiing 9.57 e he sen 9.6 Th eie esls in s 9.6 ppling in 9.5 he ibles e he: ze. ze s sen s s s 9.6 ze s s s s s ze s s s s s s ze s s 9.6 In his ppling in he fis penhesis e he: ze

In 9.6 ppling in he sen penhesis e he: ze The es f he eqin 9.6 is heefe: ze 9.6 The f he ellipi bi f he plne Me is []: enii f he bi, 6. ge sei-is = = 5,79.. Slle sei-is b 5, 79.,6 56.658.6.5,8. 5, 79.,6 55..955.6,. 5, 79.,6 69..56.65,,6 The bil pei f he h (PT) n Me (PM) n he Sn in sens e: PT, 6. PM 7, 6. 7 6 s. s. The nbe f ns h Me ( ) es n he Sn (M ) in ne en is, heefe: 7, 6. N 5, 79. 9.65 6 7, 6. Theeil ngl en f Me: 6,67.,98. 5, 79.,6 7,977. 6,67.,98. 8,. 7,. 8. 9.66,65.. 9.67 6,67.,98. 7,.,5. 9.68,6. 8,... 9.69 ppling he nei ih seel eil nbes he es f he eqin 9.6 e he: 8 8,65.,...,65. 69..56.65, 55..955.6,,5. 8,976. 9.7 Resl h e n nsie nll. 85/55

86/55 We ill bin he eliisi ngl en f he es f he eqin 9.6 in his ppling he ibles e he: ze 9.7 ze ze ze ze 9.7 7,978. 9.7 This ls eqin hs he elsie ppe f eling he spee he enine eliisi ngl en h is slle hn he heeil ngl en 9.66. The iin f he eliisi ngl en in elin he heeil ngl en is e sll n gien b: 7.5.59,,8. 7,977. 7,977. 7,978. 8. 9.7 Th enses he f he piniple f nsn f he spee f he ligh.

In eli, he eqin 9.6 pies sel eessin peihelin f Me, hih is gien b in 5, 79 5, 79 5,...,6.. 9.75 Cneing f he sen e he:,6. 5.8,..6, 7, ". 9.76 This eessin, is n epee in Nenin he is e eliisi iin f ss n eneg n is she in l bsee peessin f 5599. " 87/55

9 ne f Me s peihelin f.79 If e ie he eqin f he giinl elii eneg R eing he es f he inei eneg, he penil eneg p n he esing eneg: R p p. 9.77 eing he nseie he giinl fe is eneg is nsn. ssing hen h in 9.77 hen he is ens infinie, he spee n penil eneg ens ze, esling hen: R p 9.78 Wiing he eqin he Nen s giin eneg N hing he espnen Nen s es he 9.77: N 9.79 Whee he ineil eneg. is he inei eneg, he penil eneg n he esing eneg bee sing his 9.79 e he: 9.8 eiing 9.79 e he: N ze ze 9.8 88/55

89/55 Ming he elii eneg 9.78 eql he Nen s eneg 9.79 e he: p N R 9.8 p 9.8 In h enining he elii penil ( ) s: p 9.8 We he: 9.85 In his ne epling he ppiin: 9.86 We he: Th siplifie esls in he Nen s penil: 9.87 Repling 9.8 n he elii penil 9.85 in he elii eneg 9.78: R 9.88 We he he Nen s eneg 9.79: N

9/55 eiing he elii penil 9.85 e he he elii giinl elein ls el s in he Nen s he: Whee e he: ze N. ese he e be eie is he Nen s eneg iie b h is N h is nsn, esling hen in: In his ne ppling 9.8 e he: 9.89 The e elein is gien b 9.5:

The elii giinl elein ls 9.89 is eql he pnen f he e is () hs e he: 9.9 eing nll he nsesl elein e he: ze 9.9 ze Th is eql he eiie f he nsn ngl en ze 9.9 9.9 Reiing se eqins le esibe e he: n 9.9 9.9 e he: In his ne e 9.9 he spee f 9.8 n he ngl en e he: 9/55

9/55 G ze ze ze ze 9.95 Whee e he: 9.96 The slin he iffeenil eqin 9.95 is: s s. 9.97 Whee e nsie ze Then he is is gien b: s s 9.98 Whee is he eenii n he fs isne he ie. eiing 9.97 e he sen n s 9.99 ppling he eiies in 9.95 e he: ze ze s s s s s ze s s s s s s ze s s s s s s s

9/55 ze s s s ze s s s ze s s s ze s s 9. The effiien f he sqe -sine n be nsiee nll bese n is e lge nbe: ze s 9. Resling f he eqin 9.: ze s 9. e he nii f he eqin 9. e s he he nl slin h es i nll silnesl he penhesis n he es f he eqin, h is, e s he niqe slin f bh he flling eqins: ze n ze 9. These eqins n be ien s: b 9. 9.5 In hese nes he n e s he single slin hen e he: b 9.6 Wih 9.96 n he heeil en e he: 9.7

I is pplie in 9.5 n 9.6 esling in: b 9.8 9.9 9.8 e he he ise e in 9.5: ze,8. ze 55..955.6, 9. 9. 9.9 e he : 9. I is pplie in 9. esling in 9.: ze 9. e he: 66,67.,98. 8 55..955.6,. 6,999.999.9.599 9. Th espns he ne f Me s peihelin in ne en f: 5,79..96.,.5, 79 Clle in his :.,79 9. In ne ignei n e he 66 6.96., " sens. The ngle in sens n b he plne in ne ignei n is gien b:.96.,.96.,. If, e he egessin..96.,. If, e he n ne..96.,. 9/55

The ngl iin in sens in ne n is gien b:.96.,.96.,.96., If If ze e he egessin. ze e he n ne. In ne en e he 5,79 ns h sppl l ngl iin f: 5,79..96.,.5, 79.,79. If ze e he egessin. If ze e he n ne. Inei Igine in n infinie niese ll ep, pin O hih is he beginning f he ines f he bsee O. In he ses f he bsee O being es in nif in he l f inei eqies h he spheil elegnei es ih spee isse b se le pin O is ls bsee b O, egless f ie, ih spheil spee n heefe he nif in n es e inisingishble f eh he ein li in bh ses he l f inei. T he bsee O he eqins f elegnei he esibe he spe js lie spheil e. The ige f n bje le in O ill ls be enee n he bje iself n be f ligh eie f O ill ls ein sigh n pepenil he spheil es. Igine nhe pin O h ill be he beginning f he ines f he bsee hih hs he se ppeies s esibe f he ineil bsee O. Obisl igin pins ih n f f inein beeen he ein iniill n gehe pefel eeing he l f inei een hgh hee is nif in beeen he nl eeble e he pesene f bsees h ill be nsiee iniill in es, seing in in he he efeenil. The ininsi ppeies f hese bsees e esibe b he eqins f eliisi nsfins. Ne: he infinie niese is ne in hih n pin n be nsiee he enl pin f his niese. ( eleni nslin) Inei (lifiins) Igine in ll ep infinie niese single pin O. e he niqeness ppeies f O is f ligh eie f O s ppge ih eli. If his ppges in sigh line, hen O is efine s he igin f n ineil fe bese i is eihe es in nif eiline in. ee, in he hphesis f ppgin f he ligh being e he een f O s be inepee s he igin f n elee fe. Theefe he ppgin f f ligh is sffiien ense hehe O is he igin f n ineil fe elee fe. N igine if in he niese esibe be f he ineil efeene fe O hee is nhe ineil fe O h es n he n in f phsil inein ih O. In he bsene f n inein beeen O n O he niqeness ppeies e inilble f bh pins n s f ligh eie f O n O he he se eli. I is ipssible f he eli f ligh eie f O be iffeen f he eli f ligh eie f O bese eh efeene eiss s if he he i n eis. eing O n O he igin f ineil fes he ppgin f ligh s s in sigh line ih eli n he elins beeen ies n f eh fe e gien b ble I. 95/55

ne f Me s peihelin f.79 lle ih he Unling Relii ssing (.) ze ze. (.7) (.)....5 (.) (.).6.7.8.9.. 96/55

97/55...6..5

98/55 9.6.6.7..8....9... ns n. ns R n ns R n. R R.

99/55 R R..5.6.6 = nsn.7.....8.9 ze...

/55...5 ze R.6 ze.7 s sen s.8 ze s s s s s.9 ze s s s s s s ze s s s s s s s

/55 ze s s 6 s s s s ze s s s 6 s s s ze s s 6 ze s 6 s ze s s. ze s. ze s. ze ze s ze ze s ze ze. b. R ze b ze b.5.6

/55 b 6.7 The egessin is fnin f psiie eneg h gens he een. R 6 Regessin.8 ze b 6.9 ze ze. R ze 6 ze ze 6 ze ze This egessin is n gene b he psiie eneg

/55.6.5.5.5.5

/55...5...55.56 ns n.57 ns R n ns R n.58 R R.59 R.6 R.6

5/55.6.6.6.6.56.65.66 ze.67.68.69

6/55.7.7 ze R.7 ze.7 s sen s.8 ze s s s s s.7 ze s s s s s s ze s s s s s s s

7/55 ze s s 6 s s s s ze s s s 6 s s s ze s s 6 ze s 6 s ze s s.75 ze s.76 ze s.77 ze ze s ze ze s ze ze.78 b.79 R ze b ze b.8.8

8/55 b 6.8 The ne is fnin f negie eneg h gens he een R 6 6 ne.8 ze b 6.8 R ze ze.78 ze ze ze 6 ze.85.86 ze 6.87 ze ze.88 This ne is n gene b negie eneg ze 6.85 ze 6.89 ze 6 ze 6 6.9 ze s.77 ze s ze s.

9/55 ze s 6 ze s 6 ze s 6.9 ze s 6 ze s 6 ze s 6 9 ze s 9 R ze s 9 ze s 9 ze s.9 ze s 6.9 6 ze s 6 6 6 ze s 6 6 6 ze s 6 8 6 R

/55 ze s 8 6 R ze s 8 6 ze s 8 ze s 6 ze s 6 ze s.9 ze s 6.9 R ze s 6 ze s ze s.9 6 s s s.95

/55 neg Nenin ( N ) N N N ze N ze N ze N ze N ze N N ze s sen s ze sen s s ze s s s s

/55 ze s s s s ze s s s s ze s s s s ze s s ze s ze s ze ze N ze ze ze N N N ze N N N

Spil efin This is he spil efin. The lengh es in he efeene fe f he bsee O is gee hn he lengh h is ing ih eli elie n efeene fe he bsee O. N pe he bsee O he isne beeen O O : Ths e bin he eli :. N pe he bsee O he isne beeen O O : Ths e bin he eli :. The spee is he se bh bsees s e he: Whee ppling he elin e bin:. Whee he isne n ies inesel ih he isnes n. /55

/55 In genel, e bin (.,.): ze ze

Vibles ih line Spe n Tie en,,,, e...the e se in. Gee f spe n ie in he pln. f s. s f s. îĵî sĵ îĵî sĵ. î ĵ îs ĵîĵ î ĵ î ĵ î ĵ s s sen s g s s s î ze s. ĵ î ĵ îĵ î sĵ î sĵ s s s s s s s s s 5/55

6/55 s heeil e g g s s s s s s s s s s s s s s s s s s s s.56. s. s s eleni nslin

nsn Viinl Piniple. nsn p gngen. p Wh is he iniil eneg f he pile f ss. p çãs,,,,ze Viinl Piniple This is he eli pnen in is. S Viin f he in lng he X is. iling he ible in he nge e he seen his hen ze n hee ze e ill he he niins: ze ze ze ze ze ze Then e he ne fnin I G,,,, ze ze 7/55 n hee:,,,,,,,,

8/55 S e he,, I h pies eie: ze,,,, I ze I ze I ze ze I ze ze ze I ze ze This is he X is pnen ze ze z z This is he X is pnen z

9/55 z z z î î X is ĵ ĵ Y is z z z Z is z ĵ î z ĵ î z z ĵ î z z ĵ î

/55 z ĵ ĵ î î z ĵ î z ĵ î z ĵ î z ĵ î z ĵ î =.6 =.9 eleni nslin Viinl Piniple Cninin ns n e. nsn e nsn e ns n e T T p p p p. p p p p p p p R T p T

/55 T T p T p T T p T p T p T p p R p p p p T p T p z ĵ î z ĵ î.8 z ĵ î z ĵ î T p T p z ĵ î z ĵ î.7 z z z z ze ze p p T ze p ze p p p ze p ze

/55 ze z z.. ze z ĵ î z ĵ î z ĵ î =.9 5 gihi spil ze e.7 s sen s.8 e e e e ze e e e e e ze e e e e ze e e ze e e ze ze

/55 6 6.. e ze 6 6 6 6 ze 6 6 6 6 6 6 ze 6 6 6 6 6 6 ze 6 6 6 8 6 ze 6 6 6 ze 6 ze 6 5 gihi Spil (Cninin).7 R

/55 ze ze ze ze ze ze s sen s.8 ze s s s s s ze s s s s s s s ze s s s s s s ze s s s s s s ze s s 6 s s s s

5/55 ze s s 6 s s s s ze s s s s s s ze s s s s s s ze s s R 6 ze ) ( ) ( s ) ( ) ( s ze s s ze s s ze s s ze s s 6 ze s 6 6 s 6

6/55 ze s 6 6 s 6 ze s 6 s 6 ze s 6 s 6 ze s 6 s 6 ze s 6 s 6.6.6. 6 6 s 6 6 6 s 6 6 s 6 6 6 s 6 6 6 s 6 6 6 ze 6 8,55.,997958.,989. 6,67. 55..955.6 6 6 6 6 6 s

7/55 8 6 6 6 6 6 s 8 6 8 6 s ze s ze s 6 ze M ) ( ze ze s s ze M ze G 6 6 6 ze s s ze s ze s ze s s ze M The pesene f in he fl s ) (, lls i ls esibe spil.

8/55 5 gihi Spil Cninin II ze s s 6 ze M ) ( ze ze s 6 6 s 6 ze 6 s 6 6 s 6 ze 6 s 6 6 s 6 ze 6 s s 6 6 6 6 s 6 s 6 s 6 s s s

9/55 s s s s s s ze s ze s 6 ze M ) ( ze 6 6 6 55.6.69.568,,5659 57.99.7.,.77,89.55.,997958..,989. 6,678. 8,999.999.9. 6,999.999.9. 6,76.789..5 -.96.,.96.,. ne Regessin

/55.96., ze ne ze Regessin.59.89.5",.96., 6.59.876.997",.96., 6 5,.6.9 87,969 65,56.6. PM PT. N,99.98..7" 5,.6.9.59.89.5, N.,99.977.6." 5,.6.9.59.876.997, N. efiniin ze 6 ze M ) ( ze ze M s ze s s ze s Se If s s neg Nenin ( N ) ze s s ze s s ze ze

/55 ze ze ze ze ze N ze ze ze N ze N ze N N N

6 neen f he Peiéli f Me f,99 " Sppsing (.) ze ze. (.7) (.)....5 (.) (.).6.7.8.9.. /55

/55...6.5.5

/55.5.5.5.55.56.57.58.... e ns n e ns R n e ns R n

5/55.59.6.6.6.6.6.6.56.65.66.67 R R R R ze

6/55.68.69.7.7 5 gihi Spil nine.7 R ze

ze ze ze ze ze s sen s.8 The fis hphesis bin pil slin f he iffeenil eqin is sse he infinie is, hs bining: s zes s s ze ze R ze ze ze ze ze ze ze This esl shs h in infini he inflene f he enl ss is ze 7/55 M ze. The sen hphesis bin nhe pil slin f he iffeenil eqin is bine b bseing h he ngle f he eqin s inies he iein f he infinie is hee he inflene f he enl ss is ze M ze n heefe he iein f he ene f ss is gien b he ngle h eple in he eqin s esls in he ne eqin s h inies iein ppsie he iein f he infinie is hih is he iein f he ene f ss. s s s s s

8/55 s s s R ze ze ze ze ze 6 ze 6 ze 6 ze 6 6 6 ppling he esls f he sen hphesis in he iffeenil eqin: ze s sen s.8 ze s s s s s

9/55 ze s s s s s s s ze s s s s s s ze s s s s s s ze s s 6 s s s s ze s s 6 s s s s ze s s s s s s ze s s s s s s ze s s R ze ) ( ) ( s ) ( ) ( s ze s s

/55 ze s s ze s s ze s s 6 ze M ) ( ze ze s 6 6 s 6 ze 6 s 6 6 s 6 ze 6 s 6 6 s 6 ze 6 s s 6 6 6 6 s 6 s 6 s 6 s

s s s s s s s s Whee ppling he esl f he sen hphesis s s : Th i is n ieni ensing h he esl f he sen hphesis is e. 6 6 6 57.99.7.,,5659 55.6.69.568, 6,678..,989.,997958.8.77,89.55. /55

/55,999.999.9. 6,999.999.9. 6,76.789..5 -.96.,.96.,. ne Reess.96., ze ne ze Reess.59.89.5",.96., 6.59.876.997",.96., 6 5,.6.9 87,969 65,56.6. PM PT. N,99.98..7" 5,.6.9.59.89.5, N.,99.977.6." 5,.6.9.59.876.997, N. Nenin neg N N N N ze N

/55 ze s s s s ze s s Nenin neg N ze s s ze s s ze N ze N ze N ze N N ze s sen s ze sen s s ze s s s s ze s s s s ze s s s s

/55 ze ze ze ze ze ze ze N ze ze ze N ze N ze N N N

7 neen f Peihelin f Me f.99 "n Cniins" e s s f he eqin epessing he eqilibi f fes:.65 On he igh sie e he he giinl fe efine b Nen, n he lef sie e he he phsil esipin f e = f he Unling Relii. The phsil ppeies f eqin.65 eqie is lii hen is is ies f is gee hn ze n infinie is, s he is ies f ze <, n s e he isin bn niins. The fis bn niin is hen he is is infinie = n he giinl fe is ze, hih ens h he pile is es ih = ze n = ze n he sen bn niin is hen he is is gee hih is ze n slle hn infini ze < < hih ens h he pile is in in e he inflene f giinl fe.65 ih ze n ze. In 6 flling he llins is sbsie in.65, he eqli,.6,.69 n R, e =. fe hese sbsiins e bin he iffeenil eqin: ze 7. This eqin hs be li f he se bn niins s eqin.65, h is, i hs be li f is gee hn ze ( > ze) n infinie is (ze < ). Y slin is gien b: s 7. Whih shl e he n niins le esibe. ppling slin 7. in iffeenil eqin 7. e he: ze s sen s.8 s s s ze s s s s ze 5/55 s s s s s

6/55 ze s s s s s s ze s s s s s s ze s s 6 s s s s ze s s 6 s s s s ze s s s s s s ze s s s s s s ze s s R ze ) ( ) ( s ) ( ) ( s ze s s ze s s

s s s s ze ze 7. This eqin s he slin f he se n niins f.65. Slin f 7. f he fis bn niin hih is hen he is is infinie giinl fe is ze hih ens h he pile is es n e he = ze n = ze. =, n he ppling in 7. e ge: ( ) ( ) + + ( ) + = ze. ( ) + = ze ε = ( ) 7. qin 7. is el eql he esl f eqin 7. hen he is is infinie =, = ze n =, s shn in 7.5: = = ( ) [ + εs( )] = [ + εs( )] = + = ze 7.5 Theefe in 7. e he n e esl h esibes h in infini he eenii ε is ele he ngle f he infinie is f he pile, being ε hih ens h he in f infini ill be pbli ih ε = hpebli ih ε >. Ne h b efiniin ε > ze. Slin f 7. f he sen bn niin hih is hen he is is gee hn ze n less hn infini ze < < hih ens h he pile is in in e he inflene f giinl fe ih ze n ze. ppling = in 7. e he: s s ze 7. s s 6 6 6 s 6 6 6 s ze s 6 6 6 s s 6 s 6 ze 6 ze 6 ze 7/55

8/55 6 6 6 s 6 s 6 s 6 s s s s s s s s s

( ) + = ze ε = ( ) In he he f ni f hpeble e he ε =, eqing 7.6 e he ε = =. s( )) hih is he e fl, f he gee hlf is f hpebl. = ( ) 7.6 This esls Theefe in 7.6 e he n e esl h esibes h in he se f ze < < he eenii ε is ele he ngle f he pile, being ε hih ens h he in ill be pbli ih ε = hpebli ih ε >. Ne h b efiniin ε > ze 8 Siplifie Peielli ne Peihelin Regessin > Igine h he sn n Me e piles, ih he Sn being he igin f ine sse n Me ling pin n he plne. The e is = nneing he igin pin ill esibe Mes in in he plne. In he esipin f he een f he plne Me he bsee O espns he ibles ih line f he bsee O s ih line being se single is = n single ine sse f bh bsees. Tie is fnin f ie h is = () n ie is fnin f ie h is = ( ). = + =. + =. = =. > > =.5 = = +. =. +. = 8. The is n be nsiee fnin f ie = () ie = ( ) = [ ()] i n be nsiee fnin f ie = ( ) ie = () = [( )]. = ( ) = [ ()] = () = [( )] 8. = = + = = = = = = = = = + = = = = = = () = = = () = + = + h spees n eleins e psiie. 8. + 8. + 8.5 = = 9/55

= = = + 8.6 =. =. =. 8.7 =. = +. =. = = = =. = = = 8.8 = + = = + nsne = + = 8.9 = = = + = + 8. In his fis in eliisi inei eneg is gee hn ineil eneg Mes peihelin eee. The plne sees heie e he een. >.This ses =. = =. = =. =. = = =. = =. = =. =. = = 8. = + + = = + = ze = = + = ze = = = = = = = + = + = = = 5/55

= + = + + + + + ze + + = 8. + + + = ze + + + = ze + + + = ze + + + = ze 8. = = [ + εs( )] = ( ) = ( ) ( ) + [ + εs( )] + ( ) ( ) + + ( ) ( ) ( ) ( ) + [ + εs( )] + ( ) + + ( ) + + ( ) + [ + εs( )] = ze = ze + ( ) = ze ( ) + ε = = = + ( ) + + = ze ( ) ( ) + + ( ) + + = ze ( ) ( ) + + ( ) + = ze = 8. ( ) + + ( ) + = ze + 6 ε ε s ( ) + 6 + ε + ε + ε ε + 6 s( ) + ε ε + 6 ε + = ze ε ( ) + + ( ) 6 ( ) ( ) = + + ( ) ( ) + + ±.. = ze + + ( ) + + = ze = ze ( ) = ±. ( ) = ± 5/55

( ) = ± ( ) = = ( ) = ε ( ) = ze 8.5 hpeble eenii (ε) is efine s ε = ( ) hee ( ) is he ngle f he spe. ne f he Peielli < = + = > + = = = > = = +. =. +. = = = + = = + = = = = = ( ) = = = ( ) = + + = + + = = = = = + =. =. =. =. = + 8.6. =. = = = =. = = = 8.7 = = = + nsne = = 8.8 = = = = 8.9 In his sen in eliisi inei eneg is slle hn ineil eneg ne f Mes peihelin. The plne ell is lighe e een. 5/55 <.This ses he

=. = =. = = = = =. =. =. =. =. =. = = 8. = + + = = + = ze = = + = ze = = = = = = = + = + = = = = = + ze + = 8. + = ze + = ze + = ze + = ze 8. = = [ + εs( )] = ( ) = ( ) s( ) + ε [ + εs( )] s( ) ε [ + εs( )] ε [ + εs( )] = ze 5/55

( ) + + ( ) ( ) + + ( ) + + ( ) ( ) ( ) ( ) + + ( ) + + ( ) ( ) + + ε = = = ( ) ( ) + + + ( ) + ( ) ( ) + ( ) + ( ) + ( ) = ze ( ) + + ( ) + + = ze = ze = ze + ( ) = ze ( ) = ze ( ) ( ) + + + ( ) + = ze = 8. ( ) + ze ( ) + + + ( ) + + + ( ) 6 ( ) ( ) = + + ( ) + ( ) + ±.. = ze + ( ) + = ze + + + ( ) + = ze = ze ( ) + = ( ) = ±. ( ) = ± ( ) = ± ( ) = = ( ) = ε ( ) = ze 8. hpeble eenii (ε) is efine s ε = ( ) hee ( ) is he ngle f he spe. 5/55

The eens f he ellipses ill fs (lef) n he igin f he fe. ll ellipses e esibe b he eqin = () = = =. () ().() In hese he ngle e is (), inies he psiin f he plne Me in ll ellipses, he een f Me in he ellipses is nelise, ih he le f being he se f peihelin neen ein. The fis ellipse in ble epesens egessin f he peihelin, hee e he =.. The sen e ellipse epesens he neen f he peihelin, in his e he =.9. In his ellipse he peihelin n phelin ne in he ignei sense, h is, nelise hih is he se iein s he plnes een in he ellipse. The fifh ellipse in geen epesens sin ellipse =. ibligph "lhgh nb n en behin n pef ne beginning, n ne n begin n n ee ne en" (Chi Xie) [] lns & inn, Vl. I e II,. g lühe T, 97. [] Rbe Resni, Inç à Reliie speil,. Uni. e S. Pl e. Plígn S.., 97. []. Tes Y R. Oiz, Relii, Ci.. sps-clpe genin S.., 95. [] bh Pis, Sil é Senh... iêni e i e lbe insein,. N nei, 995. [5] Mel Rl, ísi ôi, i Téni T, 959. [6].. enz,. insein e. Minis, O Pinípi Reliie, eiçã nçã Clse Glbenin, 989. [7]. n e. ifhiz, Tei Cp, es ii i T. [8] Rbe Min isbeg, nens ísi Men,. Gnb is S.., 979. [9] M n, ísi ôi, eiçã nçã Clse Glbenin, 986. hp://.bbin.ne/phsis/fj7.h 55/55