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(2021 ೧ 4 ݆ 4 ೖण෉ɾ201X ೧ 5 ݆ 31 ೖणཀྵ)Abstract: In a previous paper, it has been reported that fingerprint pattern are related tothe distribution of veins under the skin. Since the blood transfers heat to living skin, it ispossible to infer the direction of blood flow by observing the timing of the rise intemperature of each part of the skin surface of a precooled finger using thermography. Bloodflow patterns were compared to the pattern of vein distribution of standard fingerprint
shapes, such as the loop, whorl, and arch. A similar study has been conducted for a case of
fingerprint mismatch between the left and right hands. This research clarified that theridges of fingerprints are closely related to the subcutaneous blood flow, and the
subcutaneous blood flow is related to the venous running patterns.
Keywords: fingerprint, ridge, vein, blood flow, thermography

1. ॻ‫ݶ‬ࢨ໴ͺɼଽਫ਼‫ غ‬7-12 ु໪͟ΘΉͲͶ‫׮‬੔ͤΖͳ͏ΚΗͱ͏Ζ͗[1]ɼ֦ࢨ͹ࢨ໴͹‫ܙ‬য়͗ʹ͹Γ͑͵‫ߑؽ‬ͶًҾ͢ͱรԿͤΖ͹͖ɼै෾͵ઈ໎ͺ͏Ήͫ͵͠Ηͱ͏͵͏ɽ઎Ͷըʓͺɼ੼֐તল໎ૹ஖ͳ੼֐તΩϟϧΝ༽͏ͱൿԾ͹੫ຼ෾ා͗ࢨ໴͹‫ંືͳܙ‬Ͷ‫ؖ‬࿊͢ͱ
͏Ζ͞ͳΝๅࠄͪ͢[2]ɽৼଃΏ୉ಊຼΏ୉੫ຼ͵ʹकགྷ͵१‫ح؂‬͹ഓ஖ͺɼ൅ਫ਼͹ॵ‫غ‬ஊ֌ͶҒ஖ҪఽࢢͶΓͮͱ݀ఈ͠ΗΖ͗ɼ຦୼͹໡ࡋ݄؇Ώࡋ݄؇৿ਫ਼ͺɼ݄؇಼֐͹͠Ή͡Ή͵‫ڧ؂‬ড়݇౵ͶΓͮͱรԿͤΖ͞ͳ͗எΔΗͱ͏Ζ [3, 4]ɽൿԾ͹੫ຼ෾ා͹ҩ͏͗ɼࢨ໴‫ܙ‬੔‫غ‬͹ൿԾ͹݄ӹ͹ླྀΗͶӪ‫ڻ‬Ν༫͓ͱɼ‫ܙ‬য়ͶรԿ͗ਫ਼ͣͱ͏Ζ͹Ͳͺ͵͏͖ͳ͏͑͞ͳͺ༲ҝͶ૟଀ͤΖ͞ͳ͗Ͳ͘Ζ͗ɼͨΗΝৄ໎ͤΖͶͺ݄ླྀΝ؏ࡱͲ͘Ζಝच͵ૹ஖͗චགྷͲ͍Ε[5]ɼඉ৙ͶࠖೋͳࢧΚΗͪɽ͖͢͢ɼ݄ӹ͗࢐ોΏӭ཈੔෾Ώ࿟ഉ෼͹Ά͖Ͷɼ೦ΦϋϩάʖΝӣ΁͞ͳ͗எΔΗͱ͏Ζ͞ͳ͖ΔɼϪ΢όʖ‫ݳ‬ে͹ਏஇಋ༹ [6]ɼ੼֐તγʖϠήϧϓΡʖͲ͍Δ͖ͣΌྮ‫ࢨͪ͢٭‬͹ൿේන໚Իౕ͹յ෰รԿΝ‫࣎ܨ‬దͶ؏

t 1 tࡱͤΖ͞ͳͶΓͮͱɼൿԾ͹݄ӹΏૌ৭ӹ͹ླྀΗ͹๏޴Ώଐ͠ΝਬఈͤΖ͞ͳ͗Ͳ͘Ζ͹Ͳͺ͵͏͖ͳߡ͓ΔΗͪɽࠕյɼ͍Δ͖ͣΌྮ‫ͪ͢͠٭‬Ή͡Ή͵‫ܙ‬͹ࢨ໴Ν࣍ͯࢨ͹ൿේන໚Իౕ͹‫࣎ܨ‬దรԿΝ੼֐તγʖϠήϧϓΡʖͲ؏ࡱͪ͢ͳ͞Θɼࢨ໴͹‫ͳ͟ܙ‬Ͷ໎Δ͖͵ҩ͏͗೟ΌΔΗɼͨΗͺΉͪɼൿԾ͹੫ຼ෾ාͳ΍ືંͶ‫ؖ‬࿊͢ͱ͏Ζ ͞ͳ͗໎Δ͖
Ͷ͵ͮͪɽΉͪɼҲൢͶɼࠪӊ͹ࢨ͹ࢨ໴͹‫ܙ‬͹Ҳ஗ི͗߶͏͞ͳ͗எΔΗͱ͏Ζ͗ɼಝͶࠪӊ෈Ҳ஗ྭͶͯ͏ͱ΍ಋ༹͹‫ݗ‬౾Νߨͮͪɽͨ͹݃ՎɼൿԾ੫ຼ෾ා͹ҩ͏ͶΓͮͱ
͕͞ΖൿԾ͹݄ӹΏૌ৭ӹ͹ླྀΗ͹๏޴Ώଐ͠͹ҩ͏͗ɼࢨ໴͹ཾત͹‫ܙ‬Ͷ͠Ή͡Ή͵ώϨΦʖεϥϱΝਫ਼ͣͦ͠ͱ͏Ζ΍͹ͳߡ͓ΔΗͪ͹ͲๅࠄͤΖɽ

2. ଲেͳ๏๑
2.1. ଲে
ൿේ໴ཀྵʤࢨ໴ʥ͹஦Ͳɼ࠹΍ఱ‫ܗ‬ద͵ఛয়໴ɼӖয়໴ɼ‫ٹ‬য়໴ྭ Ν؏ࡱ͢ɼർֳ‫ݗ‬౾ͪ͢ɽΉͪɼࠪӊ͹घ͹ࢨ͹ࢨ໴෈Ҳ஗ྭ̑ྭͶͯ͏ͱൿේන໚͹ԻౕรԿͳൿԾ੫ຼ෾
ාΝ؏ࡱͪ͢ɽ

2.2. ๏๑
a) ৏ࢸࢨන໚ԻౕรԿ͹؏ࡱ৏ࢸ֦ࢨΝ͍Δ͖ͣΌතਭ஦Ͷ਼෾ؔਃͪ͢͹ͬɼਭ෾Ν৩͘खΕɼγʖϠήϧϓΡʖʀγʖϠάΠ G120ʤNEC Avio ੼֐તτέόϫζʖ, ౨‫ښ‬ʥΝ༽͏ͱɼ৏ࢸ֦ࢨ͹ঢଈ຦ઇ͹න໚ԻౕรԿʤ଎ఈԻౕઅఈ൥ҕʁ20-ˈʥΝ‫࣎ܨ‬దͶࡳӪͪ͢ɽb) ৏ࢸࢨ֦຦ઇ͹ൿේ໴ཀྵ؏ࡱͳൿԾ݄؇෾ා͹؏ࡱ৏ࢸ֦ࢨঢଈ຦ઇ͹ൿේ໴ཀྵΝ೓‫ࡱ؏Ͳء‬෾ྪͪ͢‫ޛ‬ɼൿԾ͹݄؇෾ාΝۛ੼֐તड़ྙ
LED ΢ϩϝϋʖνʖʤഀௗ 850 nm, ೖ਒ుࢢ޽‫ۂ‬, ౨‫ښ‬ʥͳϜέϫϪϱθ L6000ʤΦχ Ϡ ϱ χ Ψ ϕ τ Ρ έ η ζ ϡ Ϗ ϱ , ౨ ‫ ښ‬ʥ Ν ૹ ஥ ͢ ͪ ੼ ֐ ત CCD Ω ϟ ϧ ʤ XC-E150,
SONY, ೖຌʥΝ༽͏ͱࡳӪͪ͢ɽ৏ࢸ͹֦ࢨ຦ઇ͹ഐଈʤ௼ଈʥ຤ͺ෴ଈͶ΢ϩϝϋʖνʖΝື஥ͦ͠ͱল໎͢ɼൕଲଈ͖ΔࡳӪͪ͢ʤFig. 1ʥɽ

t 2 t

Fig. 1: Analysis of dinamic infrared thermography and blood vessel distribution. The
dinamic infrared thermography of pre-cooling finger was performed using the infrared
thermographic aparatus, and the obsevation of blood vessel distribution was performed in
the dark chamber using the near infrared LED illumination and infrared CCD camera.

3. ݃Վ৏ࢸࢨ຦ઇ͹ࢨ໴͗ఛয়໴ɼӖয়໴ɼ‫ٹ;ٶ‬য়໴Ͳ͍ΖྭʤFig. 2AʥͶͯ͏ͱɼ͍Δ͖ͣΌ֦ࢨΝතਭ஦Ͳྮ‫ޛͪ͢٭‬ɼӀ͘৏͝ͱɼγʖϠήϧϓΡʖͲࢨঢଈන໚Իౕ͹յ෰รԿΝ‫࣎ܨ‬దͶ؏ࡱ͢ɼࡳӪͪ͢ʤFig. 2Bʥɽఛয়໴Ͳͺɼ੼֐તγʖϠήϧϓΡʖͲঢଈൿේන໚͹ԻౕรԿΝ؏ࡱͤΖͳɼΉͥॵΌͶࢨ຦ઇ͹஦ԟ෉ۛ຤ͺ઎୼෉ۛͲൿේන໚
Իౕ͗৏ত͢ɼ࣏͏Ͳࠪఛয়໴Ͳͺ຦ઇࠪଈͶɼӊఛয়໴Ͳͺ຦ઇӊଈͶԻౕ৏তҮ͗ड़‫ͪ͢ݳ‬ɽͨ͹ൕଲଈ྘Үͺ͢ͻΔ͚ஙΗͱൿේන໚Իౕ͗৏তͪ͢ɽۛ੼֐ޭলࣻͳ੼֐ત CCD ΩϟϧͶΓͮͱൿԾ͹݄؇෾ාΝ؏ࡱͤΖͳɼ੼֐તγʖϠήϧϓΡʖͲ઎ͶԻౕ৏ত͹‫ݡ‬ΔΗͪࢨ຦ઇ͹ࠪΉͪͺӊ୼ঢଈ୊Ҳ‫ؖ‬ઇ෉ۛ͹ൿԾͶଢ͏੫ຼ֮͗೟͠ΗͪɽӖয়໴Ͳͺɼ੼֐તγʖϠήϧϓΡʖͲঢଈൿේන໚͹ԻౕรԿΝ؏ࡱͤΖͳɼΉͥॵΌͶࢨঢଈ຦ઇ஦ԟ෨ͲԻౕ৏ত͗࢟ΉΕɼ͖ͨ͞ΔबҕͶ߁͗ΖΓ͑ͶԻౕ͗৏ত͢ɼ஦ ઇ Ͷ ޴ ͖ ͮ ͱ ർ ֳ ద ࠪ ӊ ώ ϧ ϱ η Γ ͚ Ի ౕ ৏ ত Ү ͗ ֨ ୉ ͢ ͪ ɽ ۛ ੼ ֐ ޭ ল ࣻ ͳ ੼ ֐ત
CCD ΩϟϧͶΓͮͱൿԾ͹݄؇෾ාΝ؏ࡱͤΖͳɼࢨ຦ઇ͹ঢଈ஦ԟ෨Ͷ໡ࡋ݄؇ືॄ
ͶًҾͤΖͳࢧΚΗΖࠉ͏Ӫ͗೟ΌΔΗɼͨ͞Ͷࠪӊ͖Δ̐ຌ͹ଢ͏੫ຼ͗ۛͰ͏ͱ͏ͪɽ
‫ٹ‬য়໴Ͳͺɼ੼֐તγʖϠήϧϓΡʖͲঢଈൿේන໚͹ԻౕรԿΝ؏ࡱͤΖͳɼΉͥॵΌͶࢨ຦ઇ஦ԟ෨ͲԻౕ৏ত͗࢟ΉΕɼ͖ͨ͞Δ຦ઇસରͶࠪӊώϧϱηΓ͚߁͗ͮͪ͗ɼ
͵͖͵͖஦ઇͶ߁͗Δͥɼ຦ઇͳ஦ઇ͹ؔ͹Իౕ৏ত͗ஙΗΖܑ޴͗೟ΌΔΗͪɽۛ੼֐
ޭলࣻͳ੼֐ત CCD ΩϟϧͶΓͮͱൿԾ͹݄؇෾ාΝ؏ࡱͤΖͳɼӖয়໴Ͷർ΄ͱΏΏࢨ઎୼ଈ஦ԟ෨Ͷ໡ࡋ݄؇ືॄͶًҾͤΖͳࢧΚΗΖࠉ͏Ӫ͗೟ΌΔΗɼ͖ͨ͞Δଢ͏݄
؇͗ྈଈͶ৵;ͱ͏ͪɽͪͫ͢ɼ͞͹ଢ͏݄؇Νࢨଈ໚͖Δ؏ࡱͤΖͳɼӖয়໴Ͷർ΄ͱΓΕ਄͏௼ଈͶ͍Ζ͞ͳ͗໎Δ͖Ͷ͵ͮͪɽ࣏Ͷɼࠪӊ͹ࢨ໴෈Ҳ஗ྭ ̑ྭͶͯ͏ͱɼಋ༹Ͷࢨ໴͹‫ܙ‬ɼ੼֐તγʖϠήϧϓΡʖɼۛ੼֐તলࣻͶΓΖൿԾ੫ຼ෾ාͶͯ͏ͱർֳ͢ɼͨ͹େනྭΝࣖͪ͢ʤFig. 2Cʥɽ͞͹t 3 t

ྭͲͺࠪӊ͹୊ᶚࢨ͹ࢨ໴Ͷҩ͏͗‫ݡ‬ΔΗͪɽࠪଈ͗‫ٹ‬য়໴͹஦ԟ෨Ͷϛίρφ͹͍Ζ͏
ΚΑΖ༙ଽ‫ٹ‬য়໴Ͳɼӊଈ͗ఛয়໴͹஦ԟ෨Ͷϛίρφ͹͍Ζ͏ΚΑΖ༙ଽఛয়໴Ͳ͍ͮ
ͪɽ੼֐તγʖϠήϧϓΡʖͲ͍Δ͖ͣΌྮ‫ࢨͪ͢٭‬͹ൿේන໚Իౕ͹รԿΝ؏ࡱͤΖͳɼ༙ଽ‫ٹ‬য়໴ͲͺɼॵΌͶࢨ຦ઇ஦ԟ෨ͲࣾΌͶยͮͱԻౕ৏ত͗࢟Ήͮͪ͗ɼ஦ઇଈͶ޴͖ͮͱԻౕ৏তҮͺࠪӊͶ߁͗ͮͪɽۛ੼֐તলࣻͳ੼֐ત CCD ΩϟϧͶΓͮͱൿԾ͹੫ຼ෾ාΝ؏ࡱͤΖͳɼࢨঢଈ຦ઇ͹஦ԟ෨Ͷ໡ࡋ݄؇ືॄͶًҾͤΖͳࢧΚΗΖࠉ͏Ӫ͗೟ΌΔΗɼࠪӊ͖Δ஦ԟ෨Ͷ 2ʛ3 ຌ͹݄؇͗ॄΉͮͱ͏Ζ͹͗؏ࡱ͠Ηͪɽଈ໚͖Δ
؏ࡱͤΖͳɼࠪӊͶ໎ྐ͵݄؇ͺ؏ࡱ͠ΗͥɼસରͶ෈໎ྐ͵݄؇͗ࢆࡑͪ͢ɽҲ๏ɼ༙ଽఛয়໴Ͳͺɼ͍Δ͖ͣΌྮ‫ࢨͪ͢٭‬͹ൿේන໚Իౕ͹รԿΝ੼֐તγʖϠήϧϓΡʖͲ
؏ࡱͤΖͳɼॵΌͶࢨ຦ઇ஦ԟ෨ͲࣾΌͶยͮͱԻౕ৏ত͗࢟ΉΕɼ߁͗ͮͪ͗ɼͨ͹‫ޛ‬ɼ஦ ԟ ෨ ͹ Ի ౕ ৏ ত Ү Ν ࢔ ͢ ͱ ɼ ஦ ઇ Ͷ ޴ ͖ ͮ ͱ ࠪ ӊ Ͷ ෾ ‫ ͪ ͢ ز‬ɽ ۛ ੼ ֐ ત ল ࣻ ͳ ੼ ֐ત
CCD ΩϟϧͶΓͮͱൿԾ͹੫ຼ෾ාΝ؏ࡱͤΖͳɼ຦ઇ஦ԟ෨ʤ༙ଽ໴஦ԟ෨ʥͶଢ͏݄؇͗ॐͶૺΕɼԻౕ৏তҮ͹෾‫ز‬ͶԌ͑Γ͑Ͷ۸‫ͪ͏ͱ͢ۄ‬ɽଈ໚ ͖Δ؏ࡱͤΖͳɼӊଈͶർ΄ͱࠪଈͶଢ͏݄؇͗؏ࡱ͠Ηͪɽଠ͹ࠪӊ෈Ҳ஗ 2 ྭͺɼ୊ᶚࢨ༙͗ଽఛয়໴ͳఛ
য়໴ɼ୊ᶛࢨ͗Ӗয়໴ͳ༙ଽఛয়໴͹ྭͲ͍ͮͪ͗ɼ੼֐તγʖϠήϧϓΡʖ͹݃Վɼఛয়໴ɼӖয়໴ྭͺ Fig. 2B ͹ఱ‫ͳྭܗ‬ɼΉ༙ͪଽఛয়໴ྭͺɼ෈Ҳ஗େනྭʤFig. 2Cʥ͹΍͹ͳྪࣇ͹Իౕ৏তϏνʖϱΝࣖͪ͢ɽt 4 t

Fig. 2: Fingerprint, dinamic infrared thermography, and blood vessel distribution of the palmside surface of a finger end segment. A, Fingerprint pattern. a: right loop; b: left loop; c: whorl;d: arch. B, Dinamic infrared thermography and blood vessel distribution of fingers with different
fingerprint patterns. C,

Index fingers, ᶚ, surrounded by a white line in this image: fingerprint

mismatch between the left and right fingers. Middle fingers, ᶛORRSs; ring fingers, ᶜZKRUOs.

t 5 t

4. ߡࡱ
ͽͳΕͽͳΕࢨ໴͹‫ܙ‬ͺҡ͵ͮͱ͕Εɼ੊ֆͶͽͳΕͳ͢ͱಋͣंͺ͏͵͏ͳ͠ΗΖɽ
͖͢͢ɼͨ͹ࢨ໴͹‫ʹ͗ܙ‬͹Γ͑Ͷ͢ͱ݀ΌΔΗΖ͹͖Νઈ໎ͪ͢࿨ชͺΆ ͳΞʹ͵͏ɽࢨ໴Ν੔ͤ౵ֶؔ͹ཾત‫ܙ‬੔Ͷͯ͏ͱͺɼ.ÝFNHQ ͳ Newell ͗ɼཾત‫ܙ‬੔࣎͹‫خ‬ఊࡋ๖૜͹࣏ࠫద੔ௗͳঢଈϏρχ͹ୂߨͶଲͤΖఏ߇ͶΓͮͱ༢಍͠ΗΖͳͤΖઈΝ఑঑͢ͱ͏
Ζ͗[7]ɼ෵ࡸ͵ཾત‫ܙ‬੔͹ઈ໎ͳ͢ͱै෾ͳͺ‫͏͵͓ݶ‬ɽ઎Ͷըʓͺɼࢨ໴͹‫ͳܙ‬ൿԾ͹੫ຼ෾ා͹ؔͶҲఈ͹‫ؖ‬࿊੓͍͗Ζ͞ͳΝๅࠄͪ͢ [6]ɽൿԾ͹੫ຼ෾ා͹ҩ͏͗ɼൿԾ͹݄ӹΏૌ৭ӹ͹ླྀΗ͹๏޴Ώଐ͠Ͷҩ͏Νਫ਼ͣͦ͠ɼࢨ໴͹ཾત͹ૺΖ๏޴ͶӪ‫ڻ‬Ν༫͓
ͱ͏ΖՆ೵੓Ν‫ݗ‬౾ͤΖͪΌɼ੼֐તγʖϠήϧϓΡʖΝ༽͏ͱɼ͍Δ͖ͣΌྮ‫͕ͱ͢٭‬͏֦ͪࢨঢଈ͹ൿේන໚͹ԻౕรԿΝ؏ࡱͪ͢ɽͨ͹݃Վɼఛয়໴Ͳͺɼൿේන໚Իౕ͗
૥͚৏তͤΖଈͶཾત͗ླྀΗΖΓ͑Ͷૺߨ͢ͱ͏ͪɽൿේන໚Իౕ͗૥͚৏তͤΖͳ͏͑͞ͳͺɼͨͬΔଈͶ݄ӹΏૌ৭ӹ͗ླྀΗͱ͏Ζ΍͹ͳߡ͓ΔΗΖ͗ɼͨ͞Ͷͺർֳదଢ͏
੫ຼ͗෾ා͢ͱ͏Ζ͞ͳ͗໎Δ͖Ͷ͵ͮͪɽ໡ࡋ݄؇Ώࡋ੫ຼͶɼۛ๥͹ಊຼͶ༟པͤΖ
ૌ৭ӹ͗खΕࠒΉΗͱ͕Ε[8]ɼΫϡϒϧϨʖϗρχ͖Δࡋ੫ຼͶ޴͖͑ૌ৭ӹ͹ླྀΗ͗ਫ਼ͣͱ͏Ζ΍͹ͳߡ͓ΔΗͪɽӖয়໴Ͳͺɼ຦ઇ஦ԟ෨͖ΔԻౕ͗৏ত࢟͢Όɼͨ͹৏তҮ͗຦ઇ͖Δ஦ઇ΃ͳർֳదώϧϱηΓ͚߁͗ͮͪɽ຦ઇ஦ԟ෨Ͷͺ໡ࡋ݄؇͗ືͶ෾ා͢ͱ͕Εɼ຦ઇबว෨͖Δͨ͹஦ԟ෨Ͷ݄ӹΏૌ৭ӹ͗ླྀΗࠒΊɼͨ͞ͶӖ͹஦ৼ͗Ͳ͘ͱ
͏Ζ΍͹ͳߡ͓ΔΗͪɽ‫ٹ‬য়໴Ͳͺɼ຦ઇ઎୼ଈ͹ൿේන໚ԻౕͺଐΏ͖Ͷ৏তͤΖ͗຦ઇԾଈͳ஦ઇ͹ؔ͹Իౕ৏তͺஙΗΖܑ޴Νࣖͪ͢ɽ‫ٹ‬য়໴͹຦ઇ஦ԟ෨ͶͺӖয়໴ಋ༹ ɼ໡ࡋ݄؇͗ືͶ෾ා͢ͱ͕Εɼͨ͹஦ԟ෨Ͷ݄ӹΏૌ৭ӹ͗ླྀΗࠒ΋͗ɼ͖ͨ͞ΔबวͶ‫ۋ‬౵Ͷ֨ࢆͦͥͶɼࠪӊ΃͹ླྀΗ͗‫͏ͱͮ͵͚ک‬Ζ΍͹ͳߡ͓ΔΗͪɽࠪӊ͹ࢨ໴෈Ҳ஗ྭͶͯ͏ͱ΍ɼཾત͹ૺߨͶԌ͑Γ͑ͶԻౕ৏তً͗͞Εɼ͞͹Իౕ৏তҮΝཬ෉͜ΖΓ͑ͶൿԾ੫ຼ͗෾ා͢ͱ͕Εɼ͞͹੫ຼ෾ා݄͗ӹΏૌ৭ӹ͹ླྀΗΝࡠΕɼཾત͹ૺߨͶӪ‫͏ͱ͢ڻ‬Ζ΍͹ͳߡ͓ΔΗͪɽҐ৏͹݃Վ͖Δɼ઎Ͷๅࠄͪ͢Γ͑Ͷࢨ໴͹‫ͳܙ‬ൿԾ͹੫ຼ෾ා͹ؔͶͺືં͵‫͗ܐؖ‬͍Εɼͨ͹‫ݬ‬Ҿͳ͢ͱͺɼ໡ࡋ݄؇͖Δࡋ੫ຼ΃ͳखΕࠒΉΗΖૌ৭ӹʤ݄ᕸʥ͹ླྀΗ͗ཾત͹ૺߨͶӪ‫ڻ‬Ν‫͏ͱ͢·ٶ‬Ζ΍͹ͳߡ͓ΔΗͪɽ͠Ή͡Ή͵Ҫఽබͳࢨ໴෾ා͹ยΕ͹‫ؖ‬࿊੓͗ๅࠄ͠Ηͱ͏Ζ͗[9]ɼ१‫ܧح؂‬͹઎ళదҪఽద࣮‫׳‬Ͷ൒݄͑ླྀଐౕ͹รԿΏ݄؇৿ਫ਼ҡ৙͵ʹ͗ɼൿԾ͹݄ӹΏૌ৭ӹ͹ླྀΝรԿͦ͠ɼཾતૺߨͶӪ‫ڻ‬Ν‫͏ͱ͢·ٶ‬Ζ͹Ͳͺ͵͏͖ͳߡ͓ΔΗͪɽҪఽබͲͺɼࢨ໴෾ා͹ยΕ͹Ί͵Δͥɼ‫ا‬૮͹รԿ΍ๅࠄ͠Ηͱ͕Ε[10, 11]ɼ͞ΗΔ͹‫ا‬૮͹รԿ΍ࢨ໴ͳಋ༹ɼൿԾ੫ຼ෾ාΏ݄ӹ೬ౕ͹รԿͶ൒݄͑ӹΏૌ৭ӹ͹ླྀΗ͹รԿͶًҾ͢ͱ͏Ζ͖΍͢Η͵͏ɽࠕ‫ޛ‬ɼAI ͹൅ୣ౵ͶΓͮͱɼࢨ໴Ώ‫ا‬૮͵ʹ͹‫ܙ‬ସదಝ௅͖Δɼ͠Ή͡Ή͵ҪఽదϨηέϓΟένʖΏ঳য়րવ͹ͪΌ͹৿͢͏࣑ྏ๑͗൅‫͠ݡ‬ΗΖ͞ͳ͗‫غ‬ଶ͠ΗΖɽࢂߡช‫ݛ‬[1] Seidenberg-Kajabova, H., Pospisilova, V., Vranakova, V and Varga, I., An original
histological method for studing the volar skin of the fetal hands and feet, Biomed. Pap.
Med. Fac. Univ. Palacky. Olomouc. Czech. Repub. 154, 211-218, 2010.
[2] Ԯ㟔ౌࢦ෋, ࢨ໴͹‫ͳܙ‬੫ຼ෾ා͹‫ܐؖ‬Ͷͯ͏ͱ , ‫ܙ‬͹Ռָճࢿ, ୊ 30 ‫ת‬, ୊ 2 ߺ, 104-t 6 t

114, 2015.
[3] Yashiro, K., Shiratori, H. and Hamada, H., Haemodynamics determined by a genetic
programme govern asymmetric development of the aortic arch, Nature 450, 285 -288, 2007.
[4] ᖝీദ࢚, ର͹ࠪӊඉଲস, ਫ਼ࢊͳٗढ़, ୊ 60 ‫ת‬, ୊ 3 ߺ, 77-80, 2008.
[5] Webb, R.C., Ma, Y., Krishnan, S., Li, Y., Yoon, S., Guo, X., Fe ng, X., Shi, Y., Seidel M.,Cho,N. H., Kurniawan, J., Ahad, J., Sheth, N., Kim, J., Taylor Vl, J.G., Darlington, T., Chang,K., Huang, W., Ayers, J., Gruebele, A., Pie lak, R. M., Slepian, M.J., Huang, Y., Gorbach,A. M. and Rogers, J. A., Epidermal devices for noninvasive, precise, and continuous
mapping of macrovascular and microvascular blood flow, Science Advances 1, e1500701 ,2015.
[6] ో෨Ҳ࿢, Ϫ΢όʖ‫ݳ‬েΝ༙ͤΖ᡾‫ݬ‬බ‫ं׳‬͹ྮਭෝ՛ࢾ‫ݩ‬Ͷ ΓΖγʖϠήϧϓΡ ʖ͹‫ݗ‬౾, ϨΤϜο 31 ‫ת‬, 2 ߺ, 167-173, 1991.
>@.ÝFNHQ0DQG1HZHOO$&)LQJHUSULQWIRUPDWLRQ-7KHR%LRO -83, 2005.
[8] Kobayashi, H. and Takizawa, N., Imaging of oxygen transfer among microvessels of rat
cremaster muscle, Circulation 105, 1713-1719, 2002.
[9] Okazaki, T., Relationship between deviated fingerprint distribution and vascular factor
abnormality, J. Vasc. 3, 1000125, 2018.
[10] Dimitriou, D., Leonard, H. C., Karmiloff-Smith, A., Johnson, M. H. and Thomas, M. S.,
Atypical development of configural face recognition in children with autism, Down
syndrome and Willams syndrome, J. Intellect. Disabil. Res. 59, 422 -438, 2015.
[11] Kosilek 5 3 )URKQHU 5 :ÝUW] 5 3 %HHU & 0 6FKRSRKO - 5HLQFNH 0 DQGSchneider, H. J., Diagnostic use of facial image analysis software in endocrine and genetic
disorders: review, current results and future perspectives, Eur. J. Endocrinol. 173, M39M44, 2015.

t 7 t

t 8 tմઈ࿨ช‫ܙ‬͹Ռָճࢿ ୊ 36 ‫ ת‬୊ 1 ߺब‫غ‬ഔ෾‫ز‬Ͷ͕͜Ζ࢝ͯ͹෾‫ز‬գఖࢃ޳‫ج‬ദ
ˡ-

એཁ‫ࠅࢤݬࢤݟ‬෾࣋ୈ஦ԟ --

FKDRVIUDFWDO#L&ORXGFRP)RXUELIXUFDWLRQSURFHVVHVRIWKHSHULRG-GRXEOLQJELIXUFDWLRQ<RVKLKLUR<DPDJXFKL--.RNXEXQMLGDL-&KXR,FKLKDUD&KLED--DSDQ(2021 ೧ 5 ݆ 17 ೖण෉ɾ2021 ೧ 6 ݆ 5 ೖणཀྵ)Abstract: The two-dimensional area-preserving map ܶ including one parameter ܽሺ൒ Ͳሻ isconsidered. It is assumed that ܶ is integrable at ܽ ൌ Ͳ and the Smale horseshoe ‫ ܪ‬exists atܽ ൒ ܽ௖ு . The bifurcation processes of the period-doubling bifurcation studied in this paper areclassified into the following four types (Types I-IV). Type I: The elliptic periodic orbit
undergoes only the ordinary period-doubling bifurcation and becomes a saddle orbit withreflection in ‫ܪ‬. Type II: The elliptic periodic orbit undergoes the ordinary period -doublingbifurcation, the reverse period-doubling bifurcation and the equi-period bifurcation in thisorder and becomes a saddle orbit in ‫ܪ‬. Type III: The saddle periodic orbit undergoes thereverse equi-period bifurcation and the ordinary period-doubling bifurcation in this orderand becomes a saddle orbit with reflection in ‫ ܪ‬. Type IV: The saddle periodic orbit
undergoes the reverse equi-period bifurcation, the ordinary period-doubling bifurcation, thereverse period-doubling bifurcation and the equi-period bifurcation in this order andbecomes a saddle orbit in ‫ܪ‬. Using the typical periodic orbits, we discuss the properties offour types. The code of the daughter periodic orbit appearing through the (reverse)
period-doubling bifurcation is determined. The codes of two daughter periodic orbits
appearing through the (reverse) equi-period bifurcation are also determined. The cause of
the occurrence of four types is elucidated.
Keywords: Area-preserving map, Period-doubling bifurcation, Equi-period bifurcation,
Smale horseshoe, Code1. ຌ࿨ช͹໪దຌ࿨ชͲͺ, ฑ໚৏Ͳఈٝ͠Ηͪ໚੷ฯଚࣺ଀ ܶͶ͕͜Ζब‫غ‬ഔ෾‫ز‬գఖ[1]Ν௒΄Ζ.
ܶ ‫ݕ  ׷‬௡ାଵ ൌ ‫ݕ‬௡ ൅ ݂ሺ‫ݔ‬௡ ሻǡ‫ݔ‬௡ାଵ ൌ ‫ݔ‬௡ ൅ ‫ݕ‬௡ାଵ Ǥ(1)ଶ͞ ͞ Ͳ ݂ሺ‫ݔ‬ሻ ൌ ܽሺ‫ ݔ‬െ ‫ ݔ‬ሻሺܽ ൒ Ͳሻ. ࣺ ଀ ܶͺ Φ ό ϱ ࣺ ଀ ଔ [2]Ͷ ‫ ؜‬Ή Η Ζ . ܽ ൌ ͲͲ ͺ , ࣺ ଀ ܶͺ Ն੷ ෾ ‫ ͍Ͳ ܧ‬Ζ . ܽ ൒ ܽ௖࣢ ൌ ͷǤͳ͹͸͸Ͳͷ͵͸ͻͲͶ ‫ڮ‬Ͷ ͕ ͏ ͱ, ࣺ଀ ܶͶ ͺ η ϟ ʖ ϩ ͹ ഇ ఛ ࣢͗ ଚ ࡑͤ Ζ
[3]. ͞͹͞ͳΓΕ‫ָྙߺى‬͹੔ՎΝཤ༽ͤΖ͞ͳ͗Ն೵Ͳ͍Ζ [4]. ຌ࿨ชͲͺགྷ੧ 1.1 ͹t 9 t΍ͳͲ٠࿨Νߨ͑.
གྷ੧ 1.1. ϏϧϟʖνܽΝଁՅͤΖͳ, ବԃ఼͹बΕ͹յ఺͹࢕๏ͺଁՅͤΖ .
ବ ԃ ఼ ͹ ब Ε Ν յ ఺ ͤ Ζ ब ‫ ٌ غ‬ಕ ͹ յ ఺ ͹ ࢕ ๏ Ν න ‫ ͤ ݳ‬Ζ յ ఺ ਼ ‫݌‬Ȁ‫ݍ‬Ν ಍ ೘ ͤ Ζ. յ ఺ ਼‫݌‬Ȁ‫ݍ‬ͺ, ࣺ଀ܶΝ‫ݍ‬յࡠ༽ͤΖͳब‫ٌغ‬ಕ͗ବԃ఼͹बΕΝ ‫݌‬յͫ͜յ఺ͤΖ͞ͳΝқັ͢ͱ͏Ζ. ϏϧϟʖνܽΝଁՅͤΖͳགྷ੧ 1.1 ΓΕବԃ఼͹बΕ͹յ఺͹࢕๏ͺଁՅͤΖ . յ఺਼͗ͳȀʹͶ౺ୣͤΖͳब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣΖ . ͯΉΕ, གྷ੧ 1.1 ͗ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣΖक‫ݬ‬ҾͲ͍Ζ͞ͳͺؔҩ͏͵͏࣏ Ͷ ब ‫ ٌ غ‬ಕ ͹ ત ‫ ҈ ܗ‬ఈ ੓ Ν ؈ ୱ Ͷ Ή ͳ Ό ͱ ͕ ͚ . ब ‫ݍ غ‬͹ ٌ ಕ ఼ Ν ‫ݖ‬௞ ൌ ሺ‫ݔ‬௞ ǡ ‫ݕ‬௞ ሻሺͲ ൑ ݇ ൏‫ݍ‬ሻͳͤΖ. ٌಕ఼‫ݖ‬௞ ሺൌ ܶ ௞ ‫ݖ‬଴ ሻͶ͕͜Ζત‫ܙ‬Կߨ྽ ‫ܯ‬ሺ‫ݖ‬௞ ሻͺͳ ݂Ԣሺ‫ݔ‬௞ ሻ‫ܯ‬ሺ‫ݖ‬௞ ሻ ൌ ൬൰ͳ ͳ ൅ ݂Ԣሺ‫ݔ‬௞ ሻ(2)

ͳಚΔΗΖ. ͞͞Ͳ, ݂Ԣሺ‫ݔ‬௞ ሻ ൌ ܽሺͳ െ ʹ‫ݔ‬௞ ሻ. Ήͪ, ‫ܯ‬ሺ‫ݖ‬௞ ሻ͹ߨ྽ࣞͺ 1 Ͳ͍Ζ. ͞Ηͺࣺ଀ܶ͗໚
੷ฯଚͲ͍Ζ͞ͳΝқັ͢ͱ͏Ζ .
ࣺ ଀ ܶ ௤ ͹ ત ‫ ܙ‬Կ ߨ ྽ ͺ ‫ܯ‬௤ ൌ ‫ܯ‬ሺ‫ݖ‬௤ିଵ ሻ ‫ܯ ڮ‬ሺ‫ݖ‬ଵ ሻ‫ܯ‬ሺ‫ݖ‬଴ ሻͲ ͍ Ζ . ‫ݎ‬ሺܽሻ ൌTrace‫ܯ‬௤ ͳ ͢ ͱ , ‫ ༙ ݽ‬஍ ߣΝ݀ఈͤΖ‫༙ݽ‬๏ఖࣞͺ࣏͹Γ͑Ͷॽ͜Ζ .
ߣଶ െ ‫ݎ‬ሺܽሻߣ ൅ ͳ ൌ ͲǤ(3)‫ݎ ਼ ܐ‬ሺܽሻ Ν ҈ ఈ ੓ ‫ ͜ Ͱ ໌ ͳ ਼ ܐ‬Ζ . ҈ ఈ ੓ ‫ ਼ ܐ‬Ν ΍ ͳ Ͷ ब ‫ ٌ غ‬ಕ ͹ ҈ ఈ ੓ Ν ෾ ྪ ͤ Ζ .
‫ݎ‬ሺܽሻ ൐ ʹ͵Δͻब‫ٌغ‬ಕͺγχϩ‫͍Ͳܗ‬Ε , ȁ‫ݎ‬ሺܽሻȁ ൏ ʹ͵Δͻब‫ٌغ‬ಕͺବԃ‫͍Ͳܗ‬Ε , ‫ݎ‬ሺܽሻ ൏െʹ͵Δͻब ‫ ٌغ‬ಕͺൕ ఺γ χϩ‫͍ Ͳܗ‬Ζ . ‫ݎ‬ሺܽሻ ൌ െʹΝຮͪͤ ܽ͹஍Ͳ(ൕ)ब‫غ‬ഔ ෾‫ ͗ز‬ਫ਼ͣΖ. ‫ݎ‬ሺܽሻ ൌ ʹΝຮͪͤܽ͹஍Ͳ(ൕ)ಋब‫غ‬෾‫͗ز‬ਫ਼ͣΖ.

ਦ 1ʁ࢝ͯ͹ν΢ϕ͹ब‫غ‬ഔ෾‫ز‬գఖͶ͕͜Ζ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻ͹Ϗϧϟʖν ܽғଚ੓.
ब‫غ‬ഔ෾‫ز‬գఖͳ͢ͱ࢝ͯ͹ν΢ϕΝ়ղͤΖ . ब‫غ‬ഔ෾‫ͳ͑͏ͳز‬௪৙ͺν΢ϕ I ΝқັͤΖ. ν΢ϕ II Ͷͯ͏ͱͺࢂߡช‫[ݛ‬4, 5]Ν, ν΢ϕ III Ͷͯ͏ͱࢂߡช‫[ݛ‬6]ͺ‫ݡ‬ΔΗ
ͪ͏. ν΢ ϕ IV Ͷͯ ͏ ͱͺ ຌ࿨ ช͗ ࠹ ॵ͹ ๅࠄ Ͳ͍ Ζ . ͞ ͞Ͳ ஭қ Ν͢ ͱ͕ ͪ͘ ͏ . ྭͳ͢ ͱ ν ΢ ϕ II Ͳ ͺ ఈ ੓ ‫ݎ ਼ ܐ‬ሺܽሻ͹ ‫ ݰ‬ঙ ͳ ଁ Յ ͗ ‫ ݡ‬Δ Η Ζ . ఈ ੓ ‫ݎ ਼ ܐ‬ሺܽሻ͗ ‫ ݰ‬ঙ ͢ ͱ ͏ Ζ Ϗϧϟʖν ྘Ү Ͳͺବ ԃ఼ ͹बΕ͹ յ఺ ͹࢕๏ ͺଁ Յ͢ͱ͏ Ζ . Ήͪఈ ੓‫ݎ ਼ ܐ‬ሺܽሻ͗ଁՅ ͢ͱ
t 10 t͏ΖϏϧϟʖν྘ҮͲ΍ବԃ఼͹बΕ͹ յ఺͹࢕๏ͺଁՅ͢ͱ͏Ζ (୊ 3.2 ઇΝࢂল). གྷ੧
1.1 ͹΍ͳͲବԃ఼͹ब Ε͹յ఺͹࢕๏͗ଐ͚͵ ͮͪΕங͚͵ͮͪΕͤΖ ͞ͳͺ͵͏ . ͨ͹ͪΌͶब‫غ‬ഔ෾‫ز‬գఖͺ࢝ͯ͹ν΢ϕͶ෾ྪ͠ΗΖ . ࣏Ͷ࢝ͯ͹ν΢ϕ͹ಝ௅ΝΉͳΌͱ͕͚.
ν ΢ϕ I (ਦ 1(a)). ฾ ବ ԃ‫ ܗ‬ब ‫ٌ غ‬ಕ ͗ ਫ਼ ͣͪ ‫ ޛ‬, ͨ͹ ٌ ಕ఼ ‫ݖ‬௞ ͹ ब Ε͹ յ ఺ͺ ଐ ͚ ͵Ζ .
݃Վͳ͢ͱ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻͺܽ͹ଁՅͳͳ΍Ͷ‫ݰ‬ঙ͢ , ‫ݎ‬ሺܽሻ ൌ െʹΝຮͪͤ ܽ͹஍Ͳब‫غ‬ഔ෾ ‫ز‬͗ਫ਼ͣΖ. ฾ٌಕͺ࠹शదͶൕ఺Ν൒͑γχϩ‫͵ͳܗ‬Ζ .
ν΢ϕ II (ਦ 1(b)). ҈ఈ੓‫਼ܐ‬͹΀ΖΉ͏͹࠹ॵ͹෨෾ͺν΢ϕ I ͳಋͣͲ͍Ζ. ฾ब‫غ‬ٌ ಕ ͺ ൕ ఺ Ν ൒ ͑ γ χ ϩ ‫ ޛ ͪ ͮ ͵ ͳ ܗ‬, ҈ ఈ ੓ ‫ݎ ਼ ܐ‬ሺܽሻͺ ܽ͹ ଁ Յ ͳ ͳ ΍ Ͷ ଁ Յ ͢ , ‫ݎ‬ሺܽሻ ൌെʹΝຮͪ ͤ ܽ͹஍Ͳൕ ब‫ غ‬ഔ ෾‫ز‬Ν ً͞ ͢ͱବ ԃ‫ ܗ‬Ͷ ໯Ζ . ࣏Ͷ ‫ݎ‬ሺܽሻ ൌ ʹΝຮͪ ͤ ܽ͹஍ Ͳಋब‫غ‬෾‫ز‬Νً͢͞, ࠹शదͶ฾ٌಕͺγχϩ‫͵ͳܗ‬Ζ .
ν΢ϕ III (ਦ 1(c)). ฾ब‫ٌغ‬ಕ͗ਫ਼ͣͪ௜‫ޛ‬, ͞͹ब‫ٌغ‬ಕͺγχϩ‫͍Ͳܗ‬Ζ. ҈ఈ੓‫ܐ‬਼‫ݎ‬ሺܽሻͺ ܽ͹ଁՅͳͳ΍Ͷ ଁՅ ͤΖ͗౏஦ Ͳ‫ݰ‬ঙ͢ ࢟Ό Ζ . ‫ݎ‬ሺܽሻ ൌ ʹΝຮͪͤ ܽ͹஍Ͳ ฾ٌಕͺൕಋब‫ غ‬෾‫ز‬Νً ͢͞ ବԃ‫ܗ‬Ͷ͵ Ζ . ࠹‫ޛ‬Ͷ, ฾ٌಕͺब ‫غ‬ഔ෾‫ ز‬Νً ͢͞ͱൕ఺ Ν൒͑γχϩ‫͵ͳܗ‬Ζ
ν΢ϕ IV (ਦ 1(d)). ҈ఈ੓‫਼ܐ‬͹รԿ͹઴ൔͺν΢ϕ III Ͳ, ‫ޛ‬ൔͺν΢ϕ II ͹‫ޛ‬ൔͳߡ͓ΔΗΖ.
͞͞Ͳຌ࿨ช͹໪దΝफ़΄Ζ .
໪ద 1.2. ࢝ͯ͹ब‫غ‬ഔ෾‫ز‬գఖΝཀྵմ͢ , ࢝ͯ͹෾‫ز‬գఖ͗ਫ਼ͣΖ‫ݬ‬ҾΝ໎Δ͖ͶͤΖ ࣆ.
୊ 2 ઇͲͺ, ຌ࿨ชͲ࢘༽ͤΖ਼ָద֕೨ͳಕ۫Ν಍೘ͤΖ .
୊ 3 ઇͲͺ, ྭΝ΍ͳͶ࢝ͯ͹෾‫ز‬գఖΝ‫ݗ‬౾͢ , ෾‫ز‬գఖ͗ਫ਼ͣΖ‫ݬ‬ҾΝ໎Δ͖ͶͤΖ .
୊ 4 ઇͲͺ, ΉͳΌΝߨ͑.

2. ਼ָదಕ۫
2.1. ଲ߻ͳଲসતࣺ଀ܶͺଲ߻න‫ىͲݳ‬फ़ ͠ ΗΖ [4]. ଲ߻ͶΓΖ෾մ Ν ܶ ൌ ݄ ‫͚ॽͳ݃ ל‬. ͞͞Ͳ‫ל‬ͺࣺ଀ ͹߻੔‫͍Ͳߺى‬Ζ. ‫לߺى‬͹ӊଈ͹ ݃Νࡠ༽͢, ࣏Ͷࠪଈ͹݄Νࡠ༽ͤΖ.
െ‫ݕ‬
‫ݕ‬
‫ݕ‬
െ‫ ݕ‬െ ݂ሺ‫ݔ‬ሻ݃ቀ ቁ ൌ ቀቁǡ݄ ቀ ቁ ൌ ቀ‫ ݔ‬െ ‫ݕ‬ቁǤ‫ݔ‬‫ݔ‬‫ݔ‬

ሺ4ሻଲ߻͹෈ಊ ఼͹ॄ ߻Νଲ সતͳ͏͑ . ଲ߻͖݃Δಚ ΔΗΖଲস તΝ ܵ௚ ͳॽ͘, ଲ߻ ݄͖ΔಚΔ Η Ζ ଲ স ત Ν ܵ௛ ͳ ॽ ͚ . ຌ ࿨ ช Ͳ ͺ ܵ௚ ሺ‫ ݕ‬ൌ െ݂ሺ‫ݔ‬ሻȀʹሻͳ ܵ௛ ሺ‫ ݕ‬ൌ Ͳሻ͹ ෨ ෾ Ν Ґ Ծ ͹ Γ ͑ Ͷ ఈٝͤΖ.
ܵ௚ା  ‫ ݕ ׷‬ൌ െ݂ሺ‫ݔ‬ሻȀʹሺ‫ ݔ‬൐ ͳሻǡ

(5)ܵ௚ି  ‫ ݕ ׷‬ൌ െ݂ሺ‫ݔ‬ሻȀʹሺͲ ൏ ‫ ݔ‬൏ ͳሻǡ(6)ܵ௛ା  ‫ ݕ ׷‬ൌ Ͳሺ‫ ݔ‬൐ ͳሻǡ

(7)ܵ௛ି  ‫ ݕ ׷‬ൌ ͲሺͲ ൏ ‫ ݔ‬൏ ͳሻǤ(8)͞͞Ͳଲসब‫ٌغ‬ಕͳඉଲসब‫ٌغ‬ಕ͹ఈٝΝ༫͓Ζ .

t 11 tఈٝ 2.1 (ଲসब‫ٌغ‬ಕͳඉଲসब‫ٌغ‬ಕ ). ଲসત৏Ͷٌಕ఼Ν఼್΍ͯब‫ٌغ‬ಕΝଲসब‫ٌغ‬ಕͳఈ ٝͤΖ . ຌ࿨ ชͲͺ෈ಊ ఼ͺଲস ब‫ٌ غ‬ಕͶ‫؜‬Ό ͥ෈ಊ఼ ͳ͢ ͱѽ͑ . ଲসત৏Ͷٌಕ఼Ν࣍ͪ͵͏ब‫ٌغ‬ಕΝඉଲসब‫ٌغ‬ಕͳఈٝͤΖ .
ଲসત ܵ௚ ৏͹఼ ‫ ݖ‬ൌ ሺ‫ݔ‬ǡ ‫ݕ‬ሻΝ‫ۄ‬ત ‫ ݕ‬ൌ ‫ܩ‬ሺ‫ݔ‬ሻ͗௪ գ͢ͱ͏ Ζͳ ͤΖ . ͞͹ ‫ۄ‬ત͹఼ ‫ݖ‬Ͷ͕ ͜Ζܑ͘ΝߦͳͤΖ. ͞͹‫ۄ‬તͶଲ߻ ݃Νࡠ༽ͪ͢‫ۄ‬ત ‫ ݕ‬ൌ ‫ܩ‬௚ ሺ‫ݔ‬ሻ΍ ఼‫ ݖ‬ൌ ሺ‫ݔ‬ǡ ‫ݕ‬ሻΝ௪գ͢ͱ͏Ζ. ͞͹఼Ͷ͕͜Ζܑ͘Ν ߦ௚ ͳͤΖ. ଲ߻݃Ν‫ݔ‬Ͷͯ͏ͱඏ෾ͤΖͳ ߦͳߦ௚ ͹‫͗ܐؖ‬ಚΔΗΖ.
ߦ ൅ ߦ௚ ൌ െ݂Ԣሺ‫ݔ‬ሻǤ(9)ߦͳ ߦ௚ ͗ ౵ ͢ ͚ ͵ Ζ ͹ ͺ , ྈ ं ͗ ൅ ࢆ ͤ Ζ ৖ ߻ ͳ , ߦ ൌ ߦ௚ ൌ െ݂Ԣሺ‫ݔ‬ሻȀʹΝ ຮ ͪ ͤ ৖ ߻ Ͳ ͍ Ζ . ઴ंͶͯ͏ͱͺब‫غ‬ഔ෾‫ز‬͹ྡֆ஍Ͳਫ਼ͣΖ . ͞͹য়‫ڱ‬͹ྭͺ୊ 3 ઇΝ‫ݡ‬ΔΗͪ͏. ଲসતͳଲসત͹଀ ͗ંৰ͢ ިࠫ ͤΖ͞ͳͲ γχϩό ʖχ ෾‫͗ز‬ਫ਼ͣ Ζ . Γͮͱ, γχϩόʖ χ෾‫ز‬͹ྡֆ஍Ͳ‫ंޛ‬͹য়‫͗ڱ‬ਫ਼ͣΖ͞ͳͺ໎Δ͖Ͳ͍Ζ .
ଲ স ત ܵ௛ ৏ ͹ ఼ ‫ ݓ‬ൌ ሺ‫ݔ‬ǡ ͲሻΝ ‫ ۄ‬ત ‫ ݕ‬ൌ ‫ܪ‬ሺ‫ݔ‬ሻ͗ ௪ գ ͢ ͱ ͏ Ζ ͳ ͤ Ζ . ͞ ͹ ‫ ۄ‬ત ͹ ఼ ‫ݓ‬Ͷ ͕ ͜ Ζܑ͘ΝߞͳͤΖ. ͞͹‫ۄ‬તͶଲ߻݄Νࡠ༽ͪ͢‫ۄ‬ત‫ ݕ‬ൌ ‫ܪ‬௛ ሺ‫ݔ‬ሻ΍఼‫ ݓ‬ൌ ሺ‫ݔ‬ǡ ͲሻΝ௪գ͢ͱ͏Ζ. ͞͹఼Ͷ͕͜Ζܑ͘Ν ߞ௛ ͳͤΖ. ଲ߻݄Ν‫ݔ‬Ͷͯ͏ͱඏ෾ͤΖͳ ߞͳߞ௛ ͹‫͗ܐؖ‬ಚΔΗΖ.
ߞߞ௛ ൌǤሺߞ െ ͳሻ

(10)ߞͳ ߞ௛ ͗ ౵ ͢ ͚ ͵ Ζ ͹ ͺ, ߞ ൌ ߞ௛ ൌ ʹΝ ຮ ͪ ͤ ৖ ߻ ͳ ߞ ൌ ߞ௛ ൌ ͲΝ ຮ ͪ ͤ ৖ ߻ Ͳ ͍ Ζ. ઴ ं ͺ ब‫غ‬ഔ෾‫ز‬͹ྡֆ஍Ͳਫ਼ͣΖ . ब‫غ‬ഔ෾‫ز‬͹ྭͺ୊ 3 ઇͲࣖͤ. ‫ंޛ‬ͺब‫ٌغ‬ಕ఼͗γχϩόʖχ෾‫ز‬͹ྡֆ஍Ͳਫ਼ͣΖ .
ࣺ଀ܶΝ‫ݔ‬௡ Ͳඏ෾ͤΖͳܑ͘Ͷ‫ͤؖ‬Ζࣺ଀͗ಚΔΗΖ .
ሺߦ௡ ൅ ݂Ԣሺ‫ݔ‬௡ ሻሻߦ௡ାଵ ൌ
Ǥሺߦ௡ ൅ ݂Ԣሺ‫ݔ‬௡ ሻ ൅ ͳሻ͞͞Ͳߦ௡ାଵ ൌ ݀‫ݕ‬௡ାଵ Ȁ݀‫ݔ‬௡ାଵ , ߦ௡ ൌ ݀‫ݕ‬௡ Ȁ݀‫ݔ‬௡ . Ծ‫ى‬͹‫ܐؖ‬Νཤ༽ͪ͢.
݀‫ݕ‬௡ାଵ ݀‫ݕ‬௡ାଵ ݀‫ݔ‬௡ାଵൌൌ ߦ௡ାଵ ሺߦ௡ ൅ ݂Ԣሺ‫ݔ‬௡ ሻ ൅ ͳሻǤ݀‫ݔ‬௡݀‫ݔ‬௡ାଵ ݀‫ݔ‬௡

(11)

(12)

2.2. ηϟʖϩ͹ഇఛͳ‫ߺى‬Կࣺ଀ܶͶ͕͜Ζγχϩ‫෈ܗ‬ಊ఼ ܲ ൌ ሺͲǡͲሻ͹෈҈ఈଡ༹ର ܹ௨ Ν෈ಊ఼ ܲΓΕ௧੽ͤΖ. ෈҈ఈଡ༹ରܹ௨ ͺ෈ಊ఼ܲΝड़൅͢ͱ࠹ॵͶଲসત ܵ௚ା ͳިࠫ͢, ࣏Ͷଲসત ܵ௛ା ͳިࠫͤΖ(ਦ 2). ި఼Ν ॳ Ͷ ‫ݑ‬, ‫ ͤ ͳݒ‬Ζ. ҈ ఈ ଡ༹ ର ܹ௦ ΍ಋ ͣ ި఼ Ͳ ଲ স તͳ ި ࠫͤ Ζ . ި఼ ‫ݒ ͳݑ‬ͺ, कϙ ϠέϨω ρέ ఼ͳ ‫ ͻݼ‬ΗΖ [4]. Ήͪ, ‫ܹ ܐؖ‬௦ ൌ ܹ݃௨ ͳܹ௦ ൌ ݄ܹ௨ ͗੔ Εཱི ͯ . ࢝ͯ ͹‫ ނ‬ሾܲǡ ‫ݑ‬ሿௐೠ , ‫ނ‬Ȟ௦ ൌ ሾ‫ݑ‬ǡ ‫ݒ‬ሿௐೞ , ‫ނ‬Ȟ௨ ൌ ሾ‫ݒ‬ǡ ܶ‫ݑ‬ሿௐೠ ͳ‫ނ‬ሾܶ‫ݑ‬ǡ ܲሿௐೞ ͲҕΉΗͪถ྘ҮΝ‫خ‬ຌ྘Ү ܼͳ‫( ΁ݼ‬ਦ 2 Ͷ͕͜Ζ օ ৯ ྘ Ү ). ‫ ݑ݄ ࣞ ܐ ؖ‬ൌ ݄݃‫ ݑ‬ൌ ܶ‫ ݒ ͳݑ‬ൌ ݄‫ݒ‬Γ Ε , ሾܶ‫ݑ‬ǡ ܲሿௐೞ ൌ ݄ሾܲǡ ‫ݑ‬ሿௐೠ ͳ Ȟ௦ ൌ ݄Ȟ௨ ͗ ੔ Ε ཱི ͯ .
͞ Η Δ Γ Ε ܼ ൌ ݄ܼ͗ ੔ Ε ཱི ͯ ͞ ͳ ͗ ಍ ͜ Ζ . ‫ خ‬ຌ ྘ Ү ܼΝ ࢘ ͮ ͱ η ϟ ʖ ϩ ͹ ഇ ఛ ࣢Ν ߑ ੔ ͤ
Ζ. Ϗϧ ϟʖ ν ܽΝଁ Յͤ Ζͳ ܽ ൌ ܽ௖࣢ Ͳ, ‫ି ܶނ‬ଵ Ȟ௦ ͳ‫ ނ‬Ȟ௨ ͗ ંৰ ͤΖ . ͞͹য় ‫͗ ڱ‬η ϟʖ ϩ͹ഇఛ͗‫׮‬੔Ͳ͍Ζ. ܽ ൌ ͷǤͷ ൐ ܽ௖୦௦ ͳͪ͢ਦ 3 Ͳͺ, ‫ି ܶނ‬ଵ Ȟ௦ ͳ‫ނ‬Ȟ௨ ͺ఼್ߙͳߚͲިࠫ͢ͱ͏Ζ.
͞Ηͺηϟʖϩ͹ഇఛ͗‫׮‬੔ͪ͢‫ޛ‬͹ഓ஖Ͳ͍Ζ .
ਦ 3 Ͳ ‫ ڠ‬௪྘ Ү ܼ ‫ି ܶ ת‬ଵ ܼͺ, ್ ͯ͹ ถ྘ Ү ܸ଴ ͳ ܸଵ Ͷ෾཯ ͢ ͱ ͏Ζ . ‫خ‬ຌ ྘Ү Ͷ‫ ؜‬Ή Η Ζ‫ނ‬ܶ ିଵ Ȟ௦ ͗ถ྘Ү ܸ଴ ͳܸଵ ͹‫ڧ‬ֆͳ͵ͮͱ͏Ζ. ٌಕ఼͗ถ྘Ү ܸ଴ Ͷ೘Ηͻ‫ ߺى‬0 Ν༫͓, ถ྘Үܸଵ Ͷ೘Ηͻ‫ ߺى‬1 Ν༫͓Ζ. ͞ΗΓΕҲͯ͹ٌಕͶଲ͢ͱໃ‫ݸ‬Ͷௗ͏‫ߺى‬྽ ‫͗ݏ‬ಚΔΗΖ. ब‫ݍ غ‬͹ ब ‫ ٌ غ‬ಕ ͺ ‫ ݺݍ‬͹ ‫ ߺ ى‬͹ ด ; Ͳ ಝ ௅ ෉ ͜ Δ Η Ζ . ‫ ݺݍ‬͹ ‫ ߺ ى‬͹ ด ; Ν α ʖ χ ͳ ‫ ͑ ݶ‬. αt 12 tʖ χ Ͷ ͯ ͏ ͱ ͺ ච གྷ Ͷ Ԣ ͣ ͱ ॰ յ ͤ Ζ ͞ ͳ Ν ೟ Ό Ζ . γ χ ϩ ‫ ෈ ܗ‬ಊ ఼ ܲ͹ α ʖ χ ͺ ͲͲ ͍ Ε ,ବԃ‫෈ܗ‬ಊ఼ ܳ͹αʖχͺ ͳͲ͍Ζ.

ਦ 2: γχϩ‫෈ܗ‬ಊ఼ ܲ͹҈ఈଡ༹ର ܹ௦ ͳ෈҈ఈଡ༹ ର ܹ௨ ͹‫ނ‬Νཤ༽͢ͱ ఈٝ͠ Ηͪ‫خ‬ຌ྘Ү ܼ (օ ৯ ྘ Ү ). ި ఼ ‫ݒ ͳݑ‬ͺ क ϙ Ϡ έ Ϩ ω ρ έ ఼ . ‫ ނ‬Ȟ௦ ൌ ሾ‫ݑ‬ǡ ‫ݒ‬ሿௐೞ ͳ ‫ ނ‬Ȟ௨ ൌ ሾ‫ݒ‬ǡ ܶ‫ݑ‬ሿௐೠ ͹ ఈ ٝ .
ܽ ൌ Ͷ.

ਦ 3: ηϟʖϩ͹ഇఛ ࣢͹‫׮‬੔‫ޛ‬. ܽ ൌ ͷǤͷ. (a) ‫خ‬ຌ྘Үܼͳ‫ି ܶ଀ٱ‬ଵ ܼ͹‫ܐؖ‬. (b) ถ྘Үܸ଴ ͳܸଵ ͹ఈٝ.
࣏Ͷຌ࿨ชͲ࢘༽ͤΖఈཀྵ , ఈٝ͵Δ;Ͷ੓࣯ΝΉͳΌͱ়ղͤΖ [4].
ఈཀྵ 2.2. ηϟʖϩ͹ഇఛ࣢ͶଚࡑͤΖब‫ٌغ‬ಕͳαʖχͺҲଲҲଲԢΝ͵ͤ .
ఈ ٝ 2.3(࣎ ؔ ൕ ఺ αʖχ ͳ ࣎ ؔ ൕ ఺ ଲ ). ब‫ݍ غ‬͹ ब ‫ ٌ غ‬ಕ͹ α ʖ χ Ν ‫ ݏ‬ൌ ‫ݏ‬଴ ‫ݏ‬ଵ ‫ݏ ڮ‬௤ିଵ ͳ ͤ Ζ .
‫ ߺ ى‬͹ ด ; Ν ‫ ٱ‬Ͷ ͢ ͪ α ʖ χ ‫ି ݏ‬ଵ ൌ ‫ݏ‬௤ିଵ ‫ݏ ڮ‬ଵ ‫ݏ‬଴ Ν α ʖ χ ‫ݏ‬͹ ࣎ ؔ ൕ ఺ α ʖ χ ͳ ఈ ٝ ͤ Ζ . ॰ յ͢ͱ΍‫ି ݏ ് ݏ‬ଵ ͵Δͻ, αʖχ‫ͳݏ‬αʖχ ‫ି ݏ‬ଵ Ν࣎ؔൕ఺ଲͳ໌෉͜Ζ.

t 13 t੓࣯ 2.4.
(1) αʖχ‫ؔ࣎ͳݏ‬ൕ఺αʖχ ‫ି ݏ‬ଵ Ͷͯ͏ͱ ‫ ݏ‬ൌ ‫ି ݏ‬ଵ ͗੔Εཱིͱͻ, αʖχ‫ݏ‬͹ब‫ٌغ‬ಕͺଲসब‫ٌغ‬ಕͲ͍Ζ.
(2) αʖχ‫ؔ࣎ͳݏ‬ൕ఺αʖχ ‫ି ݏ‬ଵ Ͷ͕͏ͱ, ‫ߺى‬Ν॰յ͢ͱ΍‫ି ݏ ് ݏ‬ଵ ͵Δͻαʖχ ‫ݏ‬͹ब‫ٌغ‬ಕͺඉଲসब‫ٌغ‬ಕͲ͍Ζ . αʖχ‫ݏͳݏ‬Ԣ͹ඉଲসब‫ٌغ‬ಕͺͳ΍ͶଚࡑͤΖ .
ఈٝ 2.5 (ࢍ-ଲ). ఼‫ݓ఼ͳݖ‬Ͷͯ͏ͱ ‫ ݓ‬ൌ ݃‫͗ݖ‬੔ΕཱིͯͳͤΖ. ఼‫྘͗ݖ‬Үܸ଴ ሺܸଵ ሻͶଚࡑͤΖ͵Δͻ, ఼‫ݓ‬΍ ྘Үܸ଴ ሺܸଵ ሻͶଚࡑͤΖ. ͞ΗΔΝ݃-ଲͳ໌෉͜Ζ.
ఈٝ 2.6 (αʖχ͹࠹ঘ஍න‫)ݳ‬. αʖχͲͲͳΝྭͳ͢ͱઈ໎͢, ࠹ঘ஍න‫ݳ‬Ν಍೘ͤΖ. ͞͞Ͳ ‫ ݔ‬ൌǤ ሺͲͲͳሻஶ , ‫ ݕ‬ൌǤ ሺͲͳͲሻஶ , ‫ ݖ‬ൌǤ ሺͳͲͲሻஶ ͳ ͤ Ζ . ͞ Η Δ Ν ್ ਒ ๑ Ͷ ร ‫ ͪ ͢ ׷‬஍ ͺ ܺ ൌǤ ሺͲͲͳͳͳͲሻஶ ൌ ʹȀͻ, ܻ ൌǤ ሺͲͳͳͳͲͲሻஶ ൌ ͶȀͻ, ܼ ൌǤ ሺͳͳͳͲͲͲሻஶ ൌ ͺȀͻͲ͍Ζ[4] (෉࿧ A ΝΊΓ). ͞ΗΔ͹͵͖Ͳ ܺ͗࠹ঘ஍Ͳ͍Ζ. Γͮͱ, αʖχͲͲͳΝαʖχ͹࠹ঘ஍න‫ͳݳ‬ఈٝͤΖ .
ఈٝ 2.7 (࠹ۛંٌಕ఼ଲ). αʖχΝ࠹ঘ஍න‫ݏͲݳ‬଴ ‫ݏ‬ଵ ‫ݏ ڮ‬௞ ‫ݏ ڮ‬௤ିଶ ‫ݏ‬௤ିଵ ͳॽ͚. ‫ݏߺى‬௝ ሺͲ ൑ ݆ ൑‫ ݍ‬െ ͳሻͶଲԢͤΖٌಕ఼Ν ‫ݖ‬௝ ͳͤΖ. ‫ݖ‬௞ ൌ ݃‫ݖ‬௤ିଶ (Ͳ ൑ ݇ ൑ ‫ ݍ‬െ ʹ) ͗੔ΕཱིͯͳͤΖ.
(1) ݇ ൏ ‫ ݍ‬െ ʹ͹ ৖ ߻ , ٌ ಕ ఼ ͹ ଲ ‫ݖ‬௞ ͳ ‫ݖ‬௤ିଶ Ν ‫ ڧ‬ֆ (ܶ ିଵ Ȟ௦ ) Ͷ ࠹ ۛ ં ͢ ͪ ٌ ಕ ఼ ଲ (࠹ ۛ ં ٌಕ఼ଲ)ͳ໌෉͜Ζ.
(2) ݇ ൌ ‫ ݍ‬െ ʹ͗੔Εཱིͯ৖߻, ٌಕ఼‫ݖ‬௞ Ν࠹ۛંٌಕ఼ͳ໌෉͜Ζ .
ఈٝ 2.8 (ਜ਼‫و‬න‫)ݳ‬. αʖχ͹න‫ͱ͢ͳݳ‬ఈٝ 2.7 Ͷ͕͜Ζٌಕ఼ ‫ݖ‬௞ ͹‫ߺى‬Ν઎ಆͶഓ஖ͪ͢න‫ݳ‬‫ݏ‬௞ ‫ݏ‬௞ାଵ ‫ݏ ڮ‬௤ିଶ ‫ݏ‬௤ିଵ ‫ݏ‬଴ ‫ݏ ڮ‬௞ିଵ(13)Νਜ਼‫و‬න‫͜෉໌ͳݳ‬Ζ.
ఈٝ 2.9 (αʖχ ͹ ۰‫ د‬੓ ). αʖ χͶ ‫؜‬Ή Η Ζ 1 ͹‫ ਼͗ ݺ‬۰ ਼ (‫͵ )਼د‬Δ ͻα ʖ χͺ۰(‫ͤͳ )د‬Ζ.
੓࣯ 2.10. ۰ (‫ )د‬αʖχ͹ब‫ٌغ‬ಕͺ, ഇఛ࣢͹஦Ͳγχϩ‫ܗ‬ब‫ٌغ‬ಕ (ൕ఺γχϩ‫ܗ‬ब‫ٌغ‬ಕ) Ͳ͍Ζ

2.3. ϔϫρέනࣖͳͨ͹੓࣯ࢂߡช‫[ݛ‬4]Ͳ಍೘͠ΗͪϔϫρέΝ಍೘ͤΖ .
ϔϫρέߑ੔‫و‬ଉ 2.11.
(1) յ఺਼͗ͳȀ‫ݍ‬͹৖߻͹ϔϫρέ͹ఈٝ .
ܵሺͳȀ‫ݍ‬ሻ ൌ ͳͲ௤ିଶ ͳሺ‫ ݍ‬൒ ʹሻǡ

(14)

௤ିଵ

ͳሺ‫ ݍ‬൒ ʹሻǡ(15)ͳͳሺ‫ ݍ‬൒ ʹሻǡ(16)‫ܦ‬ሺͳȀ‫ݍ‬ሻ ൌ ͳͲ௤ିଷ ͳͳሺ‫ ݍ‬൒ ͵ሻǤ(17)‫ܧ‬ሺͳȀ‫ݍ‬ሻ ൌ Ͳ‫ܨ‬ሺͳȀ‫ݍ‬ሻ ൌ Ͳ௤ିଶt 14 t(2) ‫ع‬༁෾਼‫݌‬Ȁ‫ݍ‬ሺ‫ ݌‬൒ ʹǡ ‫ ݍ‬൒ ͷሻ͹ϓΟϪ΢෾ׄΝ FPሾ‫݌‬Ȁ‫ݍ‬ሿ ൌ ሼ‫݌‬௟ Ȁ‫ݍ‬௟ ǡ ‫݌‬௥ Ȁ‫ݍ‬௥ ሽͳॽ͚. ͞͞Ͳ, ‫ ݌‬ൌ‫݌‬௟ ൅ ‫݌‬௥ , ‫ ݍ‬ൌ ‫ݍ‬௟ ൅ ‫ݍ‬௥ , ‫݌‬௥ ‫ݍ‬௟ െ ‫݌‬௟ ‫ݍ‬௥ ൌ ͳ, Ͳ ൏ ‫݌‬௟ Ȁ‫ݍ‬௟ ൏ ‫݌‬௥ Ȁ‫ݍ‬௥ ൑ ͳȀʹ. ࣏Ͷϔϫρέ͹ߑ੔‫و‬ଉΝࣖͤ .
ܵሺ‫݌‬Ȁ‫ݍ‬ሻ ൌ ܵሺ‫݌‬௥ Ȁ‫ݍ‬௥ ሻܵሺ‫݌‬௟ Ȁ‫ݍ‬௟ ሻǡ(18)‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ ൌ ‫ܧ‬ሺ‫݌‬௥ Ȁ‫ݍ‬௥ ሻܵሺ‫݌‬௟ Ȁ‫ݍ‬௟ ሻ ൌ ‫ܨ‬ሺ‫݌‬௟ Ȁ‫ݍ‬௟ ሻ‫ܧ‬ሺ‫݌‬௥ Ȁ‫ݍ‬௥ ሻǡ(19)‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ ൌ ‫ܨ‬ሺ‫݌‬௟ Ȁ‫ݍ‬௟ ሻ‫ܨ‬ሺ‫݌‬௥ Ȁ‫ݍ‬௥ ሻǡ(20)‫ܦ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ ൌ ܵሺ‫݌‬௥ Ȁ‫ݍ‬௥ ሻ‫ܦ‬ሺ‫݌‬௟ Ȁ‫ݍ‬௟ ሻ ൌ ‫ܦ‬ሺ‫݌‬௟ Ȁ‫ݍ‬௟ ሻ‫ܨ‬ሺ‫݌‬௥ Ȁ‫ݍ‬௥ ሻǤ

(21)

න 1 Ͷ, Γ͚࢘༽ͤΖϔϫρέΝΉͳΌͪ . ϔϫρέ ܵሺͳȀʹሻͳϔϫρέ ‫ܨ‬ሺͳȀʹሻ ൌ ͳͳͶͯ͏ͱ ஭ қ Ν ͢ ͱ ͕ ͚ . Ґ Ծ Ͳ ͺ ϔ ϫ ρ έ ‫ܨ‬ሺͳȀʹሻ Ν ༽ ͏ Ζ . ϔ ϫ ρ έ ͹ ߑ ੔ ‫ و‬ଉ ͹ ͪ Ό ͶܵሺͳȀʹሻ ൌ ͳͳ΍න 1 Ͷ‫؜‬Όͱ͍Ζ.
න 1: ̒ͯ͹ϔϫρέ͹ ͲͳͳͶΓΖන‫ݳ‬‫݌‬Ȁ‫ݍ‬ܵሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܦ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͳȀ͸ͳͲͲͲͲͳ

ͲͲͲͲͲͳ

ͲͲͲͲͳͳ

ͳͲͲͲͳͳ

ͳȀͷ

ͳͲͲͲͳ

ͲͲͲͲͳ

ͲͲͲͳͳ

ͳͲͲͳͳ

ʹȀͻ

ͳͲͲͳͳͲͲͲͳ

ͲͲͲͳͳͲͲͲͳ

ͲͲͲͳͳͲͲͳͳ

ͳͲͲͳͳͲͲͳͳ

ͳȀͶ

ͳͲͲͳ

ͲͲͲͳ

ͲͲͳͳ

ͳͲͳͳ

͵Ȁͳͳ

ͳͲͳͳͲͲͳͳͲͲͳ

ͲͲͳͳͲͲͳͳͲͲͳ

ͲͲͳͳͲͲͳͳͲͳͳ

ͳͲͳͳͲͲͳͳͲͳͳ

ʹȀ͹ͳͲͳͳͲͲͳ

ͲͲͳͳͲͲͳ

ͲͲͳͳͲͳͳ

ͳͲͳͳͲͳͳ

͵ȀͳͲͳͲͳͳͲͳͳͲͲͳ

ͲͲͳͳͲͳͳͲͲͳ

ͲͲͳͳͲͳͳͲͳͳ

ͳͲͳͳͲͳͳͲͳͳ

ͳȀ͵

ͳͲͳ

ͲͲͳ

Ͳͳͳ

ͳͳͳ

ͶȀͳͳ

ͳͳͳͲͳͳͲͳͳͲͳ

ͲͳͳͲͳͳͲͳͳͲͳ

ͲͳͳͲͳͳͲͳͳͳͳ

ͳͳͳͲͳͳͲͳͳͳͳ͵ȀͺͳͳͳͲͳͳͲͳ

ͲͳͳͲͳͳͲͳ

ͲͳͳͲͳͳͳͳ

ͳͳͳͲͳͳͳͳ

ͷȀͳ͵ͳͳͳͲͳͳͳͳͲͳͳͲͳ ͲͳͳͲͳͳͳͳͲͳͳͲͳ ͲͳͳͲͳͳͳͳͲͳͳͳͳ ͳͳͳͲͳͳͳͳͲͳͳͳͳ

ʹȀͷ

ͳͳͳͲͳ

ͷȀͳʹ

ͲͳͳͲͳ

Ͳͳͳͳͳ

ͳͳͳͳͳ

ͳͳͳͳͳͲͳͳͳͳͲͳ ͲͳͳͳͳͲͳͳͳͳͲͳ ͲͳͳͳͳͲͳͳͳͳͳͳ ͳͳͳͳͳͲͳͳͳͳͳͳ

͵Ȁ͹ͳͳͳͳͳͲͳ

ͲͳͳͳͳͲͳ

Ͳͳͳͳͳͳͳ

ͳͳͳͳͳͳͳ

ͶȀͻͳͳͳͳͳͳͳͲͳ

ͲͳͳͳͳͳͳͲͳ

Ͳͳͳͳͳͳͳͳͳ

ͳͳͳͳͳͳͳͳͳ

ͳȀʹ

11

Ͳͳ

11

-

2.4. ۢؔ͹ؔ͹યҢ
୊ 2.3 ઇͲ಍೘ͪ͢ϔϫρέͺ྘Ү͹໌সͲ΍͍Ζ [2]. ್ͯ͹྘Үؔ͹યҢΝયҢߨ྽ͳ͢ͱන‫͘Ͳݳ‬Ζ. યҢߨ྽͹གྷો͗ ͳ͹৖߻, યҢՆ೵Ͳ͍Ε, Ͳ͹৖߻ͺયҢͲ͘͵͏.
i) Ͳ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ‫ݎ‬Ȁ‫ ݏ‬൏ ͳȀʹǣ‫ܧۇ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܯ‬ሺ‫݌‬Ȁ‫ݍ‬ǡ ‫ݎ‬Ȁ‫ݏ‬ሻ ൌ ‫ܵۈ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܦۉ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܧ‬ሺ‫ݎ‬Ȁ‫ݏ‬ሻͳͳͲͲܵሺ‫ݎ‬Ȁ‫ݏ‬ሻͳͳͲͲt 15 t‫ܨ‬ሺ‫ݎ‬Ȁ‫ݏ‬ሻͳͳͲͲ‫ܦ‬ሺ‫ݎ‬Ȁ‫ݏ‬ሻ
ͳ
‫ۊ‬
ͳ
‫ۋ‬Ǥ
Ͳ
Ͳ
‫ی‬

(22)

ii) Ͳ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ͳȀʹǣ‫ܧ‬ሺͳȀʹሻͳͳͲͲ‫ܨ‬ሺͳȀʹሻ
ͳ
‫ۊ‬
ͳ
‫ۋ‬Ǥ
Ͳ
Ͳ
‫ی‬

(23)

‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͳͳͳͳܵሺ‫݌‬Ȁ‫ݍ‬ሻͲͲͲͲ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͳͳͳͳ‫ܦ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ
Ͳ
‫ۊ‬
Ͳ
‫ۋ‬Ǥ
Ͳ
Ͳ
‫ی‬

(24)

‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͳͳܵሺ‫݌‬Ȁ‫ݍ‬ሻͲͲ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͳͳ‫ܦ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ
Ͳ
ቍǤ
Ͳ

(25)

‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͳͳͳͳܵሺ‫݌‬Ȁ‫ݍ‬ሻͳͳͲͲ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͳͳͳͳ‫ܦ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ
ͳ
‫ۊ‬
ͳ
‫ۋ‬ǡ
Ͳ
Ͳ
‫ی‬

(26)

‫ܧ‬ሺͳȀʹሻ‫ܬ‬ሺͳȀʹǡͳȀʹሻ ൌ ൮‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻͳͳ‫ܨ‬ሺͳȀʹሻͳ൲Ǥͳ‫ܧۇ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܯ‬ሺ‫݌‬Ȁ‫ݍ‬ǡ ͳȀʹሻ ൌ ‫ܵۈ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܦۉ‬ሺ‫݌‬Ȁ‫ݍ‬ሻiii) Ͳ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ‫ݎ‬Ȁ‫ ݏ‬൏ ͳȀʹǣ‫ܧۇ‬ሺ‫ݎ‬Ȁ‫ݏ‬ሻܰሺ‫ݎ‬Ȁ‫ݏ‬ǡ ‫݌‬Ȁ‫ݍ‬ሻ ൌ ‫ܵۈ‬ሺ‫ݎ‬Ȁ‫ݏ‬ሻ‫ܨ‬ሺ‫ݎ‬Ȁ‫ݏ‬ሻ‫ܦۉ‬ሺ‫ݎ‬Ȁ‫ݏ‬ሻiv) Ͳ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ͳȀʹǣܰሺͳȀʹǡ ‫݌‬Ȁ‫ݍ‬ሻ ൌ ቌ‫ܧ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻv) Ͳ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ͳȀʹǣ‫ܧۇ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܬ‬ሺ‫݌‬Ȁ‫ݍ‬ǡ ‫݌‬Ȁ‫ݍ‬ሻ ൌ ‫ܵۈ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ‫ܦۉ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ
vi)

(27)

͞ ͞ Ͳ , ྘ Ү ͳ ϔ ϫ ρ έ Ͷ ͯ ͏ ͱ ึ ଏ ͢ ͱ ͕ ͘ ͪ ͏ . Ͳ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ͳȀʹ͹ ৖ ߻ , ࢝ ͯ ͹ ྘ Ү‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻǡ ܵሺ‫݌‬Ȁ‫ݍ‬ሻǡ ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻǡ ‫ܦ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ͗ଚࡑͤΖ. ͞ΗΔΝ΍ͳͶϔϫρέ‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻǡ ܵሺ‫݌‬Ȁ‫ݍ‬ሻǡ ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻǡ ‫ܦ‬ሺ‫݌‬Ȁ‫ݍ‬ሻΝ ఈ ٝ ͢ ͪ . ϔ ϫ ρ έ ܵሺ‫݌‬Ȁ‫ݍ‬ሻͺ ॰ յ ͤ Ζ ͳ ϔ ϫ ρ έ ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͶ ͵ Ζ . ϔ ϫ ρ έ ‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͳ ϔ ϫρ έ ‫ܨ‬ሺ‫݌‬Ȁ‫ݍ‬ሻሺܵሺ‫݌‬Ȁ‫ݍ‬ሻሻͲ ‫ ى‬फ़ ͠ Η Ζ ब ‫ ٌ غ‬ಕ ͺ ଚ ࡑ ͤ Ζ ͗ , ϔ ϫ ρ έ ‫ܦ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͲ ‫ ى‬फ़ ͠ Η Ζ ब ‫غ‬ٌಕͺଚࡑ͢͵͏.
‫݌‬Ȁ‫ ݍ‬ൌ ͳȀʹ͹ ৖ ߻ , ್ ͯ ͹ ྘ Ү ‫ܧ‬ሺͳȀʹሻͳ ܵሺͳȀʹሻ ൌ ‫ܨ‬ሺͳȀʹሻ ൌ ͳͳ͗ ଚ ࡑ ͤ Ζ . ϔ ϫ ρ έ ‫ܧ‬ሺͳȀʹሻͲ‫ى‬फ़͠ ΗΖ ब‫ٌ غ‬ಕͺ ଚ ࡑͤΖ ͗ , ϔϫ ρέ ܵሺͳȀʹሻ ൌ ‫ܨ‬ሺͳȀʹሻͲ‫ ى‬फ़͠Η Ζब ‫ ٌغ‬ಕͺଚ ࡑ ͢͵͏.
αʖχΝϔϫρέͲනͪ͢න‫ݳ‬Νϔϫρέαʖχ‫ ΁ݼ‬. યҢߨ྽͹੓࣯͖Δ੓࣯ 2.12 ͗ಚΔΗΖ[4].
੓࣯ 2.12. ϔϫρέαʖχͶ‫؜‬ΉΗΖϔϫρέ͹յ఺਼Ͷ͕͏ͱ ‫݌‬Ȁ‫࠹͗ݍ‬ঘ஍Ͳ͍ΖͳͤΖ.
ϔ ϫ ρ έ α ʖ χ ͹ ࠹ ঘ ஍ න ࣖ ͺ ‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻͲ ࢟ Ή Ζ. ͪ ͫ ͢, ࠹ ঘ ஍ න ࣖ ͹ ઎ ಆ ෨Ͷ ‫ܧ‬ሺ‫݌‬Ȁ‫ݍ‬ሻ͗ ࿊
କͤΖ͞ͳͺ͵͏.
ྭ. ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻͺ࠹ঘ஍න‫͚͵Ͳݳ‬, ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͗࠹ঘ஍න‫͍Ͳݳ‬Ζ.
ఈٝ 2.13 (‫ڒ‬༲‫)ޢࢯےͳޢ‬. ྘Ү‫͖ܣ‬Δ྘Ү‫΃ܤ‬͹યҢ͗Ն೵͵৖߻, ‫ܤܣޢ‬Ν‫ڒ‬༲‫͏͏ͳޢ‬,ͨ͑Ͳ͵͏৖߻ͺ‫ ͑͏ͳޢࢯے‬.
t 16 tྭ. ‫ܨޢ‬ሺͳȀ͵ሻܵሺͳȀ͵ሻͺ‫͍Ͳޢࢯے‬Ζ͗, ‫ܵޢ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻͺ‫ڒ‬༲‫͍Ͳޢ‬Ζ.

2.5. ෾‫ݳز‬ে
ਦ 4 Νཤ༽͢ͱ (ൕ) ब‫غ‬ഔ෾‫( ͳز‬ൕ) ಋब‫غ‬෾‫ز‬Ͷͯ͏ͱઈ໎ͤΖ . ຌ࿨ชͲͺϏϧϟʖνܽΝ৙ͶଁՅͤΖ͞ͳͲ෾‫͗ز‬ਫ਼ͣΖͳͤΖ . ବԃ‫ܗ‬ब‫ٌغ‬ಕ͹ब‫غ‬Ν ‫ݍ‬ሺ൒ ͳሻͳͤΖ .
ବԃ‫ܗ‬ब‫ٌغ‬ಕ͗ൕ఺‫ܗ‬γχϩब‫ٌغ‬ಕͶ෾‫ͤز‬Ζգఖ͗ब‫غ‬ഔ෾‫͍Ͳز‬Ζ . ͞͹෾‫Ͳز‬ब‫ݍʹ͗غ‬͹ٌ໊ಕ͗Ҳͯਫ਼ͣΖٌ໊͗ಕͺବԃ‫͍Ͳܗ‬Ζ . ͨ͹ͪΌٌ໊ಕͺब‫غ‬ഔ෾‫ز‬ΝӀ͘କ͏ͱًͤ͞. ൕ఺‫ܗ‬γχϩब‫ٌغ‬ಕ͗ବԃ‫ٌܗ‬ಕͶ෾‫ͤز‬Ζգఖ͗ൕब‫غ‬ഔ෾‫͍Ͳز‬Ζ .
͞͹෾‫Ͳز‬ब‫ݍʹ ͗غ‬͹ٌ໊ಕ͗Ҳͯਫ਼ͣΖٌ໊͗ಕͺγχϩ‫͍Ͳܗ‬Ζ .
ବԃ‫ܗ‬ब‫ٌغ‬ಕ͗γχϩब‫ٌغ‬ಕͶ෾‫ͤز‬Ζգఖ͗ಋब‫غ‬ഔ෾‫͍Ͳز‬Ζ. ͞͹෾‫Ͳز‬ब‫ݍ ͗ غ‬͹໊ ٌ ಕ ͗ ್ ͯ ਫ਼ ͣ Ζ ͗ ͞Η Δ ͹ ໊ ٌ ಕ ͺ ବ ԃ ‫ ͍Ͳ ܗ‬Ζ . ͨ ͹ ͪ Ό ໊ ٌ ಕ ͺब ‫ غ‬ഔ෾‫ز‬ΝӀ͘କ͏ͱًͤ͞. γχϩब‫ٌغ‬ಕ͗ବԃ‫ܗ‬ब‫ٌغ‬ಕͶ෾‫ͤز‬Ζգఖ͗ൕಋब‫غ‬෾‫Ͳز‬͍Ζ. ͞͹෾‫Ͳز‬ब‫ݍ ͗غ‬͹ٌ໊ಕ್͗ͯਫ਼ͣΖ͗͞ΗΔͺγχϩ‫͍Ͳܗ‬Ζ .

ਦ 4: (ൕ)ब‫غ‬ഔ෾‫(ͳز‬ൕ)ಋब‫غ‬෾‫ز‬.

3. ࢝ͯ͹෾‫ز‬գఖ
3.1. ν΢ϕ , ब‫غ‬ഔ෾‫ز‬ν΢ϕ I ͹ྭͳ͢ͱϔϫρέαʖχ ‫ܧ‬ሺͳȀͷሻ ൌ ͲͲͲͲͳͲ‫ى‬फ़͠ΗΖବԃ‫ܗ‬ब‫ٌغ‬ಕΝ༽͏Ζ .
͞͹ ब‫ ٌ غ‬ಕͺ ବԃ ‫ ෈ܗ‬ಊ఼ ܳ͗յ ఺ ෾‫ ز‬Νً ͢͞ ͱਫ਼ ͣͪ ब‫ ٌ غ‬ಕͲ ͍Ζ . ଲΝ͵ ͤ γχϩ‫ܗ‬ब‫ٌغ‬ಕ͹ϔϫρέαʖχͺ ܵሺͳȀͷሻ ൌ ͳͲͲͲͳͲ͍Ζ. ͞ΗΔ͗յ఺෾‫Ͳز‬ਫ਼ͣΖ ྡֆ஍ͺܽ௖ ሺͳȀͷሻ ൌ Ͷଶ ሺߨȀͷሻ ൌ ͳǤ͹Ͷͳͺ͹Ͳ͍Ζ[4]. ବԃ‫ܗ‬ब‫ٌغ‬ಕ͗ब‫غ‬ഔ෾‫ز‬Νًͤ͞ྡֆ஍ ܽ ൌ୮ௗܽ௖ ሺͳȀͷሻ ൌ ͳǤ͹Ͷ͵Ͳʹͺ͹ͺͲ͹ବԃ‫ܗ‬ब‫ٌغ‬ಕͳγχϩ‫ܗ‬ब‫ٌغ‬ಕ͹ٌಕ఼Νਦ 5 Ͷࣖͪ͢.

t 17 tਦ 5: ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀͷሻሺͲͲͲͲͳሻ ͹ ब ‫ ٌ غ‬ಕ ఼ ሼ‫ݖ‬଴ ǡ ‫ݖ‬ଵ ǡ ‫ݖ‬ଶ ǡ ‫ݖ‬ଷ ǡ ‫ݖ‬ସ ሽ ͳ ϔ ϫ ρ έ α ʖ χ୮ௗܵሺͳȀͷሻሺͳͲͲͲͳሻ͹ब‫ٌغ‬ಕ఼ ሼ‫ݓ‬଴ ǡ ‫ݓ‬ଵ ǡ ‫ݓ‬ଶ ǡ ‫ݓ‬ଷ ǡ ‫ݓ‬ସ ሽ. ܽ ൌ ܽ௖ ሺͳȀͷሻ ൌ ͳǤ͹Ͷ͵Ͳʹͺ͹ͺ.
ବԃ‫ܗ‬ब ‫ٌغ‬ಕ ͺଲস ત ৏Ͷ್ͯ ٌಕ఼ (‫ݖ‬ଶ ‫ܵ א‬௛ି ͳ‫ݖ‬ସ ‫ܵ א‬௚ା ) Ν΍ͯ. ಋ༹Ͷγ χϩ‫ ܗ‬ब ‫غ‬ٌಕ΍ଲসત৏Ͷ್ٌͯಕ఼ (‫ݓ‬଴ ‫ܵ א‬௛ା ͳ‫ݓ‬ଶ ‫ܵ א‬௚ି ) Ν΍ͯ. ਦ 5 ͹఼ત͹͍ͪΕͶ҈ఈଡ༹ର͹ ‫ି ܶ ނ‬ଵ Ȟ௦ ͗೘ Ε ࠒ Ίࠪ ଈ ͗ ྘ Ү ܸ଴ Ͷ, ӊ ଈ ͗ ྘Ү ܸଵ ͳ ͵ Ζ. ٌಕ ఼ Ν ॵ‫ݖ ఼ غ‬଴ ͖ Δ௧ ͑ ͞ͳͲαʖχ ͲͲͲͲͳ ൌ ‫ܧ‬ሺͳȀͷሻΝ֮೟Ͳ͘Ζ. ಋ༹Ͷॵ‫ݓ ఼غ‬଴ ͖Δ௧͑͞ͳͲαʖχ ͳͲͲͲͳ ൌ ܵሺͳȀͷሻ͗ಚΔΗΖ. ٌಕ఼‫ݖ‬ସ Ͷ͕͏ͱଲসત ܵ௚ା ͳଲসત͹଀ ܶ ଶ ܵ௛ି ͗ިࠫ͢ͱ͏Ζ͞ͳͶ஭қ͢Γ͑.
ຌ࿨ชͲͺ, ଲসતܵ௚ା Ν݃-क࣢ͳ‫;ݼ‬, ٌಕ఼‫ݖ‬ସ Ͳ݃-क࣢ͳިࠫ͢ͱ͏Ζ ଀ܶ ଶ ܵ௛ି Ν݃-෯࣢ͳ‫΁ݼ‬.
ਦ 5 ͹ٌಕ఼ ‫ݖ‬ସ Ͷ͕͜Ζ ݃-෯࣢͹ܑ͘Ͷͯ͏ͱ௒΄Ζ . ܑ͘Ͷ‫ͤؖ‬Ζࣺ଀(ࣞ(11))Νཤ༽͢ͱ ܑ ͘Ν ‫ ͤ ࢋܯ‬Ζ๏ ๑ Ν় ղ ͤΖ . Ϗϧ ϟʖ ν ܽΝ݀ Όͱ , ब‫ ٌ غ‬ಕ఼ ͹ Ғ஖ Ν݀ ఈ ͤΖ .
࠹ॵ Ͷ ‫ݖ‬ଶ ൌ ሺ‫ݔ‬ଶ ǡ Ͳሻͳଲস ત ܵ௛ି ͹ ܑ͘ ߦଶ ൌ ͲΓΕ, ܑ͘ ߦଷ ͗݀Ή Ζ. ࣏ Ͷ‫ݖ‬ଷ ͹ ‫ݔ‬-࠴ඬ‫ݔ‬ଷ ͳܑ ͘ ߦଷΓΕ ܑ͘ߦସ ͗݀ΉΖ. ಚΔΗͪ݃ՎΝਦ 6 Ͷࣖͪ͢. ϔϫρέαʖχ ‫ܧ‬ሺͳȀͷሻ͹ब‫ٌغ‬ಕ͗ਫ਼ͣͪ ྡ ֆ஍ Ͳ ͺܑ ͘ ߦሺ‫ݖ‬ସ ሻͺ, ଲস ત ܵ௚ ͹ ‫ ݔ‬ൌ ͳͶ͕ ͜Ζ ܑ͘ ܽ௖ ሺͳȀͷሻȀʹ ൌ ʹଶ ሺߨȀͷሻ ൌ ͲǤ͸ͻͲͻͺ͵୮ௗͶ ౵ ͢ ͏ . Ϗ ϧ ϟ ʖ ν ܽ͗ ଁ Յ ͤ Ζ ͳ ܑ ͘ ΍ ଁ Յ ͤ Ζ . ब ‫ غ‬ഔ ෾ ‫ ͗ ز‬ਫ਼ ͣ Ζ ྡ ֆ ஍ ܽ ൌ ܽ௖ ൌͳǤ͹Ͷ͵Ͳʹͺ͹Ͳ , ܑ ͘ ͺ ൅ ࢆ ͤ Ζ . ͨ ͹ ‫ ޛ‬, ܑ ͘ ͺ ෝ Ͷ ͵ Ε ଁ Յ ͢ , ࠸ ; ਜ਼ Ͷ ͵ Ζ . ͨ ͢ ͱ , ܑ͘ͺ৷೘͢ͱͪ͘෈҈ఈଡ༹ର͹‫ܶ ނ‬Ȟ௨ ͹ܑ͘ͶશۛͤΖ. གྷ੧ 1.1 ͺ٠࿨Ν෵ࡸԿͦ͠͵͏ͪΌͶ௪৙ཤ༽ͤΖ͗, གྷ੧ 1.1 Ν੔Εཱིͪͦͱ͏Ζ‫ݬ‬Ҿ͗෈҈ఈଡ༹ର͹‫ܶ ނ‬Ȟ௨ ͹৷೘Ͳ͍
Ζ͞ͳ͗໎Δ͖Ͷ͵ͮͪ .

t 18 tਦ 6: ଲসત͹଀ ܶ ଶ ܵ௛ି ͹‫ݖ‬ସ ‫ܵ א‬௚ା Ͷ͕͜Ζܑ͘ ߦሺ‫ݖ‬ସ ሻ. ܑ͗͘൅ࢆͤΖܽ͹஍͗ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣ
Ζྡֆ஍(ܽ ൌ ͳǤ͹Ͷ͵Ͳʹͺ͹)Ͳ͍Ζ.
࣏ Ͷ ҈ ఈ ੓ ‫ ਼ ܐ‬͹ Ϗ ϧ ϟ ʖ ν ғ ଚ ੓ Ͷ ͯ ͏ ͱ ௒ ΄ Ζ . ྡ ֆ ஍ ܽ௖ ሺͳȀͷሻ ൌ Ͷଶ ሺߨȀͷሻΓ Ε ܽΝଁՅͦ͠, ҈ఈ੓‫਼ܐ‬Ν௒΄ͪ݃ՎΝਦ 7 Ͷࣖͪ͢. ਦ(a)ͺବԃ‫ܗ‬ब‫ٌغ‬ಕ͹҈ఈ੓‫Ͳ਼ܐ‬͍Ζ. ͞ ΗΓ Ε ब ‫غ‬ഔ ෾ ‫ز‬գ ఖ ͺ ν΢ ϕ I Ͳ͍ Ζ͞ ͳ ͗ Κ͖ Ζ . ਦ(b)ͺγ χ ϩ ‫ܗ‬ब ‫ٌ غ‬ಕ͹ ҈ ఈ ੓ ‫ ͍ Ͳ ਼ ܐ‬Ζ . ବ ԃ ‫ ܗ‬ब ‫ ٌ غ‬ಕ ͹ ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀͷሻ͗ ‫ د‬α ʖ χ Ͳ ͍ Ζ ͞ ͳ Γ Ε ,͞͹ब‫ٌغ‬ಕͺഇఛ࣢͹஦Ͳͺൕ఺γχϩ‫ܗ‬ब‫ٌغ‬ಕͲ͍Ζ͞ͳ͗಍͖ΗΖ . Ҳ๏͹ϔϫρέαʖ χ ܵሺͳȀͷሻ͗۰Ͳ͍ Ζ͞ ͳ ΓΕ , ͞͹ ٌಕͺ ഇఛ ͹ ஦Ͳͺ γχ ϩ‫ܗ‬ब ‫ ٌغ‬ಕ Ͳ͍Ζ ͞ͳ͗Κ͖Ζ.

ਦ 7: ҈ఈ੓‫਼ܐ‬. (a) ϔϫρέα ʖ χ ‫ܧ‬ሺͳȀͷሻ͹ब‫ ٌغ‬ಕ . ब‫غ‬ഔ෾‫ ͗ز‬ਫ਼ͣΖ ྡֆ஍ͺ ܽ ൌ୮ௗܽ௖ ൌ ͳǤ͹Ͷ͵Ͳʹͺ͹. (b) ϔϫρέαʖχ ܵሺͳȀͷሻ͹ब‫ٌغ‬ಕ.
ब ‫غ‬ഔ ෾ ‫ ͗ ز‬ਫ਼ͣ Ζ ྡ ֆ ஍ Ͳ ݃-෯࣢ ͹ܑ ͘ ͗ ൅ ࢆ ͤ Ζ͞ ͳ Ν ࣖ ͢ͱ ͕ ͚ . ݃-෯࣢ Ͷ ଲ ߻ ݃Νࡠ༽ͤΖ.
݃ሺܶ ଶ ܵ௛ି ሻ ൌ ܶ ିଶ ݃ܵ௛ି ൌ ܶ ିଶ ݄݃ܵ௛ି ൌ ܶ ିଷ ܵ௛ି ൌ ܶ ିହ ሺܶ ଶ ܵ௛ି ሻǤ͞ ͹ ‫ ܐ ؖ‬Γ Ε , ݃-෯ ࣢ Ͷ ଲ ߻ ݃Ν ࡠ ༽ ͤ Ζ ͞ ͳ ͺ ݃-෯ ࣢ Ͷ ܶΖ.
t 19 tିହ(28)Νࡠ༽ͤΖ͞ͳͳಋ஍Ͳ͍

௣ௗਦ 8: ݃-क࣢, ݃-෯࣢ͳ଀݃ሺܶ ଶ ܵ௛ି ሻ͹ഓ஖. (a) ܽ ൌ ܽ௖ െ ͲǤʹ. (b) ܽ ൌ ܽ௖ . (c) ܽ ൌ ܽ௖ ൅ ͲǤͳ. (d)௣ௗܽ ൌ ܽ௖ ൅ ͲǤʹ. ܽ௖ ൌ ͳǤ͹Ͷ͵Ͳʹͺ͹.
ਦ 8 Ͳͺ, ݃-෯࣢ͺࠉͲඵ͘, ݃-෯࣢͹଀݃ሺܶ ଶ ܵ௛ି ሻͺ੪Ͳඵ͏ͪ. ਦ(a)ͺब‫غ‬ഔ෾‫ز‬͹઴͹য়‫ڱ‬Νන͢ͱ͏Ζ. ٌಕ఼ ‫ݖ‬ଶ ͹ۛ๥Ͷ ݃-෯࣢ͳ ݃-෯࣢͹଀ͳ͹ި఼ͺ͵͏. ྡֆ஍͹য়‫Ͳڱ‬͍Ζਦ(b)Ͷ͕͏ͱ΍ಋ༹Ͳ͍Ζ. ͞͹৖߻ͺ, ݃-෯࣢ͳ ݃-෯࣢͹଀ٌ͗ಕ఼ ‫ݖ‬ଶ Ͳં͢ͱ͏Ζ .
ͯΉΕ, ݃-෯࣢ͳ݃-෯࣢͹ ଀͹ܑ͘ ͺͳ΍Ͷ ൅ࢆ͢ ͱ͏Ζ . ब‫غ‬ഔ෾ ‫ز‬͹‫ ޛ‬͹য়‫͍ Ͳڱ‬Ζਦ(c)ͳ (d)Ͷ ͕ ͏ ͱ , ݃-෯ ࣢ ͳ ݃-෯ ࣢ ͹ ଀ ͹ ި ఼ ͗ ਫ਼ ͣ Ζ . ٌ ಕ ఼ ‫ݖ‬ଶ ͹ ࠪ Ծ ๏ ͹ ٌ ಕ ఼ Ν ‫ݑ‬ଶ ͳ
ͤ Ζ ͳ, ‫ݖ‬ଶ ͹ ࠪ ৏ ๏ ͹ ٌ ಕ ఼ ͺ ‫ ଻ݑ‬ሺൌ ݃‫ݑ‬ଶ ሻͲ ͍ Ζ. ͞ Η Δ ͺ ٌ ಕ ఼ ‫ݖ‬ଶ Γ Ε ਫ਼ͣ ͪ ໊ ٌ ಕ ఼ Ͳब‫غ‬ͺ 10 Ͳ͍Ζ. ಚΔΗͪ݃ՎΝ੓࣯ 3.1 ͳͤΖ.
੓࣯ 3.1. ݃-क࣢৏͹ٌಕ఼‫͗ݖ‬ब‫غ‬ഔ෾‫ز‬Νً͢͞ͱਫ਼ٌ໊ͣͪಕ఼͗ ݃-෯࣢৏Ͷ͍Ζ৖߻ ,ब‫غ‬ഔ෾‫ز‬͹ྡֆ஍Ͳٌಕ఼ ‫ݖ‬Ͷ͕͜Ζ݃-෯࣢͹ܑ͘ͺ൅ࢆͤΖ.
࠹ۛંٌಕ఼ଲ ‫ݖ‬଴ ͳ‫ݖ‬ଷ ሺൌ ݃‫ݖ‬଴ ሻ͖Δٌ໊ಕ఼͗ਫ਼ͣΖ༹ࢢΝਦ 9 Ͷࣖͪ͢. ͞ΗΔͺͳ΍Ͷ྘Үܸ଴ ͶଚࡑͤΖ. ٌಕ఼‫ݖ‬଴ ͖Δਫ਼ٌͣͪಕ఼ 0 ͺӊ๏޴ͶҢಊ͢྘Ү ܸଵ Ͷ೘Ζ. Ҳ๏͹ٌಕ
఼ 5 ͺࠪ๏޴ͶҢಊ͢྘Үܸ଴ ͶཻΉΖ. ٌಕ఼‫ݖ‬ଷ ͖Δਫ਼ٌͣͪಕ఼ 3 ͺӊ๏޴ͶҢಊ͢྘Ү ܸଵͶ೘Ζ. Ҳ๏͹ٌಕ఼ 8 ͺࠪ๏޴ͶҢಊ͢྘Ү ܸ଴ ͶཻΉΖ. ͞͞Ͳٌಕ఼ 0 (5) Ͷ݃Νࡠ༽ͤΖͳ, ٌಕ఼ͺٌಕ఼ 3 (8) ͶҢΖ. ٌಕ఼‫ݖ‬଴ Ͷܶ ଷ Νࡠ༽ͪ͢଀ٌ͗ಕ఼ ‫ݖ‬ଷ Ͳ͍Ζ. ٌಕ఼‫ݖ‬଴͹ӊͶ͍Ζٌಕ఼ 0 Ͷܶ ଷ Νࡠ༽ͪ͢଀ͺٌಕ఼ 3 ͳ͵Ζ͗, ͞͹఼ͺ ٌಕ఼‫ݖ‬ଷ ͹ӊͶ͍Ζ .
Ήͪ, ٌಕ఼‫ݖ‬଴ ͹ࠪͶ͍Ζٌಕ఼ 5 Ͷܶ ଷ Νࡠ༽ͪ͢଀ͺٌಕ఼ 8 ͳ͵Ζ͗, ͞͹఼ͺٌಕ఼ ‫ݖ‬ଷt 20 t͹ࠪͶ͍Ζ. ࣺ଀ܶ ଷ ͹ࡠ༽ Ͳࠪӊ͹‫ ܐؖ‬ͺฯଚ͠ Η Ζ . ͞͹Γ͑͵ٌಕ ఼͹ ҢಊΝͳ΍͵ ͑ब‫غ‬ഔ෾‫ز‬Ν๏޴ฯଚ‫͍Ͳܗ‬Ζͳන‫ͤݳ‬Ζ .

ਦ 9: ฾ٌಕ఼ሼ‫ݖ‬଴ ǡ ‫ݖ‬ଵ ǡ ‫ݖ ڮ‬ସ ሽͳٌ໊ಕ఼ሼͲǡͳǡ ‫ͻ ڮ‬ሽ. (a) ܽ ൌ ܽ௖ ሺͳȀͷሻ ൅ ͲǤͶ ൌ ͳǤ͹ͺͳͻ͹. (b)ܽ ൌ ͵Ǥʹ.

ਦ 10: ब ‫ غ‬ഔ ෾ ‫ ͗ ز‬ฯ ଚ ‫ ͍ Ͳ ܗ‬Ζ ৖ ߻ Ͷ ͕ ͜ Ζ , ໊ ٌ ಕ ͹ α ʖ χ ͹ ݀ ఈ ๏ ๑ . ࠹ ‫ ޛ‬͹ ߨ ͹Ͳǡͳǡ ‫ͻ ڮ‬ͺਦ 9 Ͷ͕͜Ζٌ໊ಕ఼ Ͳǡͳǡ ‫ͻ ڮ‬Νන͢ͱ͏Ζ.
ब‫غ‬ഔ෾‫͗ز‬๏޴ฯଚ‫͍Ͳܗ‬Ζ͞ͳΝ೨ಆͶ͕͏ͱਫ਼ٌ໊ͣͪಕ͹αʖχΝ݀ఈ͢Γ͑.
ͨ͹ͪΌͶ฾ٌ ಕ͹αʖ χ͹࠹ঘ஍න‫( ݳ‬ਜ਼‫و‬න‫Ͳݳ‬΍͍Ζ) Ν್ͯ࿊କ ͢ͱॽ͚ (ਦ 10).
઎Ͷफ़΄ͪΓ͑Ͷٌಕ఼ 0 ͳ 3 ͗྘Үܸଵ ͶҢಊͤΖ. ͞ΗΓΕٌಕ఼ Ͳͳ͵͹αʖχΝ 1 ͳॽ
͘ ௜ ͤ . ಚ Δ Η ͪ α ʖ χ ͳͲͲͳͳ ‫ͳͲͲͲͲ ڄ‬Ν ϔ ϫ ρ έ Ͳ න ࣖ ͤ Ζ ͳ ‫ܦ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀͷሻ͗ ಚ Δ Η Ζ (࠹ঘ஍න‫Ͳݳ‬ͺ ‫ܧ‬ሺͳȀͷሻ‫ܦ‬ሺͳȀͷሻ). ࠹ۛંٌಕ఼͖Δਫ਼ٌ໊ͣͪಕ͹Ңಊ͹࢕๏Ν΍ͳͶٌ໊ಕ͹αʖχΝ݀ఈ๏๑Ν, ؈݁Ͷ࠹ۛંٌಕ఼‫ߺى‬஖͘‫͓׷‬๑ͳ໌෉͜Ζ .
฾ٌಕ͗ν΢ϕ I ͹෾‫ز‬գఖΝًͤ͞ড়݇Ν࣏Ͷࣖͤ .
ড়݇ 3.2. ฾ब‫ٌغ‬ಕ͗ਫ਼ͣͪ௜‫ޛ‬Ͷ͕͏ͱବԃ‫د͖ͯͲܗ‬αʖχͲ͍Ζ͞ͳ .
࠹‫ޛ‬ͶึଏΝफ़΄ͱ͕͚ . Ҳ࣏‫͠ڂݜͲ଀ࣺݫ‬Ηͪब‫غ‬ഔ෾‫ز‬ͺν΢ϕ I Ͳ͍Ζ. Ҳ࣏‫ݫ‬ࣺ଀Ͷ͕͜Ζब‫غ‬ഔ෾‫ز‬͹࿨ชͺඉ৙Ͷଡ͚͍Ζ͗ , ϓΟ΢ΰϱώΤϞ (M. J. Feigenbaum)ͶΓͮͱॽ͖Η್ͪͯ͹࿨ช͗ब‫غ‬ഔ෾‫ز‬͹΍ͯ૮ࣇ੓Ͷ‫ͤؖ‬Ζ‫ڂݜ‬͹ड़൅ͳ͵ͮͪժ‫غ‬ద͵࿨ชͲ͍Ζ[7, 8]. Ҳ࣏‫଀ࣺݫ‬Ͷ͕͜Ζब‫غ‬ഔ෾‫ز‬Ͷ‫ͤؖ‬Ζ೘໵ॽͳ͢ͱࢂߡช‫[ ݛ‬9, 10]Νಣ΋ ͞ͳΝ ‫ש‬Ό Ζ . Ή ͪ, ್࣏‫଀ࣺ ݫ‬Ͷ ͕͜ Ζ ब‫غ‬ഔ ෾‫ز‬Ͷ ͯ͏ ͱͺࢂ ߡช‫[ ݛ‬11]͗ༀ Ͷͪͯ.
t 21 t3.2. ν΢ϕ ,, ब‫غ‬ഔ෾͖Δൕब‫غ‬ഔ෾‫ ز‬ಋब‫غ‬෾‫΃ز‬ν ΢ ϕ II ͹ ྭ ͳ ͢ ͱ γ χ ϩ ό ʖ χ ෾ ‫ Ͳ ز‬ਫ਼ ͣ ͪ ब ‫ ٌ غ‬ಕ Ν ༽ ͏ Ζ . ϔ ϫ ρ έ α ʖ χ‫ܧ‬ሺͳȀͶሻ‫ܧ‬ሺͳȀ͵ሻ ൌ ͲͲͲͳ ‫ܧ ͳͳͲͲ ڄ‬ሺͳȀͶሻ‫ܦ‬ሺͳȀ͵ሻ ൌ ͲͲͲͳ ‫ͳͳͳ ڄ‬͹ ब ‫ ٌ غ‬ಕ ͗ ଲ Ͳ ਫ਼ ͣ Ζ . ϔ ϫ ρ έ αʖ χ ‫ܧ‬ሺͳȀͶሻ‫ܧ‬ሺͳȀ͵ሻͺ ۰ Ͳ ͍ Ζ . ͞ ͹ ब ‫ ٌ غ‬ಕ ͺ , γ χ ϩ ό ʖ χ ෾ ‫ ز‬͹ ௜ ‫ Ͳ ޛ‬ͺ γ χ ϩ ‫͍ Ͳ ܗ‬Ε ഇఛ ࣢͹ ஦ Ͳ΍ γ χ ϩ‫͍ Ͳ ܗ‬Ζ . ΍ ͑Ҳ ͯ ͹ ϔϫ ρ έα ʖ χ ‫ܧ‬ሺͳȀͶሻ‫ܦ‬ሺͳȀ͵ሻ͹ब ‫ ٌ غ‬ಕ͗ ਫ਼ͣ ͪ ௜ ‫ ޛ‬ͺ ବ ԃ ‫ ͍ Ͳ ܗ‬Ζ . ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀͶሻ‫ܧ‬ሺͳȀ͵ሻͺ ۰ Ͳ ͍ Ζ ͪ Ό Ͷ , Ϗ ϧ ϟ ʖ ν ͹ ଁՅͶͳ΍͵ ͏͞͹ब ‫ ٌغ‬ಕͺब‫غ‬ഔ ෾‫ز‬Ν࠹ ॵͶ ً͢͞ , ࣏Ͷൕब‫ غ‬ഔ෾ ‫ز‬Νً͢͞ , ࠹‫ޛ‬Ͷಋ ब‫ غ‬෾ ‫ز‬Ν ً ͢͞ ͱ γχ ϩ‫͵ ͳ ܗ‬Ζ . ͯΉ Ε ϔϫ ρέ α ʖχ ‫ܧ‬ሺͳȀͶሻ‫ܧ‬ሺͳȀ͵ሻ͹ ब‫ ٌ غ‬ಕͺ
ν΢ϕ II ͹෾‫ز‬գఖΝًͤ͞͞ͳ͗Κ͖Ζ .
ࣺ଀ܶ ଻ ͹ଲ߻න‫ݳ‬Ν಍೘ͤΖ.
ܶ ଻ ൌ ݄ ‫ ଺ ܶ݃ ל‬Ǥ(29)ଲ߻݃ܶ ଺ Ͷ‫෈ͱؖ͢‬ร͵ॄ߻ͺܶ ିଷ ܵ௚ Ͳ͍Ζ. ͯΉΕ,݃ܶ ଺ ሺܶ ିଷ ܵ௚ ሻ ൌ ݃ܶ ଷ ܵ௚ ൌ ܶ ିଷ ݃ܵ௚ ൌ ܶ ିଷ ܵ௚ Ǥଲসતܵ௛ି Ν݄-क࣢ͳ໌෉͜, ଀ܶ ିଷ ܵ௚ି Ν݄-෯࣢ͳ໌෉͜Ζ. ଀ܶ ିଷ ܵ௚ Ͷଲ߻݄Νࡠ༽ͤΖͳ,݄ܶ ିଷ ܵ௚ ൌ ܶ ଷ ݄ܵ௚ ൌ ܶ ଷ ݄݃ܵ௚ ൌ ܶ ସ ܵ௚ ൌ ܶ ଻ ሺܶ ିଷ ܵ௚ ሻ
͗ಚΔΗΖ. ͯΉΕ, ݄-෯࣢Ͷ݄Νࡠ༽ͤΖ͞ͳͺ ݄-෯࣢Ͷࣺ଀ ܶ ଻ Νࡠ༽ͤΖ͞ͳͳಋ஍Ͳ͍
Ζ.
࣏Ͷࣺ଀ܶ ଻ ͹พ͹ଲ߻න‫ݳ‬Νߡ͓Γ͑ .
ܶ ଻ ൌ ܶ ଺ ݄ ‫݃ ל‬Ǥ଺ଷଲ ߻ ܶ ݄Ͷ ‫ ෈ ͱ ͢ ؖ‬ร ͵ ॄ ߻ ͺ ܶ ܵ௛ Ͳ ͍ Ζ .

ଲ স ત ܵ௚ି Ν ݃-क ࣢ ͳ ໌ ෉ ͜ ,(30)
଀ ܶ ଷ ܵ௛ି Ν ݃-෯ ࣢ ͳ໌෉͜Ζ. ଀ܶ ଷ ܵ௛ Ͷଲ߻݃Νࡠ༽ͤΖͳ݃ܶ ଷ ܵ௛ ൌ ܶ ିଷ ݃ܵ௛ ൌ ܶ ିଷ ሺ݄݃ሻܵ௛ ൌ ܶ ିସ ܵ௛ ൌ ܶ ି଻ ሺܶ ଷ ܵ௛ ሻ
͗ಚΔΗΖ. ݃-෯࣢Ͷ݃Νࡠ༽ͤΖ͞ͳͺ ݃-෯࣢Ͷࣺ଀ܶ ି଻ Νࡠ༽ͤΖ͞ͳͳಋ஍Ͳ͍Ζ .
ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀͶሻ‫ܦ‬ሺͳȀ͵ሻ͹ ब ‫ ٌ غ‬ಕ ͳ ‫ܧ‬ሺͳȀͶሻ‫ܧ‬ሺͳȀ͵ሻ͹ ब ‫ ٌ غ‬ಕ ͗ γ χ ϩ ό ʖ χ ෾ ‫Ͳ ز‬
ਫ਼ͣΖ༹ࢢΝਦ 11 Ͷࣖͪ͢. ਦ(a)Ͳͺ, ଀ܶ ସ ܵ௚ି ͗ଲসત ܵ௛ି ͳં͢ͱ͏Ζ. ͞͹য়‫͗ڱ‬γχϩ ό ʖ χ ෾ ‫ ز‬͹ ྡ ֆ ஍ Ͳ ਫ਼ ͣ Ζ . Ϗ ϧ ϟ ʖ ν ͹ ଁ Յ Ͷ ͳ ΍ ͵ ͏ ٌ ಕ ఼ ‫ݖ‬ଵ ͳ ‫ݓ‬ଵ ͺ ଲ স ત ܵ௛ି ৏Ν ࠪ ӊ Ͷ ཯ Η ͱ ͏ ͚ . ٌ ಕ ఼ ‫ݖ‬ଵ ͳ ‫ݓ‬ଵ ͹ ؔ Ͷ ෈ ҈ ఈ ଡ ༹ ର ͹ ‫ܶ ނ‬Ȟ௨ ͗ ৷ ೘ ͢ ͱ ͏ Ζ ͞ ͳ Ͷ ஭ қ͢ͱ Ά ͢ ͏ . ଲ স੓ Γ Ε ҈ఈ ଡ ༹ ର͹ ‫ି ܶ ނ‬ଵ Ȟ௦ ሺൌ ݄ሺܶȞ௨ ሻሻ΍৷ ೘ ͢ ͱ͏ Ζ . ٌಕ ‫ݖ‬ଵ ͺ ϔϫ ρέα ʖ χ ‫ܧ‬ሺͳȀͶሻ‫ܦ‬ሺͳȀ͵ሻ͹ ब ‫ ٌغ‬ಕ ఼ Ͳ ͍ Ε , ਫ਼ ͣ ͪ ௜ ‫ޛ‬ͺ ବ ԃ ‫ ͍ Ͳ ܗ‬Ζ . Ҳ ๏͹ ٌ ಕ ‫ݓ‬ଵ ͺ ϔ ϫρέαʖχ ‫ܧ‬ሺͳȀͶሻ‫ܧ‬ሺͳȀ͵ሻ͹ब‫ٌغ‬ಕ఼Ͳ͍Ε, ਫ਼ͣͪ௜‫ޛ‬ͺγχϩ‫͍Ͳܗ‬Ζ .

t 22 tਦ 11: (a) γχϩόʖχ෾‫ز‬͹ྡֆ஍ ܽ ൌ ͵Ǥʹͳ͹ͲͲͲ͵ͳʹ. ଀ܶ ସ ܵ௚ି ͗ଲসતܵ௛ି ͳં͢ͱ͏Ζ. (b)γχ ϩό ʖ χ෾ ‫ ͗ز‬ਫ਼ͣ ͪ௜ ‫ ޛ‬. ٌ ಕ఼ ‫ݖ‬ଵ ͗ବ ԃ ‫ ܗ‬ब‫ٌ غ‬ಕ Ͳ , ٌಕ఼ ‫ݓ‬ଵ ͗ γ χϩ ‫ܗ‬ब ‫ٌغ‬ಕ. ܽ ൌ ͵ǤͶ.

ਦ 12: ବ ԃ ‫ ٌ ܗ‬ಕ ͹ ҈ ఈ ੓ ‫ ਼ ܐ‬͹ Ϗ ϧ ϟ ʖ ν ғ ଚ ੓ . γ χ ϩ ό ʖ χ ෾ ‫ ز‬͹ ྡ ֆ ஍ ܽ ൌ͵Ǥʹͳ͹ͲͲͲ͵ͳʹ . (a) ब ‫ غ‬ഔ ෾ ‫ ز‬͹ ྡ ֆ ஍ ܽ ൌ ͵Ǥʹͳ͹ͲͲ͸͵ʹ . (b) ൕ ब ‫ غ‬ഔ ෾ ‫ ز‬͹ ྡ ֆ ஍ ܽ ൌͶǤͲ͵͹ʹ͹ʹͻʹ, ಋब‫غ‬෾‫ز‬͹ྡֆ஍ ܽ ൌ ͶǤͲ͵ͺʹͻʹͺʹ. ‫ܮߺى‬ଷ ͳܴଷ Ͷͯ͏ͱͺ, ୊ 3.2 ઇ͹ ࠹‫ޛ‬
Ͳઈ໎ͤΖ.
ϔϫ ρέ αʖ χ ‫ܧ‬ሺͳȀͶሻ‫ܦ‬ሺͳȀ͵ሻ͹ବԃ ‫ ܗ‬ब‫ٌ غ‬ಕ ͹҈ ఈ੓ ‫ ਼ܐ‬Ν ௒΄ Ζ . ҈ఈ ੓‫਼ ܐ‬ͺ Ϗ ϧϟ ʖ ν ܽ͹ ଁ Յ Ͷ ͳ ΍ ͵ ͏ ‫ݎ‬ሺܽሻ ൌ ʹΓ Ε ‫ ݰ‬ঙ ͢ ͱ ‫ݎ‬ሺܽሻ ൌ െʹͶ ୣ ͢ ߍ Ͷ ‫ ݰ‬ঙ ͤ Ζ ( ਦ 12(a)).
‫ݎ‬ሺܽሻ ൌ െʹͲब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣΖ. ‫ݎ‬ሺܽሻ ൏ െʹ͗੔ΕཱིͯϏϧϟʖν྘ҮͲͺ , ब‫ٌغ‬ಕͺൕ఺Ν ൒ ͑ γ χ ϩ ‫ ͍ Ͳ ܗ‬Ζ. ҈ ఈ ੓ ‫ ਼ ܐ‬ͺ ౏ ஦ Ͳ ଁ Յ Ͷ ఺ ͣ, ࠸ ; ‫ݎ‬ሺܽሻ ൌ െʹͶ ͵ Ζ. ͯ Ή Ε, ൕब‫غ‬ഔ෾‫͗ ز‬ਫ਼ͣ ͱବԃ ‫ܗ‬Ͷ໯Ζ . ࠹शదͶ, ͞͹ब‫ٌغ‬ಕ ͺ ‫ݎ‬ሺܽሻ ൌ ʹΝຮͪͤ ܽ͹஍Ͳಋब‫ غ‬෾ ‫ ز‬Ν ً ͞ ͢ ͱ γχ ϩ ‫ ͵ ͳ ܗ‬Ζ(ਦ 12(b)). Ґ ৏ Γ Ε ϔ ϫ ρ έ α ʖχ ‫ܧ‬ሺͳȀͶሻ‫ܦ‬ሺͳȀ͵ሻ͹ ब ‫غ‬ٌಕͺν΢ϕ II ͹෾‫ز‬գఖΝًͤ͞͞ͳ֮͗೟Ͳͪ͘ .
ਦ 13(a)Ͷ ͺ ϔ ϫ ρέ α ʖ χ ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻ͹ ฾ٌ ಕ Ν ඵ ͘ , ਦ 13(b)Ͷͺ ϔ ϫ ρ έ α ʖ χ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻ͹ ٌ໊ ಕ Ν ඵ͏ ͪ . ਦ(a)Ͳ, ್ͯ ͹ ٌಕ ఼ ‫ݖ‬଴ ͳ ‫ݖ‬ଵ ሺൌ ݃‫ݖ‬଴ ሻͺ ‫ ڠ‬Ͷ྘Ү ܸଵ Ͷ ͍ Ε , ͖ ͯ ‫ ڧ‬ֆ Ͷ ࠹ ۛ ં ͢ ͪ ٌ ಕ ఼ Ͳ ͍ Ζ . ब ‫ غ‬ഔ ෾ ‫ ޛ ز‬, ͞ Η Δ ͹ ఼ Γ Ε ਫ਼ ͣ ͪ ٌ໊ ಕ ఼ ͹Ң ಊ ͹ ࢕ ๏ Ν ௒ ΄ Ζ . ٌ ಕ఼ ‫ݖ‬ଵ ͖ Δ ଲ স ત ܵ௛ି ৏ Ͷ ໊ ٌಕ ఼ ‫ݑ‬ଵ ͳ ‫ ͗ ଼ݑ‬ਫ਼ ͣ Ζ . ͞ Η Δ
t 23 tͺ, ਦ(b)Ͳͺ 1 ͳ 8 ͳ‫͠ࡎى‬Ηͱ͏Ζ. ٌಕ఼‫ݑ‬ଵ ͺࠪͶҢಊ͢, ྘Үܸ଴ Ͷ೘Ζ. ٌಕ఼‫ ଼ݑ‬ͺӊͶҢಊͤΖ͗྘Ү ܸଵ ͶཻΉΖ (ਦ(b)). ਦ 13 Ͳ, ॐͶ೘ͮͱ͏Ζ఼ત͹͍ͪΕͶ҈ఈଡ༹ର͹ ‫ି ܶ ނ‬ଵ Ȟ௦ ͗৷ ೘ ͤ Ζ . ٌ ಕ ఼ ‫ݑ‬ଵ ͳ ݃-ଲΝ ͵ ͤ ٌ ಕ ఼ ‫ݑ‬଴ ΍྘ Ү ܸ଴ Ͷ Ңಊ ͤ Ζ . ͢ ͖ ͢, ٌ ಕ ఼‫݃ ͳ ଼ݑ‬-ଲ Ν ͵ ͤ ٌ ಕ ఼ ‫ ଻ݑ‬ͺ ྘ Ү ܸଵ Ͷ ཻ Ή Ζ . ฾ ٌ ಕ ఼ ‫ݖ‬଴ ͹ ࠪ Ͷ ໊ ٌ ಕ ఼ ‫ݑ‬଴ ͗ ͍ Ε , ӊ Ͷ ‫͗ ଻ݑ‬͍Ζ. Ήͪ, ฾ٌ ಕ఼ ‫ݖ‬ଵ ͹ ࠪͶ ٌ໊ ಕ ఼ ‫ݑ‬ଵ ͍͗ Ε, ӊ Ͷ ‫͍͗ ଼ݑ‬Ζ. ࠪӊ ͹ ‫Ͳܶ ଀ࣺ ͗ ܐؖ‬ฯ ଚ͠Ηͱ͏Ζ. ͯΉΕ, ๏޴ฯଚ‫ܗ‬͹ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣͱ͏Ζ .

ਦ 13: (a) ฾ٌಕ. ϔϫρέαʖχ ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻሺͳͳͳ ‫ͳͲͲͲ ڄ‬ሻ. (b) ब‫غ‬ഔ෾‫Ͳز‬ਫ਼ٌ໊ͣͪ
ಕ . ੖ ਼ ݇ͺ ఼ ‫ݑ‬௞ Ν න ͤ . ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻሺͲͲͳ ‫ͳͲͲͲ ڄ ͳͳͳ ڄ ͳͲͲͲ ڄ‬ሻ.
ܽ ൌ ͵Ǥ͵.

ਦ 14: ฾ٌಕ఼‫ݖ‬ସ ΓΕٌ໊ಕ఼ ‫ݑ‬ସ ͳ‫ݑ‬ଵଵ ͗ਫ਼ͣΖ༹ࢢ. (a) ܽ ൌ ͵Ǥʹͳ͹ͲͲͲ͵͵. (b) ܽ ൌ ͵Ǥʹͳ͹ͲͲʹͷ.
(c) ܽ ൌ ͵Ǥʹͳ͹ͲͲ͸͵ʹ. (d) ܽ ൌ ͵Ǥʹͳ͹Ͳʹ.
t 24 t͞ ͹ ब ‫ غ‬ഔ ෾ ‫ ز‬Ν ‫ ش‬Ձ ָ ద Ͷ ཀྵ մ ͤ Ζ ͪ Ό Ͷ , ఼ ‫ݖ‬ସ ‫ܵ א‬௚ି Ͷ ͕ ͜ Ζ ଀ ܶ ଷ ܵ௛ି ͳ ଀ ܶ ିସ ܵ௛ି ሺൌ݃ܶ ଷ ܵ௛ି ሻ͹‫ܐؖ‬Ν௒΄Ζ. ਦ 14 Ν‫ͱݡ‬ཋ͢͏. ਦ(a)Ͳͺ, ଀ܶ ଷ ܵ௛ି ͳ଀ܶ ିସ ܵ௛ି ሺൌ ݃ܶ ଷ ܵ௛ି ሻ͗ଲসતܵ௚ି ͳં͢ͱ͏Ζ. ͞Ηͺ γχϩόʖχ෾‫ ز‬͹ྡֆ ஍Ͳ͹য়‫͍Ͳڱ‬Ζ . γχϩόʖχ෾‫Ͳز‬ਫ਼ ͣ ͪ ್ ఼ ‫ݖ‬ସ ͳ ‫ݓ‬ସ ͺ ଲ স ત ܵ௚ି ৏ Ͷ ͍ Ε ‫ ͏ ޕ‬Ͷ ཯ Η ͱ ͏ ͚ (ਦ(b)). ਦ (c)Ͳ ͺ, ଀ ܶ ଷ ܵ௛ି ͳ ଀ܶ ିସ ܵ௛ି ͗࠸; ં͢ ͱ͏Ζ . ఼‫ݖ‬ସ Ͳ, ྈं͹ܑ͘ ͺ൅ ࢆ͢ ͱ ͏Ζ͞ ͳͶ ஭қ͢ Γ͑ . ͨ͹‫ޛ‬, ਦ(d)Ͳͺ఼‫ݖ‬ସ ΓΕब‫غ‬ഔ෾‫Ͳز‬ਫ਼ٌ໊ͣͪಕ఼ ‫ݑ‬ସ ͳ‫ݑ‬ଵଵ ሺൌ ݄‫ݑ‬ସ ሻ͗ඵ͖Ηͱ͏Ζ. Ϗϧϟʖν͹ଁՅͶͳ΍͵͏఼್ ‫ݑ‬ସ ͳ‫ݑ‬ଵଵ ͺଲসત ܵ௚ି ৏Ν఼‫ݖ‬ସ ΓΕ཯Ηͱ͏͚.

ਦ 15: ब‫غ‬ഔ෾‫ز‬઴‫ޛ‬Ͷ͕͜Ζ଀ ܶ ଷ ܵ௛ି ͹ٌಕ఼ ‫ݖ‬ସ Ͷ͕͜Ζܑ͘ ߦሺ‫ݖ‬ସ ሻ͹รԿ.
଀ܶ ଷ ܵ௛ି ͹఼‫ݖ‬ସ Ͷ͕͜Ζܑ͘ ߦሺ‫ݖ‬ସ ሻΝਦ 15 Ͷࣖͪ͢. γχϩόʖχ෾‫Ͳ఼ز‬ͺܑ͘ͺଲসતܵ௚ି ͹ܑ͘Ͷ౵͚͢ෝ Ͳ͍Ζ . ब‫غ‬ഔ෾‫ز‬͹ྡ ֆ஍Ͳ ൅ࢆ͢ , ྡֆ஍Νգ͙Ζ ͳܑ͘ͺਜ਼ͳ͵Ζ.

ਦ 16: ൕब‫غ‬ഔ෾‫ز‬. (a) ܽ ൌ ͶǤͲ͵͹ʹ͹ʹͻʹ. ఼‫ݖ‬ଵ Ν௪գ͢ͱ͏Ζ௜તͺܑ͘ 2 ͹௜ત. (b) ܽ ൌͶǤʹ.
࣏Ͷ ϔϫ ρέ αʖ χ ‫ܧ‬ሺͳȀͶሻ‫ܦ‬ሺͳȀ͵ሻ͹ब ‫ ٌغ‬ಕ͗ ൕब ‫غ‬ഔ ෾‫ ز‬Ν ً͞ ͤգ ఖΝ ‫ݗ‬౾ ͤΖ . ൕब ‫ غ‬ഔ ෾ ‫ Ͳ ز‬ͺ ٌ ಕ ఼ ‫ݖ‬ସ ͖ Δ ್ ͯ ͹ ໊ ٌ ಕ ఼ ͗ ଲ স ત ܵ௚ି ৏ Ͷ ਫ਼ ͣ Ζ . ͨ ͹ ͪ Ό ൕ ब ‫ غ‬ഔ ෾‫ ͗ز‬ਫ਼ ͣΖ ͞ ͳ Ν֮ ೟ ͤ Ζͪ Ό Ͷͺ ଲ স ત ܵ௛ି ৏͹ ఼ ‫ݖ‬ଵ Ͳ ܶ ସ ܵ௚ି ͳܶ ିଷ ܵ௚ି ͹ ‫ ܐؖ‬Ν ؏ ࡱͤ Ζ ච གྷ͍͗Ζ. ਦ 16(a)Ͳͺ, ܶ ସ ܵ௚ି ͳܶ ିଷ ܵ௚ି ఼͗‫ݖ‬ଵ Ͷ͕͏ͱඉԥஇదިࠫΝ͢ͱ͏Ζ . ྈं͹ܑ͘ͺ
ͳ΍Ͷ 2 Ͳ͍Ζ. ఼‫ݖ‬ଵ Ν௪գͤΖܑ͘ 2 ͹௜તΝਦ (a)Ͷ௧Յ͢ͱ͍Ζ. ͞͹য়‫͗ڱ‬ൕब‫غ‬ഔ
t 25 t෾‫ز‬͹ྡֆ஍Ͳਫ਼ͣΖ. ਦ 16(b)Ͳͺ, ఼‫ݖ‬ଵ ͖Δਫ਼ٌ໊ͣͪಕ఼ ‫ݒ‬ଵ ͳ‫ ଼ݒ‬ሺൌ ݄‫ݒ‬ଵ ሻ͗؏ࡱͲ͘Ζ.

ਦ 17: (a) ଀ܶ ସ ܵ௚ି ͹ٌಕ఼‫ݖ‬ଵ Ͷ͕͜Ζܑ͘ ߦሺ‫ݖ‬ଵ ሻͺ, ෝ͖Δਜ਼΃ͳรԿͤΖ. (b) ܑ͗͘ 2 Ͷ͵ͮͪϏϧϟʖν஍Ͳൕब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣΖ .
଀ܶ ସ ܵ௚ି ͹఼‫ݖ‬ଵ Ͷ͕͜Ζܑ͘͹Ϗϧϟʖνғଚ੓Νਦ 17 Ͷࣖͪ͢. ఼‫ݖ‬ଵ ͗γχϩόʖχ෾‫ز‬Ͳਫ਼ͣͪ௜‫ޛ‬͹ܑ͘ͺෝͲ͍Ζ. ܑ͗͘൅ࢆͪ͢‫ޛ‬, ܑ͘ͺਜ਼Ͷ͵Ζ. ͨ͹‫ޛ‬, ܑ͘ͺ‫ݰ‬ঙ͢, ܑ͗͘ 2 Ͷ͵ͮͪϏϧϟʖν஍Ͳൕब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣΖ . Ґ৏͹݃ՎΝΉͳΌͱ੓࣯ 3.3
ͳͤΖ.
੓࣯ 3.3. ݄-क࣢৏͹ٌಕ఼‫͗ݖ‬ब‫غ‬ഔ෾‫ز‬Νً͢͞ͱਫ਼ٌ໊ͣͪಕ఼͗ ݄-෯࣢৏Ͷ͍Ζ৖߻,(ൕ)ब‫غ‬ഔ෾‫ز‬͹ྡֆ஍Ͳٌಕ఼ ‫ݖ‬Ͷ͕͜Ζ ݄-෯࣢͹ܑ͘ͺ 2 Ͳ͍Ζ.

ਦ 18: (a) ฾ٌಕ఼. ϔϫρέαʖχ ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻሺͳͳͳ ‫ͳͲͲͲ ڄ‬ሻ. (b) ൕब‫غ‬ഔ෾‫Ͳز‬ਫ਼໊ͣͪ ٌ ಕ ఼ . ੖ ਼ ݇ͺ ‫ݒ‬௞ Ν қ ັ ͤ Ζ . ϔ ϫ ρ έ α ʖ χ ܵሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻ‫ܨ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻሺͲͳͳ ‫ڄ ͳͲͲͲ ڄ‬ͳͲͳ ‫ͳͲͲͲ ڄ‬ሻ. ܽ ൌ ͶǤ͸.
ൕब‫غ‬ഔ෾‫ز‬Νًͤ͞฾ٌಕΝਦ 18(a)Ͷඵ͘, ൕब‫غ‬ഔ෾‫Ͳز‬ਫ਼ٌ໊ͣͪಕΝਦ 18(b)Ͷඵ͏ͪ. ॐͶ೘ͮͱ͏Ζ఼ત͹͍ͪΕͶ҈ఈଡ༹ର͹‫ି ܶ ނ‬ଵ Ȟ௦ ͗৷೘͢, ఼ત͹ࠪଈ͗྘Үܸ଴ ͳ ͵ Ε ӊ ଈ ͗ ྘ Ү ܸଵ ͳ ͵ Ζ . ൕ ब ‫ غ‬ഔ ෾ ‫ ز‬Ͷ ͕ ͏ ͱ ͺ , ‫ ଼ݒ‬ൌ ݃‫ݒ‬଴ ͳ ‫ ଻ݒ‬ൌ ݃‫ݒ‬ଵ ͗ ੔ Ε ཱི ͯ . ‫ݖ‬଴͹ࠪ Ͷ ‫ݒ‬଴ ͗ ͍Ε, ӊ Ͷ ‫ ͍͗ ଻ݒ‬Ζ. ͖͢ ͢, ‫ݖ‬ଵ ͹ ࠪ Ͷ ‫ ͍͗ ଼ݒ‬Ε, ӊͶ ‫ݒ‬ଵ ͍͗ Ζ. ࠪӊ ͹ ‫ࣺ͗ ܐؖ‬
t 26 t଀ܶͲൕ఺͢ͱ͏Ζ. ͯΉΕ, ൕब‫غ‬ഔ෾‫Ͳز‬ͺ๏޴ൕ఺‫͗ܗ‬ਫ਼ͣΖ . Ґ৏ΓΕ੓࣯ 3.4 ͗ಚΔΗΖ.
੓࣯ 3.4. ब‫غ‬ഔ෾‫͗ز‬઎Ͷਫ਼ͣ, ࣏Ͷൕब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣΖ৖߻ , ब‫غ‬ഔ෾‫ز‬͹ν΢ϕͺ๏޴ฯଚ‫Ͳܗ‬, ൕब‫غ‬ഔ෾‫ز‬͹ν΢ϕͺ๏޴ൕ఺‫͍Ͳܗ‬Ζ .
฾ٌಕͺ࠹‫ޛ‬Ͷಋब‫غ‬෾‫ز‬Νً͢͞ͱγχϩ‫ܗ‬ब‫ٌغ‬ಕͳ͵Ζ . ब‫ ͗غ‬7 ͹್ͯ͹ٌ໊ಕ
͗ਫ਼ͣΖ͗͞ΗΔͺඉଲসब‫ٌغ‬ಕͲ͍Ζ .

ਦ 19: ٌ໊ಕ͹‫ߺى‬Կ‫و‬ଉ. (a) ब‫غ‬ഔ෾‫ز‬. (b) ൕब‫غ‬ഔ෾‫ز‬.
ࢀͯ͹෾‫Ͳز‬ਫ਼ٌ໊ͣͪಕ͹ϔϫρέαʖχΝ݀ΌΓ͑ . ब‫غ‬ഔ෾‫Ͳز‬ਫ਼ٌ໊ͣͪಕ͹ٌಕΝඵ͏ͪਦ 13(b)Ν‫ݡ‬Γ͑. ड़൅఼‫ݑ‬଴ ΓΕٌಕ఼Ν௧͑͞ͳͲٌಕ఼͹‫ߺى‬Νॳ࣏݀ΌΖ͞ͳ͗Ͳ͘Ζ. ͞͞Ͳ‫ݑ‬ଵ ൌ ݃‫ݑ‬଴ Ͳ͍Ζ. ఼‫ݑ‬଴ ͗ਫ਼ͣͪ௜‫ޛ‬ͺ఼ ‫ݖ‬ଵ ͹͚ۛͶ͍Ζ͗, Ϗϧϟʖν͹ଁ ՅͶ ͳ΍ ͵ ͏ࠪ ͶҢ ಊ͢ ࠹श దͶ ྘ Ү ܸ଴ Ͷ೘Ζ . Ҳ๏͹ ఼ ‫ݑ‬ଵ ΍ಋ ༹Ͷ Ңಊ Ν͢ ͱ྘ Ү ܸ଴Ͷ೘Ζ. ਜ਼‫و‬න‫ܦ ݳ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻΝ್ͯ࿊କͪ͢න‫ݳ‬Ν༽қͤΖ (ਦ 19(a)). ఼್‫ݑ‬଴ ͳ‫ݑ‬ଵ ͹‫ߺى‬1 Ν 0 Ͷॽ͘‫͓׷‬Ζ͞ͳͲٌ໊ಕ͹αʖχ͗ಚΔΗΖ . ಚΔΗͪαʖχΝϔϫρέαʖχනࣖͲॽ͚.
ͲͲͳͲͲͲͳ ‫ ͳͲͲͲͳͳͳ ڄ‬ൌ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻ ‫ܦ ڄ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻǤ(31)࣏ Ͷ ൕ ब ‫ غ‬ഔ ෾ ‫ Ͳ ز‬ਫ਼ͣ ͪ ໊ ٌ ಕ ͹ ٌ ಕ Ν ඵ ͏ͪ ਦ 18(b)Ν ‫ ݡ‬Γ ͑. ͞ ͞ Ͳ ‫ ଼ݒ‬ൌ ݃‫ݒ‬଴ Ͳ ͍
Ζ. ఼ ‫ݒ‬଴ ͗ ਫ਼ͣ ͪ ௜ ‫ ޛ‬ͺ ఼ ‫ݖ‬ଵ ͹ ۛ ͚ Ͷ ͍Ζ ͗ , Ϗ ϧ ϟ ʖ ν ͹ ଁՅ Ͷ ͳ ΍ ͵ ͏ ࠪ Ͷ Ң ಊ ͢࠹ शదͶ྘Ү ܸ଴ Ͷ೘Ζ. Ҳ๏͹఼‫ ଼ݒ‬΍ಋ༹ͶҢಊΝ͢ͱ྘Үܸ଴ Ͷ೘Ζ. ਜ਼‫و‬න‫ܦݳ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻΝ್ͯ࿊କͪ͢න‫ݳ‬Ν༽қͤΖ (ਦ 19(b)). ఼್‫ݒ‬଴ ͳ‫ ଼ݒ‬͹‫ ߺى‬1 Ν 0 Ͷॽ͘‫͓׷‬Ζ͞ͳͲٌ໊ಕ͹αʖχ͗ಚΔΗΖ. ಚΔΗͪαʖχΝϔϫρέαʖχͲॽ͚ .
ͳͲͳͲͲͲͳ ‫ ͳͲͲͲͳͲͳ ڄ‬ൌ ܵሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻ ‫ܨ ڄ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻǤ(32)͞͹න‫ݳ‬͹઴ൔ͹αʖχ ܵሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻ ൌ ͳͲͳ ‫ͳͳͲͲͲ ڄ‬, ‫ޛ‬ൔ͹αʖχ ‫ܨ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͶሻ ൌ Ͳͳͳ ‫ͳͲͲͲ ڄ‬ͺ࣎ؔൕ఺ ଲͲ͍Ζ . ‫ߺى‬Ν॰յ͢ ͱ΍್ͯ ͹α ʖχͺҲ஗ ͢͵͏͞ ͳΓ Ε , ͞ΗΔ͹αʖχͲ‫ى‬फ़͠ΗΖब‫ٌغ‬ಕ͗ඉଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳ͗Κ͖Ζ . ͞ΗΔͺಋब‫غ‬෾‫Ͳز‬ਫ਼್ͣͪͯ͹ٌ໊ಕ͹ϔϫρέαʖχͲ͍Ζ . Ґ৏ͲಚΔΗͪ݃ՎΝΉͳΌͱ੓࣯ 3.5 ͳͤΖ.
੓࣯ 3.5. ൕब‫غ‬ഔ෾‫Ͳ ز‬ਫ਼ͣͪ ٌ໊ ಕ͹ϔ ϫρ έαʖχ Ν ܲ௟ ܲ௥ ͳͤΖ. ܲ௟ ͳ ܲ௥ ͺಋब‫غ‬෾ ‫ز‬Ͳਫ਼್ͣͪͯ͹໊ඉଲসब‫ٌغ‬ಕ͹ϔϫρέαʖχͲ͍Ζ .
ඉଲসब‫ٌغ‬ಕΝ૮ฑ໚Ͳඵ͚͞ͳ͗Ͳ͘͵͏͹Ͳ , αʖχΝཤ༽͢ͱ‫ߺى‬ฑ໚Ͳ ಋब‫غ‬෾‫Ͳز‬ਫ਼್ͣͪͯ͹ٌ໊ ಕΝඵ͚(ਦ 20). ฾ٌಕ͹ٌಕ఼ ‫ݖ‬Ƹଵ ΓΕ‫ݑ‬ොଵ ͳ‫ݒ‬ොଵ ͗ਫ਼ͣΖ. ͞ΗΔͺ
t 27 tଲসત ܵመ௛ Ͷ‫ ͱ͢ ؖ‬ଲস ͵Ғ ஖Ͷ͍ Ζ . ‫ߺى‬ฑ໚ Ͳఈ ٝ͠Η ࣺͪ ଀ , ଲ߻ ͵Δ ;Ͷब ‫ ٌغ‬ಕ͹ඵ͘๏Ͷͯ͏ͱͺ෉࿧ A Ͳઈ໎ͪ͢.

ਦ 20: ‫ߺى‬ฑ໚৏Ͷ͕ ͜Ζ್ͯ͹ඉଲসٌ໊ಕ ͹ٌಕ఼ (ͲͳͳͲͲͲͳʁ੼‫؛‬, ͳͲͳͲͲͲͳʁ੪‫)؛‬ͳ, ଲসब‫ٌغ‬ಕ఼ (ͳͳͳͲͲͲͳʁࠉ‫)؛‬.
࠸ౕ, ҈ఈ੓‫਼ܐ‬͹ਦ 12 Ν‫ͱݡ‬ཋ͢͏. ҈ఈ੓‫ݰ਼͗ܐ‬ঙ͢ͱ ‫ݎ‬ሺܽሻ ൌ െͳͳިࠫͤΖ఼Ν‫ܮ‬ଷͳ ໌ ෉ ͜ Ζ (ਦ (a)). ҈ ఈ ੓ ‫ ଁ ͗ ਼ ܐ‬Յ ͢ ͱ , ࠸ ౕ ‫ݎ‬ሺܽሻ ൌ െͳͳ ި ࠫ ͤ Ζ ఼ Ν ܴଷ ͳ ͤ Ζ (ਦ(b)). ‫ݎ‬ሺܽሻ ൌ െͳ ൌ ʹ ሺʹߨȀ͵ሻΓΕ, ‫ܮ‬ଷ ͺࣺ଀ܶ ଻ ΝୱҒͳ͢ͱյ఺਼ͳȀ͵͹ब‫ٌغ‬ಕ͗յ఺෾‫ز‬Ͳ ਫ਼ ͣ Ζ ྡ ֆ ఼ Ͳ ͍ Ζ ͞ ͳ ͺ ؈ ୱ Ͷ ཀྵ մ Ͳ ͘ Ζ . ໲ ୌ ͺ ܴଷ Ͳ ͍ Ζ . մ ए ͹ ࢕ ๏ ͺ ̐ ௪ Ε ͍
Ζ . ‫ݎ‬ሺܽሻ ൌ െͳ ൌ ʹ ሺʹ ൈ ሺെͳሻߨȀ͵ሻͳ ߡ ͓ Η ͻ , ‫ܮ‬ଷ Ͳ ͺ յ ఺ ਼ ሺെͳȀ͵ሻ͹ ब ‫ ٌ غ‬ಕ ͗ յ ఺ ෾ ‫Ͳز‬ਫ਼ͣͪͳմएͲ͘Ζ. ͖͢͢, ͞͹մएͲͺյ఺਼͗‫ݰ‬ঙ͢ͱ͏Ζ͖Δ , ͞͹մएͺགྷ੧ 1.1
ͶൕͤΖ. གྷ੧ 1.1 ͹΍ͳͲͺ ‫ݎ‬ሺܽሻ ൌ െͳ ൌ ʹ ሺʹ ൈ ʹߨȀ͵ሻͳ͢ͱ, ܴଷ Ͳͺյ఺਼ሺʹȀ͵ሻ͹ब‫ٌغ‬ಕ͗յ఺෾ ‫Ͳز‬ਫ਼ͣ ͪͳ մएͤΖච གྷ͍͗Ζ . ͯΉΕ, ҈ఈ੓‫ଁ ਼͗ܐ‬Յ ͢ͱ͏ΖϏ ϧϟʖν ྘Ү Ͳ ͺ , յ ఺਼ ͳȀʹ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ͳȀͳ͹ ब ‫ٌ غ‬ಕ ͗ յ ఺ ෾‫ Ͳ ز‬ਫ਼ ͣ ͱ ͏Ζ . Ґ ৏ Γ Εյ ఺ ෾ ‫ز‬͹յ఺਼ۢؔ͗ ሺͲǡͳȀʹሻ ‫ ׫‬ሺͳȀʹǡͳሻͶ֨௃͠Ηͪ.
฾ٌಕ͗ν΢ϕ II ͹෾‫ز‬գఖΝًͤ͞ড়݇Νࣖͤ .
ড়݇ 3.6. ฾ब‫ٌغ‬ಕ͗ਫ਼ͣͪ௜‫ޛ‬Ͷ͕͏ͱବԃ‫͖ͯͲܗ‬۰αʖχͲ͍Ζ͞ͳ .

3.3. ν΢ϕ ,,, ൕಋब‫غ‬෾‫͖ز‬Δब‫غ‬ഔ෾‫ز‬
3.3.1. ౛  ࢧࢻࣝࣀ࣮ࢻศᒱ࡛⏕ࡌࡓ࿘ᮇ㌶㐨
ν΢ϕ III ͹࠹ॵ͹ྭͳ͢ͱγχϩόʖχ෾‫Ͳز‬ਫ਼ͣͪब‫ٌغ‬ಕΝཤ༽ͤΖ . ϔϫρέαʖ χ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻͳ ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ͹ ब ‫ ٌ غ‬ಕ ͹ ෾ ‫ ز‬գ ఖ Ν ௒ ΄ Ζ [6]. α ʖ χ ͺͳ΍ Ͷ‫ ͍Ͳ د‬Ζ͖ Δ , ͞ ΗΔ ͺഇ ఛ ࣢͹஦Ͳ ͺൕ ఺ γχ ϩ‫ ܗ‬ब‫ٌ غ‬ಕ Ͳ͍ Ζ . γχϩ όʖχ෾‫Ͳز‬ਫ਼ͣͪ௜‫ޛ‬͹҈ఈ੓‫਼ܐ‬Ν਼஍‫Ͳࢋܯ‬௒΄ͪ݃Վ , ϔϫρέαʖχ
E(1/3)E(1/2)E(1/2)͹ब‫ٌغ‬ಕ͗γχϩ‫ Ͳܗ‬, E(1/3)F(1/2)F(1/2)͹ब‫ٌغ‬ಕ͗ବԃ‫͍Ͳܗ‬t 28 tΖ ͞ ͳ ͗ ಚ Δ Η ͪ . Γ ͮ ͱ , ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ͹ ब ‫ ٌ غ‬ಕ ͗ ν ΢ ϕ III ͹෾‫ز‬գఖΝࣖ͢, ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ͹ब‫ٌغ‬ಕͺν΢ϕ I ͹෾‫ز‬գఖΝࣖͤ͞ͳ͗Κ͖Ζ .
͞͞Ͳब‫ غ‬7 ͹ࣺ଀ܶ ଻ ͹ଲ߻න‫(ࣞͲݳ‬30)Νཤ༽ͤΖ. ݃-क࣢ (ܵ௚ା )ͳંৰ͢ިࠫͤΖ ݃-෯࣢ͺ଀ܶ ଷ ܵ௛ି Ͳ͍Ζ. γχϩόʖχ෾‫್ͯͲز‬͹ब‫ٌغ‬ಕ఼͗ਫ਼ͣΖ༹ࢢΝਦ 21 Ͷࣖͪ͢. ਦ͹఼ ‫ ͗ܣ‬ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ͹ ब ‫ ٌ غ‬ಕ ͹ ٌ ಕ ఼ Ͳ , ఼ ‫ܧ ͗ܤ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ͹ ब ‫ ٌ غ‬ಕ ͹ ٌ ಕ ఼ Ͳ ͍ Ζ . ͬ ͵ Ί Ͷ ఼ ‫ܥ‬ͺ ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀ͹ሻ͹ ब ‫ ٌ غ‬ಕ ͹ ٌ ಕ ఼ Ͳ͍Ζ.

ਦ 21: ఼ ‫ܣ‬ͺ ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ͹ ब ‫ ٌ غ‬ಕ ఼ Ͳ , ఼ ‫ ܤ‬ͺ ϔ ϫ ρ έ α ʖ χ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ͹ब‫ٌغ‬ಕ఼. ఼‫ܥ‬ͺϔϫρέαʖχ ‫ܧ‬ሺͳȀ͹ሻ͹ब‫ٌغ‬ಕ͹ٌಕ఼. ܽ ൌ ͶǤͻ.

ਦ 22: (a) ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ͹ ब ‫ ٌ غ‬ಕ ఼ . ܽ ൌ ͷ. (b) ϔ ϫ ρ έ α ʖ χ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ͹ ब ‫ ٌ غ‬ಕ ͗ ब ‫ غ‬ഔ ෾ ‫ ( ز‬๏ ޴ ൕ ఺ ‫ ) ܗ‬Ν ً ͞ ͢ ͱ ਫ਼ ͣ ͪ ໊ ٌ ಕ ఼ߙ௞ ሺͲ ൑ ݇ ൑ ͳ͵ሻ. ໊ ٌ ಕ ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻሺͳͳ ‫ͳͲͲ ڄ ͳͳ ڄ ͳͲ ڄ ͳͲͲ ڄ ͳͲ ڄ‬ሻ͹ ٌಕ఼ͺ, ਦͲͺ 0 ͖Δ 13 ͳ‫ى‬फ़͠Ηͱ͏Ζ. ब‫غ‬ഔ෾‫ز‬͹ྡֆ஍ͺ ܽ ൌ ͶǤ͹ͺͶͺʹͳʹ͸. ఼ત͹ࠪଈ͗྘Ү ܸ଴ ͳ͵Εӊଈ͗྘Ү ܸଵ ͳ͵Ζ.

t 29 tϔϫρέαʖχ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ͹ब‫ٌغ‬ಕ఼͹સରΝਦ 22(a)Ͷࣖͪ͢. ͞͹ब‫ٌغ‬ಕͺ, ࠹ॵͶൕಋब‫ غ‬෾‫ ز‬Νً͢͞ͱ γχϩ‫͖ ܗ‬Δ ବԃ‫ܗ‬Ͷร ΚΖ . ࣏Ͷब ‫غ‬ഔ෾‫ز‬Ν ً͢͞ൕ఺γχϩ‫ܗ‬Ͷ͵Ζ. ब‫غ‬ഔ෾‫Ͳز‬ਫ਼ٌ໊ͣͪಕΝ ਦ 22(b)Ͷඵ͏ͪ. ਦ(a)Ͳ, ྘Үܸ଴ ͳ ܸଵ͹ ‫ ڧ‬ֆ Ͷ ࠹ ۛ ં ͢ ͱ ͏ Ζ ್ ఼ ‫ݖ‬଴ ͳ ‫ݖ‬ଶ ሺൌ ݃‫ݖ‬଴ ሻ ͺ ྘ Ү ܸ଴ Ͷ ͍ Ζ . ٌ ಕ ఼ ‫ݖ‬଴ Γ Ε ໊ ٌ ಕ ఼ ߙ଴ (ਦͲͺ 0 ͳॽ͖Ηͱ͏Ζ)ͳߙ଻ ͗ਫ਼ͣΖ. ఼ߙ଴ ͺӊ๏޴ͶҢಊ͢྘Ү ܸଵ Ͷ೘Ζ. ఼ߙ଻ ͺࠪ๏޴Ͷ Ң ಊ ͢ ྘ Ү ܸ଴ Ͷ ཻ Ή Ζ . ٌ ಕ ఼ ‫ݖ‬ଶ Γ Ε ໊ ٌ ಕ ఼ ߙଶ ͳ ߙଽ ͗ ਫ਼ ͣ Ζ . ఼ ߙଽ ͺ ӊ ๏ ޴ Ͷ Ң ಊ ͢ ྘Ү ܸଵ Ͷ೘ Ε, ఼ߙଶ ͺ ࠪ๏ ޴ Ͷ Ңಊ ͢ ྘Ү ܸ଴ Ͷ ཻ ΉΖ . ߙଶ ൌ ݃ߙ଻ ͳ ߙଽ ൌ ݃ߙ଴ ͗੔ Ε ཱིͯ . ࣺ ଀ ܶ ଶͶΓΖٌಕ఼͹ಊ͖͘Δ , ͞͹Ңಊ͗๏޴ൕ఺‫͍Ͳܗ‬Ζ͞ͳ͗Κ͖Ζ . ͞ΗΝ੓࣯ 3.7 ͳͤΖ.
੓࣯ 3.7. ൕಋब‫غ‬෾‫͗ز‬ਫ਼ͣͪ‫ޛ‬͹ब‫غ‬ഔ෾‫ز‬ͺ๏޴ൕ఺‫͍Ͳܗ‬Ζ .
ϔϫρέαʖχ ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ͹ब‫ٌغ‬ಕ఼Νਦ 23(a)Ͷࣖͪ͢. ‫ڧ‬ֆͶ࠹ۛં͢ͱ͏
Ζ ٌ ಕ ఼ ͺ ‫ݓ‬଴ ͳ ‫ݓ‬ଶ ሺൌ ݃‫ݓ‬଴ ሻͲ ͍ Ζ . ͞ Η Δ ͺ ྘ Ү ܸଵ Ͷ ͍ Ζ . ͞ ͹ ৖ ߻ , ٌ ಕ ఼ ‫ݓ‬଴ Γ Ε ໊ ٌ ಕ఼ ߚ଴ ͳ ߚ଻ ͗ਫ਼ ͣ Ζ (ਦ 23(b)Ͷ ͕͜Ζ 0 ͳ 7). ٌಕ ఼ ‫ݓ‬ଶ ΓΕ ໊ ٌ ಕ ఼ ߚଶ ͳߚଽ ͗ ਫ਼ ͣ Ζ (ਦ23(b)Ͷ͕͜Ζ 2 ͳ 9). ఼್ߚ଻ ͳߚଽ ͺӊ๏޴ͶҢಊ͢྘Ү ܸଵ ͶཻΉΕ, ఼್ߚ଴ ͳߚଶ ͺࠪ๏޴ͶҢ ಊ ͢ ྘ Ү ܸ଴ Ͷ ೘ Ζ. ߙଶ ൌ ݃ߙ଻ ͳ ߙଽ ൌ ݃ߙ଴ ͗ ੔ Ε ཱི ͯ. ࣺ ଀ ܶ ଶ Ͷ Γ Ζ ٌ ಕ ఼ ͹ ಊ ͘ ͺ ๏ ޴ ฯଚ‫͍Ͳܗ‬Ζ.

ਦ 23: (a) ϔ ϫ ρ έ α ʖ χ ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ͹ ब ‫ ٌ غ‬ಕ ఼ . ܽ ൌ ͷ. (b) ϔ ϫ ρ έ α ʖ χ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ ͹ ब ‫ ٌ غ‬ಕ ͗ ब ‫ غ‬ഔ ෾ ‫ ( ز‬๏ ޴ ฯ ଚ ‫ ) ܗ‬Ν ً ͞ ͢ ͱ ਫ਼ ͣ ͪ ໊ ٌ ಕ ఼ߚ௞ ሺͲ ൑ ݇ ൑ ͳ͵ሻ. ਦͲͺ 0 ͖Δ 13 ͳ‫ى‬फ़͠Ηͱ͏Ζ. ϔϫρέαʖχ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻሺͲͳ ‫ͳͲͲ ڄ ͳͳ ڄ ͳͳ ڄ ͳͲͲ ڄ ͳͲ ڄ‬ሻ͹ ब ‫ غ‬ഔ ෾ ‫ ز‬͹ ྡ ֆ ஍ ͺ ܽ ൌ ͶǤ͹͵ͷͺͶͶͶ͵ͷ. ఼ ત ͹ࠪଈ͗྘Ү ܸ଴ ͳ͵Εӊଈ͗྘Ү ܸଵ ͳ͵Ζ.
ϔϫρέαʖχ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ͹ब‫ٌغ‬ಕͳ, ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ͹ब‫ٌغ‬ಕ͹҈ఈ੓‫ ਼ ܐ‬Ν ਦ 24 Ͷ ࣖ ͢ ͪ . ͞ Η Δ Γ Ε ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ͹ ब ‫ ٌ غ‬ಕ ͗ ν ΢ ϕ
III ͹ ෾ ‫ ز‬Ν ࣖ ͤ ͞ ͳ ͳ , ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ͹ ब ‫ ٌ غ‬ಕ ͗ ν ΢ ϕ I ͹ ෾ ‫ز‬Νࣖͤ͞ͳ֮͗೟Ͳ͘Ζ .

t 30 tਦ 24: (a) ϔϫρέαʖχ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ͹ब‫ٌغ‬ಕ͹҈ఈ੓‫਼ܐ‬. (b) ϔϫρέαʖχ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ͹ब‫ٌغ‬ಕ͹҈ఈ੓‫ ਼ܐ‬.
ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ͹ ब ‫ ٌ غ‬ಕ ͹ ٌ ಕ ఼ ‫ݖ‬ଵ Ͷ ͕ ͜ Ζ ݃-෯ ࣢ (ܶ ଷ ܵ௛ି ) ͹ ܑ ͘ߦሺ‫ݖ‬ଵ ሻΝ௒΄Ζ. ݃Վͺਦ 25 Ͷࣖͪ͢. ਦ(a)Ͷ͕͏ͱܑ͘ͺਜ਼Ͳ͍Ζ . Ϗϧϟʖν͹ଁՅͶͯ Η ͱ ܑ ͘ ͺ ଁ Յ ͢ , ਦ (b)Ͷ ͕ ͏ ͱ ܑ ͘ ߦሺ‫ݖ‬ଵ ሻ͗ ൅ ࢆ ͤ Ζ . ͞ ͹ য় ‫ ͗ ڱ‬ब ‫ غ‬ഔ ෾ ‫ ز‬͹ ྡ ֆ ஍Ͳਫ਼ ͣΖ . ਦ(c)͹Γ ͑Ͷ ܑ͘ ߦሺ‫ݖ‬ଵ ሻ͗ෝ Ͷ ͵ͮ ͪ‫ ޛ‬, ਦ (d)Ͳ ࣖ͢ ͪΓ ͑ Ͷܑ ͘ ߦሺ‫ݖ‬ଵ ሻͺਜ਼ ͳ͵
Ζ . ݃-෯ ࣢ ͹ ܑ ͘ ͺ ৷ ೘ ͢ ͱ ͘ ͪ ෈ ҈ ఈ ଡ ༹ ର ͹ ܶȞ௨ ͹ ܑ ͘ Ͷ શ ۛ ͤ Ζ . ν ΢ ϕ III ͹ ෾ ‫ز‬գఖ͹‫ݬ‬Ҿͺ ܶȞ௨ ͹৷೘Ͳ͍Ζ.

ਦ 25: ఼‫ݖ‬ଵ Ͷ͕͜Ζ݃-෯࣢͹ܑ͘. (a) ܽ ൌ ͶǤ͹ͷ. γχϩόʖχ෾‫ܽ( ز‬௖ୱ௡ ൌ ͶǤ͹͵ͷͺ͵ͻͷͶ) ͹௜‫ޛ‬. (b) ܽ ൌ ͶǤ͹ͺͶͺʹͳʹ͸. ब‫غ‬ഔ෾‫ز‬͹ྡֆ஍. (c) ܽ ൌ ͶǤͻ. (d) ܽ ൌ ͷǤʹ.

t 31 tਦ 26: ‫ߺى‬Կ‫و‬ଉ. ਦ͹࠹Ծ෨Ͷඵ͏ͪ൬ߺͺٌಕ఼ ߙ௞ ሺͲ ൑ ݇ ൑ ͳ͵ሻ͹శ͓࣊Ͳ͍Ζϔϫ ρέ α ʖχ ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ͹ब ‫ ٌ غ‬ಕ͗ ब‫ غ‬ഔ ෾‫ ز‬Νً ͞ ͢ ͱਫ਼ ͣͪ ໊ ٌಕ ͹αʖχΝ݀ఈ͢Γ͑. ਦ 26(a)͹Γ͑Ͷ, ਜ਼‫و‬න‫ͳͲͲͳͲͳͲݳ‬Ν࿊କ͢ͱॽ͚. ٌಕ఼ߙ଴ ͺ྘Ү ܸ଴Ͷ Ң ಊ ͤ Ζ . ٌ ಕ ఼ ߙଽ ΍ ಋ ༹ ͹ Ң ಊ Ν ͤ Ζ . ਦ ͹ ࠹ Ծ ෨ Ͷ ඵ ͏ ͪ ൬ ߺ ͺ ٌ ಕ ఼ ߙ௞ ሺͲ ൑ ݇ ൑ͳ͵ሻ͹శ͓࣊Ͳ͍Ζ. ୊ 0 ൬ͳ୊ 9 ൬͹‫ߺى‬Ν 1 ͶรߍͤΖͳ, ٌ໊ಕ͹αʖχ͗ಚΔΗΖ .
͞ Η Ν ϔ ϫ ρ έ α ʖ χ Ͳ ‫ ى‬फ़ ͤ Ζ ͳ , ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ ‫ܧ ڄ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ͗ ಚ Δ Η Ζ .
ൕ ಋ ब ‫ غ‬෾ ‫ Ͳ ز‬ਫ਼ ͣ ͪ ್ ͯ ͹ ໊ ٌ ಕ ͹ ϔ ϫ ρ έ α ʖ χ ͺ , ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻͳ ‫ܧ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻͲ͍Ζ͞ͳ͗ࣙષͶ಍͖ΗΖ .
࣏ Ͷ , ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ͹ ब ‫ ٌ غ‬ಕ ͗ ब ‫ غ‬ഔ ෾ ‫ ز‬Ν ً ͞ ͢ ͱ ਫ਼ ͣ ͪ ໊ ٌಕ͹αʖχ͹݀ఈ͹࢕๏Ν , ਦ 26(b)Ͷࣖͪ͢. ͞͹৖߻Ͷ΍ਜ਼‫و‬න‫ͳͲͲͳͳͳͳ ݳ‬Ν್ͯ࿊݃͢
ͪ. ࣏Ͷ, ఼್ߚ଴ ͳߚଶ ͗྘Үܸ଴ ͶҢಊͤΖ͞ͳ͖Δ, ͞ΗΔ͹‫ߺى‬Ν 0 ͶรߍͤΖ. ٌ໊ಕ͹α ʖ χ ͺ ͲͳͲͳͲͲͳ ‫ ͳͳͲͲͳͳͳͳ ڄ‬ಚ Δ Η, ͞ Η ͹ ϔ ϫ ρ έ α ʖ χ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ ‫ܨ ڄ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ͗ಚΔ ΗΖ . ઴ൔͳ ‫ޛ‬ൔ ͺγχ ϩό ʖχ ෾‫Ͳ ز‬ਫ਼ ್ͣͪ ͯ͹ ฾਎ ͹ϔ ϫρ έαʖ χͲ͍Ζ.
࠹‫ޛ‬Ͷ, ൕಋब‫ غ‬෾‫ Ͳز‬ਫ਼್ͣͪͯ ͹ٌ໊ ಕ͹ϔ ϫρέαʖ χΝ΍ ͳͶ , ͞ΗΔ͹ٌ ಕΝ‫ ߺ ى‬ฑ ໚ Ͳ ඵ ͏ ͪ (ਦ 27). ٌ ಕ ఼ ‫ݑ‬ො௞ ሺͲ ൑ ݇ ൑ ͸ሻ͗ ϔ ϫ ρ έ α ʖ χ ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ͹ ٌಕ఼Ͳ, ٌಕ఼‫ݒ‬ො௞ ሺͲ ൑ ݇ ൑ ͸ሻ͗ϔϫρέαʖχ ‫ܧ‬ሺͳȀʹሻ‫ܨ‬ሺͳȀʹሻ‫ܧ‬ሺͳȀ͵ሻ͹ٌಕ఼Ͳ͍Ζ.

ਦ 27: ‫ߺى‬ฑ໚Ͳ͹ٌಕ . ٌಕ఼‫ݖ‬Ƹ௞ ሺͲ ൑ ݇ ൑ ͸ሻ͗ൕಋब‫غ‬෾‫ز‬Νً͢͞ͱਫ਼್ͣͪͯ͹ٌ໊ಕ఼ ‫ݑ‬ො௞ ͳ‫ݒ‬ො௞ .
t 32 t3.3.2. ౛  ᅇ㌿ศᒱ࡛⏕ࡌࡓ࿘ᮇ㌶㐨
್ͯ໪͹ྭͳ͢ ͱ , ϔϫρέαʖχ ‫ܧ‬ሺͳȀ͵ሻͲ‫ى‬फ़͠ΗΖ ब‫ٌغ‬ಕ͗յ఺ ෾‫ز‬Νً ͢͞ͱਫ਼ͣ ͪ ब ‫ ٌ غ‬ಕ ͹ ब ‫ غ‬ഔ ෾ ‫ ز‬Ν ௒ ΄ Ζ . ͞ ͹ ঘ ઇ Ͳ ͺ ϔ ϫ ρ έ ͹ յ ఺ ਼ ͳȀ͵Ν ঴ ͚ . ࣺ ଀ ܶ ଷ ΝୱҒͳͪ͢ब‫ غ‬3 ͹ब‫ٌغ‬ಕ͹ϔϫρέαʖχͺ ‫͍Ͳܧܦܧͳܨܵܧ‬Ζ. ਼஍‫ࢋܯ‬ΓΕ, յ఺෾‫ز‬͗ ਫ਼ ͣ ͪ ௜ ‫ ޛ‬Ͷ ͕ ͏ ͱ ϔ ϫ ρ έ α ʖ χ ‫ܨܵܧ‬͹ ब ‫ ٌ غ‬ಕ ͺ ବ ԃ ‫ ͍ Ͳ ܗ‬Ε, ϔ ϫ ρ έ α ʖ χ ‫ܧܦܧ‬͹ब‫ٌغ‬ಕ ͺγχϩ ‫ ͍ Ͳܗ‬Ζ͞ͳ͗ ಚΔΗͪ . ͞ΗΔ͹αʖ χ͹۰‫ د‬੓ͺ ‫͍Ͳد‬Ζ . ͨ͹݃Վ, ϔ ϫρ έα ʖ χ ‫ܨܵܧ‬͹ब ‫ ٌغ‬ಕͺ ν ΢ϕ I ͹ब ‫غ‬ഔ ෾‫ ز‬Ν ً͞ ͢ , ϔϫ ρέ αʖ χ ‫ܧܦܧ‬͹ब‫ٌغ‬ಕͺν΢ϕ III ͹ब‫غ‬ഔ෾‫ز‬Νًͤ͞͞ͳ͗Κ͖Ζ. ҐԾͲͺϔϫρέαʖχ‫ܧܦܧ‬͹ब‫ٌغ‬ಕ͹෾‫ز‬գఖΝ௒΄Ζ .
฾ٌಕ‫͗ܧܦܧ‬ब‫غ‬ഔ෾‫ز‬Νً͢͞ͱਫ਼ٌ໊ͣͪಕΝਦ 28 Ͷඵ͏ͪ. ਦ 28 Ͳͺ, ಝͶଲসતܵ௛ି ͹ब วͶ ͍Ζٌ ಕ఼ ͹ ΊΝඵ ͏ͪ . ٌಕ఼ ‫ݖ‬ଵ ͳ‫ ଻ݖ‬ͺ, ྘Ү‫ܧ‬Ͷ‫؜‬Ή ΗΖ ͹Ͳϔ ϫρέ ͺ ‫ܧ‬Ͳ ͍ Ζ . ϔ ϫ ρ έ ͹ յ ఺ ਼ ͳȀ͵ͺ ঴ ͏ ͪ . ٌ ಕ ఼ ‫ݖ‬ସ ͺ ྘ Ү ‫ܦ‬Ͷ ͍ Ζ ͹ Ͳ , ϔ ϫ ρ έ α ʖ χ ͗ ‫ܦ‬Ͳ͍Ζ. ٌಕ఼‫ݖ‬ସ ͹ٌಕ఼ ΓΕ್ͯ͹ٌ໊ಕ఼ ‫ݑ‬ସ ͳ‫ݑ‬ଵଷ ͗ ਫ਼ͣΖ . ٌಕ఼‫ݑ‬ସ ͺଲসત ܵ௛ି ͹Ծ๏Ͷ Ң ಊ ͢ ྘ Ү ܵͶ ೘ Ζ . ٌ ಕ ఼ ‫ݑ‬ଵଷ ͺ ଲ স ત ܵ௛ି ͹ ৏ ๏ Ͷ Ң ಊ ͢ ྘ Ү ‫ܨ‬Ͷ ೘ Ζ . ΍ ͢ ͞ Η Δ ͗ ྘Ү‫ܦ‬ͶཻΉΖͳٌ໊ಕ͹ϔϫρέαʖχͺ ‫͵ͳܧܦܧ ڄ ܧܦܧ‬Ζ. ͞Ηͺ฾ϔϫρέαʖχ͹‫܃‬Εศ͢Ͳ͍Ζ͖Δ໅८Ͳ͍Ζ . Ґ৏ΓΕٌ໊ಕ͹ϔϫρέαʖχ͗݀ΉΖ .
‫ܧܨܧ ڄ ܧܵܧ‬Ǥ

(33)

ਦ 19(b)ͳਦ 26(a)ͳಋͣΓ͑Ͷ༽қͪ͢઴ൔ͹‫ͳߺى‬ଲԢͤΖ‫ޛ‬ൔ͹‫ߺى‬͹ྈंͲ 0 ͳ 1
͹ ೘ Η ଺ ͓ Ν ߨ ͵ ͑ ͞ ͳ Ͳ ໊ ٌ ಕ ͹ α ʖ χ ͗ ݀ Ή Ζ . ਦ 28(b) Ν ‫ ͱ ݡ‬Ά ͢ ͏ . ‫ݑ ࣞ ܐ ؖ‬ସ ൌ݃‫ݑ‬ଵଶ , ‫ݑ‬ଵଷ ൌ ݃‫ݑ‬ଷ ͳ ‫ݖ‬ସ ൌ ݃‫ݖ‬ଷ ͗ ੔ Ε ཱི ͯ . ٌ ಕ ఼ ‫ݑ‬ଷ ͖ Δ ‫ݑ‬ସ ΃ Ӏ ͏ ͪ ໾ һ ͗ , ٌ ಕ ఼ ‫ݑ‬ଵଶ ͖ Δ ‫ݑ‬ଵଷ ΃Ӏ͏ͪ໾һͳިࠫͤΖ. Γͮͱ, ͞͹৖߻͹ब‫غ‬ഔ෾‫ز‬΍๏޴ൕ఺‫͍Ͳܗ‬Ζ .

t 33 tਦ 28: (a) ϔϫρέαʖχ‫ܧܦܧ‬͹ٌಕ఼ʁ ‫ݖ‬ଵ ՜ ‫ݖ‬ସ ՜ ‫ ଻ݖ‬՜ ‫ݖ‬ଵ . ϔϫρέαʖχ ‫ܧܨܧ ڄ ܧܵܧ‬͹ٌಕ఼ʁ ‫ݑ‬ଵ ՜ ‫ݑ‬ସ ՜ ‫ ଻ݑ‬՜ ‫ݑ‬ଵ଴ ՜ ‫ݑ‬ଵଷ ՜ ‫ݑ‬ଵ଺ ՜ ‫ݑ‬ଵ . (b) ब‫غ‬ഔ෾‫ز‬ͺ๏޴ൕ఺‫͍Ͳܗ‬Ζ . ܽ ൌ ͶǤͳͷ.
ྭ 1 ͳྭ 2 Ͷ͕͏ͱٌ໊ಕ͹‫ߺى‬Կ͗ҡ͵ͮͱ͏Ζ . ྭ 1 ͹৖߻, ࠹ۛંٌಕ఼‫ߺى‬஖͘‫͓׷‬๑Νཤ༽͢ͱ‫ߺى‬Ν೘Η଺͓ͪ . ྭ 2 Ͷͯ͏ͱ΍ಋ༹͹घॳΝཤ༽͢ͱΊΓ͑ . ྭ 2 Ͳ
ͺ, ࠹ۛંٌಕ ఼ͺ ‫ݖ‬଴ ͳ‫͍ Ͳ ଻ݖ‬Ζ. ब‫غ‬ഔ෾ ‫͗ز‬๏ ޴ ൕ఺‫ٌ ໊ͱ͢ ͳܗ‬ಕ͹ α ʖχΝ݀ ΌΖ .
઎ͥ฾ٌಕΝ್ͯ࿊݃͢, ୊ 0 ൬໪ͳ୊ 16ሺൌ ͹ ൅ ͻሻ൬໪͹ 0 Ν 1 ͶรߍͤΖ.
ͲͲͳͳͳͳͲͲͳ ‫ ͳͲͲͳͳͳͳͲͲ ڄ‬՜ ͳͲͳͳͳͳͲͲͳ ‫ͳͳͲͳͳͳͳͲͲ ڄ‬Ǥ(34)͞ Η Ν ϔ ϫ ρ έ α ʖ χ Ͳ ‫ ى‬फ़ ͤ Ζ ͳ ܵ‫ ͗ܨܦܧ ڄ ܧܦ‬ಚ Δ Η Ζ . ͢ ͖ ͢ , ͞ ͹ α ʖ χ ͺ ‫ޢ ࢯ ے‬‫(ܵܨ‬຦ඎ͹‫ͳܨ‬઎ಆ͹ ܵ)Ν‫؜‬΋. ྭ 2 ͹৖߻, ٌ໊ಕ͹αʖχΝ݀ΌΖͪΌͶ࠹ۛંٌಕ఼‫ى‬ߺ஖͘‫͓׷‬๑Ν͓࢘͵͏ .
฾ϔϫρέαʖχΝ ‫ܦܧ‬ሶ ‫͚ॽͳܧ‬. ͪͫ͢, ‫ܦ‬ሶ ͺͳͳሶͳΝන͢ͱ͏Ζ. ‫ͳߺى‬ሶͲ‫ى‬फ़͠ΗΖٌಕ఼‫ݖ‬ସ ͗ଲসત ܵ௛ି ৏Ͷ͍ Ζ͞ͳ Ν қັ͢ͱ ͏Ζ . ٌಕ఼ ‫ݖ‬ସ ͖ Δ෾‫ ٌ໊ͪ ͢ز‬ಕ఼ͺ ଲ সત ܵ௛ି ΓΕ཯Η྘Ү ܵͳ‫΃ܨ‬ҢಊͤΖ. ͞ΗΓΕԾ‫ى‬͹Γ͑Ͷ஖͘‫͓׷‬Νߨ͑͞ͳͲٌ໊ಕ͹ϔϫρέα
ʖχ͗݀ΉΖ.
‫ܦܧ‬ሶ ‫ܦܧ ڄ ܧ‬ሶ ‫ ܧ‬՜ ‫ܧܨܧ ڄ ܧܵܧ‬Ǥ

(35)͞͹஖͘‫͓׷‬Ν ‫ܨܵ ڄ ܦ‬-஖͘‫͓׷‬๑ͳ໌෉͜Ζ.

3.3.3. ࡲ࡜ࡵ฾ٌಕ͗ν΢ϕ III ͹෾‫ز‬գఖΝًͤ͞ড়݇Νড়݇ 3.8 ͳͤΖ. ࠹‫ޛ‬Ͷν΢ϕ III ͹෾‫ز‬գఖͲਫ਼ٌ໊ͣͪಕ͹‫ߺى‬Կ‫و‬ଉΝ‫و‬ଉ 3.9 ͳ͢ͱΉͳΌͱ͕͚.
ড়݇ 3.8. ฾ब‫ٌغ‬ಕ͗ਫ਼ͣͪ௜‫ޛ‬Ͷ͕͏ͱγχϩ‫د͖ͯͲܗ‬αʖχͲ͍Ζ͞ͳ .
‫و‬ଉ 3.9. ฾ٌ ಕ ͹ϔ ϫ ρ έα ʖ χ ͗ , ϔϫ ρ έ ‫ܦ‬ሶ ሺ‫݌‬Ȁ‫ݍ‬ሻΝ ‫؜‬Ή ͵ ͏ ৖߻ ͺ [I] Ν ཤ༽ ͢ , ϔϫρέ‫ܦ‬ሶ ሺ‫݌‬Ȁ‫ݍ‬ሻΝ‫؜‬΋৖߻ͺ [II] Νཤ༽ͤΖ.
[I] ࠹ۛંٌಕ఼‫ߺى‬஖͘‫͓׷‬๑Νཤ༽͢ͱ‫ߺى‬͹೘Η଺͓Νߨ͑ .
[II] ‫ܨܵ ڄ ܦ‬-஖͘‫͓׷‬๑Νཤ༽͢ͱϔϫρέαʖχ͹೘Η଺͓Νߨ͑ .

3.4. ν΢ϕ ,9 ൕಋब‫غ‬෾‫͖ز‬Δब‫غ‬ഔ෾‫ز‬ɼൕब‫غ‬ഔ෾‫ز‬ɼಋब‫غ‬෾‫΃ز‬ν΢ϕ IV ͹ྭͳ͢ͱγχϩόʖχ෾‫Ͳز‬ਫ਼ͣͪब‫ غ‬14 ͹ब‫ٌغ‬ಕΝѽ͑. ͨ͹ͪΌࣺ଀ܶ ଵସ ͹ଲ߻න‫ݳ‬Ν಍೘ͤΖ.
ܶ ଵସ ൌ ܶ ଵଷ ݃ ‫݃ ؠ ݄ ל‬ො ‫݄ ל‬Ǥ

(36)

଻

ଲ߻݃ොͶ‫෈ͱؖ͢‬ร͵ॄ߻ͺܶ ܵ௛ Ͳ͍Ζ.
ܶ ଵଷ ݃ሺܶ ଻ ܵ௛ ሻ ൌ ܶ ଺ ݃ܵ௛ ൌ ܶ ଺ ݄݃ܵ௛ ൌ ܶ ଻ ܵ௛ Ǥ(37)ଲসતܵ௛ି , ଀ܶ ଻ ܵ௛ି ͳ‫ܶނ‬Ȟ௨ ͹Ғ஖‫ܐؖ‬Νਦ 29 Ͷࣖͪ͢. ਦ(a)ͺγχϩόʖχ෾‫ز‬઴͹ഓ஖
Ͳ͍Ζ. γχϩόʖχ෾‫ز‬͹ྡֆ஍Ͳ, ଀ܶ ଻ ܵ௛ି ͗ଲসતܵ௛ି ͳંͤΖ(ਦ(b)). γχϩόʖχ෾‫ ز‬͹ ‫ ޛ‬, ٌ ಕ ఼ ‫ݖ‬ସ ͳ ‫ݓ‬ସ ͗ ਫ਼ ͣ Ζ (ਦ (c)). ଀ ܶ ଻ ܵ௛ି ͹ ઎ ಆ ෨ ෾ ͺ ࠪ Ͷ ‫ ͗ ۄ‬Ε , ෈ ҈ ఈ ଡ ༹ ର ͹‫ܶ ނ‬Ȟ௨ Ͷ Ԍ ͮ ͱ ͹ ; ͱ ͏ ͚ . ଀ ܶ ଻ ܵ௛ି ͹ ઎ ಆ ෨ ෾ ͺ ࠸ ; ଲ স ત ܵ௛ି Ͷ ં ͢ ި ఼ ߙସ ͳ ߚସ ͗ ਫ਼ ͣ Ζ(ਦ(d)).
t 34 tϔϫρέαʖχ ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ٌಕ఼Νਦ 30(a)Ͷࣖͪ͢. ྘Үܸ଴ ͳܸଵ ͹‫ڧ‬ֆͶۛ͏ٌಕ఼ ‫ݖ‬ଷ ͳ‫ݖ‬ସ ͹‫ߺى‬ͺ 1 Ͳ͍Ζ. Ήͪ, ٌಕ఼‫ݖ‬଴ ͳ‫ ଻ݖ‬͹‫ߺى‬ͺ 0 Ͳ͍Ζ͞ͳ͗Κ͖Ζ .
͞ Η Δ Γ Ε α ʖ χ ͲͲͳ ‫ ͳͲͲͲͲ ڄ ͳͲͲ ڄ ͳͳͳ ڄ‬ൌ ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͷሻ͗ ֮ ೟ Ͳ ͘ Ζ . ϔ ϫ ρ έαʖχ‫ܧ‬ሺͳȀͷሻܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ͹ٌಕ఼Νਦ 30(b)Ͷඵ͏ͪ. ٌಕ఼‫ݓ‬ଷ ͳ‫ݓ‬ସ ͹‫ߺى‬ͺ 1 Ͳ,
ٌ ಕ ఼ ‫ݓ ͳ ଺ݓ‬ଵ ͹ ‫ ߺ ى‬ͺ 0 Ͳ ͍ Ζ . Γ ͮ ͱ α ʖ χ ͳͲͳ ‫ ͳͲͲͲͲ ڄ ͳͳͲ ڄ ͳͳͳ ڄ‬ൌ ܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͷሻ֮͗೟Ͳ͘Ζ.

ਦ 29: ଲসતܵ௛ି , ଀ܶ ଻ ܵ௛ି ͳ෈҈ఈଡ༹ର ܶȞ௨ ͹ഓ஖. (a) γχϩόʖχ෾‫ز‬͹઴. ܽ ൌ ͵ǤͲʹ. (b)γχϩόʖχ෾‫ز‬͹ྡֆ஍ . ܽ ൌ ͵ǤͲ͵ͷͳͻ͸͵ͲͻͲ. (c) γχϩόʖχ෾‫ز‬͹‫ޛ‬. ܽ ൌ ͵ǤͲͷ. (d) ܽ ൌ͵Ǥͳ͹.

ਦ 30: ଀ܶ ଻ ܵ௛ି , ଀ܶ ି଻ ܵ௛ି ͳ෈҈ఈଡ ༹ର ܶȞ௨ ͹ഓ஖. ܽ ൌ ͵Ǥʹ. (a) ٌಕ ఼‫ݖ‬௞ ሺͲ ൑ ݇ ൑ ͳ͵ሻͺϔϫρέαʖχ ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͷሻሺͲͲͳ ‫ͳͲͲͲͲ ڄ ͳͲͲ ڄ ͳͳͳ ڄ‬ሻ͹ٌಕ఼. (b) ٌಕ఼‫ݓ‬௞ ሺͲ ൑ ݇ ൑ͳ͵ሻͺϔϫρέαʖχ ܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀͷሻሺͳͲͳ ‫ͳͲͲͲͲ ڄ ͳͳͲ ڄ ͳͳͳ ڄ‬ሻ͹ٌಕ఼.
t 35 tਦ 31: ϔϫρ έαʖ χ ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ब‫ ٌغ‬ಕ͹҈ఈ ੓‫ ਼ܐ‬. (a) સରਦ. (b)֨୉ਦ.
ϔϫρέαʖχ ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ब‫ٌغ‬ಕ͹҈ఈ੓‫਼ܐ‬Νਦ 31(a)Ͷࣖͪ͢. ਦ
31(b)Ͷ ͺ , γ χ ϩ ό ʖ χ ෾ ‫ ͗ ز‬ਫ਼ ͣ ͪ ௜ ‫ ޛ‬͹ ҈ ఈ ੓ ‫ ਼ ܐ‬͹ Ϗ ϧ ϟ ʖ ν ғ ଚ ੓ Ν ࣖ ͢ ͪ . ͞Η Δ ͹ ਦ Γ Ε , ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ ब ‫ ٌ غ‬ಕ ͗ ν ΢ ϕ IV ͹ ෾ ‫ ز‬գఖΝًͤ͞͞ͳ֮͗೟Ͳ͘Ζ .
ϔϫρέαʖχ ‫ܧ‬ሺͳȀͷሻܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ͹ब‫ٌغ‬ಕ͹҈ఈ੓‫਼ܐ‬Νਦ 32(a)Ͷࣖͪ͢. ਦ32(b)Ͷ ͺ , γ χ ϩ ό ʖ χ ෾ ‫ ͗ ز‬ਫ਼ ͣ ͪ ௜ ‫ ޛ‬͹ ҈ ఈ ੓ ‫ ਼ ܐ‬͹ ৾ Ζ ෥ ͏ Ν ࣖ ͢ ͪ . ࠫ ܽ െ ܽ௖ୱ௡ ͗ඉ ৙ Ͷ ঘ ͠ ͏ ͹ Ͳ , ਦ(b)͹ ԥ ࣢ ͺ ଵ଴ ሺܽ െ ܽ௖ୱ௡ ሻͳ ͢ ͪ . ਦ (a)ͳ ਦ (b)Ν ‫ ݡ‬Ζ ͳ , ϔ ϫ ρ έ αʖχ‫ܧ‬ሺͳȀͷሻܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ͹ब‫ٌغ‬ಕͺν΢ϕ II ͹෾‫ز‬գఖΝًͤ͞͞ͳ͗Κ͖Ζ .

ਦ 32: ϔ ϫρ έ αʖ χ ‫ܧ‬ሺͳȀͷሻܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ͹ब ‫ ٌ غ‬ಕ͹ ҈ ఈ੓ ‫ ਼ ܐ‬. (a) સର ਦ. (b)֨୉ਦ.
ϔ ϫ ρ έ α ʖ χ ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ ब ‫ ٌ غ‬ಕ ͹ ٌ ಕ ఼ ‫ݖ‬ସ Ͷ ͕ ͜ Ζ ଀ ܶ ଻ ܵ௛ି ͹ ܑ ͘ߦሺ‫ݖ‬ସ ሻΝ௒΄ͪ݃ՎΝਦ 33 Ͷࣖͪ͢. γχϩόʖχ෾‫ز‬௜‫ޛ‬͹ܑ͘ͺਜ਼Ͳ͍Ζ . ͨ͹‫ޛ‬, ܑ͘ͺ ൅ ࢆ ͢ ෝ Ͷ ร Κ Ζ (ਦ (a)). ࠸ ; ൅ ࢆ ͢ ਜ਼ Ͷ ͵ Ζ (ਦ (b)). ܑ ͘ ͺ ‫ ݰ‬ঙ ͢ ሺ൅ʹሻͶ ͵ Ζ ͳ ൕब ‫ غ‬ഔ ෾ ‫ ͗ ز‬ਫ਼ ͣ Ζ . ࠹ श ద Ͷ ܑ ͘ ͺ ෈ ҈ ఈ ଡ ༹ ର ͹ ‫ܶ ނ‬Ȟ௨ ͹ ܑ ͘ Ͷ શ ۛ ͤ Ζ . ܑ ͘ ߦሺ‫ݖ‬ସ ሻ͹ร Կ Γ Ε, ν ΢ ϕ IV ͹ ෾ ‫ ز‬գ ఖ ͹ ‫ ݬ‬Ҿ ͺ ෈ ҈ ఈ ଡ ༹ ର ͹ ‫ܶ ނ‬Ȟ௨ ͹ ৷ ೘ Ͳ ͍ Ζ ͞ ͳ ͗ ཀྵ մ Ͳ
t 36 t͘ΖͲ͍Θ͑.

ਦ 33: ଀ܶ ଻ ܵ௛ି ͹ٌಕ఼‫ݖ‬ସ Ͷ͕͜Ζܑ͘ ߦሺ‫ݖ‬ସ ሻ͹Ϗϧϟʖνғଚ੓.
฾ ٌ ಕ ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͖ Δ ਫ਼ ͣ ͪ ໊ ٌ ಕ ͹ α ʖ χ Ν ݀ ఈ ͢ Γ ͑ . ࠹ ॵ ͹ ब ‫ غ‬ഔ෾‫ز‬ͺൕಋ ब‫غ‬෾‫ ز‬͹‫ ޛ‬Ͷਫ਼ͣΖ͹ Ͳ , ब‫غ‬ഔ෾ ‫ز‬ͺ๏޴ൕ ఺‫ ͍Ͳܗ‬Ζ . Ήͪൕब‫ غ‬ഔ෾‫ز‬͹‫ޛ‬Ͷಋब‫غ‬෾‫͗ز‬ਫ਼ͣΖ͖Δ , ൕब‫غ‬ഔ෾‫ز‬΍๏޴ൕ఺‫͍Ͳܗ‬Ζ .
฾ ٌ ಕ ͹ ϔ ϫ ρ έ α ʖ χ ͺ ‫ܧ‬ሶ ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሶ ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ ൌ ͲͲሶͲͲͳ ‫ͳ ڄ ͳͲͲ ڄ‬ሶͳͳ ‫ ͜ ॽ ͳͳͲͲ ڄ‬Ζ . ϔϫρέ‫ܦ‬ሶ ሺͳȀ͵ሻΝ‫؜‬΋ͪΌ, ν΢ϕ III Ͳ಍೘ͪ͢‫و‬ଉ 3.9[II]Ͷॊ͏ϔϫρέΝ೘Η଺͓Ζ . ਦ
34(a)Ͷ͕͏ͱ୊ 4 ൬໪ͳ୊ 17 ൬໪͗݃-ଲΝ͵ͤ. ͞ΗΔ͹‫ߺى‬Ν 1 ͖Δ 0 ΃รߍͤΖ. รߍ ͢ ͪ ݃ Վ Ν ϔ ϫ ρ έ Ͳ න ‫ ͢ ݳ‬, ࣏ Ͷ ࠹ ঘ ஍ න ‫ ݳ‬Ͷ ͤ Ζ ͳ ࣞ (38)͗ ಚ Δ Η Ζ . ͞ ͹ න ‫Ͳ ݳ‬೘Η଺͓ͪϔϫρέͶͺԾતΝӀ͏ͪ .
‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻܵሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ ‫ܧ ڄ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ

(38)

ਦ 34: (a) ฾ٌಕ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͗ब‫غ‬ഔ෾‫ز‬Νً͢͞ͱਫ਼ٌ໊ͣͪಕ͹‫ߺى‬Կ .
(b) ฾ٌಕ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͗ൕब‫غ‬ഔ෾‫ز‬Νً͢͞ͱਫ਼ٌ໊ͣͪಕ͹‫ߺى‬Կ .
࣏Ͷൕ ब‫غ‬ഔ ෾‫Ͳ ز‬ਫ਼ ͣ ٌ໊ͪ ಕ͹α ʖχ ͺ , ν ΢ϕ II Ͷ͕͜ Ζൕब ‫ غ‬ഔ෾‫Ͳ ز‬ਫ਼ͣ ٌ໊ͪ ಕ͹ α ʖχ ͹ ݀ఈ Ͳ ࢘༽ ͪ͢ ๏ ๑ (ਦ 19(b)) Νཤ ༽ͤ Ζ . ‫ߺى‬͹ ೘ Η଺ ͓घ ॳ ͺ , ਦ34(b)Ͷࣖͪ͢. ਜ਼‫و‬න‫ͳͲͲͳͳͳͳͲͲݳ‬Ν್ͯ࿊݃͢ͱ, ࠹ॵ͖Δٌಕ͹൬ߺΝͯͪ͜ . ୊ 0 ൬
໪ͳ୊ 21 ൬໪͗݃-ଲΝ͵ͤ. ͞ΗΔ͹‫ߺى‬Ν 0 ͖Δ 1 ΃รߍͪ͢න‫ݳ‬ΝϔϫρέͲ‫ى‬फ़͢࠹t 37 tঘ஍න ‫ݳ‬Ͷͤ Ζͳ , ൕब ‫غ‬ഔ෾ ‫Ͳز‬ਫ਼ ͣͪ ٌ໊ಕ ͹ϔϫ ρέα ʖχ ͗ಚΔ ΗΖ. ࣞ(39)Ͷ͕͏ͱ೘Η଺͓ͪϔϫρέͶͺԾતΝӀ͏ͪ .
‫ܧ‬ሺͳȀͷሻܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ ‫ܧ ڄ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻǤ(39)

৏ ‫( ࣞ ͪ ͢ ࣖ Ͳ ى‬38)ͳ ࣞ (39)Ґ ֐ ͹ α ʖ χ Ͳ ‫ ى‬फ़ ͠ Η Ζ ໊ ٌ ಕ ͗ ଚ ࡑ ͢ ͵ ͏ ͞ ͳ Ν ࣖ ͤ .
͞ ͞ Ͳ, ‫ܧ‬ሺͳȀͷሻ‫ܧ ڄ ܧܦܧ‬ሺͳȀͷሻ‫ܧܦܧ‬Ν༽ қ ͤ Ζ (‫ ܧ‬ൌ ‫ܧ‬ሺͳȀ͵ሻǡ ‫ ܦ‬ൌ ‫ܦ‬ሺͳȀ͵ሻ). ઴ ൔ ෨͹ ‫ܧܦܧ‬Ν ܵ‫ͳܧܦ‬ॽ͘‫͓׷‬Ζͳ, ઴ൔ෨͹ ‫ܧܦܧ‬ͺሺܵ‫ܧܦ‬ሻିଵ ൌ ‫͵ͳܨܦܧ‬Ζ. ಚΔΗͪϔϫρέαʖχͺ ‫ܧ‬ሺͳȀͷሻܵ‫ڄ ܧܦ‬‫ܧ‬ሺͳȀͷሻ‫ Ͳܨܦܧ‬, ͞ Η ͺ ࣞ (39)Ͳ ͍ Ζ . ઴ ൔ ෨ ͹ ‫ܧܦܧ‬Ν ‫ ͤ ͳܨܦܧ‬Ζ ͳ , ‫ ޛ‬ൔ ͺ ܵ‫ ͍ Ͳܧܦ‬Ζ . ͞ Η΍ ࣞ (39)ͳ ಋ ͣ Ͳ ͍ Ζ . ઴ ൔ ෨ ͹ ‫ܧܦܧ‬Ν ‫ ͓ ׷ ͘ ॽ ͳܧܵܧ‬Ζ ͳ , ઴ ൔ ෨ ͹ ‫ܧܦܧ‬ͺ ሺ‫ܧܵܧ‬ሻିଵ ൌ ‫ܧܨܧ‬ͳ͵Ζ. ͞ΗΓΕಚΔΗΖϔϫρέαʖχͺࣞ (38)Ͳ͍Ζ. ઴ൔ෨͹ ‫ܧܦܧ‬Ν‫ͤͳܧܨܧ‬Ζͳ, ‫ޛ‬ൔͺ‫͍Ͳܧܵܧ‬Ζ. ͞Η΍ࣞ(38)ͳಋͣͲ͍Ζ.
઴ൔ෨͹‫ܧܦܧ‬ΝҐԾ͹Γ͑Ͷ஖͘‫͓׷‬Ζ͞ͳͺͲ͘͵͏ .
‫ ܧܦܧ‬՜ ‫ܧܦܦ‬ǡ ‫ܦܦܧ‬ǡ ‫ܵܦܧ‬ǡ ‫ܧܦܨ‬Ǥ(40)

ཀྵ༟ͺ, ‫ܦܦޢࢯے‬ǡ ‫ܵܦ‬ǡ ‫ܦܨ‬Ν‫؜‬΋͖ΔͲ͍Ζ.
ब‫غ‬ഔ෾‫Ͳز‬ͺ, ฾ٌಕ ఼͖Δਫ਼ٌ໊ͣͪಕ఼͹ ݃-ଲΝ͵఼್ͤͫ͗͜྘ Ү ܸ଴ ሺܸଵ ሻ͖Δ྘Ү ܸଵ ሺܸ଴ ሻ΃Ңಊ ͤ Ζ . ͞Η ͺ , ઴ൔ ෨͹ ϔ ϫρ έΝ Ҳ ͯ รߍ ͤΖ ͳ ࣎ؔ ൕ఺ ଲস ੓Γ Ε‫ ޛ‬ൔ෨͹ϔϫρέ͗Ҳͯรߍ͠ΗΖ͞ͳΝқັͤΖ . ͞͹ଲস੓Νຮٌ໊ͪͤಕ͹ϔϫρέαʖχͺࣞ(38)ͳࣞ(39)ͫ͜Ͳ͍Ζ͞ͳ͗ࣖ͠Ηͪ .

ਦ 35: (a) ฾ٌಕ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ٌಕ఼. (b) ٌ໊ಕ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻܵሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ ‫ڄ‬‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ٌಕ఼. ੖਼݇ͺٌಕ఼ ߱ෝ௞ ΝқັͤΖ.
৏‫Ͳى‬ಚΔΗͪ‫ߺى‬Կ͗ਜ਼͢͏͞ͳΝ࣏Ͷ͖֮ΌΖ . ͨ͹ͪΌͶब‫غ‬ഔ෾‫Ͳز‬ਫ਼ͣΖٌ໊ಕ఼ͳൕब‫غ‬ഔ෾‫Ͳز‬ਫ਼ͣΖٌ໊ಕ఼Ν௒΄Ζ . ब‫ غ‬28 ͹ٌ໊ಕ఼Ν‫ߺى‬ฑ໚Ͳඵ͏ͱ໊
ٌಕ఼͹੓࣯Νཀྵմ͢Γ͑ . ฾ٌಕ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ٌಕ఼Νਦ 35(a)Ͷ, ٌ໊ಕ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻܵሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ ‫ܧ ڄ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ٌಕ఼Νਦ 35(b)Ͷඵ͏ͪ. ଲসતܵመ௛ ৏Ͷٌಕ఼ ‫ݖ‬Ƹସͳ‫ݖ‬Ƹଵଵ͍͗Ζ. ٌಕ఼‫ݖ‬Ƹସ ͖Δਫ਼ٌͣͪಕ఼߱ෝସ ͳ߱ෝଵ଼ ͹ଲসત ܵመ௛ ͖Δ͹֐Ηͺ໎ྐͲ͍Ζ . ͞ΗΔ͹ٌಕ఼ͺਦ(b)Ͳͺ 4 ͳ 18 ͳ͢ͱ͍Ζ.
࣏Ͷ, ฾ٌಕ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ٌಕ఼Νਦ 36(a)Ͷ, ٌ໊ಕ‫ܧ‬ሺͳȀͷሻܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ ‫ܧ ڄ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ͹ٌಕ఼Νਦ 36(b)Ͷඵ͏ͪ. ଲসતܵመ௛ ৏Ͷٌಕ఼ ‫ݖ‬Ƹସͳ‫ݖ‬Ƹଵଵ͍͗Ζ. ٌಕ఼‫ݖ‬Ƹସ ͖Δਫ਼ٌͣͪಕ఼෡ ସ ͳȳ෡ ଵ଼ ͹ଲসત ܵመ௛ ͖Δ͹֐Ηͺඉ৙ͶΚ͖ͥͲ͍Ζ . ͞ΗΔ͹ٌಕ఼͹ഓ஖ΓΕ , ब‫غ‬ഔ෾ȳt 38 t෡ ସ ͳȳ෡ଵ଼ ͗‫Ͳޛ‬ਫ਼ͣͪͳ൓இͲ͘
‫ٌ໊Ͳز‬ಕ఼߱ෝସ ͳ߱ෝଵ଼ ͗઎Ͷਫ਼ͣ, ൕब‫غ‬ഔ෾‫ٌ໊Ͳز‬ಕ఼ȳΖ. Ҳൢ Ͷ, ઎Ͷ ਫ਼ ͣͪ ٌಕ ఼ ͺ ฾ٌ ಕ ఼ ͖Δ ԗ ͚Ͷ ͍ Ε , ‫ Ͳޛ‬ਫ਼ ͣͪ ٌ ಕ఼ ͺ ฾ ٌಕ ఼ ͹͚ۛͶ͍Ζ͞ͳΝཤ༽ͪ͢ . ͞͹੓࣯ͺٌ໊ಕ͹αʖχΝ݀ఈͤΖͪΌͶཤ༽Ͳ͘Ζ .

ਦ 36: (a) ฾ٌಕ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ٌಕ఼. (b) ٌ໊ಕ‫ܧ‬ሺͳȀͷሻܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ ‫ڄ‬෡ ௞ ΝқັͤΖ.
‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ͹ٌಕ఼. ੖਼݇ͺٌಕ఼ ȳ૮ฑ໚Ͷ͕͜Ζ͞ΗΔ͹ٌಕ఼Νਦ 37 Ͷࣖͪ͢. ઎Ͷਫ਼఼್ͣͪ ߱ସ ͳ߱ଵ଼ ͺ฾ٌಕ఼‫ݖ‬ସ ͖Δԗ͚΃Ңಊ͢ͱ͏Ζ͗ , ‫Ͳޛ‬ਫ਼఼್ͣͪ ȳସ ͳȳଵ଼ ͺ฾ٌಕ఼ ‫ݖ‬ସ ͹͚ۛͶ͍Ζ. Ґ৏Ͳٌ໊ಕ
͹ϔϫρέαʖχ͹֮೟Νश͓Ζ .

ਦ 37: ఼ ‫ݖ‬ସ ͺ ฾ ٌ ಕ ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ ٌ ಕ ఼ . ఼ ߱ସ ͳ ఼ ߱ଵ଼ ͺ ໊ ٌ ಕ ‫ܧ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻܵሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ ‫ܧ ڄ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ͹ ٌ ಕ ఼ . ఼ ȳସ ͳ ఼ ȳଵ଼ ͺ ໊ ٌ ಕ ‫ܧ‬ሺͳȀͷሻܵሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܧ‬ሺͳȀ͵ሻ ‫ܧ ڄ‬ሺͳȀͷሻ‫ܧ‬ሺͳȀ͵ሻ‫ܦ‬ሺͳȀ͵ሻ‫ܨ‬ሺͳȀ͵ሻ ͹ ٌ ಕ ఼ . (a) ܽ ൌ ͵ǤͲʹ . (b) ܽ ൌ ͵ǤͲ͵ͷͳͻ͸͵Ͳͻ . (c) ܽ ൌ͵ǤͲͷ. (d) ܽ ൌ ͵Ǥͳ.
t 39 tਦ 38 Ͷඵ͏ͪড়݇Νຮͪͤब‫ٌغ‬ಕ఼ ߚΝ΍ͯब‫ٌغ‬ಕ͗ν΢ϕ IV ͹෾‫ز‬գఖΝًͤ͞͞ ͳ ͗ Κ ͖ ͮ ͱ ͏ Ζ . ͞ ͹ ब ‫ ٌ غ‬ಕ ͹ ϔ ϫ ρ έ α ʖ χ ͺ ච ͥ ‫ܦ‬ሶ ሺ‫݌‬Ȁ‫ݍ‬ሻሺͲ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ͳȀʹሻΝ ‫ ؜‬΋ .
͞ΗΝ΍ͳͶ͢ͱν΢ϕ IV ͹෾‫ز‬գఖΝًͤ͞ড়݇ 3.10 ͗ಚΔΗΖ.
ড় ݇ 3.10. ฾ ब ‫ ٌ غ‬ಕ ͗ ਫ਼ ͣ ͪ ௜ ‫ ޛ‬Ͷ ͕ ͏ ͱ γ χ ϩ ‫ ͍ Ͳ ܗ‬Ε , ϔ ϫ ρ έ α ʖ χ ͗ ϔ ϫ ρ έ‫ܦ‬ሶ ሺ‫݌‬Ȁ‫ݍ‬ሻሺͲ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ͳȀʹሻΝ‫؜‬Ί͖ͯ۰αʖχͲ͍Ζ͞ͳ .

ਦ 38: ଲসત͹଀͗, ࠹ॵͶଲসત ܵ௛ି ͳި఼͢ߙͳߚ͗ਫ਼ͣΖ. ࣏Ͷި఼ ߛͳߜ͗ਫ਼ͣΖ. ި఼ߚΝब‫ٌغ‬ಕ఼ͳ͢ͱ΍ͯ ଲসब‫ٌغ‬ಕͺν΢ϕ IV ͹෾‫ز‬գఖΝࣖͤ.
ਦ 38 Ͷࣖͪ࢝ͯ͢͹ި఼͹ઈ໎Νߨ͑ . ଲসત͹଀͗, ࠹ॵͶଲসત ܵ௛ି ͳިࠫ͢, ި఼ߙͳߚ͗ਫ਼ͣΖ. ࣏Ͷଲসત͹଀ͺ෈҈ఈଡ༹ର͹‫ܶ ނ‬Ȟ௨ Νᷘյͪ͢‫ޛ‬, ࠸ౕଲসતܵ௛ି ͳިࠫ ͢ި఼ߛͳߜ͗ਫ਼ͣΖ . ‫ܶނ‬Ȟ௨ ͗ި఼ߚͳߛ͹ؔͶ೘Εࠒ΋͞ͳͶ஭қ͢Γ͑ . ި఼ߙΝब‫ٌغ‬ಕ఼ͳ ͢ ͱ ΍ ͯ ଲ স ब ‫ ٌ غ‬ಕ ͺ ν ΢ ϕ II ͹ ෾ ‫ ز‬գ ఖ Ν ࣖ ͤ . ި ఼ ߛΝ ब ‫ ٌ غ‬ಕ ఼ ͳ ͢ ͱ ΍ ͯ ଲস ब ‫ ٌغ‬ಕ ͺ γ χ ϩ ‫ܗ‬ब ‫ ٌ غ‬ಕ఼ Ͳ ෾ ‫ ݳ ز‬ে Ν ً ͞ ͠ ͵͏ . ި ఼ ߜΝ ब ‫ ٌغ‬ಕ ఼ ͳ͢ ͱ ΍ ͯଲসब‫ٌغ‬ಕͺν΢ϕ I ͹෾‫ز‬գఖΝࣖͤ.

4. ݃‫ޢ‬Ҳ࣎‫( غ‬1970 ೧େ‫ޛ‬ൔ͖Δ 1980 ೧େ઴ൔ), ΩΨη΃ࢺΖϩʖφ͹Ҳͯͳ͢ͱब‫غ‬ഔ෾‫ز‬ͺ‫ޭٯ‬Νཌྷ ;ͪ . ͖͢͢, ΩΨη΃͹ ‫ ͗ৼؖ‬മΗ ΖͶͯΗͱ ब‫غ‬ഔ෾ ‫ز‬͹ ‫ڂݜ‬΍‫͢ͱ ͮݰ‬Ή
ͮͪ. ජंͺ, ब‫غ‬ഔ෾‫͗ز‬ଠ͹‫ݳ‬েͳ‫͏ͱཱི͢ݿ‬Ζ৖߻͹ν΢ϕ I Ͷ‫ͤؖ‬Ζ‫ڂݜ‬ͺ΍͑श྅͢ͱ͏Ζͳߡ͓ͱ͏Ζ .
ຌ࿨ชͲࣖͪ͢Γ͑Ͷब‫غ‬ഔ෾‫ز‬Νพ͹؏఼͖Δ‫ݡ‬Ζͳ‫͘΄ͤڂݜ‬՟ୌ͗ଡ͚͍Ζ . ಝͶν΢ϕ IV Ͷͯ͏ͱͺຌ࿨ช͗ॵΌ͹ๅࠄͲ͍Ζ . ν΢ϕ II ͖Δ IV Ͷͯ͏ͱͺ‫΄ͤڂݜ‬͘໲ୌ͗࢔ͮͱ͏Ζͳߡ͓ͱ͏Ζ . ब‫غ‬ഔ෾‫ͳز‬ಋब‫غ‬෾‫ز‬Ν‫ؖ‬࿊෉͜ͱ٠࿨ͤΖ͞ͳͺ৿͢͏๏޴੓Ͳ͍Ζ. ͞ΗΔ͹෾‫ز‬գఖΝ٠࿨ͤΖͪΌͶͺ҈ఈଡ༹ରͳ෈҈ఈଡ༹ର͹৷೘
ΝໃࢻͤΖ͞ͳͺͲ͘͵͏ . ࠕ‫ޛ‬ͺ, ͞͹Γ͑͵ߡ͓ͶΓΖ‫ڂݜ‬Ν๮Ίͪ͏ .

ःࣛࠬಣंΓΕଡ͚͹αϟϱφΝࣂΕΉͪ͢ . ৼΓΕ‫״‬ः͏ͪ͢Ήͤ.
t 40 tࢂߡช‫ݛ‬[1] ब‫غ‬ഔ෾‫ز‬Ͷ‫ͤؖ‬Ζմઈ . http://www.scholarpedia.org/article/Period_doubling>@+ËQRQ01XPHULFDOVWXG\RITXDGUDWLFDUHD -preserving mappings. Quart. Appl. Math.,
27 (1969), 291t312.
[3] Smale, S., Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747 -817.
https://www.ams.org/journals/bull/1967-73-06/S0002-9904-1967-11798-1/S0002-99
04-1967-11798-1.pdf[4] ࢃ޳‫ج‬ദ, ୫ઔੜཾ, ഇఛ΃͹ಕ (‫ཱིڠ‬ड़൝, 2016).
[5] Yamaguchi, Y. and Tanikawa, K., A new interpretation of the symbolic codes for the+ËQRQ

PDS

,,

3URJ

7KHRU

3K\V



(2011),

435-471.

https://doi.org/10.1143/PTP.125.435[6] ࢃ޳ ‫ج‬ദ , ୫ ઔ ੜཾ , ब‫ غ‬ഔ ෾ ‫ Ͳز‬ਫ਼ ͣ ͪ ଲ স ब‫ ٌ غ‬ಕ ͹ ‫ ߺى‬ଉ , ࠅ ཱི ళช ୈ ๅ ୊ 21‫( ת‬2021), 1-20. https://doi.org/10.50916/rnaoj.21.0_1[7] Feigenbaum, M. J., Qualitative universality for a class of nonlinear transformations. J.
Stat. Phys. 19 (1978), 25-52. https://doi.org/10.1007/BF01020332[8] Feigenbaum, M. J., Universal metric properties of nonlinear transformations. J. Stat.
Phys. 21 (1979), 669-706. https://doi.org/10.1007/BF0110790[9] Devaney, R.,

An Introduction to Chaotic Dynamical Systems , (Westview Press, 2003).

๞༃:ΩΨηྙָ‫ܧ‬೘໵ ୊̐൝ ৿ఙ൝ (‫ޛ‬౽‫ݓ‬Ҳͨ͹ଠ༃), 2003, ‫ཱིڠ‬ड़൝‫ࣞה‬ճऀ.
[10] ௗౣ߄߀, ഇ৖ྒྷ࿪, ΩΨη೘໵, ക෫‫ؙ‬, (1992).
[11] MacKay, R. S.,

Renormalisation in area-preserving maps (World Scientific,1993).

t 41 t෉࿧ $ ‫ߺى‬ฑ໚Ͷ͕͜Ζࣺ଀ͳ‫ߺى‬ฑ໚Ͳब‫ٌغ‬ಕΝඵ͚ϕϫήϧϞα ʖ χ ‫ݏ‬͹ ۰ ‫ د‬੓ ͗ ‫ د‬͹ ৖ ߻ , ‫ݏ ޢ‬Ԣ ൌ ‫ݏݏ‬Ν ༽ қ ͤ Ζ . ۰ ͹ ৖ ߻ , ‫ݏ ޢ‬Ԣ ൌ ‫ ͤ ͳݏ‬Ζ . ྭ ͳ ͢ ͱ ,‫ ݏ‬ൌ ͲͲͳ͹ ৖ ߻ ͺ ‫ݏ‬Ԣ ൌ ͲͲͳͲͲͳͲ , ‫ ݏ‬ൌ Ͳͳͳ͹ ৖ ߻ ͺ ‫ݏ‬Ԣ ൌ ͲͳͳͲ ͍ Ζ . Ґ Ծ Ͳ ͺ ‫ݏ‬Ԣ ൌ ͲͲͳͲͲͳΝ ΍ ͳͶ‫ߺى‬྽ΝԾ‫ى‬͹Γ͑Ͷॽ͚ . ͞͞Ͳঘ਼఼Νഓ஖ͪ͢.
‫ͳͲͲͳͲͲ ڮ‬Ȉ ͲͲͳͲͲͳ ‫ڮ‬Ǥঘ ਼ ఼ Ȉ Γ Ε ӊ ๏ ޴ Ͷ ಣ ΋ . ͞ Η Ν ‫ ݔ‬ൌȈ ሺͲͲͳͲͲͳሻஶ ͳ ॽ ͚ . ঘ ਼ ఼ Γ Ε ࠪ ๏ ޴ Ͷ ಣ Ί ,‫ ݕ‬ൌȈ ሺͳͲͲͳͲͲሻஶ ͳ ॽ ͚. ͞ ͞ Ͳ ‫ݕ ͳݔ‬Ͷ ͯ ͏ ͱ ್ ਒ ๑ ΃ ร ‫ ͤ ׷‬Ζ (α ʖ χ ͹ ್ ਒ ๑ ΃ ͹ ร ‫ ׷‬घକ͘ A.1 ΝࢂߡͶͦΓ). ร‫ޛ׷‬͹න‫ݳ‬Ν࣏Ͷࣖͤ.
ܺ଴ ൌȈ ሺͲͲͳͳͳͲሻஶ ൌ ሺ͹Ȁ͵ʹሻȀሺͳ െ ͳȀ͸Ͷሻ ൌ ʹȀͻǡܻ଴ ൌȈ ሺͳͳͳͲͲͲሻஶ ൌ ሺ͹ȀͺሻȀሺͳ െ ͳȀ͸Ͷሻ ൌ ͺȀͻǤॵ ‫ ఼غ‬ሺܺ଴ ǡ ܻ଴ ሻͶ Ծ ‫ ى‬͹ࣺ ଀ ܶ෠ Ν ࡠ ༽ ͢ͱ , ͨ ͹ ‫ޛ‬͹ ٌ ಕ ఼ Ν ݀ఈ ͤ Ζ . ‫ ߺى‬ฑ ໚ Ͳ ͺ , ‫ݕ‬ො࠴ඬ͹৏ԾΝൕ఺͢ͱٌಕ఼Νඵ͚. ͞ΗͶΓͮͱ૮ฑ໚͹ٌಕ఼ͳ͹ଲԢ͗෾͖ΕΏ͚ͤ͵
Ζ.
‫ߺى‬ฑ໚ (Ͳ ൑ ‫ݔ‬ොǡ ‫ݕ‬ො ൑ ͳ) Ͷ͕͜Ζࣺ଀ܶ෠Ν়ղͤΖ.
‫ݔ‬ො௡ାଵ ൌ ʹ‫ݔ‬ො௡ ǡ‫ݔ‬ො௡ାଵ ൌ ʹ െ ʹ‫ݔ‬ො௡ ǡ‫ݕ‬ො௡ାଵ ൌ ‫ݕ‬ො௡ ȀʹǡሺͲ ൑ ‫ݔ‬ො௡ ൑ ͳȀʹሻ‫ݕ‬ො௡ାଵ ൌ ͳ െ ‫ݕ‬ො௡ ȀʹǤሺͳȀʹ ൏ ‫ݔ‬ො௡ ൑ ͳሻ

(41)
(42)

ࣺ଀ܶ෠ͺ, ଲ߻݄෠ͳ݃ො͹੷Ͳ‫ى‬फ़Ͳ͘Ζ.
݄෠ሺ‫ݔ‬ොǡ ‫ݕ‬ොሻ ൌ ሺ‫ݕ‬ොǡ ‫ݔ‬ොሻǡሺ‫ݕ‬ොȀʹǡʹ‫ݔ‬ොሻǡ݃ොሺ‫ݔ‬ොǡ ‫ݕ‬ොሻ ൌ ൜ሺͳ െ ‫ݕ‬ොȀʹǡʹ െ ʹ‫ݔ‬ොሻǤ

(43)
ሺͲ ൑ ‫ݔ‬ො ൑ ͳȀʹሻሺͳȀʹ ൏ ‫ݔ‬ො ൑ ͳሻ(44)‫ߺى‬ฑ໚Ͳ͹ଲসતΝ࣏Ͷࣖͤ .
ܵመ௛ ǣ ‫ݕ‬ො ൌ ‫ݔ‬ොǡ‫ݕ‬ො ൌ ʹ‫ݔ‬ොǡܵመ௚ ǣ ൜‫ݕ‬ො ൌ ʹ െ ʹ‫ݔ‬ොǤሺͲ ൑ ‫ݔ‬ො ൑ ͳȀʹሻሺͳȀʹ ൏ ‫ݔ‬ො ൑ ͳሻ

(45)
(46)

͞ΗΔ͹ଲসતͺસͱ͹‫ߺى‬ฑ໚͹ਦͲඵ͖Ηͱ͏Ζ .
αʖχ͹್਒๑΃͹ร‫׷‬घକ͘ A.1.
α ʖ χ Ν ‫ݏ‬଴ ‫ݏ‬ଵ ‫ݏ ڮ‬௤ିଵ ͳ ॽ ͚ . α ʖ χ ͹ ۰ ‫ د‬੓ ͗ ۰ ͵ Δ ͻ ร ‫ ׷‬઴ ͹ α ʖ χ Ν ‫ݏ‬଴ ‫ݏ‬ଵ ‫ݏ ڮ‬௤ିଵ ͳ ͢ ,ร‫ ޛ׷‬ͺ ‫ݐ‬଴ ‫ݐ‬ଵ ‫ݐ ڮ‬௤ିଵ ͳͤΖ. ‫ ͵د‬Δ ͻ ‫ݏ‬଴ ‫ݏ‬ଵ ‫ݏ ڮ‬௤ିଵ ‫ݏ‬଴ ‫ݏ‬ଵ ‫ݏ ڮ‬௤ିଵ Ν༽ қ͢ , శ࣊ ͗࿊ କ ͤΖΓ ͑Ͷ ॽ͘௜͢, ร‫׷‬઴͹αʖχΝ‫ݏ‬଴ ‫ݏ‬ଵ ‫ݏ ڮ‬ଶ௤ିଵ ͳͤΖ. ร‫ޛ׷‬ͺ‫ݐ‬଴ ‫ݐ‬ଵ ‫ݐ ڮ‬ଶ௤ିଵ ͳͤΖ. ร‫׷‬઴͹αʖχͶଲ͢ͱҐԾ͹घॳΝࣰߨͤΖ .
[1] ‫ݐ‬଴ ൌ ‫ݏ‬଴ .
[2] ͳ ൑ ݇ ൑ ‫ ݍ‬െ ͳΉͪͺͳ ൑ ݇ ൑ ʹ‫ ݍ‬െ ͳ͹৖߻, ‫ݐ‬௞ ΝԾ‫ى‬͹‫و‬ଉͶॊͮͱ݀ΌΖ .
(a) ‫ݏ‬଴ ͖Δ‫ݏ‬௞ିଵ ΉͲ͹ͳ͹‫਼د਼͗ݺ‬͹৖߻. ‫ݐ‬௞ ൌ ͳ െ ‫ݏ‬௞ .
(b) ‫ݏ‬଴ ͖Δ‫ݏ‬௞ିଵ ΉͲ͹ͳ͹‫਼͗ݺ‬۰਼͹৖߻. ‫ݐ‬௞ ൌ ‫ݏ‬௞ .
[3]௤ିଵ۰‫د‬੓͗۰͹৖߻: ܺ ൌ ሺσ௞ୀ଴ ‫ݐ‬௞ Ȁʹ௞ ሻȀሺͳ െ ʹ௤ ሻ.
ଶ௤ିଵ۰‫د‬੓͗‫د‬͹৖߻: ܺ ൌ ሺσ௞ୀ଴ ‫ݐ‬௞ Ȁʹ௞ ሻȀሺͳ െ ʹଶ௤ ሻ.
࠹‫ޛ‬Ͷ, ‫ߺى‬ฑ໚৏Ͳब‫ٌغ‬ಕΝඵ͚ Mathematica ˕R ͹ϕϫήϧϞΝࡎͦͪ.

t 42 tt 43 tt 44 t࿊ࡎߪ࠴‫ܙ‬͹Ռָճࢿ ୊ 36 ‫ ת‬୊ 1 ߺColaboratory Ͳ‫ܙ‬͹ՌָΝֺ͢΍͑t ୊ 3 յ ϧϱξϞΤΧʖέʁؔҩͮͪ৚ๅͺ֨ࢆͤΖ t
ฑీཾ߀
ෳҬ୉ָ

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ΉΗͱ͏ΖΆ͑͗ɼླྀා͢Ώͤ͏ɽ͠ͱɼҋ͏࿫ୌ͗ଡ͏αϫψՔ͹ԾͲɼ໎Ζ͏࿫ୌͳ͢ͱखΕ৏͝ΔΗΖ͞ͳ͗ଡ͏͹͗ɼ౽Ҭ૳ଢ̐‫̏̐̎̐(ץ‬೧݆̏̕ೖ఼࣎)͹‫׈‬༄Ͳ͍Ζɽᰞ૚ͳ‫ݳ‬Ηͪ੪೧‫͗࢞ػ‬ɼ‫ک‬తΝ଩ͬෝ͖͢ɼ‫ى͏ݻ‬࿧Ν࣏ʓͳ଩ͬഃͮͱ͏༹͚ͺɼ‫ͬ࣍ـ‬Ν໎Ζ͚͢ͱ͚ΗΖɽΉͪҲ๏ಋ࣎Ͷɼϗτϧϱ‫͗࢞ػ‬ɼ౽Ҭ૳ଢΝ଩ͬෝ͖ͤͳɼ୉͚͘஭໪͠ΗɼϓΟϱ͖Δ
࣍ͬ৏͝ΔΗΖɽҲྭͳ͢ͱ࠹ۛͲͺɼ਄Ӟ 9 ஊ͗Ԩ࠴ક͹ຌકͲ౽Ҭ૳ଢ 2 ‫ץ‬ΝԾ͢ɼʰগ‫ػ‬੗ਕ͹৷ྲྀ͖Δஏ‫ځ‬Νगͮͪʱɼʰஏ‫ځ‬େනͺ਄Ӟ 9 ஊʱͳϋρφ͹ωϣʖηͲखΕ৏͝ΔΗͪɽ͞Ηͺɼʲਕؔ཯ΗͤΖΆʹ‫͙ͤک‬Ζ౽Ҭ૳ଢͺʰগ‫ػ‬੗ਕʱͲ͍Ζɽʳͳ͏͑઴఑͗ഐ
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Ͳ͍Ζɽ௙ࢃ໎͹ʰχϧβϱϚʖϩʱ͗ϔʖϞ͵ͮͱ͕Εɼγ΢Ϣਕ͗ஏ‫ځ‬Ͷ৷ྲྀͶ͘ͱɼकਕޮ͹ଛ‫ۯޝ‬ʤஏ‫ځ‬үͬ͹γ΢Ϣਕʥͪͬ͗ஏ‫ځ‬ΝगΖͳ͏͑ఴ֋͹ͳ͘Ͳ͍ͮͪɽஏ
‫ځ‬େනͳ͢ͱɼஏ‫ځ‬͹າཔΝ୙ͪ͢͹ͺɼγ΢Ϣਕ͹ଛ‫ͱ͍ͮͲۯޝ‬ɼਕؔͳ͢ͱͺ࠹‫ک‬͹έϨϨϱͲͺ͵͖ͮͪɽ౲࣎ɼӍਫ਼ 9 ஊ͍͗ΉΕͶ΍‫͙ͤک‬Ζ͹Ͳɼࣰͺ‫ۯޝ‬͹Γ͑Ͷগ‫ػ‬੗͖ΔૻΕࠒΉΗͪʰগ‫ػ‬੗ਕʱͲ͍Ζͳ͏͑ɼϓΟϱ͹ໟ૟Ͳ͍Ζɽͨ͞Ͳͺɼʰ΍͢ɼগ‫ػ‬੗ਕ͗ஏ‫ځ‬Ν৷ྲྀͶ͘ͱɼஏ‫ځ‬େනͳકͮͱɼஏ‫ځ‬ଈ͗ෝͪ͜Δɼ৷ྲྀ͠ΗΖɽ

t 45 tͨ͹ͳ͘͹ஏ‫ځ‬ଈ͹େනͳ͢ͱӣໍΝ୙ͤ͹ͺɼʰগ‫ػ‬੗ਕʱ͹Ӎਫ਼ͫΘ͑ɽ਄Ӟͺஏ‫ځ‬ਕ࠹‫͍Ͳک‬Ζ͗ɼஏ‫ځ‬େනͺ਄ӞͲͺ͵͏ͫΘ͑ɽʱͳ͏͑ɼ࿫Ͳ͍ͮͪɽ΍͢গ‫ػ‬੗ਕ͗৷ྲྀ͢ͱͪ͘Δɼʰஏ‫ځ‬େනͺɼগ‫ػ‬੗ਕͲ͍Ζɼգ‫ڊ‬ͺӍਫ਼ 9 ஊͲ͍Εɼࠕͺ౽Ҭ૳ଢ 2‫͍Ͳץ‬Εɼ਄Ӟ 9 ஊͲͺ͵͏ͺͥͲ͍Ζʱɽ͖͢͢ɼ৚ๅ͗ఽΚͮͱ͏͚͑ͬͶɼร༲͢ͱ͢Ή͏ɼஏ‫ځ‬େන਄Ӟ 9 ஊͳ৯ࢶͶॽ͏ͱ͚ΗͳབΉΗΖΓ͑Ͷ͵ͮͱ͢Ήͮͪɽ਄ӞʻέϨϨϱ͹ཱིͬҒ஖ͺɼਕؔେනͲ͍Ζ͗ɼஏ‫ځ‬େනͲͺ͵͏ɽ೨͹ҟɼʰஏ‫ځ‬͹ଚ๤ͺɼέϨϨϱͲͺ͵͚ɼ‫ۯޝ‬Ͷ୙ͪ͢͏ͫΘ͑ɽʱͳ͏͑ϋρφ৏͹ϋν࿫Ͳ͍ͮͪɽ
֨ࢆ‫ݳ‬েΝϝέϫ͵ࢻ఼͖ΔΊͪϠυϩͳ͢ͱ͹ϔϧΤϱӣಊ[1]Ͷ‫ͱؖ͢‬΍ɼΓ͚ࣇͪҵ࿫͗ଚࡑͤΖ[2]ɽϔϧΤϱӣಊ΍ɼϔϧΤϱ͗‫ݨ‬ඏ‫Ͳں‬ɼ՘ฆΝ؏ࡱ͢ͱ͏ͱ՘ฆ͹ಊ
͖͘Δ൅‫͓ఽͳͪ͢ݡ‬ΔΗͱͪ͘ɽ͖͢͢ਜ਼֮Ͷͺɼ՘ฆͲͺ͵͚՘ฆ͖Δ๎ड़͠Ηཽͪࢢ͹ಊ͖͘Δ൅‫ͪ͢ݡ‬͹Ͳ͍ΖɽೖຌͲ΍ɼ‫ݨ‬ඏ‫Ͳں‬՘ฆ͹ಊ͘Ν؏଎ͤΖͳɼϔϧΤϱӣಊΝ؏଎Ͳ͘Ζͳ়ղ͠Ηͱͪ͘ɽਫ਼෼༟པ͹՘ฆ͗೦ΑΔ͙ͳ͢ͱɼ͍͖ͪ΍ਫ਼෼͹Γ͑Ͷಊ͚ͳ͏͑͹͗ϛ΢ϱφͲɼοϥʖέ͹ฆ͵ʹͲͺ໚പ͚͵͏ɽͨΗΑ͓ɼʰ՘ฆ͗ϔϧΤϱӣಊͤΖ͹ͺɼ՘ฆ͗ਫ਼෼Ͳ͍Ζ͖ΔͲͺ͵͚ɼ೦༵Δ͙ͶΓΖ΍͹Ͳ͍Ζɽʱͳ͏͑͹Ν‫ک‬௒ͪ͢͏͗ͪΌɼ෼ཀྵָंͺʰϔϧΤϱӣಊ͹ཽࢢ͹ྭͳ͢ͱɼ՘ฆΝ༽͏ͪ
͗Ζʱ͹Ͳͺ͵͏ͫΘ͖͑ʃͳ͏͑͹͗ɼஸं͹ਬ଎Ͳ͍Ζɽࠕյ͹ߪ࠴ͲͺɼϔϧΤϱӣಊΝखΕ৏͝ɼ֨ࢆͤΖཽࢢ͹εϝϣϪʖεϥϱࣰ‫ݩ‬ΝࢾΊΖɽ͞͞ͲͺɼColaboratory Ͳ Python Ν࢘͏ϧϱξϞΤΧʖέʤ਴ิʥ͹εϝϣϪʖεϥϱΝࢾΊΖɽ

2. 3\WKRQ ͲϧϱξϞΤΧʖέ
2.1. ֨ࢆ๏ఖࣞͳϧϱξϞΤΧʖέϧϱξϞΤΧʖέ͹εϝϣϪʖεϥϱΝ͕͞͵͑઴Ͷɼ֨ࢆ๏ఖࣞΝΊͱ͕͑͞ɽୱ९
ԿͤΖͪΌɼ֨ࢆ‫਼ܐ‬ͺɼ‫ؔۯ‬Ώ࣎ؔͶΓΔͥҲఈͲ͍ΖͳͤΖɽ ֨ࢆ๏ఖࣞͺɼ߲߶ൌ ‫׏ܦ‬ଶ ߶߲‫ݐ‬ͳ͏͑ત‫ܙ‬͹ยඏ෾๏ఖࣞͶ͵Ζɽ͞͞ͲɼСͺ֨ࢆ෼࣯͹ືౕɼt ͺ࣎ؔɼ˺(ψϔϧ)ͺ‫ؔۯ‬ඏ෾ࡠ༽ોɼD ͺ֨ࢆ‫͍Ͳ਼ܐ‬Ζɽ͵͕ɼ֨ࢆ๏ఖࣞΝඏࢻదͶཽࢢ͹ࢻ఼͖ΔΊΖͳɼϧϱξϞΤΧʖέͶଲԢͤΖɽ

2.2.

ਜ਼‫و‬ཛྷ਼

෰सͳϧϱξϞΤΧʖέ͹६ඍͳ͢ͱɼ୊ 1 յߪ࠴[3]Ͳ΍়ղͪ͢ Colaboratory ΃Γ͑ͨ͞[4]Ͳࣖ͠Ηͱ͏Ζγϱϕϩʰυʖνγ΢Φϱηʱ͹ཛྷ਼Ν‫͑͞͏ͱݡ‬ɽ

2.2.1. ஘ᩘࢆ‽ഛࡍࡿϔϧΤδΝ࢘ͮͱɼColaboratory ͹ HP ͶΠέιη͢ͱɼυʖνγ΢Φϱη͹ͳ͞ΘͶߨͮͱΊΓ͑ɽಋ࣎Ͷɼmatplotlib[5]͹γϱϕϩϕϫήϧϞΝࢂߡͶͤΖɽγϱϕϩϕϫ
ήϧϞΝঙ͢ร͓ͱɼཛྷ਼͹੓࣯ΝΊͱ͏͑͞ɽΉͥɼਜ਼‫و‬෾ාʤฑ‫̎ۋ‬ɾ෾ࢆʤඬ६ย

t 46 tࠫʥ̏ʥͶॊ͑ਜ਼‫و‬ཛྷ਼Ν 200 ‫ݺ‬ਫ਼੔͢ͱΊΓ͑ɽϕϫήϧϞϨηφ̏
import numpy as np
from matplotlib import pyplot as plt
np.random.seed(19680801)
number_of_data = 200
data = np.random.randn(1,number_of_data)x = [x for x in range(number_of_data)]
fig, axs = plt.subplots(1, 2, figsize=(11, 5))
axs[0].plot(x, data[0])axs[1].hist(data[0])plt.show()ਦ̏: ཛྷ਼͹࣎‫ܧ‬྽ͳළౕ෾ාɽ

ਦ 1 ͹ළౕ෾ාΝ‫ݡ‬Ζͳɼਜ਼‫و‬෾ාͶ‫͖͏͵͓ݡ‬΍͢Η͵͏ɽ൅ਫ਼ͦ͠Ζཛྷ਼͹‫਼ݺ‬
number_of_data = 200 Νร͓ͱɼnumber_of_data = 20000

ͳ͢ͱɼཛྷ਼͗ਜ਼‫و‬෾ාͶ

ॊͮͱ͏Ζ͖͖֮ΌͱΊΓ͑ɽ൅ਫ਼ͤΖ‫਼ݺ‬ΝଁΏͤͳɼਜ਼‫و‬෾ාͶͪ͗͑͢ཛྷ਼Ͷ͵ͮͱ͏Ζ͹͗໎֮Ͷ͵Ζɽbin ͹‫਼ݺ‬ΝଁΏͤͳɼΓΕ໎֮Ͷਜ਼‫و‬෾ාͶ͵ͮͱ͏Ζ͹͗෾
͖ΖɽϕϫήϧϞϨηφ̐
import numpy as np
from matplotlib import pyplot as plt

t 47 tnp.random.seed(19680801)
number_of_data = 1000000
number_of_bins = 100
data = np.random.randn(1,number_of_data)x = [x for x in range(number_of_data)]
fig, axs = plt.subplots()
axs.hist(data[0],number_of_bins)
plt.show()ਦ̐: ཛྷ਼Ν 1000000 ‫ݺ‬൅ਫ਼ͦ͠ͱਜ਼‫و‬෾ාͶ͵Ζ͹Ν‫ݡ‬Γ͑ɽ bin ͹‫਼ݺ‬ͺ 100 Ͳ͍Ζɽ࣏Ͷɼਫ਼੔ͪ͢ਜ਼‫و‬ཛྷ਼྽Ν࢘ͮͱɼϧϱξϞΤΧʖέΝ͕͞͵͑ɽ

2.3. ϧϱξϞΤΧʖέ
2.3.1.  ḟඖࡢࣛࣥࢲ࣒࢛࣮࢘ࢡ
࣏̏‫ݫ‬͹ϧϱξϞΤΧʖέʤ਴ิʥ͖Δ‫͑͞͏ͱݡ‬ɽϠυϩΝ؈ྲྀԿ͢ͱߡ͓ΖͪΌɼ࣎ؔητρϕΝ཯ࢆ஍᷐t ͳͤΖɽ͵͕ɼҢಊ‫཯ړ‬ͺ࿊କ஍᷐x ͳ͢ͱɼ࣎ؔητρϕ᷐tຘͶɼ᷐x ͫࠪ͜ӊͶҢಊͤΖͳͤΖɽ͞͞Ͳɼ᷐x ͺɼ઴হͲࣖͪ͢ਜ਼‫و‬ཛྷ਼Ν༽͏Ζɽ
1 ࣏‫ݫ‬͹ϧϱξϞΤΧʖέ͹εϝϣϪʖεϥϱʤϕϫήϧϞϨηφ̑ʥΝਦ̒Ͷࣖͤɽ͵͕ɼήϧϓ͵ʹ͹ඵժͶͺɼmatplotlib Ν࢘༽ͤΖ[5]ɽ

t 48 tਦ 3: 1 ࣏‫ݫ‬͹ϧϱξϞΤΧʖέɽϕϫήϧϞϨηφ̑
import numpy as np
from matplotlib import pyplot as plt
np.random.seed(20210601) # ஘ᩘࡢ✀ࢆタᐃ
number_of_data = 100000
number_of_bins = 50
noise = np.random.randn(1,number_of_data)x = [x for x in range(number_of_data)]
position = noise.tolist() # numpy 㓄ิࢆ Python ࡢ㓄ิ࡟ኚ᥮
position[0][0] = 0# ึᮇ఩⨨ࢆ㸮࡟ࡍࡿ

def randomwalk():
for i in range(number_of_data - 1):
position[0][i + 1] = position[0][i] + noise[0][i]returnrandomwalk()fig, ax = plt.subplots()
ax.plot(x, position[0])ax.grid()ax.set_xlabel('Steps')
ax.set_ylabel('Position')ax.set_title('1D Random Walk')
plt.show()t 49 tਦ̒: 1 ࣏‫ݫ‬͹ϧϱξϞΤΧʖέɽ࣎ؔητρϕͶଲ͢ͱ x ࣢৏͹Ғ஖Νϕϫρφɽ

2.3.2.  ḟඖࡢࣛࣥࢲ࣒࢛࣮࢘ࢡ
࣏Ͷɼ2 ࣏‫ݫ‬͹ϧϱξϞΤΧʖέʤϕϫήϧϞϨηφ̒ʥΝ‫ͱݡ‬ΊΓ͑ɽεϝϣϪʖεϥϱ݃ՎΝਦ̓ͶࣖͤɽϕϫήϧϞϨηφ̒
import numpy as np
from matplotlib import pyplot as plt
np.random.seed(20210601)
number_of_data = 1000
number_of_bins = 30
noise = np.random.randn(2,number_of_data)t = [t for t in range(number_of_data)]#

2 ͯ͹ਜ਼‫و‬ཛྷ਼# t ͺ࣎ؔ͹ร਼

position = noise.tolist() # numpy ഓ྽Ν Python
position[0][0] = 0# ॵ‫غ‬Ғ஖ x Ν̎ͶͤΖ

position[1][0] = 0# ॵ‫غ‬Ғ஖ y Ν̎ͶͤΖ

def randomwalk():
for i in range(number_of_data - 1):

t 50 t͹ഓ྽Ͷร‫׷‬position[0][i + 1] = position[0][i] + noise[0][i]position[1][i + 1] = position[1][i] + noise[1][i]returnrandomwalk()fig, ax = plt.subplots(1,3,figsize=(16,5))ax[0].plot(t, position[0])ax[0].grid()ax[1].plot(t, position[1])ax[1].grid()ax[2].plot(position[0], position[1])ax[2].grid()ax[0].set_xlabel('Steps')ax[1].set_xlabel('Steps')ax[0].set_title('x Position')ax[1].set_title('y Position')ax[2].set_title('Trajectory of the Random Walker in 2-D')
plt.show()ਦ 5: 2 ࣏‫ݫ‬͹ϧϱξϞΤΧʖέɽࠪͺ x vs. time ϕϫρφɼ஦ԟͺ y vs. time ϕϫρφɼӊͺϧϱξϞΤΧʖΩʖ͹ 2 ࣏‫ݫ‬ฑ໚৏͹ٌ੽ɽ

2.4. ‫ܨ‬գ࣎ؔͳϧϱξϞΤΧʖΩʖ͹౺ୣ‫཯ړ‬
2.4.1.  ḟඖᖹ㠃࡛ࡢࢩ࣑࣮ࣗࣞࢩࣙࣥ
ϧϱξϞΤΧʖέ͹‫ܨ‬գ࣎ؔͳ౺ୣ‫཯ړ‬Ν௒΄Ζɽ࣎ࠃ̎Ͳɼ࠴ඬ͹‫఼͖ݬ‬Δड़൅ͦ͠ͱɼ࣎ؔ‫ܨ‬գͶଲ͢ͱɼ‫఼͖ݬ‬Δ͹‫཯ړ‬Ν௒΄ͪɽ2 ࣏‫࠴ݫ‬ඬͲ͹εϝϣϪʖεϥϱ݃Վt 51 tΝਦ̔Ͷࣖͤɽਦ͹ࠪ͗ϧϱξϞΤΧʖΩʖ͹ٌ੽ɼ஦ԟͶ‫఼͖ݬ‬Δ͹‫཯ړ‬ɼӊͶ‫఼͖ݬ‬Δ͹࠹ԗ౺ୣ‫཯ړ‬Νϕϫρφͪ͢΍͹ΝࣖͤɽϕϫήϧϞϨηφ̓
import numpy as np
from matplotlib import pyplot as plt
np.random.seed(20210601)
number_of_steps = 10000
reachdistance = 0# ࠹߶౺ୣ‫཯ړ‬noise = np.random.randn(2,number_of_steps)t = [t for t in range(number_of_steps)]#2 ͯ͹ਜ਼‫و‬ཛྷ਼# t ͺ࣎ؔ͹ร਼

position = noise.tolist() # Walker ͹Ғ஖Ν Numpy ഓ྽͖Δ Pyhon ͹ഓ྽Ͷร‫׷‬
position[0][0] = 0# ॵ‫غ‬Ғ஖ x Ν̎ͶͤΖ

position[1][0] = 0# ॵ‫غ‬Ғ஖ y Ν̎ͶͤΖ

distance = [0]# ‫఼͖ݬ‬Δ͹‫ܧ࣎཯ړ‬྽͹ഓ྽

maxreach = [0]# ࠹ԗ౺ୣ‫཯ړ‬͹࣎‫ܧ‬྽͹ഓ྽

def randomwalk():
reach = 0
for i in range(number_of_steps - 1):
position[0][i + 1] = position[0][i] + noise[0][i]position[1][i + 1] = position[1][i] + noise[1][i]rr = reachd(position[0][i+1], position[1][i+1])
if rr >= reach :reach = rr
distance.append(rr)maxreach.append(reach)
returndef reachd(xp, yp):r = np.sqrt(xp*xp + yp*yp)
return(r)randomwalk()fig, ax = plt.subplots(1,3,figsize=(16,5))

t 52 tax[0].plot(position[0], position[1])ax[0].grid()ax[1].plot(t,distance)ax[2].plot(t,maxreach)ax[0].set_xlabel('x position')ax[0].set_ylabel('y position')ax[1].set_xlabel('Steps')ax[2].set_xlabel('Steps')ax[0].set_title('Trajectory of the Random Walker in 2-D')ax[1].set_title('Timeseries of Distance of Walker')ax[2].set_title('Timeseries of Max Reach Distance')
plt.show()ਦ̔: 2 ࣏‫ݫ‬͹ϧϱξϞΤΧʖέ͹εϝϣϪʖεϥϱɽࠪͺ 2 ࣏‫ٌݫ‬ಕɼ஦ԟͺ‫఼͖ݬ‬Δ଎ͮͪ‫཯ړ‬͹࣎‫ܧ‬྽ɼӊͺ࠹ԗ౺ୣ‫཯ړ‬͹࣎‫ܧ‬྽ɽਦ͖̔Δɼ֨ࢆ๏ఖࣞͳϧϱξϞΤΧʖέ͹‫ܐؖ‬ɼϧϱξϞΤΧʖΩʖ͹࠹ԗ౺ୣ‫཯ړ‬̓ͺɼ࣎ؔ t Ͷଲ͢ͱ ŕt 0.5 Ͳ͍Ζ͞ͳΝɼεϝϣϪʖεϥϱͲࣰ‫͘Ͳ״‬ΖͲͺ͵͏ͫΘ͖͑ɽ෵਼‫ݺ‬͹ϧϱξϞΤΧʖΩʖΝ࢘ͮͪεϝϣϪʖεϥϱΝ͕͞͵͑ͳɼҲ૜໎֮Ͷ͖֮ΌΖ͞ͳ͗Ͳ͘Ζ͹Ͳேક͢ͱΊͱΆ͢͏ɽ

2.4.2. ࣛࣥࢲ࣒࢛࣮࣮࢘࢝ࡢ఩⨨ศᕸࡢ᫬㛫ኚ໬෵਼‫ݺ‬͹ϧϱξϞΤΧʖΩʖΝ‫఼͖ݬ‬Δड़൅ͦ͠ɼ࣎ؔ‫ܨ‬գͳͳ΍ ͶɼϧϱξϞΤΧʖΩʖ͹‫఼͖ݬ‬Δ͹‫཯ړ‬෾ාΝ௒΄ͪɽ͞͞Ͳͺɼ1 ࣏‫ݫ‬͹ϧϱξϞΤΧʖέ͹εϝϣϪʖεϥϱ݃ՎΝࣖͤɽϕϫήϧϞϨηφ̔
import numpy as np
from matplotlib import pyplot as plt
np.random.seed(20210601)
t 53 tnumber_of_steps = 8000

# ࠹ௗ͹࣎ؔητρϕnwalker = 1000

# ϧϱξϞΤΧʖΩʖ͹‫਼ݺ‬#ਜ਼‫و‬ཛྷ਼͹‫਼ݺ‬ʻ࣎ؔητρϕʹQZDONHU ͹‫਼ݺ‬noise = np.random.randn(number_of_steps, nwalker)
position = noise.tolist() # Walker ͹Ғ஖Ν Numpy ഓ྽͖Δ Pyhon ͹ഓ྽Ͷร‫׷‬
def randomwalk():
for i in range(number_of_steps - 1):
for j in range(nwalker):position[i + 1][j] = position[i][j] + noise[i][j]returnfor i in range(nwalker):position[0][i] = 0# ॵ‫غ‬஍Ν̎ͶͤΖrandomwalk()
range1 =(-100,100)
number_of_bins = 30
fig, ax = plt.subplots(1,5,figsize=(16,5))ax[2].set_title('Histograms of the Position of Walker in 1-D')
for i in range(5):ax[i].set_ylim([-1,1100])for i in range(4):ax[i].hist(position[20**i -1],bins=number_of_bins,range=range1)ax[4].hist(position[20**i -1],
bins=number_of_bins,histtype='step',stacked=True,fill=False,range=range1)
plt.show()t 54 tਦ̕: ළౕ෾ා͹รયɽϧϱξϞΤΧʖΩʖ͹Ғ஖͹෾ාͺɼਜ਼‫و‬෾ාʤN(И, О 2 )ʥͶॊ͑ɽ࣎ؔ‫ܨ‬գͳͳ΍Ͷɼਜ਼‫و‬෾ා͹ฑ‫ۋ‬Иʻ0 ͺ͖ΚΔͥɼ෾ࢆО 2 ͗୉͚͘͵ͮͱ͏͚ɽͨ͹݃Վɼै෾Ͷ࣎ؔ͗‫ܨ‬գͤΖͳɼϧϱξϞΤΧʖΩʖ͗‫ؔۯ‬ͶҲ༹Ͷ෾ා͢ɼҲ༹෾ාͶۛࣇͲ͘ΖΓ͑Ͷ͵ΖɽColaboratory Ͳ Python ͳϧ΢ϔϧϨ matplotlib Ν࢘͑ͳɼϔϧΤϱӣಊͳ֨ࢆ๏ఖࣞ͹ ຌ Ͷ ࢘ ༽ Ͳ ͘ Ζ ਦ ʤ ྭ ͓ ͻ ɼ ธ ୖ ෍ ඔ ࢢ ͹ ຌ ͹ න ࢶ ʥ ͗ ؈ ୱ ͵ ϕ ϫ ή ϧ Ϟ Ͳ ࡠ Η Ζɽ
matplotlib.org ͹ Gallery Ͷͺɼ໚പ͏γϱϕϩϕϫήϧϞ͗ଡ͚͍Ζ[5]ɽ
ʤAnimation ͹ΩτβϨʖͶɼ2 ॑৾Εࢢ͵ʹ͹ಊժ΍͍ΖɽʥGallery ͹ URL ΝҐԾͶ‫͚͕ͱ͢ى‬ɽ
https://matplotlib.org/stable/gallery/index.html
ϕϫήϧϞ͹ॽ͘‫͓׷‬ͶேકͤΖ͞ͳΝ௪͢ͱɼPython ͹ϕϫήϧϝϱήͶ΍‫׵‬ΗΖͫ
Θ͑͞ͳ͗‫غ‬ଶ͠ΗΖɽ঱ΔͥͶɼ͏Θ͏Θͳࢾ͢ͱֺ͢ΞͲΆ͢͏ɽ
͠ͱɼPython ͹ϧ΢ϔϧϨ matplotlib Ν࢘͑͞ͳͶΓΕɼ؈ୱͶήϧϓΝࡠΗΖ͹ΝΊ
ͱͪ͘ɽ͞͹Γ͑͵ϧ΢ϔϧϨ͗ैࣰ͢ͱ͏Ζ͹΍ Python ͹ಝ௅͹ͽͳͯͲ͍Εɼสཤ͵ϧ΢ϔϧϨΝ‫ͯ͜ݡ‬Ζ͹΍ Python ϕϫήϧϝϱή͹ֺ͢ΊͲ͍ΖɽPython ͺɼָ;Ώͤ͏ϕϫήϧϝϱή‫͍Ͳޢݶ‬Εʤྭ͓ͻɼࢂߡॽ[6]Νࢂলʥɼ͠ΔͶ Colaboratory Ν࢘͑͞ͳͶΓΕɼͤ͛ͶϕϫήϧϝϱήΝ࢟ΌΔΗΖɽͧͽɼη ϜʖφϓΧϱ͵ʹΝ࢘͏घܲͶϕϫήϧϝϱήΝֺ͢ΞͲ௘ͪ͜Δࢧ͑ɽΉͪɼϕϫήϧϞͺ‫ܙ‬͹Ռָճ͹ HP ͖Δ
download Ͳ͘ΖΓ͑ͶͤΖ༩ఈͲ͍Ζɽʴʤ஭ʥࡠ੔ͪ͢ਦͺຌཔΩϧʖͲ͍ΕɼപࠉͲһࡰͤΖͳ‫ݡ‬Ͷ͚͏৖߻͍͗Ε Ήͤ ɽʵःࣛ‫ߚݬ‬ΝಣΞͲ༙ӻ͵αϟϱφΝ͢ͱ͚ͫͮͪ͠∃ీभध‫گ‬दΝͺͣΌͳͤΖෳҬ୉ָ୉ָӅ޽ָ‫ڂݜ‬Ռஎ೵εητϞ޽ָ઒ߊඉત‫ܙ‬Ռָ‫࣪ڂݜ‬͹ϟϱώʖͶ‫״‬ः͏ͪ͢Ήͤɽࢂߡช‫ݛ‬[1]ธୖ෍ඔࢢʁϔϧΤϱӣಊ෼ཀྵָ One Point 27 ‫ת‬ɼ‫ཱིڠ‬ड़൝ɼpp. 160(1986)ɿϔϧΤϱӣಊͶ‫ͱؖ͢‬ɼε ϝϣ Ϫʖεϥϱ݃ՎΝࣖ ͢͵ ͗Δɼ෼ཀྵָΝ઒໵ ͳ͢ ͵͏๏Ͷ΍෾ΕΏͤ͏ࢂߡॽ

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ja.wikipedia.org/wiki/ϔϧΤϱӣಊͶΉͯΚΖ‫ޣ‬մ[3]ฑీཾ߀ʁColaboratory Ͳ‫ܙ‬͹ՌָΝֺ͢΍͑ʷαρϙ‫ۄ‬તΝඵ͏ͱΊΓ͑ʷ ɼ‫ܙ‬͹Ռָճࢿɼ35 ‫ת‬ɼ1 ߺɼ75-80ɼʤ2020ʥ

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Google Colaboratory

https://colab.research.google.com/notebooks/intro.ipynb#[5]matplotlib.org[6]έζϧඊߨ‫ط‬ʁκϫ͖ΔΏ͚͢͠ͺͣΌΖ Python ೘໵ɼϜ΢ψϑड़൝ɼpp.255ɼʤ2018ʥ

https://matplotlib.orgߪ࠴Ͷ‫͠ࡎܟ‬ΗͪϕϫήϧϞͺɼ‫ܙ‬͹Ռָճ͹ HP ͹ҐԾ͖ΔξΤϱϫʖχͲ͘Ήͤɽ
https://katachi-jp.com/journal36ΩρφˏϘʖηφͲ Colaboratory ͹αʖχιϩͶహΕ෉͜ͱɼૺΔͦͱΊͱ͚ͫ͠͏ɽ

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