ᙧࡢ⛉Ꮫ఍ㄅ ➨  ᕳ ➨  ྕ 

┠

ḟ

࠙ㄽᩥࠚ
ᅄ㔜ྍ㏫㠃✚ಖᏑ෗ീ࡟࠾ࡅࡿᅄ㔜ᑐ⛠࿘ᮇ㌶㐨
ᒣཱྀ ႐༤ ·····································································

࠙ㄽᩥࠚᶆ‽෗ീ࡟࠾ࡅࡿ␗ᖖᅇ㌿ศᒱᒣཱྀ ႐༤ ····································································· ࠙ㄽᩥࠚ
3\WKRQ ࢆጞࡵ࡚㸯 ᪥࡛ࢤ࣮࣒ࢆసࢀࡿࡼ࠺࡟࡞ࡿࡢ࠿㸽̽ 㸴㸯ṓ࠿ࡽࡢᣮᡓ㸸㸮࠿ࡽጞࡵࡿ 3\WKRQ3URJUDPPLQJ ̽ᖹ⏣ 㝯ᖾ ····································································· ࠙㐃㍕ㅮᗙࠚ
&RODERUDWRU\ ࡛ᙧࡢ⛉Ꮫࢆᴦࡋࡶ࠺̽ ➨  ᅇ ࢥࢵ࣍᭤⥺ࢆᥥ࠸࡚ࡳࡼ࠺ ̽ᖹ⏣ 㝯ᖾ ····································································· ࠙఍࿌࡞࡝ࠚ
఍࿌···································································································· ཎ✏ເ㞟 ···························································································· ࿨ช‫ܙ‬͹Ռָճࢿ ୊ 35 ‫ ת‬୊ 1 ߺ࢝॑Ն‫ٱ‬໚੷ฯଚࣺ଀Ͷ͕͜Ζ࢝॑ଲসब‫ٌغ‬ಕࢃ޳‫ج‬ദ
ˡ-

એཁ‫ࠅࢤݬࢤݟ‬෾࣋ୈ஦ԟ --

FKDRVIUDFWDO#L&ORXGFRP

4XDGUXSO\V\PPHWULFSHULRGLFRUELWV
LQWKHTXDGUXSO\UHYHUVLEOHDUHD-SUHVHUYLQJPDS<RVKLKLUR<DPDJXFKL--.RNXEXQMLGDL-&KXR,FKLKDUD&KLED--DSDQ(2020 ೧ 7 ݆ 23 ೖण෉ɾ2020 ೧ 9 ݆ 15 ೖणཀྵ)Abstract: The Harper map is a typical example of the quadruply reversible area -preserving
maps defined on torus. In this map, there exist the singly symmetric periodic orbits, thedoubly symmetric orbits (DSPOs) and the quadruply symmetric periodic orbits (QSPOs).
The period ‫ ݍ‬of QSPO is an even number, but not a multiple of four. In order tocharacterize QSPOs, the map ܶ ௤ is separated into the three operations; the left-involution,
the identity map and the right-involution. Using the left-involution and the right-involution,
we can derive the condition that QSPO exists. The identity map also gives the same condition
for the existence of QSPO. The stability coefficient ‫ݎ‬ሺܽሻ to determine the property of theelliptic periodic orbit is introduced. We obtain that ‫ݎ‬ሺܽሻ as a function of ܽ for DSPO has a
U-shaped structure and ‫ݎ‬ሺܽሻ for QSPO has a W-shaped structure. Let ‫ ܫ‬be the rotation
number interval that the elliptic periodic orbit undergoes the rotation bifurcation. Theinterval ‫ ܫ‬is determined as follows : ‫ ܫ‬ൌ ሺͲǡͳሻ for DSPO and ‫ ܫ‬ൌ ሺͲǡʹሻ of QSPO.
Keywords: Harper map, Quadruply symmetric periodic orbit, Involution representation,
Stability coefficient, Bifurcation

1. ຌ࿨ช͹໪దࢂߡช‫[ݛ‬1]Ͷ͕͏ͱ, ජंͺඬ६ࣺ଀ͶଚࡑͤΖ್॑ଲসब‫ٌغ‬ಕ͹੓࣯Ν٠࿨ͪ͢ . ್॑ଲসब‫ٌغ‬ಕΝಝ௅෉͜Ζ৿͢͏ଲ߻න‫ݳ‬Ν಍೘͢ , ৿͢͏ଲ߻න‫ݳ‬͹େ਼ߑଆͳ‫ش‬Ձָద͵қັΝ໎Δ͖Ͷͪ͢. ࣏Ͷ, ବԃ‫್॑ܗ‬ଲসब‫ٌغ‬ಕ͗௪৙͹ब‫غ‬ഔ෾‫ز‬Νًͥ͞͠յ఺෾‫ز‬Ν ًͤ͞ͳ ͏͑ ৿͢͏෾‫ݳ ز‬েΝ‫ ݡ‬ड़͢ , ෾‫ز‬͹੓࣯Νմ໎ ͪ͢. ଲ߻න‫ݳ‬͹ ֨௃Νߨ͏, ଲস੓ͨ͹΍͹͗ब‫ٌغ‬ಕ͹෾‫ݳز‬েΝ੏‫͏ͱ͢ݸ‬Ζ͞ͳΝࣖͪ͢ .
2004 ೧Ͷජं౵ͺύʖϏʖࣺ଀ (Harper map)[2]͹ଲসब‫ٌغ‬ಕ͹෾ྪͶ‫ͤؖ‬ΖๅࠄΝߨͮͪ[3]. ͖͢͢, ଲস ब‫ٌغ‬ಕ ͹੓࣯͹ ‫ݗ‬౾ͺ ߨΚ͵͖ ͮͪ . ύʖϏʖ ࣺ଀ Ͷͺ, ඬ६ࣺ଀ͲଚࡑͤΖ್॑ଲসब‫ٌغ‬ಕΓΕߍͶଲস੓͹߶͏࢝॑ଲসब‫ٌغ‬ಕ͗ଚࡑͤΖ . ଲস੓͗߶͚͵ͮͪ͞ͳ͗෾‫ݳز‬েͶʹ͹Γ͑͵Ӫ‫ڻ‬Ν΍ͪΔͤ͹ͫΘ͖͑ . ඬ६ࣺ଀͹್॑ଲসt 1 tब‫ ٌغ‬ಕ͹ ੓࣯ ΓΕ ߍͶ ‫ັڷ‬਄ ͏੓ ࣯͗ ύʖ Ϗ ʖࣺ଀ ͹ଲ সब ‫ ٌغ‬ಕͲ ‫ ͖ͯݡ‬Ζ͹ Ͳͺ͵͏͖ͳ‫غ‬ଶ͠ΗΖ. ࣺ଀ͺҡ͵Ζ͗ຌ࿨ช͹಼༲ͺࢂߡช‫[ ݛ‬1]͹କ͘Ͳ͍Ζ.
ຌ ࿨ ช Ͳ ѽ ͑ φ ʖ ϧ η ໚ (Ͳ ൑ ‫ ݔ‬൏ ʹߨǡ Ͳ ൑ ‫ ݕ‬൏ ʹߨ)Ͳ ఈ ٝ ͠ Η ͪ Ն ‫ ٱ‬໚ ੷ ฯ ଚ ๏ ޴ ฯ ଚ ύʖϏʖࣺ଀ ܶ Ν়ղͤΖ.
ܶ ‫ݕ  ׷‬௡ାଵ ൌ ‫ݕ‬௡ ൅ ݂ሺ‫ݔ‬௡ ሻሺ‫ߨʹ݀݋‬ሻǡ‫ݔ‬௡ାଵ ൌ ‫ݔ‬௡ ൅ ݂ሺ‫ݕ‬௡ାଵ ሻሺ‫ߨʹ݀݋‬ሻǤሺ1ሻ͞͞Ͳ, ݂ሺ‫ݔ‬ሻ ൌ ܽ‫͵ ݔ‬Δ;Ͷ ݂ሺ‫ݕ‬ሻ ൌ ܽ‫ݕ‬. ͪͫ͢, ܽ ൒ Ͳ. ජंͺ, ύʖϏ ʖࣺ଀͹Ն ‫ٱ‬੓ Ͷ ‫ ͍ ͗ ৼ ؖ‬Ζ . ͨ ͹ ͪ Ό , ࢘ ༽ ͤ Ζ ࣺ ଀ ܶͳ ͢ ͱ ύ ʖ Ϗ ʖ ࣺ ଀ ͹ ஦ Ͳ ΍ ࠹ ΍ ଲ স ੓ ͗ Γ ͏৏‫ى‬͹‫ࣞܙ‬Νཤ༽ͤΖ. Ϗϧϟʖν͗ ܽͫ͜Ͳ͍Ζ͞ͳΓΕ‫ݳ‬ে͹ཀྵմ͗͢Ώͤ͏ .
ܲ଴଴ ൌ ሺͲǡͲሻͳ ܲଵଵ ൌ ሺߨǡ ߨሻ͗ γ χ ϩ ‫ ෈ ܗ‬ಊ ఼ Ͳ , ܳଵ଴ ൌ ሺߨǡ Ͳሻͳ ܳ଴ଵ ൌ ሺͲǡ ߨሻ͗ ବ ԃ ‫ ෈ ܗ‬ಊ ఼ Ͳ ͍
Ζ. ͞ΗΔͺ Ͳ ൏ ܽ ൏ ʹͲͺବ ԃ‫ Ͳ ܗ‬, ܽ ൐ ʹͲͺൕ఺ Ν൒͑ γχ ϩ‫ ͍Ͳܗ‬Ζ . ବԃ‫෈ ܗ‬ಊ ఼ͺܽ ൌ ʹͲब‫غ‬ഔ෾‫ز‬Νًͤ͞. φʖϧη໚Ͷ͕͜ΖٌಕΝਦ 1 Ͷඵ͏ͪ. ब‫ٌغ‬ಕͳଲসતΝ‫ݡ‬Ώ͚ͤනࣖͤΖͪΌͶ , φʖϧη໚Ν Ͳ ൑ ‫ ݔ‬൑ ʹߨǡ െߨ ൑ ‫ ݕ‬൑ ߨͳͪ͢.

ਦ 1ʁφʖϧη໚ (Ͳ ൑ ‫ ݔ‬൑ ʹߨǡ െߨ ൑ ‫ ݕ‬൑ ߨ) Ͷ͕͜Ζٌಕ. (a) ܽ ൌ ͲǤ͹ͷ. (b) ܽ ൌ ͳǤͷ.
ࣺ ଀ ܶͺ Ґ Ծ ͹ Γ ͑ Ͷ ࢝ झ ͹ ଲ ߻ න ‫ ى Ͳ ݳ‬फ़ ͠ Η Ζ . ଲ ߻ Ͷ Γ Ζ ෾ մ Ν ܶ ൌ ‫ܪ‬௜ ‫ܩ ל‬௜ ሺ݅ ൌͳǡʹǡ͵ǡͶሻͳॽ͚. ͞͞Ͳ ‫ל‬ͺࣺ଀ ͹ ߻੔‫͍ Ͳߺ ى‬Ζ . ‫ߺى‬͹ ӊଈ͹ ‫ܩ‬௜ Νࡠ ༽͢ , ࣏Ͷࠪ ଈ͹ ‫ܪ‬௜ Ν࣏Ͷࡠ༽ͤΖ.
‫ݕ‬‫ݕ‬‫ݕ‬‫ ݕ‬൅ ݂ሺ‫ݔ‬ሻ‫ܪ‬ଵ ቀ ቁ ൌ ቀെ‫ ݔ‬൅ ݂ሺ‫ݕ‬ሻቁǡ‫ܩ‬ଵ ቀ ቁ ൌ ቀ
ቁǤ
‫ݔ‬
‫ݔ‬
െ‫ݔ‬െ‫ݕ‬‫ݕ‬‫ݕ‬െ‫ ݕ‬െ ݂ሺ‫ݔ‬ሻ‫ܪ‬ଶ ቀ ቁ ൌ ቀ‫ ݔ‬െ ݂ሺ‫ݕ‬ሻቁ ǡ‫ܩ‬ଶ ቀ ቁ ൌ ቀ
ቁǤ
‫ݔ‬
‫ݔ‬
‫ݔ‬െ‫ ݔ‬൅ ߨെ‫ ݔ‬െ ݂ሺ‫ ݕ‬൅ ݂ሺ‫ݔ‬ሻሻ ൅ ߨ‫ݕ‬‫ݕ‬‫ܪ‬ଷ ቀ ቁ ൌ ቀെ‫ ݕ‬൅ ߨቁ ǡ‫ܩ‬ଷ ቀ ቁ ൌ ൬
൰Ǥ
‫ݔ‬
‫ݔ‬
െ‫ ݕ‬െ ݂ሺ‫ݔ‬ሻ ൅ ߨ‫ݔ‬െߨ‫ ݔ‬൅ ݂ሺ‫ ݕ‬൅ ݂ሺ‫ݔ‬ሻሻ െ ߨ‫ݕ‬‫ݕ‬‫ܪ‬ସ ቀ ቁ ൌ ቀ‫ ݕ‬൅ ߨቁ ǡ‫ܩ‬ସ ቀ ቁ ൌ ൬
൰Ǥ
‫ݔ‬
‫ݔ‬
‫ ݕ‬൅ ݂ሺ‫ݔ‬ሻ ൅ ߨ

ሺ2ሻ
ሺ3ሻ
ሺ4ሻ
ሺ5ሻ

͞͞ Ͳ, ‫ܩ‬௜ ‫ܩ ל‬௜ ൌ ‫ܪ‬௜ ‫ܪ ל‬௜ ൌ ‫݀ܫ‬Ǥ. ଲ߻ͺ ໚੷ ฯ ଚͲ ͍Ζ ͗๏ ޴ൕ ఺ࣺ ଀ Ͳ͍ Ζ . ࣺ଀ ܶͲͺร਼‫ͳݕ‬ร਼‫ݔ‬Ͷ ‫ ߨʹ݀݋‬Νࡠ༽ͤΖ . ͨ͹ͪΌଲ߻͹න‫ Ͳݳ‬൅ߨΝെߨͳ͢, െߨΝ൅ߨͳ஖͘‫͓ ׷‬
ͱ΍Γ͏.

t 2 t

ਦ 2ʁφʖϧη໚ (Ͳ ൑ ‫ ݔ‬൑ ʹߨǡ െߨ ൑ ‫ ݕ‬൑ ߨ)Ͷ͕͜Ζ, ࠪӊ৏Ծ͹ଲসત (ਦ(a))ͳࣾΌࠪӊ৏Ծ͹ଲসત (ਦ(b)). ܽ ൌ ͳ.
ଲ߻͹෈ಊ఼͹ॄ߻͗ଲসતͲ͍Ζ . ଲ߻ͶΓΖब‫ٌغ‬ಕ఼͹ಊ͘ͺଲসતͶ‫ͤؖ‬Ζൕ఺
Ͳ͍ Ζ. Γͮ ͱଲ ߻ ͶΓ Ζब ‫ٌ غ‬ಕ ఼ ͹ಊ ͘ ΓΕ , ܵுభ ͳܵீభ Ν ࠪӊ ͹ ଲস ત ͳ‫ ; ݼ‬, ܵுమ ͳ ܵீమΝ৏Ծ͹ଲসત ͳ‫ ΁ݼ‬. สٕదͶ ܵுయ ͳܵீయ ΝࣾΌࠪӊ͹ ଲসતͳ‫ ;ݼ‬, ܵுర ͳ ܵீర ΝࣾΌ৏Ծ͹ଲসતͳ‫΁ݼ‬. ͞ΗΔ͹ଲসતΝҐԾͶࣖͪ͢ . Ήͪ, ଲসતΝਦ 2 Ͷඵ͏ͪ͹ͲࢂߡͶ͢ͱΆ͢͏.
ࠪӊ͹ଲসત.
ܵுభ ǣ‫ ݔ‬ൌ ݂ሺ‫ݕ‬ሻȀʹǡ‫ ݔ‬ൌ ݂ሺ‫ݕ‬ሻȀʹ ൅ ߨǡ

ሺ6ሻ

ܵீభ ǣ

‫ ݔ‬ൌ Ͳǡ‫ ݔ‬ൌ ߨǤ

ሺ7ሻ

ܵுమ ǣ

ܵுమ ǣ ‫ ݕ‬ൌ Ͳǡ‫ ݕ‬ൌ ߨǡ

ሺ8ሻ

ܵீమ ǣ

‫ ݕ‬ൌ െ݂ሺ‫ݔ‬ሻȀʹǡ‫ ݕ‬ൌ െ݂ሺ‫ݔ‬ሻȀʹ ൅ ߨǤ

ሺ9ሻ

ܵுయ ǣ

‫ ݕ‬ൌ െ‫ ݔ‬൅ ߨǡ

(10)

ܵீయ ǣ

‫ ݕ‬ൌ െ‫ ݔ‬െ ݂ሺ‫ݔ‬ሻ ൅ ߨǤ

(11)

ܵுర ǣ

‫ ݕ‬ൌ ‫ ݔ‬െ ߨǡ

(12)

ܵீర ǣ

‫ ݕ‬ൌ ‫ ݔ‬െ ݂ሺ‫ݔ‬ሻ െ ߨǤ(13)৏Ծ͹ଲসત.

ࣾΌࠪӊ͹ଲসત.

ࣾΌ৏Ծ͹ଲসત.

ࣺ଀͗ଲ߻͹੷Ͳන‫͠ݳ‬ΗΖ৖߻ , ‫ܧ‬ͺώʖαϓ͹қັͲ͹Ն‫ٱ‬੓Ν΍ͯͳ͏͑ [4]. ৏‫ى‬͹Γ͑Ͷ࢝ झྪ͹ҡ ͵ͮ ͪଲ߻͹੷ Ͳන‫ ͠ݳ‬ΗΖ ࣺ଀ͺ , ࢝॑Ն‫ͻݼ ͳ଀ ࣺٱ‬ΗΖ . ࢝॑Ն‫଀ࣺٱ‬Ͷͺ, Ҳ॑ଲসब‫ٌغ‬ಕ, ್॑ଲসब‫ٌغ‬ಕ, ࢝॑ଲসब‫ٌغ‬ಕ͗ଚࡑͤΖ. ଲসતΝཤ༽͢ͱ͞ΗΔ͹ଲসब‫ٌغ‬ಕΝఈٝͤΖ .
ఈٝ 1.1.
[i] ଲসब‫ٌغ‬ಕͳඉଲসब‫ٌغ‬ಕ .
t 3 tଲসત৏Ͷٌಕ఼Ν΍ٌͯಕΝଲসٌಕͳ͏͑ . ଲসٌಕ͗ଲসત৏Ͷ۰਼‫ݺ‬͹ٌಕ఼Ν΍ͯ͵Δͻ, ͞͹ٌಕͺ ଲসब‫ ٌغ‬ಕͳ‫ ͻݼ‬ΗΖ . ຌ࿨ชͲͺ෈ಊ఼ ͺଲ সब‫ٌغ‬ಕ Ͷ‫؜‬Όͥ, ෈ಊ఼ͳ͢ͱѽ͑. ଲসત৏Ͷٌಕ఼Ν࣍ͪ͵͏ब‫ٌغ‬ಕΝඉଲসब‫ٌغ‬ಕͳఈٝͤΖ .
[ii] Ҳ॑ଲসब‫ٌغ‬ಕ.
৏Ծ͹ଲস ત͹Ί Ͷٌಕ ఼Ν΍ͯब ‫ٌغ‬ಕ Ν , Ҳ॑৏Ծଲস ब‫ ٌغ‬ಕͳఈ ٝͤΖ . ࠪӊ͹ଲসત͹Ί Ͷٌಕ఼ Ν΍ ͯब‫ٌ غ‬ಕ Ν, Ҳ॑ࠪӊ ଲসब‫ ٌغ‬ಕͳఈٝ ͤΖ . ࣾΌ৏Ծ͹ଲসત͹ΊͶٌ ಕ఼Ν΍ ͯब ‫ٌغ‬ಕΝ , Ҳ॑ࣾΌ ৏Ծ ଲসब‫ ٌغ‬ಕͳఈٝ ͤΖ . ࣾΌࠪӊ͹ଲসત͹ΊͶٌ ಕ఼Ν΍ ͯब ‫ٌغ‬ಕΝ , Ҳ॑ࣾΌ ࠪӊ ଲসब‫ ٌغ‬ಕͳఈٝ ͤΖ . ͞ΗΔΝΉͳΌͱҲ॑ଲসब‫ٌغ‬ಕͳ‫ ΁ݼ‬.
[iii] ್॑ଲসब‫ٌغ‬ಕ.
৏Ծͳࠪӊ͹ଲসત৏Ͷٌಕ఼Ν΍ͯब‫ٌغ‬ಕΝ , ್॑৏Ծࠪӊଲসब‫ٌغ‬ಕͳఈٝͤΖ .
ࣾΌ৏ԾͳࣾΌࠪӊ͹ଲসત৏Ͷٌಕ఼Ν΍ͯब‫ٌغ‬ಕΝ , ್॑ࣾΌ৏Ծࠪӊଲসब‫ٌغ‬ಕ
ͳఈٝͤΖ. ͞ΗΔΝΉͳΌͱ್॑ଲসब‫ٌغ‬ಕͳ‫ ΁ݼ‬.
[iv] ࢝॑ଲসब‫ٌغ‬ಕ.
৏Ծͳࠪӊ͹ଲসત৏Ͷٌಕ఼Ν΍ͬ , ͖ͯࣾΌ৏ԾͳࣾΌࠪӊ͹ଲসત৏Ͷٌಕ఼Ν΍ͯब‫ٌغ‬ಕΝ, ࢝॑ଲসब‫ٌغ‬ಕͳఈٝͤΖ .
͞͞Ͳଲসब‫ٌغ‬ಕͶͯ͏ͱึଏΝफ़΄ͱ͕͚ . Ҳ॑ଲসब‫ٌغ‬ಕͺଲসત৏Ͷ್ٌͯಕ఼Ν΍ͯ. ఼್͗Ҳ఼Ͷ͵ͮͪ৖߻͗෈ಊ ఼Ͳ͍Ζ. ್॑ଲসब‫ٌغ‬ಕͺ್ͯ͹৖߻͍͗Ζ .
ࠪӊ͹ଲস ત৏Ͷ್ ٌͯ ಕ఼Ν΍ͬ , ͖ͯ৏Ծ͹ ଲসત৏Ͷ ್ٌͯಕ ఼Ν ΍ͯ৖߻ͳ , ࣾΌࠪӊ͹ଲসત৏Ͷ್ٌͯಕ఼Ν΍ͬ , ͖ͯࣾΌ৏Ծ͹ଲসત৏Ͷ್ٌͯಕ఼Ν΍ͯ৖߻Ͳ͍
Ζ. ఼࢝͗Ҳ఼Ͷ͵ͮͪ৖߻͗෈ಊ఼Ͳ͍Ζ. ࠹‫ޛ‬Ͷ࢝॑ଲসब‫ٌغ‬ಕͺ ৏Ծࠪӊ͹ଲসત৏Ͷٌ࢝ͯಕ఼Ν΍ͬ, ͖ͯࣾΌ৏Ծࠪӊ͹ଲসત৏Ͷٌ࢝ͯಕ఼ Ν࣍ͯ. ൂ఼͗Ҳ఼Ͷ͵ͮͪ৖߻͗෈ಊ఼Ͳ͍Ζ .
ຌ࿨ชͲ٠࿨ͤΖ໲ୌΝ়ղͤΖ .
໲ୌ 1.2.
(1) ࢝॑ଲসब‫ٌغ‬ಕΝಝ௅෉͜Ζ ଲ߻න‫ݳ‬ͺՁ͖.
(2) ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕͺ , ʹ͹Γ͑͵෾‫ز‬Νॳً࣏ͤ͞͹͖ .
ຌ࿨ช͹୊ 2 ઇҐ‫ޛ‬͹಼༲Νࣖͤ.
୊ 2 ઇͲͺ, ຌ࿨ชͲ࢘༽ͤΖ‫֕ͳߺى‬೨Ν়ղͤΖ .
୊ 3 ઇͲͺ, Ҳ॑ଲসब‫ٌغ‬ಕͳ್॑ଲসब‫ٌغ‬ಕͶ‫ͤؖ‬Ζ੓࣯ΝΉͳΌΖ .
୊ 4 ઇͲͺ, ࢝॑ଲসब‫ٌغ‬ಕΝಝ௅෉͜Ζଲ߻න‫ݳ‬͹େ਼ߑଆΝ٠࿨ͤΖ .
୊ 5 ઇͲͺ, ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ͹෾‫ߑز‬ଆΝ௒΄Ζ .
୊ 6 ઇͲͺ, ຌ࿨ชͲಚΔΗͪ෾‫ز‬Ͷ‫ͤؖ‬ΖմएΝ༫͓Ζ .
୊ 7 ઇͲͺ, ຌ࿨ชͲಚΔΗͪ݃ՎΝΉͳΌΖ .

t 4 t

2. ६ඍ
2.1. ब‫ٌغ‬ಕ͹҈ఈ੓ब‫ ٌغ‬ಕ͹ ҈ఈ ੓Ν ൓இ ͤΖ๏ ๑Ν ়ղ ͤΖ . ͞͞Ͳ ‫ݍ‬ሺ൒ ͳሻΝब‫ٌ غ‬ಕ ͹ब ‫ͤ ͳغ‬Ζ . ब‫ݍغ‬͹ब‫ٌغ‬ಕ఼Ν ‫ݖ‬௞ ൌ ሺ‫ݔ‬௞ ǡ ‫ݕ‬௞ ሻሺͲ ൑ ݇ ൑ ‫ ݍ‬െ ͳሻͳͤΖ. ٌಕ఼‫ݖ‬௞ Ͷ͕͜Ζત‫ܙ‬Կߨ྽ ‫ܯ‬ሺ‫ݖ‬௞ ሻͺͳ
݂ ᇱ ሺ‫ݔ‬௞ ሻ‫ܯ‬ሺ‫ݖ‬௞ ሻ ൌ ൬ ᇱ൰(14)݂ ሺ‫ݕ‬௞ାଵ ሻ ͳ ൅ ݂ ᇱ ሺ‫ݔ‬௞ ሻ݂ ᇱ ሺ‫ݕ‬௞ାଵ ሻ
ͳಚΔΗΖ. ͞͞Ͳ, ݂Ԣሺ‫ݔ‬௞ ሻ ൌ ܽ ‫ݔ‬௞ , ݂Ԣሺ‫ݕ‬௞ାଵ ሻ ൌ ܽ ‫ݕ‬௞ାଵ . ໚੷ฯଚࣺ଀Ͳ͍Ζ͖Δ , ‫ܯ‬ሺ‫ݖ‬௞ ሻ͹ߨ྽ࣞͺ 1 Ͳ͍Ζ. ࣺ଀ܶ ௤ ሺ‫ ݍ‬൒ ͳሻ͹ત‫ܙ‬Կߨ྽Ν ‫ܯ‬௤ ൌ ‫ܯ‬ሺ‫ݖ‬௤ିଵ ሻ‫ܯ‬ሺ‫ݖ‬௤ିଶ ሻ ‫ܯ ڮ‬ሺ‫ݖ‬଴ ሻͳͤΖͳ, ‫ݽ‬༙஍ߣΝ݀ఈͤΖ๏ఖࣞߣଶ െ ‫ݎ‬ሺܽሻߣ ൅ ͳ ൌ Ͳ(15)

͗ಚΔΗΖ. ͞͞Ͳ, ‫ݎ‬ሺܽሻ ൌTrace‫ܯ‬௤ . ‫ݎ਼ܐ‬ሺܽሻΝ҈ఈ੓‫͜෉໌ͳ਼ܐ‬Ζ.
ࣞ(15)͹մͲ ͍Ζ‫༙ ݽ‬஍ Ν ߣേ ͳॽ ͚. ҈ఈ ੓‫ݎ ਼ ܐ‬ሺܽሻΝཤ༽͢ ͱ ब‫ݍ غ‬͹ब‫ ٌ غ‬ಕ͹҈ ఈ੓ Ν ఈ Ό Ζ . ൓ พ ࣞ ͺ ‫ ܦ‬ൌ ‫ ݎ‬ଶ ሺܽሻ െ ͶͲ ͍ Ζ . ‫ݎ‬ሺܽሻ ൐ ʹ͵ Δ ͻ , ‫ ༙ ݽ‬஍ ͺ Ͳ ൏ ߣି ൏ ͳ ൏ ߣା Ν ຮ ͪ
ͤ. ब‫ݍغ‬͹ब‫ٌغ‬ಕ఼ͺγχϩ‫҈෈Ͳܗ‬ఈͲ͍Ζ . ȁ‫ݎ‬ሺܽሻȁ ൏ ʹ͵Δͻ, ‫༙ݽ‬஍ߣേ ͺ෵ો‫ڠ‬ༀͲ͍ Ζ . Γ ͮ ͱ , ब ‫ݍ غ‬͹ ब ‫ ٌ غ‬ಕ ఼ ͺ ବ ԃ ‫ ҈ Ͳ ܗ‬ఈ Ͳ ͍ Ζ . ‫ݎ‬ሺܽሻ ൏ െʹ͵ Δ ͻ , ‫ ༙ ݽ‬஍ ͺ ߣି ൏െͳ ൏ ߣା ൏ ͲΝ ຮ ͪ ͤ . ब ‫ݍ غ‬͹ ब ‫ ٌ غ‬ಕ ఼ ͺ ൕ ఺ Ν ൒ ͑ γ χ ϩ ‫ ҈ ෈ Ͳ ܗ‬ఈ Ͳ ͍ Ζ . ब ‫ غ‬ഔ ब‫غ‬෾‫( ز‬ൕब ‫غ‬ഔ ब‫ غ‬෾ ‫ ) ز‬͹෾ ‫Ͳ ఼ز‬ͺ ‫ݎ‬ሺܽሻ ൌ െʹ ͗੔ Εཱིͬ , ‫ ༙ݽ‬஍ͺ ߣേ ൌ െͳͲ͍ Ζ.
ಋब‫غ‬෾‫( ز‬ൕಋब‫غ‬෾ ‫ ) ز‬͹෾‫Ͳ఼ز‬ͺ ‫ݎ‬ሺܽሻ ൌ ʹ ͗੔Εཱིͬ, ‫༙ݽ‬஍ͺߣേ ൌ ͳͲ͍Ζ. ͞͞Ͳफ़΄ͪ෾‫ز‬͹৆ࡋͺࢂߡช‫[ ݛ‬5]Ν‫ݡ‬ΔΗͪ͏.

2.2. յ఺෾‫ͳز‬ब‫غ‬ഔ෾‫ز‬ବԃ‫෈ܗ‬ಊ఼ͳ͢ͱ ܳଵ଴ Νཤ༽͢ͱյ఺෾‫ز‬౵͹ઈ໎Νߨ͑ . Ϗϧϟʖν ܽΝ 0 ͖ΔଁՅͤ
Ζͳ, ବԃ‫෈ܗ‬ಊ఼ ܳଵ଴ ۛ๥Ͷ͕͜Ζٌಕ఼͹յ఺͹࢕๏͗ଐ͚͵Ζ . յ఺͹๏޴ͺ࣎‫ܯ‬յΕͲ͍ Ζ. ͞͹ յ ఺͹ ࢕ ๏ ͹ࢨ ඬ ͳ͢ ͱ , ‫ع‬༁ ෾ ਼ Ͳ͍ Ζ յ఺ ਼ ‫݌‬Ȁ‫ݍ‬Ν಍ ೘ ͤΖ . ͞͞ Ͳ ‫ݍ‬ͺब‫غ‬. ब‫ٌغ‬ಕ͗ 1 ब‫غ‬͹ؔͶ ܳଵ଴ ͹बΕΝյ఺ͤΖյ਼͗ ‫͍Ͳ݌‬Ζ. յ఺਼‫݌‬Ȁ‫ݍ‬ͺҲյ͹ࣺ଀Ͳٌಕ͗ฑ‫ ߨʹ ֱౕۋ‬ൈ ‫݌‬Ȁ‫ܳͫ͜ݍ‬ଵ଴ ͹बΕΝյ఺ͤΖ͞ͳΝқັͤΖ . Ϗϧϟʖν ܽ͗ܽ௖ ሺ‫݌‬Ȁ‫ݍ‬ሻ ൌ ʹሺߨ‫݌‬Ȁ‫ݍ‬ሻሺͲ ൏ ‫݌‬Ȁ‫ ݍ‬൑ ͳȀʹሻ(16)

Ν ຮ ͪ ͤ ͳ , ଲ স ત ৏ Ͷ ٌ ಕ ఼ Ν ࣍ ͯ յ ఺ ਼ ‫݌‬Ȁ‫ݍ‬͹ ब ‫ ٌ غ‬ಕ ͗ ܳଵ଴ Γ Ε ਫ਼ ͣ Ζ . Ͳ ൏ ‫݌‬Ȁ‫ ݍ‬൏ͳȀʹΝຮͪͤ৖߻Νյ఺෾‫ ;ݼͳز‬, ‫݌‬Ȁ‫ ݍ‬ൌ ͳȀʹΝຮͪͤ৖߻Νब‫غ‬ഔ෾‫ ΁ݼͳز‬.
ࣞ (16)Ν ಍ ͚ . ෈ ಊ ఼ ܳଵ଴ ͹ ब Ε Ͳ ત ‫ ܙ‬մ ੵ Ν ߨ ͑ ͳ , ત ‫ ܙ‬Կ ߨ ྽ ‫ܯ‬ଵ ͗ Ծ ‫ ى‬͹ Γ ͑ Ͷ ಚ Δ
ΗΖ.
‫ܯ‬ଵ ൌ ቀ

ͳ
ܽ

െܽ
ቁǤ
ͳ െ ܽଶ

(17)್ͯ͹‫༙ݽ‬஍ ߣേ ͺ, Ͳ ൏ ܽ ൏ ʹͲͺ෵ો਼ͳ͵Ζ.
ߣേ ൌሺʹ െ ܽଶ ሻ േ ݅ξͶܽଶ െ ܽସ‫ ؠ‬ሺʹߨ‫݌‬Ȁ‫ݍ‬ሻ േ ݅ሺʹߨ‫݌‬Ȁ‫ݍ‬ሻǤʹ(18)

յ఺ ਼ ‫݌‬Ȁ‫ݍ‬Ν ݀Ό ͱ ࣞ (18)Ν ܽͶͯ ͏ ͱ մ͜ ͻ ྡ ֆ ஍ ܽ௖ ሺ‫݌‬Ȁ‫ݍ‬ሻ͗‫ ٽ‬Ή Ζ . ͞͞ Ͳߨ ͵ ͮ ͪ‫ܯ‬ࢋΝ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻΝ༽͏ͱ࠸ౕߨ͑ͳ, յ఺਼‫݌‬Ȁ‫҈ͳݍ‬ఈ੓‫ݎ ਼ܐ‬ሺܽሻ͹‫͗ܐؖ‬ಚΔΗΖ.
‫ݎ‬ሺܽሻ ൌ ʹ ሺʹߨ‫݌‬Ȁ‫ݍ‬ሻǤ

t 5 t

(19)

2.3. ܑ͘Ͷ‫ͤؖ‬Ζࣺ଀෈ಊ ఼ ܳଵ଴ Ν௪գ ͤΖ ଲস ત Ν ‫ ݕ‬ൌ ‫ܨ‬ሺ‫ݔ‬ሻͳॽ͚. ‫ܨ‬ሺߨሻ ൌ ͲͲ͍Ζ ͞ͳ Ν ஭қ ͤΖ . ෈ಊ఼ܳଵ଴Ͷ ͕ ͜ Ζ ଲ স ત ͹ ܑ ͘ Ν ߦ଴ ൌ ݀‫ܨ‬ሺߨሻȀ݀‫ ͤ ͳݔ‬Ζ . ࣺ ଀ ܶͶ Γ ͮ ͱ ଲ স ત ͹ ܑ ͘ ͗ ร Կ ͤ Ζ . ܑ͘͹รԿͺҲ࣏‫଀ࣺݫ‬ߦ௡ାଵ ൌߦ௡ െ ܽ
ܽሺߦ௡ െ ܽሻ ൅ ͳͲ݀ఈ͠ΗΖ. ͞͞Ͳࣞ(20)͹಍ड़๏๑Νઈ໎ͤΖ. ࣺ଀ܶΝ‫ݔ‬௡ Ͳඏ෾ͤΖͳ݀‫ݕ‬௡ାଵ ݀‫ݔ‬௡ାଵ ݀‫ݕ‬௡ାଵ ݀‫ݕ‬௡ൌൌ൅ ܽ ‫ݔ‬௡ ǡ݀‫ݔ‬௡݀‫ݔ‬௡ ݀‫ݔ‬௡ାଵ ݀‫ݔ‬௡݀‫ݔ‬௡ାଵ
݀‫ݕ‬௡ାଵ݀‫ݕ‬௡ൌ ͳ ൅ ܽ ‫ݕ‬௡ାଵൌ ͳ ൅ ܽ ‫ݕ‬௡ାଵ ൬൅ ܽ ‫ݔ‬௡ ൰
݀‫ݔ‬௡
݀‫ݔ‬௡
݀‫ݔ‬௡

(20)

(21)
(22)

͗ ಚ Δ Η Ζ . ‫ݔ‬௡ ൌ ߨǡ ‫ݕ‬௡ାଵ ൌ Ͳͳ ͢ ͱ , ࣞ (22)Ν ࣞ (21)Ͷ େ ೘ ͤ Ζ . ܑ ͘ ߦ௡ ൌ ݀‫ݕ‬௡ Ȁ݀‫ݔ‬௡ ͳߦ௡ାଵ ൌ݀‫ݕ‬௡ାଵ Ȁ݀‫ݔ‬௡ାଵ Ν಍೘͢, ͞ΗΔΝ༽͏Ζͳࣞ(20)͗಍͖ΗΖ.

2.4. ϛΠϱΩϪࢨ਼ϛΠϱΩϪࢨ਼Νఈٝ͢ , ϛΠϱΩϪࢨ਼Ͷ‫ͤؖ‬ΖఈཀྵΝ়ղͤΖ [5]. ϛΠϱΩϪࢨ਼ͺ‫Ͳނׇ‬ҕΞͲන‫ͤݳ‬Ζ.
ఈٝ 2.1 (ϛΠϱΩϪࢨ਼͹ఈٝ).
ࣺ଀ܶ ௤ ሺ‫ ݍ‬൒ ͳሻͶ͕͜Ζ෈ಊ఼ ‫ݖ‬͹ϛΠϱΩϪࢨ਼ΝҐԾ͹Γ͑ͶఈٝͤΖ .
(1) ෈ಊ఼‫͗ݖ‬γχϩ‫ܗ‬ब‫ٌغ‬ಕ఼͵Δ , ͨ͹ϛΠϱΩϪࢨ਼Ν ሺെͳሻͳͤΖ(2) ෈ಊ఼‫͗ݖ‬ବԃ‫ܗ‬ब‫ٌغ‬ಕ఼͵Δ , ͨ͹ϛΠϱΩϪࢨ਼Ν ሺ൅ͳሻͳͤΖ.
(3) ෈ಊ఼‫͗ݖ‬ൕ఺‫ܗ‬γχϩब‫ٌغ‬ಕ఼͵Δ , ͨ͹ϛΠϱΩϪࢨ਼Ν ሺ൅ͳሻͳͤΖ.
ఈ ཀྵ 2.2 (ϛ Π ϱ Ω Ϫ ࢨ ਼ ͹ ૱ ࿪ ฯ ଚ ଉ ). ෾ ‫ ز‬͹ ઴ ‫ Ͳ ޛ‬ϛ Π ϱ Ω Ϫ ࢨ ਼ ͹ ૱ ࿪ ͺ ฯ ଚ ͠ Η
Ζ.

2.5.  ౕյ఺ࡠ༽
ྭͳ͢ͱ್ͯ͹ଲ߻͹੷ ‫ܪ‬ଵ ‫ܪ‬ଶ Ν‫ͤࢋܯ‬Ζͳ‫ݕ‬െ‫ݕ‬‫ܪ‬ଵ ‫ܪ‬ଶ ቀ ቁ ൌ ቀ ቁ
‫ݔ‬
െ‫ݔ‬

(23)

͗ಚΔΗΖ. ͯΉΕ, ‫ܪ‬ଵ ‫ܪ‬ଶ ͺ 180 ౕյ఺ࡠ༽Νන‫͏ͱ͢ݳ‬Ζ . ࣞ(23)ͺ, ܲ଴଴ ൌ ሺͲǡͲሻΝ஦ৼͳ
ͪ͢ 180 ౕյ ఺Ν қັ ͢ͱ ͏Ζ . ͞͹ ࣞ͹ ӊว ͹ െ‫ݔ‬Νെ‫ ݔ‬൅ ʹߨͳͤ Ηͻ , ܳଵ଴ ൌ ሺߨǡ ͲሻΝ஦ ৼ
ͳͪ͢ 180 ౕյ఺Ν༫͓Ζ. 180 ౕյ఺ࡠ༽ܴ͹නࣞͳ੓࣯ΝΉͳΌ, ੓࣯ 2.3 ͳͤΖ.
੓࣯ 2.3.
(1) 180 ౕյ఺ࡠ༽ ܴͺҐԾ͹Γ͑Ͷ್ͯ͹ଲ߻͹੷Ͳන‫͠ݳ‬ΗΖ .
ܴ ൌ ‫ܪ‬ଵ ‫ܪ‬ଶ ൌ ‫ܪ‬ଶ ‫ܪ‬ଵ ൌ ‫ܪ‬ଷ ‫ܪ‬ସ ൌ ‫ܪ‬ସ ‫ܪ‬ଷ ൌ ‫ܩ‬ଵ ‫ܩ‬ଶ ൌ ‫ܩ‬ଶ ‫ܩ‬ଵ ൌ ‫ܩ‬ଷ ‫ܩ‬ସ ൌ ‫ܩ‬ସ ‫ܩ‬ଷ Ǥ(24)(2) ଲ স ब ‫ ٌ غ‬ಕ ͹ ್ ͯ ͹ ٌ ಕ ఼ ‫ݖ‬଴ ͳ ‫ݖ‬௞ ሺ݇ ് Ͳሻ͗ ಋ ͣ ଲ স ત ৏ Ͷ ͍ Ε , ‫ݖ‬௞ ൌ ܴ‫ݖ‬଴ Ν ຮ ͪ ͤ ͳ
ͤΖ. ͞͹ଲসब‫ٌغ‬ಕ͹ब‫ݍ غ‬ͺ‫ ݍ‬ൌ ʹ݇Ͳ͍Ζ. Γͮͱ, 180 ౕյ఺ࡠ༽ܴͺ ܴ ൌ ܶ ௤Ȁଶ Ήͪͺܴ ൌ ܶ ି௤Ȁଶ ͳ͖͜Ζ.
t 6 t

3. Ҳ॑ଲসब‫ٌغ‬ಕͳ್॑ଲসब‫ٌغ‬ಕ
3.1. Ҳ॑ଲসब‫ٌغ‬ಕ͹ଲ߻න‫ݳ‬ύʖϏʖࣺ଀Ͷ͕͜ΖҲ॑ଲসब‫ٌغ‬ಕ͹ଲ߻න‫ݳ‬ΝΉͳΌͱ͕͚ .
ܶ ଶ௞ ൌ ܶ ଶ௞ିଵ ‫ܪ‬௜ ‫ܩ ל‬௜ ൌ ‫ܪ‬௜ ‫ܩ ל‬௜ ܶ ଶ௞ିଵ ǡ
ܶ

ଶ௞ାଵ

ଶ௞

(25)

ଶ௞

ൌ ܶ ‫ܪ‬௜ ‫ܩ ל‬௜ ൌ ‫ܪ‬௜ ‫ܩ ל‬௜ ܶ Ǥ(26)͞͞Ͳ, ݅ ൌ ͳǡʹǡ͵ǡͶ. ࣞ(25)͹৖߻, ब‫ݍغ‬ͺ ‫ ݍ‬ൌ ʹ݇ሺ݇ ൒ ͳሻͲ͍Ε, ࣞ(26)͹৖߻, ब‫ݍغ‬ͺ ‫ ݍ‬ൌʹ݇ ൅ ͳሺ݇ ൒ ͳሻͲ ͍ Ζ . ύ ʖ Ϗ ʖ ࣺ ଀ Ͷ ͺ ଡ ͚ ͹ Ҳ ॑ ଲ স ब ‫ ٌ غ‬ಕ ͗ ͍ Ζ [3]. ͞ Η Δ ͹ ੓ ࣯Ͷͯ͏ͱͺ͞͞Ͳͺ٠࿨͢͵͏ .
ྭͳ͢ͱ ܶ ଶ௞ ൌ ܶ ଶ௞ିଵ ‫ܪ‬௜ ‫ܩ ל‬௜ ͹න‫Ͳݳ‬, ܶ ଶ௞ିଵ ‫ܪ‬௜ Νࠪଲ߻, ‫ܩ‬௜ Νӊଲ߻ͳ‫΁ݼ‬. ࠪଲ߻ͳӊଲ߻Ͷͯ͏ͱ෈ร͵ॄ߻Νන 1 Ͷࣖͤ. ͞ΗΔ͹෈รॄ߻ͺଲসતͳ‫ͻݼ‬ΗΖ .
න̏ʁଲ߻ͳ෈รॄ߻
ଲ߻

‫ܪ‬௜

‫ܩ‬௜

ܶ ଶ௞ିଵ ‫ܪ‬௜‫ܩ‬௜ ܶ ଶ௞ିଵܶ ଶ௞ ‫ܪ‬௜‫ܩ‬௜ ܶ ଶ௞

෈รॄ߻

ܵு೔

ܵீ೔

ܶ ௞ ܵீ೔ܶ ି௞ ܵு೔ܶ ௞ ܵு೔ܶ ି௞ ܵீ೔

ྭͳ͢ͱܶ ௞ ܵீ೔ ͗ࠪଲ߻ ܶ ଶ௞ିଵ ‫ܪ‬௜ Ͷͯ͏ͱ෈รॄ߻Ͳ͍Ζ͞ͳΝ֮೟͢Γ͑ .
ܶ ଶ௞ିଵ ‫ܪ‬௜ ‫ ܶ ל‬௞ ܵீ೔ ൌ ܶ ௞ିଵ ‫ܪ‬௜ ܵீ೔ ൌ ܶ ௞ିଵ ‫ܪ‬௜ ‫ܩ‬௜ ܵீ೔ ൌ ܶ ௞ ܵீ೔ Ǥ(27)఼‫ݖ‬଴ ΓΕड़൅ͤΖҲ॑ଲসब‫ٌغ‬ಕ ܱሺ‫ݖ‬଴ ሻ͗ଚࡑͤΖ৖߻, ܴܱሺ‫ݖ‬଴ ሻ ൌ ܱሺܴ‫ݖ‬଴ ሻ΍ଚࡑͤΖ.

3.2. ್॑ଲসब‫ٌغ‬ಕ͹෾ྪࢂߡช‫[ݛ‬1]Ν΍ͳͶ, ύʖϏʖࣺ଀Ͷ͕͜Ζ್॑ଲসब‫ٌغ‬ಕͺൂझྪͶ෾ྪ͠ΗΖ . ෾ྪΝන 2 ͳ͢ͱ༫͓Ζ. ࣏Ͷ್॑ଲসब‫ٌغ‬ಕ͹ଲ߻න‫ݳ‬Νන 3 Ͷࣖͤ. 1
න̐ʁ್॑ଲসब‫ٌغ‬ಕ͹෾ྪ෾ྪ1ଲসતͳٌಕ఼

ब‫غ‬:‫ݍ‬ড়݇a‫ܪ‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬ଶ௞ ሿ

Ͷ݇

݇൒ͳ

b

‫ܪ‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬ଶ௞ାଵ ሿͶ݇ ൅ ʹ

݇൒Ͳc‫ܩ‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬ଶ௞ିଵ ሿͶ݇ െ ʹ݇൒ʹd‫ܩ‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬ଶ௞ ሿ

Ͷ݇

݇൒ͳ

e

‫ܪ‬ଷ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬ଶ௞ ሿ

Ͷ݇

݇൒ͳ

f

‫ܪ‬ଷ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬ଶ௞ାଵ ሿͶ݇ ൅ ʹ݇൒ͳg‫ܩ‬ଷ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬ଶ௞ିଵ ሿͶ݇ െ ʹ݇൒ʹh‫ܩ‬ଷ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬ଶ௞ ሿ

Ͷ݇

݇൒ͳ

͞͞Ͳͺࢂߡช‫[ݛ‬3]͹಼༲Ν੠ືԿͪ݃͢ՎΝ‫ ͪ͢ࡎܟ‬. ‫ޛ‬ड़͹න 4 ͳන 5 Ͷ͕͏ͱ΍

ಋ༹Ͳ͍Ζ.
t 7 t

න̑ʁ್॑ଲসब‫ٌغ‬ಕ͹ଲ߻න‫ݳ‬෾ྪଲ߻න‫ࠪ[ݳ‬ଲ߻, ߅౵ࣺ଀, ӊଲ߻]

ড়݇ 1

ড়݇ 2

a

ܶ ସ௞ ൌ ܶ ଶ௞ ‫ܪ‬ଵ ‫ܩ ל‬ଵ ܶ ଶ௞ିଵ ‫ܪ‬ଶ ‫ܪ ל‬ଶܶ ௞ ܵுభ ‫ܵ ת‬ுమ ് ‫׎‬ܶ ௞ ܵுమ ‫ܵ ת‬ுభ ് ‫׎‬bܶ ସ௞ାଶ ൌ ܶ ଶ௞ ‫ܪ‬ଵ ‫ܩ ל‬ଵ ܶ ଶ௞ ‫ܪ‬ଶ ‫ܩ ל‬ଶܶ ௞ ܵுభ ‫ீܵ ת‬మ ് ‫׎‬ܶ ௞ାଵ ܵீమ ‫ܵ ת‬ுభ ് ‫׎‬cܶ ସ௞ିଶ ൌ ܶ ଶ௞ିଵ ‫ܪ‬ଵ ‫ܩ ל‬ଵ ܶ ଶ௞ିଶ ‫ܪ‬ଶ ‫ܪ ל‬ଶܶ ௞ ܵீభ ‫ܵ ת‬ுమ ് ‫׎‬ܶ ௞ିଵ ܵுమ ‫ீܵ ת‬భ ് ‫׎‬dܶ ସ௞ ൌ ܶ ଶ௞ିଵ ‫ܪ‬ଵ ‫ܩ ל‬ଵ ܶ ଶ௞ିଵ ‫ܪ‬ଶ ‫ܩ ל‬ଶܶ ௞ ܵீభ ‫ீܵ ת‬మ ് ‫׎‬ܶ ௞ ܵீమ ‫ீܵ ת‬భ ് ‫׎‬eܶ ସ௞ ൌ ܶ ଶ௞ ‫ܪ‬ଷ ‫ܩ ל‬ଷ ܶ ଶ௞ିଵ ‫ܪ‬ସ ‫ܪ ל‬ସܶ ௞ ܵுయ ‫ܵ ת‬ுర ് ‫׎‬ܶ ௞ ܵுర ‫ܵ ת‬ுయ ് ‫׎‬fܶ ସ௞ାଶ ൌ ܶ ଶ௞ ‫ܪ‬ଷ ‫ܩ ל‬ଷ ܶ ଶ௞ ‫ܪ‬ସ ‫ܩ ל‬ସܶ ௞ ܵுయ ‫ீܵ ת‬ర ് ‫׎‬ܶ ௞ାଵ ܵீర ‫ܵ ת‬ுయ ് ‫׎‬gܶ ସ௞ିଶ ൌ ܶ ଶ௞ିଵ ‫ܪ‬ଷ ‫ܩ ל‬ଷ ܶ ଶ௞ିଶ ‫ܪ‬ସ ‫ܪ ל‬ସܶ ௞ ܵீయ ‫ܵ ת‬ுర ് ‫׎‬ܶ ௞ିଵ ܵுర ‫ீܵ ת‬య ് ‫׎‬hܶ ସ௞ ൌ ܶ ଶ௞ିଵ ‫ܪ‬ଷ ‫ܩ ל‬ଷ ܶ ଶ௞ିଵ ‫ܪ‬ସ ‫ܪ ל‬ସܶ ௞ ܵீయ ‫ீܵ ת‬ర ് ‫׎‬ܶ ௞ ܵீర ‫ீܵ ת‬య ് ‫׎‬ब‫ غ‬2 Ͷͯ͏ͱͺ஭қͤΖචགྷ͍͗Ζ . න 2 ͹෾ྪ b Ͷ͕͜Ζब‫ غ‬2 ͹ब‫ٌغ‬ಕͺ್॑ଲসब‫ٌغ‬ಕͲ͍Ζ. ҲൢͶ್॑ଲসब‫ٌغ‬ಕͺଲসત৏Ͷ఼ٌ࢝ಕ఼Ν΍ͯ͗ , ෾ྪ b Ͷ͕͜Ζब‫ غ‬2 ͹ब‫ٌغ‬ಕͺଲসત ܵுభ ͳܵீమ ͹ި఼Ͷٌಕ఼Ν΍ͯ. ͞͹৖߻, ި఼Ν್‫਼ͳݺ‬͓Ηͻ߻‫͵ͳݺ࢝ܯ‬Ζ.
෾ྪ c ͳ f Ͳͺब‫ غ‬2 ͹ब‫ٌغ‬ಕͺਫ਼ͣ͵͏. ෾ྪ g ͹৖߻Ͷ͕͏ͱՆ೵͵ब‫ غ‬2 ͹ब‫غ‬ٌಕͺ࢝॑ଲসब‫ٌغ‬ಕͲ͍Ζ . ͨ͹ͪΌড়݇Ν ݇ ൒ ʹͳ͢ͱब‫ غ‬2 ͹ब‫ٌغ‬ಕΝঈ͏ͱ͍Ζ .
͞͹৖߻Ͷͯ͏ͱ؈ୱͶઈ໎Ν௧ՅͤΖ . ҲൢͶ࢝॑ଲসब‫ٌغ‬ಕͺଲসત৏Ͷൂ఼ٌಕ఼
Ν΍ͯ. ͞͞Ͳ, ‫ޛ‬ड़͹ਦ 3(a)Ν‫ͱݡ‬ཋ͢͏. ࢝ͯ͹ଲসત ܵுభ , ܵீమ , ܵீయ , ܵுర ͗ިࠫ͢ͱ͏Ζ఼್͍͗ͯΖ͞ͳ͗෾͖ΖͲ͍Θ͑ . ͞ΗΔ͹ި఼͗ब‫ غ‬2 ͹ٌಕ఼Ͳ͍Ζ. ͯΉΕ, ब‫ غ‬2͹ٌಕ఼ͺ࢝झྪ͹ଲসત͹ި఼Ͷ͵ͮͱ͏Ζ͹Ͳ , ͽͳͯި఼Ν࢝‫਼͓ͳݺ‬Ηͻ߻‫ݺൂܯ‬ͳ͵Ζ.
න 2 ͳන 3 Ͷ͕͜Ζ෾ྪ a Νྭͳ͢ͱઈ໎ͤΖ. න‫ܪݳ‬ଵ ሾ‫ݖ‬଴ ሿͺٌಕ఼‫ݖ‬଴ ͗ଲসતܵுభ ৏Ͷ͍Ζ
͞ ͳ Ν න ‫ ͏ ͱ ͢ ݳ‬Ζ. ࣏ ͹ න ‫ܪ ݳ‬ଶ ሾ‫ݖ‬௞ ሿͺ ٌ ಕ ఼ ‫ݖ‬௞ ͗ ଲ স ત ܵுమ ৏ Ͷ ͍ Ζ ͞ ͳ Ν න ‫͢ ݳ‬, ࠹ ‫ ޛ‬͹න ‫ܪ ݳ‬ଵ ሾ‫ݖ‬ଶ௞ ሿͺ ٌ ಕ ఼ ‫ݖ‬ଶ௞ ͗ ଲ স ત ܵுభ ৏ Ͷ ͍ Ζ ͞ ͳ Ν න ‫ ͏ ͱ ͢ ݳ‬Ζ . ‫ܪ ߺ ى‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬௞ ሿͺ , ‫ݖ‬ଵ
͖ Δ ‫ݖ‬௞ିଵ Ή Ͳ ͹ ٌ ಕ ఼ ͗ ଲ স ત ৏ Ͷ ͵ ͏ ͞ ͳ ΍ қ ັ ͢ ͱ ͏ Ζ . ‫ܪ ߺ ى‬ଶ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬ଶ௞ ሿͶ ͯ ͏ ͱ΍ಋ༹Ͳ͍Ζ.
න 2 ͹ब‫غ‬ΓΕ, ್॑ଲসब‫ٌغ‬ಕ͹ब‫͗غ‬۰਼Ͳ͍Ζ͞ͳ͗෾͖Ζ . ෾ྪ a ͹න 3 Ͷ͕͜Ζড়݇ 1 ͺ, ଲসત͹଀ ܶ ௞ ܵுభ ͳଲসતܵுమ ͹ި఼್͗॑ଲসब‫ٌغ‬ಕ͹ٌಕ఼Ͳ͍Ζ͞ͳΝқັ͢ͱ͏Ζ. ͪͫ͢, ި఼ͳ͢ͱ෈ಊ఼ͺঈ͚. ড়݇ 2 ΍ಋ༹͹͞ͳΝқັ͢ͱ͏Ζ . ড়݇ 1 Νཤ༽͢ͱ΍ড়݇ 2 Νཤ༽͢ͱ΍ಋ್ͣ॑ଲসब‫ٌغ‬ಕ͗݀ΉΖ .
್॑ଲসब‫ٌغ‬ಕͳ୊ 4 ઇͲ٠࿨ͤΖ࢝॑ଲসब‫ٌغ‬ಕͳ͹૮ҩΝ೼ѴͤΖͪΌͶ , ್॑ଲসब‫ٌغ‬ಕ͹ྭΝਦ 3 Ͷࣖͪ͢. ਦ 3(a), (b),ǥ,(h)͹ٌಕͺͨΗͩΗන 2 ͳන 3 ͹෾ྪ
a,b,ǥ,h Ͷ଒ͤΖब‫ٌغ‬ಕྭͲ͍Ζ . ਦ 3(a)͹৖߻, ॵ‫ݖ఼غ‬଴ (ਦͲͺ 0)ͺଲসતܵுభ ৏Ͷ͍Ε,‫ݖ‬ଵ ͺܵுమ ৏Ͷ͍Ζ. ब‫غ‬ͺ 4 Ͳ͍Ζ. ଠ͹ਦ΍ಋ༹Ͳ͍Ζ.
ब‫غ‬Ν‫͢ͳݍ‬, ෾ྪ a ͖Δ෾ྪ h Ͷ͕͏ͱࣺ଀ ܶ ௤ Νࢀͯ͹ࡠ༽Ͷ෾մͪ͢. ͖ࠪΔॳͶࠪଲ߻, ߅౵ࣺ଀, ӊଲ߻ͳ‫[΁ݼ‬1]. ͨΗͩΗ͹න‫ݳ‬Ͷͯ͏ͱͺන 3 Ͷࡎͦͪ. ࠹ॵͶ, ෾ྪ aͳ෾ྪ c Ͷ͕͜Ζब‫ݍ غ‬͹಍ड़Νߨ͑. ࢔Ε͹৖߻͹಍ड़ͺ঴͚.
෾ྪ a. ͞͹৖߻, ब‫ٌغ‬ಕ͗ଲসત ܵுభ ৏ͳܵுమ ৏Ͷٌ ಕ఼Ν΍͖ͯΔ ್॑ଲসब‫ٌغ‬ಕͲt 8 t

͍Ζ. ॵ‫ݖ఼غ‬଴ ͗ܵுభ Ͷ͍Ε, ‫ݖ‬௞ ͗ଲসત ܵுమ ৏Ͷ͍Ζ͞ͳ͖Δब‫غ‬Ν‫ٽ‬Ό Ζ. ‫ܪ‬ଵ ሾ‫ݖ‬଴ ሿΓΕ‫ܪ‬ଵ ‫ݖ‬଴ ൌ‫ݖ‬଴ ͗ ಚ Δ Η , ‫ܪ‬ଶ ሾ‫ݖ‬௞ ሿΓ Ε ‫ܪ‬ଶ ‫ݖ‬௞ ൌ ‫ݖ‬௞ ͗ ಚ Δ Η Ζ . ‫ ं ޛ‬Ν ‫ܪ‬ଶ ‫ݖ‬ଶ௞ ൌ ‫ݖ‬଴ ͳ ͵ ͕ ͢ , ઴ ं Ͷ େ ೘ ͤ Ζ ͳ‫ܪ‬ଵ ‫ݖ‬଴ ൌ ‫ܪ‬ଶ ‫ݖ‬ଶ௞ ͗‫ٽ‬ΉΖ. ͞ΗΝ ‫ܪ‬ଶ ‫ܪ‬ଵ ‫ݖ‬଴ ൌ ‫ݖ‬ଶ௞ ͳॽ͘, ߍͶ 180 ౕյ఺ࡠ༽ܴ ൌ ‫ܪ‬ଶ ‫ܪ‬ଵ Νཤ༽ͤΖͳ ܴ‫ݖ‬଴ ൌ ‫ݖ‬ଶ௞ ൌ ܶ ଶ௞ ‫ݖ‬଴ ͗ ಚ Δ Η Ζ . ߍ Ͷ ܶ ଶ௞ Ν ‫ݖ‬ଶ௞ Ͷ ࡠ ༽ ͤ Ζ ͳ ‫ݖ‬଴ Ͷ ໯ Ζ ͹ Ͳ , ब ‫ݍ غ‬ͺ ‫ ݍ‬ൌ Ͷ݇ͳಚΔΗΖ.
෾ྪ c. ͞͹৖ ߻ , ब‫ٌ غ‬ಕ͗ଲ সત ܵீభ ৏ͳ ܵுమ ৏Ͷٌ ಕ఼Ν΍ ͖ͯΔ ್॑ଲ সब‫ ٌغ‬ಕͲ͍Ζ. ॵ‫ݖ఼غ‬଴ ͗ܵீభ Ͷ͍Ε , ‫ݖ‬௞ ͗ଲসતܵுమ ৏Ͷ͍Ζ͞ͳ͖Δब‫غ‬Ν‫ٽ‬ΌΖ . ‫ܩ‬ଵ ሾ‫ݖ‬଴ ሿΓΕ ‫ܩ‬ଵ ‫ݖ‬଴ ൌ‫ݖ‬଴ ͗ ಚ Δ Η , ‫ܪ‬ଶ ሾ‫ݖ‬௞ ሿΓ Ε ‫ܪ‬ଶ ‫ݖ‬௞ ൌ ‫ݖ‬௞ ͗ ಚ Δ Η Ζ . ‫ ं ޛ‬Ν ‫ܪ‬ଶ ‫ݖ‬ଶ௞ ൌ ‫ݖ‬଴ ͳ ॽ ͘ ͵ ͕ ͢ , ઴ ं Ͷ େ ೘ ͤ
Ζ ͳ ‫ܩ‬ଵ ‫ݖ‬଴ ൌ ‫ܪ‬ଶ ‫ݖ‬ଶ௞ ͗ ‫ ٽ‬Ή Ζ . ྈ ว Ͷ ‫ܩ‬ଶ Ν ࡠ ༽ ͤ Ζ . ࠪ ว ͺ ‫ܩ‬ଶ ‫ܩ‬ଵ ‫ݖ‬଴ ൌ ܴ‫ݖ‬଴ Ͳ , ӊ ว ͺ ‫ܩ‬ଶ ‫ܪ‬ଶ ‫ݖ‬ଶ௞ ൌܶ ିଵ ‫ݖ‬ଶ௞ ൌ ‫ݖ‬ଶ௞ିଵ . ܴ‫ݖ‬଴ ൌ ‫ݖ‬ଶ௞ିଵ ΓΕ, ‫ ݍ‬ൌ Ͷ݇ െ ʹ͗ಚΔΗΖ.

t 9 t

ਦ 3ʁ್॑ଲসब‫ٌغ‬ಕ͹ྭ. ੖਼݇ͺٌಕ఼‫ݖ‬௞ Νනͤ. (a) ܽ ൌ ʹǤͷ. (b) ܽ ൌ ͵. (c) ܽ ൌ ʹǤͷ.
(d) ܽ ൌ ʹǤͷ. (e) ܽ ൌ ʹ. (f) ܽ ൌ ͵. (g) ܽ ൌ ʹǤͷ. (h) ܽ ൌ ʹ.

t 10 t෾ྪ a-h Ͷ͕͜Ζྡֆ஍ͳড়݇͹ઈ໎Νߨ͑ .
෾ྪ a. ͞͹৖߻, ࠪଲ߻ܶ ଶ௞ ‫ܪ‬ଵ Ͷͯ͏ͱ෈ร͵ॄ߻ͺ ܶ ௞ ܵுభ Ͳ͍Ζ. Ήͪӊଲ߻ ‫ܪ‬ଶ Ͷͯ͏ͱ෈ร͵ॄ߻ͺ ܵுమ Ͳ͍Ζ. ͯΉΕ, ܶ ௞ ܵுభ ͳ ܵுమ ͹ި఼ͺ್॑ଲসब‫ٌغ‬ಕ͹ٌಕ఼Ͳ͍Ζ (ި఼ͳ͢ͱ෈ಊ఼ͺ঴͚). ͞Η͗ড়݇ 1 Νන͢ͱ͏Ζ . ࣏Ͷ߅౵ࣺ଀ͺ, ෾ྪ a ͹ٌಕ఼‫ݖ‬௜ Ͷଲ͢ͱ‫ܩ‬ଵ ܶ ଶ௞ିଵ ‫ܪ‬ଶ ‫ݖ‬௜ ൌ ‫ݖ‬௜
ଶ௞

(28)
ଶ௞

௞

Νຮͪͤ. ͞ΗΝܶ ‫ܪ‬ଶ ‫ݖ‬௜ ൌ ܶ‫ܩ‬ଵ ‫ݖ‬௜ ൌ ‫ܪ‬ଵ ‫ݖ‬௜ ͳॽ͘௜ͤ. ܶ ‫ܪ‬ଶ Ͷͯ͏ͱ෈ร͵ॄ߻ͺ ܶ ܵுమ Ͳ͍Ζ.
‫ܪ‬ଵ Ͷͯ ͏ͱ ෈ร ͵ॄ ߻ ͺ ܵு Ͳ͍Ζ. ͯΉΕ, ٌಕ ఼ ‫ݖ‬௜ ͺ, ܶ ௞ ܵுమ ͳ ܵு ͹ި ఼Ͳ ͍Ζ . ͞Η ͗̏̏

ড়݇ 2 Νқັ͢ͱ͏Ζ. ଲসતܵுభ ͳܵுమ ͹ި఼ͺ෈ಊ఼͹ΊͲ͍Ζ . Γͮͱ, ݇ ൌ ͳ͹৖߻Ν௒΄Ζ. ͞͞Ͳͺড়݇ 2 Νཤ༽ͤΖ. ଲসતܵுమ ͹ܳଵ଴ Ͷ͕͜Ζܑ͘ͺ ͲͲ͍Ζ. ͞ΗΝ, ߦ଴ ൌ Ͳͳ
ॽ͚. ݇ ൌ ͳͲ͍ Ζ͖ Δ , ࣞ(20)Νཤ༽͢ ͱ ߦଵ ͗‫ٽ‬Ή Ζ . Ҳ๏Ͳ ܵுభ ͹ ܳଵ଴ Ͷ͕͜ Ζܑ ͘ͺ ʹȀܽͲ͍Ζ. ߦଵ ൌ ʹȀܽΓΕྡֆ஍͗‫ٽ‬ΉΖ.
ߦଵ ൌ

െܽ
ʹ
ൌ Ǥ
ଶ
ͳെܽ
ܽ

(29)

͞ΗΓΕྡֆ஍͗ ܽ ൌ ξʹͳಚΔΗΖ. ݇Ͷͯ͏ͱ͹ড়݇ ݇ ൒ ͳ͗ࣖ͠Ηͪ.
෾ྪ b. ͞͹৖߻, ݇ ൌ ͲΝຮͪͤब‫ٌغ‬ಕ͗ଚࡑͤΖ͞ͳΝࣖͤ . ܽΝଁՅͤΖͳ, ଲসતܵுభ ͳܵீమ ͗ંͤΖ. ͞͹ྡֆ஍Ν݀ΌΖࣞͺ, ʹȀܽ ൌ ܽȀʹͲ͍Ζ. ࠪว͗ଲসતܵுభ ͹ܳଵ଴ Ͷ͕͜
Ζܑ͘Ͳ, ӊว͗ ܵீమ ͹ܳଵ଴ Ͷ͕͜Ζܑ͘Ͳ͍Ζ. ͞ΗΓΕྡֆ஍ͺܽ ൌ ʹͲ͍Ζ. ࠹ঘ͹ब‫غ‬ͺ 2 Ͳ͍Ζ. ͞͹ब‫ غ‬2 ͹ब‫ٌغ‬ಕͺ್॑ଲসब‫ٌغ‬ಕͲ͍Ζ .
෾ྪ c. ͞͹৖ ߻ , ܶܵீభ ͺ ‫ ݕ‬ൌ ‫ ݔ‬െ ߨͳॽ͜Ζ. ͞Ηͳ ‫ܵ( ࣢ݔ‬ுమ )ͺ෈ಊ఼Ґ ֐Ͳ ިࠫ ͢͵ ͏ .
͞Ηͺब‫ غ‬2 ͗ਫ਼ͣ͵͏͞ͳΝқັͤΖ . Γͮͱ, ݇ ൌ ʹ ͹৖߻Ν௒΄Ζ. ܶܵீభ ͹ܳଵ଴ Ͷ͕͜Ζ ܑ ͘ ͺ ߦଵ ൌ ͳͲ ͍ Ζ . ࣏ Ͷ ܶ ଶ ܵீభ ͹ ܳଵ଴ Ͷ ͕ ͜ Ζ ܑ ͘ ߦଶ ൌ ሺͳ െ ܽሻȀሺܽሺͳ െ ܽሻ ൅ ͳሻ͗ ಚ Δ Η Ζ .
͞Η͗ 0 Ͷ͵Ζ͞ͳΓΕ, ྡֆ஍ͺܽ ൌ ͳͲ͍Ζ. ݇Ͷͯ͏ͱ͹ড়݇ ݇ ൒ ʹ͗ࣖ͠Ηͪ.
෾ྪ d. ͞͹৖߻, ܶܵீభ ͹ܑ͘ͺ 1 Ͳ͍Ζ. Ҳ๏͹ܵீమ ͹ܑ͘ͺܽȀʹͲ͍Ζ. Γͮͱྡֆ஍ͺܽ ൌ ʹ͗಍͖ΗΖ. ݇Ͷͯ͏ͱ͹ড়݇ ݇ ൒ ͳ͗ࣖ͠Ηͪ.
෾ྪ e. ଲসતܵுయ ͳܵுర ͺ෈ಊ఼Ґ֐Ͳިࠫ͢͵͏͹Ͳ ݇ ൌ ͳͳͤΖ. ଲসતܵுయ ͹ܳଵ଴ Ͷ͕͜Ζܑ͘ͺߦ଴ ൌ െͳΝཤ༽͢ͱ, ߦଵ Ν‫ٽ‬ΌΖ. ଲসતܵுర ͹ܳଵ଴ Ͷ͕͜Ζܑ͘ͺ ͳͲ͍Ζ.
ܽ൅ͳ
ߦଵ ൌ
ൌͳ
ܽሺܽ ൅ ͳሻ െ ͳΓΕ, ྡֆ஍ͺܽ ൌ ξʹͳಚΔΗΖ.
෾ྪ f. ଲসતܵுయ ͳܵீర ͺ෈ಊ఼Ґ֐Ͳިࠫ͢͵͏ . ଲসતܵீర ͹ܳଵ଴ Ͷ͕͜Ζܑ͘ͺ ܽ ൅ ͳͲ͍
Ζ. ݇ ൌ ͳͳ͢ͱ, ܽΝ݀ΌΖࣞ
ߦଵ ൌ

ܽ൅ͳ
ൌܽ൅ͳ
ܽሺܽ ൅ ͳሻ െ ͳ͗ಚΔΗΖ. ͞ΗΝմ͏ͱྡֆ஍ͺ ܽ ൌ ͳͳ͵Ζ.

t 11 t෾ྪ g. ݇ ൌ ͳͳͤΖͳब‫ غ‬2 ͹ब‫ٌغ‬ಕ͗ಚΔΗΖ. ͖͢͢, ͞Ηͺ࢝॑ଲসब‫ٌغ‬ಕͲ͍Ζ. ͨ ͹ ͪ ΌͶ ড় ݇ ͳ ͢ ͱ ݇ ൒ ʹͳ ͢ ͪ. ଲস ત ܵீయ ͹ ܳଵ଴ Ͷ ͕ ͜ Ζ ܑ ͘ ͺ ߦ଴ ൌ ܽ െ ͳͲ͍ Ζ. ଲসતܵுర ͹ܳଵ଴ Ͷ͕͜Ζܑ͘ͺ ͳͲ͍Ζ. ݇ ൌ ʹͳ͢ͱߦଶ ൌܽଶ െ ܽ െ ͳൌͳܽଷ െ ܽଶ െ ʹܽ ൅ ͳ͗ಚΔΗΖ. ͞ΗΓΕ ྡֆ஍ ܽ ൌ ͳ͗‫ٽ‬ΉΖ.
෾ྪ h. ଲসતܵீయ ͳܵீర ͺ෈ಊ఼Ґ֐Ͳިࠫ͢͵͏ . ଲসતܵீయ ͹ܳଵ଴ Ͷ͕͜Ζܑ͘ͺ ߦ଴ ൌ ܽ െ ͳͲ͍ Ε, ଲ সત ܵீర ͹ ܳଵ଴ Ͷ͕ ͜ Ζܑ ͘ͺ ܽ ൅ ͳͲ ͍Ζ . ݇ ൌ ͳͳ͢ ͱ, ߦଵ ൌ ͳȀሺܽ െ ͳሻ ൌ ܽ ൅ ͳΝմ͏ͱ, ྡֆ஍͗ܽ ൌ ξʹͳಚΔΗΖ.
࠹‫ޛ‬Ͷ್॑ଲসब‫ٌغ‬ಕ͹੓࣯Νఈཀྵ 3.1 ͳ͢ͱΉͳΌͱ͕͚. ್॑ଲসब‫ٌغ‬ಕ͹੓࣯ͺࢂߡช‫[ݛ‬1]ΝΊΔΗͪ͏.
ఈཀྵ 3.1. ್॑ଲসब‫ٌغ‬ಕͶͯ͏ͱ (1)͖Δ(5)ΉͲ͗੔Εཱིͯ.
(1) ब‫ݍغ‬ͺ, 2 Ґ৏͹۰਼ .
(2) ࣺ଀ܶ ௤ ͺࠪଲ߻, ߅౵ࣺ଀(ߛ), ӊଲ߻͹ࢀͯ͹ࡠ༽Ͷ෾մ͠ΗΖ .
(3) ࠪ ଲ ߻ Ͷ ͯ ͏ ͱ ෈ ร ͵ ॄ ߻ Ν ܵ௟ ͳ ͢, ӊ ଲ ߻ Ͷ ͯ ͏ ͱ ෈ ร ͵ ॄ ߻ ܵ௥ ͳ ͤ Ζ . ‫ ݖ ܐ ؖ‬ൌ ܵ௟ ‫ת‬ܵ௥ Νຮ఼ͪͤ ‫ݖ‬ͺ್॑ଲসब‫ٌغ‬ಕ͹ٌಕ఼Ͳ͍Ζ . ఼‫ݖ‬ͺ෈ಊ఼Ͳ͵ ͏ͳͤΖ.
(4) ড় ݇ ߛ‫ ݓ‬ൌ ‫ݓ‬Ν ຮ ͪ ͤ ఼ ‫ݓ‬ͺ ್ ॑ ଲ স ब ‫ ٌ غ‬ಕ ͹ ٌ ಕ ఼ Ͳ ͍ Ζ . ఼ ‫ݓ‬ͺ ෈ ಊ ఼ Ͳ ͵ ͏ ͳ ͤΖ.
(5) ್॑ଲসब‫ٌغ‬ಕͺ 180 ౕյ఺ଲস੓Ν΍ͯ.

3.3. 8 ࣊‫ܙ‬෾‫ز‬࠹ॵͶ್॑ब‫غ‬ഔ෾‫್॑ͳز‬ಋब‫غ‬෾‫ز‬͹ఈٝΝ༫͓Ζ . ್॑ಋब‫غ‬෾‫ز‬Ͷͯ͏ͱͺ࢝॑ଲসब‫ٌغ‬ಕ͹෾‫ز‬ΝཀྵմͤΖͪΌͶචགྷͳ͵Ζ .
ఈٝ 3.2.
(i) ब‫غ‬ഔ෾‫ͳز‬ൕब‫غ‬ഔ෾‫͗ز‬ಋ࣎Ͷਫ਼ͣΖ৖߻ , ͞͹෾‫ز‬Ν್॑ब‫غ‬ഔ෾‫ͳز‬ఈٝͤΖ .
(ii) ಋ ब ‫ غ‬෾ ‫ ͳ ز‬ൕ ಋ ब ‫ غ‬෾ ‫ ͗ ز‬ಋ ࣎ Ͷ ਫ਼ ͣ Ζ ৖ ߻ , ͞ ͹ ෾ ‫ ز‬Ν ್ ॑ ಋ ब ‫ غ‬෾ ‫ ͳ ز‬ఈ ٝ ͤΖ.
ໍୌ 3.3 ͺࢂߡช‫[ݛ‬1]ͲಚΔΗͪ݃ՎͲ͍Ζ.
ໍୌ 3.3.
(i) ବԃ‫ ܗ‬Ҳ॑ ଲ সब ‫ٌ غ‬ಕ ͺब ‫غ‬ഔ ෾‫ ز‬Ν ً͞ ͢ , ਫ਼ͣ ໊ͪ ब ‫ ٌغ‬ಕͺ Ҳ॑ ଲস ब‫ٌ غ‬ಕͲ͍Ζ.
(ii) ବ ԃ ‫ ॑ ್ ܗ‬ଲ স ब ‫ ٌ غ‬ಕ ͺ ࠹ ॵ Ͷ ್ ॑ ब ‫ غ‬ഔ ෾ ‫ ز‬Ν ً ͞ ͢ , ࣏ Ͷ ಋ ब ‫ غ‬෾ ‫ ز‬Ν ً ͞ ͤ .
͞ΗΔ͹෾‫Ͳز‬ਫ਼໊ͣͪब‫ٌغ‬ಕͺͳ΍ͶҲ॑ଲসब‫ٌغ‬ಕͲ͍Ζ .
ύʖϏʖࣺ଀Ͷ͕͜Ζବԃ‫ܗ‬Ҳ॑ଲসब‫ٌغ‬ಕ͹෾‫ز‬͹ྭΝਦ 4(a)Ͷࣖͪ͢. ବԃ‫ܗ‬Ҳ॑

t 12 tଲসब‫ٌغ‬ಕͺब‫غ‬ഔ෾‫ز‬͹ΊΝًͤ͞ . ବԃ‫್॑ܗ‬ଲসब‫ٌغ‬ಕ(෾ྪ e)͹෾‫ز‬Νਦ 4(b)Ͷࣖ ͢ ͪ . ͞͹ ब ‫ٌ غ‬ಕ ͺ࠹ ॵ Ͷ ್॑ ब ‫ غ‬ഔ෾ ‫ ز‬Νً ͞ ͢ , ࣏Ͷ ಋ ब‫ غ‬෾ ‫ز‬Ν ً ͞ ͤ . ಝͶ
ਦ 4(b)Ͷ͕͜Ζ҈ఈ੓‫ݎ਼ܐ‬ሺܽሻ͹‫ܙ‬য়Ν U ࣊‫ ΁ݼͳܙ‬. U ࣊‫ܙ‬Ͷॊ͑෾‫ز‬Ν U ࣊‫ܙ‬෾‫໌ͳز‬
෉͜Ζ. ବԃ‫್॑ܗ‬ଲসब‫ٌغ‬ಕͺ U ࣊‫ܙ‬෾‫ز‬Νًͤ͞.
ਦ 4(a)Ͳࣖͪ͢ब‫غ‬ഔ෾‫ز‬͹ྭͺΓ͚எΔΗͱ͏Ζ͹Ͳ , ͞͞Ͳͺਦ 4(b)͹ U ࣊‫ܙ‬෾‫ز‬͹ྭΝ়ղͤΖ. ್॑ब‫غ‬ഔ෾‫Ͳز‬ਫ਼ͣͪब‫ٌغ‬ಕ͹ྭΝਦ 5 Ͷࣖͪ͢. ब‫ غ‬4 ͹ବԃ‫್॑ܗ‬ଲসब‫ٌغ‬ಕͺ୉͘͵‫Ͳ؛‬ඵ͘ , ෾‫Ͳز‬ਫ਼ͣͪब‫ غ‬8 ͹ٌಕ఼ͺঘ͠͵‫Ͳ؛‬ඵ͏ͪ . ਦ 5(a)ͳਦ 5(b)Ͳඵ͏ͪब‫ غ‬8 ͹ब‫ٌغ‬ಕ͗ਫ਼ͣͪ௜‫ޛ‬ͺବԃ‫͍Ͳܗ‬Ζ . ਦ 5(a)Ͳͺٌಕ఼ ‫ݖ‬଴ ΓΕਫ਼఼್ͣͪ͗ଲসત ܵுయ ৏Ͷ͍Ε, ਦ 5(b)Ͳͺٌಕ఼ ‫ݖ‬ଶ ΓΕਫ਼఼್ͣͪ͗ଲসત ܵுయ ৏Ͷ͍Ζ .
ਦ 5(c)ͳਦ 5(d)Ͳඵ͏ͪब‫ غ‬8 ͹ब‫ٌغ‬ಕ͗ਫ਼ͣͪ௜‫ޛ‬ͺγχϩ‫͍Ͳܗ‬Ζ . ਦ 5(c)Ͳͺٌಕ఼ ‫ݖ‬ଵ ΓΕਫ਼ ͣͪ ఼್ ͗ଲ স ત ܵுర ৏Ͷ͍ Ε , ਦ 5(d)Ͳͺٌ ಕ఼ ‫ݖ‬ଷ ΓΕ ਫ਼ͣ ್ͪ ఼ ͗ଲ সત ܵுర৏Ͷ͍Ζ.

ਦ 4 ʁ (a) ବ ԃ ‫ ܗ‬Ҳ ॑ ଲ স ब ‫ ٌ غ‬ಕ ͹ ෾ ‫ ز‬͹ ྭ . ‫ݖ‬଴ ‫ܵ א‬ுర , ‫ݖ‬ଶ ‫ீܵ א‬ర , ‫ ݍ‬ൌ ͷ , ܽ௖ ሺͳȀͷሻ ൌʹሺߨȀͷሻ ൌ ͳǤͳ͹ͷͷ. (b) ବԃ‫್॑ܗ‬ଲসब‫ٌغ‬ಕ͹෾‫ز‬͹ྭ . ෾ྪ e. ‫ݖ‬଴ ‫ܵ א‬ுయ , ‫ݖ‬ଵ ‫ܵ א‬ுర , ‫ ݍ‬ൌ Ͷ,ܽ௖ ሺͳȀͶሻ ൌ ξʹ.

ਦ 5ʁब‫ غ‬4 ͹್॑ଲসब‫ٌغ‬ಕ͹ٌಕ఼ ‫ݖ‬௞ ሺͲ ൑ ݇ ൑ ͵)(୉͘͵‫؛‬һ)ΓΕਫ਼ͣͪब‫ غ‬8 ͹ٌಕ఼(ঘ͠͵‫؛‬һ). (a)ͳ(b)͹ঘ͠͵‫؛‬һ͹ब‫ٌغ‬ಕ͗ਫ਼ͣͪ௜‫ޛ‬ͺବԃ‫ ܗ‬. (c)ͳ(d)͹ঘ͠͵‫؛‬һ͹ब‫ٌغ‬ಕ͗ਫ਼ͣͪ௜‫ޛ‬ͺγχϩ‫ ܗ‬. ܽ ൌ ʹǤͶ.
t 13 t್॑ब‫غ‬ഔ ෾‫Ͳز‬ͺ , ବԃ‫ܗ‬ब‫ ٌغ‬ಕͳγχ ϩ‫ ܗ‬ब‫ٌغ‬ಕ͗ ‫ڠ‬Ͷ್ͯ ਫ਼ͣ Ζ . ࣺ଀ܶ ଼ ͹΍ͳͲ, ବԃ‫ܗ‬ब‫ٌغ‬ಕ͹ϛΠϱΩϪࢨ਼ͺ ሺʹ ൈ ሺ൅ͳሻሻͲ, Ҳ๏͹γχϩ‫ܗ‬ब‫ٌغ‬ಕ͹ϛΠϱΩϪࢨ਼ͺሺʹ ൈ ሺെͳሻሻͲ͍Ζ. ್॑ब‫غ‬ഔ෾‫ز‬͹઴, ब‫ غ‬4 ͹ब‫ٌغ‬ಕͺବԃ‫͍Ͳܗ‬Ζ . ࣺ଀ܶ ସͲब ‫ ٌ غ‬ಕ఼ ͗ ବԃ ఼ ͵ Δͻ , ͞ ͹ٌ ಕ ఼ ͺࣺ ଀ ܶ ଼ Ͳ΍ ಋ ༹ Ͳ͍ Ζ . Γͮ ͱ , ෾‫ ز‬઴͹ ϛΠϱ Ω Ϫ ࢨ ਼ ͹ ૱ ࿪ ͺ ሺ൅ͳሻͲ ͍ Ζ . ϛ Π ϱ Ω Ϫ ࢨ ਼ ͹ ฯ ଚ ଉ Γ Ε , ෾ ‫ ޛ ز‬΍ ሺ൅ͳሻ ൌ ሺ൅ͳሻ ൅ሺʹ ൈ ሺ൅ͳሻሻ ൅ ሺʹ ൈ ሺെͳሻሻͲ͍Ζ. ͞ΗΓΕब‫ غ‬4 ͹ब‫ٌغ‬ಕ͹ϛΠϱΩϪࢨ਼ͺ෾‫ز‬͹઴‫Ͳޛ‬ร Կ ͢ ͵ ͏ ͞ ͳ ͗ ಍ ͖ Η Ζ . ͞ ͹ ͞ ͳ ͺ , ҈ ఈ ੓ ‫ݎ ਼ ܐ‬ሺܽሻ͗ ‫ ݎ‬ൌ െʹͳ ં ͢ ͱ ͏ Ζ ͞ ͳ ͹ ๥ ৄͳ͵Ζ.

4. ࢝॑ଲসब‫ٌغ‬ಕ͹੓࣯
4.1. ଲ߻͹੷Ͷ‫ͤؖ‬Ζ‫ܐؖ‬್ͯ͹ଲ߻͹੷Ͷ‫ͤؖ‬Ζ‫ܐؖ‬Νࣞ (30)͖Δࣞ(32)ͶΉͳΌͱ͕͚.
‫ܪ‬ଵ ‫ܪ‬ଷ ൌ ‫ܩ‬ଵ ‫ܩ‬ଷ ൌ ‫ܪ‬ଶ ‫ܪ‬ସ ൌ ‫ܪ‬ସ ‫ܩ‬ଵ ൌ ‫ܪ‬ଷ ‫ܩ‬ଶ ൌ ‫ܩ‬ଶ ‫ܩ‬ସൌ െ‫ܪ‬ଵ ‫ܪ‬ସ ൌ െ‫ܪ‬ଷ ‫ܩ‬ଵ ൌ െ‫ܪ‬ସ ‫ܩ‬ଶ ൌ െ‫ܩ‬ଵ ‫ܩ‬ସ ൌ െ‫ܪ‬ଶ ‫ܪ‬ଷ ൌ െ‫ܩ‬ଶ ‫ܩ‬ଷ ǡ(30)‫ܩ‬ସ ‫ܪ‬ଵ ൌ ‫ܩ‬ଷ ‫ܪ‬ଶ ൌ െ‫ܩ‬ସ ‫ܪ‬ଶ ൌ െ‫ܩ‬ଷ ‫ܪ‬ଵ ǡ(31)‫ܩ‬ଵ ‫ܪ‬ଶ ൌ ‫ܩ‬ଶ ‫ܪ‬ଵ ൌ ‫ܩ‬ଷ ‫ܪ‬ସ ൌ ‫ܩ‬ସ ‫ܪ‬ଷ Ǥ(32)ྭͳ͢ͱ, ‫ܪ‬ଵ ‫ܪ‬ଷ ൌ ‫ܩ‬ଵ ‫ܩ‬ଷ ͺ‫ܪ‬ଷ ‫ܪ‬ଵ ൌ ‫ܩ‬ଷ ‫ܩ‬ଵ ΍қັͤΖ. ଲ߻Νཤ༽͢ͱ‫ͤࢋܯ‬Ζ͞ͳͲ͞ΗΔ͹‫ؖ‬‫ ܐ‬Ν ࣖ ͤ ͞ ͳ ͗ Ͳ ͘ Ζ . Γ ͮ ͱ , ࣞ (30)͖ Δ ࣞ (32)͹ ಍ ड़ ͺ ঴ ͚ . ಝ Ͷ ࣞ (32)ͺ ್ ॑ ଲ স ब‫ٌغ‬ಕͶ͕͏ͱ΍੔ཱིͤΖ . ҐԾ͹٠࿨Ͳͺ৏‫ى‬͹‫ܐؖ‬ΝචགྷͶԢͣͱཤ༽ͤΖ .

4.2. ࢝॑ଲসब‫ٌغ‬ಕ͹෾ྪ࢝॑ଲসब‫ٌغ‬ಕͺҲब‫غ‬͹ؔͶҡ͵ͮͪ࢝ͯ͹ଲসતΝ௪գͤΖ . ௪գ͹࢕๏Ͳ࢝॑ଲ
সब‫ٌغ‬ಕͺൂझྪͶ෾ྪͲ͘Ζ [3]. ൂझྪΝන 4 ͳන 5 Ͷࣖͪ͢.
න̒ʁ࢝॑ଲসब‫ٌغ‬ಕ͹෾ྪ I
෾ྪ ଲসતͳٌಕ఼

ब‫غ‬:‫ݍ‬ড়݇

A1 ‫ܪ‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬ଶ௞ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬ଷ௞ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬ସ௞ାଵ ሿͺ݇ ൅ ʹ

݇൒ͳ

B1 ‫ܪ‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬ଶ௞ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬ଷ௞ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬ସ௞ାଵ ሿͺ݇ ൅ ʹ

݇൒Ͳ

C1 ‫ܪ‬ଶ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬ଶ௞ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬ଷ௞ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬ସ௞ାଵ ሿͺ݇ ൅ ʹ

݇൒ͳ

D1 ‫ܪ‬ଶ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬ଶ௞ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬ଷ௞ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬ସ௞ାଵ ሿͺ݇ ൅ ʹ

݇൒ͳ

E1 ‫ܪ‬ଷ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬ଶ௞ିଵ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬ଷ௞ିଵ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬ସ௞ିଵ ሿͺ݇ െ ʹ

݇൒ͳ

F1 ‫ܪ‬ଷ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬ଶ௞ିଵ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬ଷ௞ିଵ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬ସ௞ିଵ ሿͺ݇ െ ʹ

݇൒ʹ

G1 ‫ܪ‬ସ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬ଶ௞ିଵ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬ଷ௞ିଵ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬ସ௞ିଵ ሿͺ݇ െ ʹ

݇൒ͳ

H1 ‫ܪ‬ସ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬௞ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬ଶ௞ିଵ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬ଷ௞ିଵ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬ସ௞ିଵ ሿͺ݇ െ ʹ

݇൒ͳt 14 tන̓ʁ࢝॑ଲসब‫ٌغ‬ಕ͹෾ྪᶚ෾ྪ ଲসતͳٌಕ఼

ब‫غ‬:‫ݍ‬ড়݇

A2 ‫ܩ‬ଶ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬ଶ௞ାଵ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬ଷ௞ାଵ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬ସ௞ାଵ ሿͺ݇ ൅ ʹ

݇൒ͳ

B2 ‫ܩ‬ଶ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬ଶ௞ାଵ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬ଷ௞ାଵ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬ସ௞ାଵ ሿͺ݇ ൅ ʹ

݇൒Ͳ

C2 ‫ܩ‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬ଶ௞ାଵ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬ଷ௞ାଵ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬ସ௞ାଵ ሿͺ݇ ൅ ʹ

݇൒ͳ

D2 ‫ܩ‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬ଶ௞ାଵ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬ଷ௞ାଵ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬ସ௞ାଵ ሿͺ݇ ൅ ʹ

݇൒ͳ

E2 ‫ܩ‬ସ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬ଶ௞ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬ଷ௞ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬ସ௞ିଵ ሿͺ݇ െ ʹ

݇൒ͳ

F2 ‫ܩ‬ସ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬ଶ௞ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬ଷ௞ ሿ ՜ ‫ܩ‬ସ ሾ‫ݖ‬ସ௞ିଵ ሿͺ݇ െ ʹ

݇൒ʹ

G2 ‫ܩ‬ଷ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ଶ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬ଶ௞ ሿ ՜ ‫ܪ‬ଵ ሾ‫ݖ‬ଷ௞ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬ସ௞ିଵ ሿͺ݇ െ ʹ

݇൒ͳ

H2 ‫ܩ‬ଷ ሾ‫ݖ‬଴ ሿ ՜ ‫ܩ‬ଵ ሾ‫ݖ‬௞ ሿ ՜ ‫ܪ‬ସ ሾ‫ݖ‬ଶ௞ ሿ ՜ ‫ܪ‬ଶ ሾ‫ݖ‬ଷ௞ ሿ ՜ ‫ܩ‬ଷ ሾ‫ݖ‬ସ௞ିଵ ሿͺ݇ െ ʹ

݇൒ͳ࠹ॵͶ A1 Ν΍ͳͶන͹‫ߺى‬͹қັΝઈ໎ͤΖ . ब‫ݍغ‬ͺ‫ ݍ‬ൌ ͺ݇ ൅ ʹͲ͍Ζ. ॵ‫ݖ఼غ‬଴ ͺଲসતܵுభ ৏Ͷ͍Ε, ٌಕ఼ ‫ݖ‬௞ ͺଲসતܵுయ Ͷ͍Ζ. ‫ܪ ߺى‬ଵ ሾ‫ݖ‬଴ ሿ ՜ ‫ܪ‬ଷ ሾ‫ݖ‬௞ ሿ͗, ͞ ͹͞ͳΝන‫͏ͱ͢ ݳ‬
Ζ. ٌಕ఼ ‫ݖ‬ଵ ͖Δ‫ݖ‬௞ିଵ ΉͲͺଲ সતͶ͵͏ . ࣏Ͷٌಕ఼ ‫ݖ‬ଶ௞ ͺଲসત ܵீమ ৏Ͷ͍Ε, ٌಕ఼‫ݖ‬ଷ௞ ͺଲসતܵீర Ͷ͍Ζ. ब‫غ‬͹஺ౕൔ෾Ͳ࠸;ٌಕ఼ͺଲসત ܵுభ Ͷ໯Ζ. ͯΉΕ, ‫ݖ‬ସ௞ାଵ ‫ܵ א‬ுభ ͗੔Εཱིͯ. ͞͞Ͳ‫ݖ‬଴ ് ‫ݖ‬ସ௞ାଵ Ͳ͍Ζ. A1 Ͷ‫؜‬ΉΗΖ࢝॑ଲসब‫ٌغ‬ಕ͹࠹ঘब‫غ‬ͺ ‫ ݍ‬ൌ ͳͲሺ݇ ൌ ͳሻͲ͍Ζ.
ब‫ٌغ‬ಕ͗௪գͤΖଲসત͹ॳ൬ͺ , න 4 ͹෾ྪͲͺ ‫ܪ‬௜ ሺ݅ ൌ ͳǡʹǡ͵ǡͶሻ͖Δ࢟ΉΖ. ͞͹න‫ݳ‬Ν॰յ͢, ٌಕ఼͹ॳ൬ΝรߍͤΖͳන 5 ͗ಚΔΗΖ. න 4 ͳන 5 ͺಋͣ৚ๅΝ༫͓Ζ. ॵ‫غ‬఼ ͗ ܵு೔ ሺ݅ ൌ ͳǡʹǡ͵ǡͶሻ͗ ͍ Ζ ৖ ߻ ͳ , ܵீ೔ ሺ݅ ൌ ͳǡʹǡ͵ǡͶሻͶ ͍ Ζ ৖ ߻ Ͷ ෾ ͜ ͱ ͕ ͚ ͳ න ͗ ཤ ༽ ͢ Ώ ͤ͏.
ൂझྪ͹࢝॑ଲসब‫ٌغ‬ಕ͹ྭΝਦ 5 Ͷࣖͤ. ਦ 5(A)Ͳ, ‫ݖ‬଴ ‫ܵ א‬ுభ (‫ݖ‬଴ ͺ, ਦͲͺ 0 ͳන‫͠ݳ‬Ηͱ͏Ζ)͖Δड़൅ͪ͢৖߻͗෾ྪ A1 Ͷ͵Ε, ‫ݖ‬ଶ Νॵ‫ͪ͢ͳ఼غ‬৖߻͗෾ྪ A2 Ͳ͍Ζ. ਦ
5(A)͹ྭͲͺ, ब‫غ‬ͺ‫ ݍ‬ൌ ͳͲͲ, 8 ‫ݺ‬͹ٌಕ఼͗ଲসત৏Ͷ͍Ζ .

t 15 tਦ 6ʁ࢝॑ଲসब‫ٌغ‬ಕ͹ྭ.੖਼݇ͺٌಕ఼‫ݖ‬௞ Νනͤ. (A) ܽ ൌ ʹǤͷ. (B) ܽ ൌ ͳ. (C) ܽ ൌ ͳ. (D)ܽ ൌ ʹǤͷ. (E) ܽ ൌ ͳǤͷ. (F) ܽ ൌ ʹ. (G) ܽ ൌ Ͷ. (H) ܽ ൌ ʹ.

t 16 t෾ྪ A1 ͹৖߻, ଲসતܵுభ Ͷ͍Ζॵ‫ݖ ఼غ‬଴ ͖Δड़൅͢, ౏஦Ͳଲসત ܵுయ , ܵீమ , ܵீర Ν‫ܨ‬༟͢ͱ࠸;ଲসત ܵுభ Ͷ໯Ζ. ͞͹ब‫ٌغ‬ಕͺ, ࠪӊ͹ଲসત ܵுభ , ৏Ծ͹ଲসતܵீమ , ࣾΌࠪӊ͹ଲস ત ܵுయ , ͨ ͢ ͱ ࣾ Ό ৏ Ծ ͹ ଲ স ત ܵீర Ͷ ٌ ಕ ఼ Ν ΍ ͯ ͖ Δ ࢝ ॑ ଲ স ब ‫ ٌ غ‬ಕ Ͳ ͍ Ζ . ͞ ͹ ब‫ٌغ‬ಕ͗‫ܨ‬༟͢ͱ͏͚ॳ൬Ν΍ͳͶ , ब‫͗غ‬ͺ݇ ൅ ʹͲ͍Ζ͞ͳΝৄ໎ͤΖ.
A1 ͹৖߻. ࠹ॵͶब‫غ‬Ν݀ఈ͢, ࣏Ͷ͞͹ब‫ٌغ‬ಕ͗ଚࡑͤΖͪΌ͹ড়݇Ν಍͚ . ٌಕ఼‫ݖ‬଴ͳ ‫ݖ‬௞ Ͷ ͯ ͏ ͱ , ‫ݖ‬଴ ‫ܵ א‬ுభ ͖ ͯ ‫ݖ‬௞ ‫ܵ א‬ுయ ͗ ੔ Ε ཱི ͯ . ࠹ ॵ Ͷ ‫ݖ‬ଶ௞ ‫ீܵ א‬మ Ν ಍ ͚ . ‫ݖ‬௞ ‫ܵ א‬ுయ Γ Ε , ‫ݖ‬௞ ൌ‫ܪ‬ଷ ‫ݖ‬௞ ͗ ੔ Ε ཱི ͯ . ྈ ว Ͷ ܶ ௞ Ν ࡠ ༽ ͤ Ζ . ࠪ ว ͺ ‫ݖ‬ଶ௞ Ͳ , ӊ ว ͺ ܶ ௞ ‫ܪ‬ଷ ‫ݖ‬௞ ൌ ‫ܪ‬ଷ ܶ ି௞ ‫ݖ‬௞ ൌ ‫ܪ‬ଷ ‫ݖ‬଴ . ‫ܐ ؖ‬‫ݖ‬ଶ௞ ൌ ‫ܪ‬ଷ ‫ݖ‬଴ ͹ ྈ ว Ͷ ‫ܩ‬ଶ Ν ࡠ ༽ ͤ Ζ ͳ , ‫ܩ‬ଶ ‫ܪ‬ଷ ‫ݖ‬଴ ൌ ‫ܩ‬ଶ ‫ݖ‬ଶ௞ ͗ ಚ Δ Η Ζ . ‫ܩ ࣞ ܐ ؖ‬ଶ ‫ܪ‬ଷ ൌ ‫ܪ‬ଷ ‫ܪ‬ଵ Ν ཤ ༽ ͤΖ(ࣞ(30)Ν‫ݡ‬Γ). ‫ܪ‬ଵ ‫ݖ‬଴ ൌ ‫ݖ‬଴ Ͳ͍ Ζ͖Δ , ‫ܪ‬ଷ ‫ܪ‬ଵ ‫ݖ‬଴ ൌ ‫ܪ‬ଷ ‫ݖ‬଴ ൌ ‫ݖ‬ଶ௞ . Γͮͱ, ‫ݖ‬ଶ௞ ൌ ‫ܩ‬ଶ ‫ݖ‬ଶ௞ ੔Ε ཱིͯ .
͞Ηͺ‫ݖ‬ଶ௞ ‫ீܵ א‬మ ΝқັͤΖ.
࣏ Ͷ ‫ݖ‬ଶ௞ ൌ ‫ܩ‬ଶ ‫ݖ‬ଶ௞ ͹ ྈ ว Ͷ ܶ ି௞ Ν ࡠ ༽ ͤ Ζ . ࠪ ว ͺ ‫ݖ‬௞ Ͳ , ӊ ว ͺ ܶ ି௞ ‫ܩ‬ଶ ‫ݖ‬ଶ௞ ൌ ‫ܩ‬ଶ ܶ ௞ ‫ݖ‬ଶ௞ ൌ ‫ܩ‬ଶ ‫ݖ‬ଷ௞ .
ಚ Δ Η ͪ ‫ܩ‬ଶ ‫ݖ‬ଷ௞ ൌ ‫ݖ‬௞ Ν ‫ݖ‬ଷ௞ ൌ ‫ܩ‬ଶ ‫ݖ‬௞ ͳ ॽ ͘ ௜ ͢ , ྈ ว Ͷ ‫ܩ‬ସ Ν ࡠ ༽ ͤ Ζ . ࠪ ว ͺ ‫ܩ‬ସ ‫ݖ‬ଷ௞ Ͳ , ӊ ว ͺ‫ܩ‬ସ ‫ܩ‬ଶ ‫ݖ‬௞ . ͞͹ӊวΝҐԾ͹Γ͑Ͷॽ͘‫͓׷‬Ζ .
‫ܩ‬ସ ‫ܩ‬ଶ ‫ݖ‬௞ ൌ ‫ܩ‬ଷ ‫ܩ‬ଵ ‫ݖ‬௞ ൌ ‫ܩ‬ଷ ‫ܩ‬ଵ ‫ܪ‬ଵ ‫ܪ‬ଵ ‫ݖ‬௞ ൌ ‫ܩ‬ଷ ܶ ିଵ ‫ܪ‬ଵ ‫ݖ‬௞ൌ ‫ܩ‬ଷ ܶ ିଵ ‫ିݖ‬௞ ൌ ‫ܩ‬ଷ ‫ିݖ‬௞ିଵ ൌ ‫ܩ‬ଷ ‫ܪ‬ଷ ‫ܪ‬ଷ ‫ିݖ‬௞ିଵ ൌ ܶ ିଵ ‫ݖ‬ଷ௞ାଵ ൌ ‫ݖ‬ଷ௞

(33)

͞ ͞ Ͳ , ‫ܩ‬ସ ‫ܩ‬ଶ ൌ ‫ܩ‬ଷ ‫ܩ‬ଵ , ‫ܪ‬ଵ ‫ݖ‬௞ ൌ ‫ିݖ‬௞ , ‫ܪ‬ଷ ‫ିݖ‬௞ିଵ ൌ ‫ݖ‬ଷ௞ାଵ ͹ ‫ ܐ ؖ‬Ν ॳ Ͷ ཤ ༽ ͢ ͪ . Ή ͳ Ό ͱ , ‫ݖ‬ଷ௞ ൌ‫ܩ‬ସ ‫ݖ‬ଷ௞ ͗ಚΔΗͪ. ͞Ηͺ‫ݖ‬ଷ௞ ‫ீܵ א‬ర ΝқັͤΖ.
ब‫غ‬Ν಍͚. ‫ܩ‬ସ ‫ݖ‬ଷ௞ ൌ ‫ݖ‬ଷ௞ ͹ࠪวΝҐԾ͹Γ͑Ͷॽ͘‫͓׷‬Ζ .
‫ܩ‬ସ ‫ݖ‬ଷ௞ ൌ ‫ܩ‬ସ ‫ܩ‬ଶ ‫ݖ‬௞ ൌ ‫ܩ‬ସ ‫ܩ‬ଶ ‫ܪ‬ଷ ‫ݖ‬௞ ൌ ‫ܩ‬ଷ ‫ܩ‬ଵ ‫ܪ‬ଷ ‫ݖ‬௞ ൌ ‫ܩ‬ଷ ‫ܪ‬ସ ‫ܪ‬ଵ ‫ݖ‬௞ൌ ‫ܩ‬ଷ ‫ܪ‬ସ ‫ିݖ‬௞ ൌ ‫ܩ‬ଷ ‫ܪ‬ଷ ‫ܪ‬ଷ ‫ܪ‬ସ ‫ିݖ‬௞ ൌ ܶ ିଵ ܴ‫ିݖ‬௞ ൌ ‫ିݖ‬௞ିଵା௤Ȁଶ Ǥ͞ ͞ Ͳ , ‫ܩ‬ଶ ‫ݖ‬௞ ൌ ‫ݖ‬ଷ௞ , ‫ݖ‬௞ ൌ ‫ܪ‬ଷ ‫ݖ‬௞ , ‫ܩ‬ସ ‫ܩ‬ଶ ൌ ‫ܩ‬ଷ ‫ܩ‬ଵ , ‫ܩ‬ଵ ‫ܪ‬ଷ ൌ ‫ܪ‬ସ ‫ܪ‬ଵ , ‫ܪ‬ଵ ‫ݖ‬௞ ൌ ‫ିݖ‬௞ , ܴ ൌ ‫ܪ‬ଷ ‫ܪ‬ସ , ܴ ൌ ܶ

(34)
௤Ȁଶ

Ν

ॳͶཤ༽ͪ͢. Ґ৏Ͳ‫ݖ‬ଷ௞ ൌ ‫ିݖ‬௞ିଵା௤Ȁଶ ͗಍͖Η, ब‫ ݍ͗غ‬ൌ ͺ݇ ൅ ʹͳ݀ΉΖ.
‫ݖ ܐ ؖ‬ସ௞ାଵ ‫ܵ א‬ுభ Ν ࣖ ͤ . ‫ݖ‬଴ ൌ ‫ܪ‬ଵ ‫ݖ‬଴ ͹ ྈ ว Ͷ ܶ ିସ௞ିଵ Ν ࡠ ༽ ͤ Ζ ͳ , ‫ିݖ‬ସ௞ିଵ ൌ ܶ ିସ௞ିଵ ‫ܪ‬ଵ ‫ݖ‬଴ ൌ‫ܪ‬ଵ ܶ ସ௞ାଵ ‫ݖ‬଴ ൌ ‫ܪ‬ଵ ‫ݖ‬ସ௞ାଵ . ब ‫ ݍ ͗ غ‬ൌ ͺ݇ ൅ ʹͲ ͍ Ζ ͖ Δ , ‫ିݖ‬ସ௞ିଵ ൌ ‫ݖ‬ସ௞ାଵ . Γ ͮ ͱ , ‫ݖ‬ସ௞ାଵ ൌ ‫ܪ‬ଵ ‫ݖ‬ସ௞ାଵ ͗੔Εཱིͯ.
݇ ൌ ͲΝຮͪͤब‫ غ‬2 ͗ଚࡑ͢͵͏͞ͳΝࣖͤ . ݇ ൌ Ͳͺ, ଲসતܵுభ ͳܵுయ ͗ܳଵ଴ Ґ֐Ͳިࠫͤ
Ζ ͞ ͳ Ν қ ັ ͤ Ζ . ͢ ͖ ͢ , ଲ স ત ܵுభ ͹ ܳଵ଴ Ͷ ͕ ͜ Ζ ܑ ͘ ʹȀܽ͗ ਜ਼ Ͳ , Ҳ ๏ ͹ ܵுయ ͹ ܑ ͘ െͳͺ ෝ Ͳ͍ Ζ . Γ ͮ ͱ, ܵுభ ͳ ܵுయ ͗ ܳଵ଴ Ґ ֐Ͳ ި ࠫ ͤ Ζ ͞ ͳͺ ͵ ͏ . ࣏ Ͷ , ݇ ൌ ͳͺ Ն ೵ Ͳ ͍ Ζ ͞ͳΝࣖͤ. ܵுభ ͹ܑ͘ߦ଴ ൌ ʹȀܽ͹଀ ߦଵ ൌ ሺܽଶ െ ʹሻȀሺܽଷ െ ͵ܽሻ ͗ 0 Ͷ͵Ζড়݇ͺ, ܽ ൌ ξʹͲ͍Ζ.
B1 ͹৖߻. ଲসતܵுభ ͹ܳଵ଴ Ͷ͕͜Ζܑ͘ͺ ʹȀܽͲ͍Ζ. ଲসતܵுర ͹ܑ͘ͺ 1 Ͳ͍Ζ͖Δ, ྡֆ஍ͺ 2 ͳಚΔΗΖ. ͞͹৖߻, ݇ ൒ ͲͲ͍Ζ. ͯΉΕ, ͞͹৖߻Ͷ͹Ίब‫ غ‬2 ͗ଚࡑͤΖ.
C1 ͹৖߻. ଲসત ܵுమ ͹ܑ͘ͺߦ଴ ൌ ͲͲ, ଲসત ܵுయ ͹ܑ͘ͺെͳͲ͍Ζ͖Δ, ݇ ൌ Ͳͺ੔Εཱིͪ͵͏. Γͮͱ, ݇ ൌ ͳͳͤΖ. ߦଵ Ν‫ٽ‬Όͱ, ߦଵ ൌ െͳͳ஖͚.
ܽ
ൌ െͳǤ
ଶ
ܽ െͳ

ሺ35ሻ

͞ΗΓΕྡֆ஍ͺ ሺξͷ െ ͳሻȀʹͲ͍Ζ. ͞͹৖߻, ݇ ൒ ͳͲ͍Ζ.
D1 ͹৖߻. ଲসતܵுమ ͹ܑ͘ͺ ߦ଴ ൌ ͲͲ, ଲসતܵுర ͹ܑ͘ͺ 1 Ͳ͍Ζ͖Δ, ݇ ൌ Ͳͺ੔Εཱིͪt 17 t͵͏. Γͮͱ, ݇ ൌ ͳͳͤΖ. C1 ͹৖߻ͳಋ༹Ͷ͢ͱܽൌͳܽଶ െ ͳ

(36)

͗ಚΔΗΖ. ͞ΗΓΕྡֆ஍ͺ ሺξͷ ൅ ͳሻȀʹͲ͍Ζ. ͞͹৖߻, ݇ ൒ ͳͲ͍Ζ.
E1 ͹৖߻. ଲসત ܵுయ ͹ܑ ͘ͺ ߦ଴ ൌ െͳͲ, ଲস ત ܵுభ ͹ܑ͘ͺ ʹȀܽͲ͍Ζ͖ Δ , ݇ ൌ Ͳͺ੔Ε ཱིͪ͵͏. ݇ ൌ ͳͳ͢ͱߦଵ Ν‫ٽ‬ΌΖͳྡֆ஍Ν݀ΌΖࣞ
ܽ൅ͳ
ʹ
ൌ
ଶ
ܽ ൅ܽെͳ ܽ

ሺ37ሻ

͗ಚΔΗΖ. ྡֆ஍ͺͳͲ͍Ζ. ͞͹৖߻, ݇ ൒ ͳͲ͍Ζ.
F1 ͹৖߻. ଲসત ܵுయ ͹ܑ ͘ͺ ߦ଴ ൌ െͳͲ, ଲসત ܵுమ ͹ܑ͘ͺ ͲͲ͍Ζ ͖Δ , ݇ ൌ Ͳͺ੔Ε ཱིͪ͵͏ . ݇ ൌ ͳͳ͢ ͱ, ߦଵ Ν ‫ٽ‬Ό Ζ ͳ ߦଵ ൌ ሺܽ ൅ ͳሻȀሺܽଶ ൅ ܽ െ ͳሻ͗ಚ ΔΗ Ζ . ͖͢ ͢ , ߦଵ ൌ Ͳͺ੔ ཱི͢͵͏. Γͮͱ, ߦଶ Ν‫ٽ‬ΌΖ.
ߦଶ ൌܽଷ ൅ ܽଶ െ ʹܽ െ ͳǤܽସ ൅ ܽଷ െ ͵ܽଶ െ ʹܽ ൅ ͳ(38)͞͹ࣞ͹෾ࢢΝ 0 ͳ͢ͱྡֆ஍͗ 1.24698 ͳಚΔΗΖ. ড়݇ͺ݇ ൒ ʹͳ͵Ζ.
G1 ͹৖߻. ݇ ൌ ͳͺ, ब‫ غ‬2 ͹ٌಕ͖Δਫ਼ͣͪब‫ غ‬3 ͹ब‫ٌغ‬ಕΝ༫͓Ζ(ਦ 5(G)). ͯΉΕ,ब‫غ‬ͺ͸ ൌ ʹ ൈ ͵. ਼஍‫ Ͳࢋܯ‬෾‫͗ز‬ਫ਼ͣΖྡֆ஍ͺ 3.6963 ͳಚΔΗΖ.
H1 ͹৖߻. ଲসતܵுర ͹ܑ͘ͺߦ଴ ൌ ͳΝཤ༽͢ͱߦଵ Ν‫ٽ‬ΌΖ.
ܽെͳߦଵ ൌ ଶ
Ǥܽ െܽെͳ

(39)ଲসતܵுమ ͹ܑ͘ͺ 0 Ͳ͍Ζ͖Δ, ྡֆ஍ͺ 1. Ήͪ݇ ൒ ͳͲ͍Ζ.
ໍୌ 4.1. ࢝॑Ն‫Ͳ଀ࣺٱ‬ͺ, 180 ౕյ఺ଲস੓Ν΍ͯब‫ٌغ‬ಕͺ್॑ଲসब‫ٌغ‬ಕͳ࢝॑ଲ
সब‫ٌغ‬ಕͶ‫ݸ‬ΔΗΖ.
ৄ໎. ್॑ଲসब‫ٌغ‬ಕ͹৖߻͹क௃ͺఈཀྵ 3.1 Ͳ‫ع‬Ͷफ़΄ΔΗͱ͏Ζ . ࢝॑ଲসब‫ٌغ‬ಕ͹৖߻͹क௃Νৄ໎ͤΖ . ͞͞Ͳͺන 4 Ͷ͕͜Ζ A1 ͹৖߻Νৄ໎ͤΖ.
࠹ॵͶ‫ݖ‬௤Ȁଶ ൌ ܴ‫ݖ‬଴ Νࣖͤ. A1 ͹৖߻ͺ, ܴ‫ݖ‬଴ ൌ ‫ܪ‬ଶ ‫ܪ‬ଵ ‫ݖ‬଴ Νཤ༽ͤΖ. ‫ݖ‬଴ ‫ܵ א‬ுభ ΓΕ, ܴ‫ݖ‬଴ ൌ ‫ܪ‬ଶ ‫ݖ‬଴ ͗
ಚΔΗΖ. ࣏Ͷ‫ݖ‬ଶ௞ ‫ீܵ א‬మ ΓΕ, ‫ܩ‬ଶ ‫ݖ‬ଶ௞ ൌ ‫ݖ‬ଶ௞ ͗੔Εཱིͯ. ྈวͶ‫ܪ‬ଶ Νࡠ༽ͤΖ. ‫ܪ‬ଶ ‫ܩ‬ଶ ‫ݖ‬ଶ௞ ൌ ‫ݖ‬ଶ௞ାଵ ΓΕ, ‫ݖ‬ଶ௞ାଵ ൌ ‫ܪ‬ଶ ‫ݖ‬ଶ௞ ͗ಚΔΗΖ. ͞ΗΝ, ‫ݖ‬ସ௞ାଵ ൌ ‫ܪ‬ଶ ‫ݖ‬଴ ͳॽ͘௜ͤͳ, ܴ‫ݖ‬଴ ൌ ‫ܪ‬ଶ ‫ݖ‬଴ ൌ ‫ݖ‬ସ௞ାଵ ͗‫ٽ‬ΉΖ .
‫ݍ‬Ȁʹ ൌ Ͷ݇ ൅ ͳ Ͳ ͍ Ζ ͖ Δ , ‫ݖ‬௤Ȁଶ ൌ ܴ‫ݖ‬଴ ͗ ࣖ ͠ Η ͪ . ࣏ Ͷ ‫ݖ‬௤Ȁଶ ൌ ܴ‫ݖ‬଴ ͹ ྈ ว Ͷ ܶ Ν ࡠ ༽ ͤ Ζ ͳ ,‫ݖ‬௤Ȁଶାଵ ൌ ܴܶ‫ݖ‬଴ ൌ ܴܶ‫ݖ‬଴ ൌ ܴ‫ݖ‬ଵ ͗ಚΔΗΖ. ͞ΗΝ‫܃‬Εศͤ͞ͳͲ, ࢝॑ଲসब‫ٌغ‬ಕ͗ 180 ౕյ఺ଲস੓Ν࣍ͯ͞ͳ͗ࣖ͠ΗΖ . Ґ৏Ͳৄ໎Νश͓Ζ.

4.3. ଲ߻න‫ݳ‬࢝॑ଲসब‫ٌغ‬ಕΝಝ௅෉͜Ζଲ߻න‫ݳ‬Ν಍͑͞ . ݃Վͺන 6 ͳන 7 Ͷࡎͦͪ. ͞͞Ͳͺ
A1 ͹ଲ߻න‫ݳ‬Ν಍ड़ͤΖ. Ήͥ, ܶ ଼௞ାଶ Ν࣏͹Γ͑Ͷ෾մͤΖ͞ͳ͖Δ࢟ΌΖ .
ܶ ଼௞ାଶ ൌ ܶ ଶ௞ ‫ܪ‬ଵ ‫ܩ ל‬ଵ ܶ ଺௞ାଵ ‫ܪ‬ଷ ‫ܪ ל‬ଷ Ǥ࣏Ͷ୊್൬໪͹ࡠ༽Νॽ͘௜ͤ .
t 18 t(40)‫ܩ‬ଵ ܶ ଺௞ାଵ ‫ܪ‬ଷ ൌ ‫ܩ‬ଶ ‫ܩ‬ଶ ‫ܩ‬ଵ ܶ ଺௞ାଵ ‫ܪ‬ଷ ൌ ‫ܩ‬ଶ ܴܶ ଺௞ାଵ ‫ܪ‬ଷൌ ‫ܩ‬ଶ ܶ ିସ௞ିଵ ܶ ଺௞ାଵ ‫ܪ‬ଷ ൌ ‫ܩ‬ଶ ܶ ଶ௞ ‫ܪ‬ଷ Ǥॽ ͘ ‫ ͓ ׷‬͹ ౏ ஦ Ͳ ܴΝ ܴ ൌ ܶ

ିସ௞ିଵ

(41)
ଶ௞

Ͳ ஖ ͘ ‫ ͪ ͓ ׷‬. ୊ ್ ͹ ࡠ ༽ ‫ܩ‬ଶ ܶ ‫ܪ‬ଷ Ͷ , ࠪ ଲ ߻ ͳ ӊ ଲ ߻ Ν ߻Κͦͱଲ߻න‫͗ݳ‬ಚΔΗΖ .
ܶ ଼௞ାଶ ൌ ܶ ଶ௞ ‫ܪ‬ଵ ‫ܩ ל‬ଶ ܶ ଶ௞ ‫ܪ‬ଷ ‫ܪ ל‬ଷ Ǥ(42)න̔ʁଲ߻න‫ݳ‬ᶙ෾ྪ ଲ߻න‫ࠪ[ݳ‬ଲ߻, ߅౵ࣺ଀, ӊଲ߻]
଼௞ାଶ

ଶ௞

ଶ௞

ଚࡑড়݇ 1௞ଚࡑড়݇ 2௞A1

ܶൌ ܶ ‫ܪ‬ଵ ‫ܩ ל‬ଶ ܶ ‫ܪ‬ଷ ‫ܪ ל‬ଷܶ ܵுభ ‫ܵ ת‬ுయ ് ‫׎‬ܶ ܵுయ ‫ீܵ ת‬మ ് ‫׎‬

B1ܶ ଼௞ାଶ ൌ ܶ ଶ௞ ‫ܪ‬ଵ ‫ܩ ל‬ଶ ܶ ଶ௞ ‫ܪ‬ସ ‫ܪ ל‬ସܶ ௞ ܵுభ ‫ܵ ת‬ுర ് ‫׎‬ܶ ௞ ܵுర ‫ீܵ ת‬మ ് ‫׎‬C1

ܶ ଼௞ାଶ ൌ ܶ ଶ௞ ‫ܪ‬ଶ ‫ܩ ל‬ଵ ܶ ଶ௞ ‫ܪ‬ଷ ‫ܪ ל‬ଷܶ ௞ ܵுమ ‫ܵ ת‬ுయ ് ‫׎‬ܶ ௞ ܵுయ ‫ீܵ ת‬భ ് ‫׎‬D1ܶ ଼௞ାଶ ൌ ܶ ଶ௞ ‫ܪ‬ଶ ‫ܩ ל‬ଵ ܶ ଶ௞ ‫ܪ‬ସ ‫ܪ ל‬ସܶ ௞ ܵுమ ‫ܵ ת‬ுర ് ‫׎‬ܶ ௞ ܵுర ‫ீܵ ת‬భ ് ‫׎‬E1

ܶ ଼௞ିଶ ൌ ܶ ଶ௞ ‫ܪ‬ଷ ‫ܩ ל‬ସ ܶ ଶ௞ିଶ ‫ܪ‬ଵ ‫ܪ ל‬ଵܶ ௞ ܵுయ ‫ܵ ת‬ுభ ് ‫׎‬ܶ ௞ିଵ ܵுభ ‫ீܵ ת‬ర ് ‫׎‬F1ܶ ଼௞ିଶ ൌ ܶ ଶ௞ ‫ܪ‬ଷ ‫ܩ ל‬ସ ܶ ଶ௞ିଶ ‫ܪ‬ଶ ‫ܪ ל‬ଶܶ ௞ ܵுయ ‫ܵ ת‬ுమ ് ‫׎‬ܶ ௞ିଵ ܵுమ ‫ீܵ ת‬ర ് ‫׎‬G1ܶ ଼௞ିଶ ൌ ܶ ଶ௞ ‫ܪ‬ସ ‫ܩ ל‬ଷ ܶ ଶ௞ିଶ ‫ܪ‬ଵ ‫ܪ ל‬ଵܶ ௞ ܵுర ‫ܵ ת‬ுభ ് ‫׎‬ܶ ௞ିଵ ܵுభ ‫ீܵ ת‬య ് ‫׎‬H1ܶ ଼௞ିଶ ൌ ܶ ଶ௞ ‫ܪ‬ସ ‫ܩ ל‬ଷ ܶ ଶ௞ିଶ ‫ܪ‬ଶ ‫ܪ ל‬ଶܶ ௞ ܵுర ‫ܵ ת‬ுమ ് ‫׎‬ܶ ௞ିଵ ܵுమ ‫ீܵ ת‬య ് ‫׎‬න̕ʁଲ߻න‫ ݳ‬II෾ྪଲ߻න‫ࠪ[ݳ‬ଲ߻, ߅౵ࣺ଀, ӊଲ߻]ൌ ‫ܩ‬ସ ܶൌ ‫ܩ‬ଷ ܶଶ௞‫ܩ ל‬ସ ܶିଶ௞ିଶ‫ܩ ל‬ଷ ܶ

ିଶ௞ିଶ

B2

ܶ

ି଼௞ିଶ

C2

ܶ ି଼௞ିଶ ൌ ‫ܩ‬ସ ܶ ଶ௞ ‫ܩ ל‬ସ ܶ ିଶ௞ିଶ ‫ܪ‬ଶ ‫ܩ ל‬ଵܶ ି௞ ܵீర ‫ீܵ ת‬భ ് ‫׎‬ܶ ି௞ିଵ ܵுమ ‫ீܵ ת‬ర ് ‫׎‬D2ܶ ି଼௞ିଶ ൌ ‫ܩ‬ଷ ܶ ଶ௞ ‫ܩ ל‬ଷ ܶ ିଶ௞ିଶ ‫ܪ‬ଶ ‫ܩ ל‬ଵܶ ି௞ ܵீయ ‫ீܵ ת‬భ ് ‫׎‬ܶ ି௞ିଵ ܵுమ ‫ீܵ ת‬య ് ‫׎‬E2ܶ ି଼௞ାଶ ൌ ‫ܩ‬ଶ ܶ ଶ௞ ‫ܩ ל‬ଶ ܶ ିଶ௞ ‫ܪ‬ଷ ‫ܩ ל‬ସܶ ି௞ ܵீమ ‫ீܵ ת‬ర ് ‫׎‬ܶ ି௞ ܵுయ ‫ீܵ ת‬మ ് ‫׎‬F2ܶ ି଼௞ାଶ ൌ ‫ܩ‬ଵ ܶ ଶ௞ ‫ܩ ל‬ଵ ܶ ିଶ௞ ‫ܪ‬ଷ ‫ܩ ל‬ସܶ ି௞ ܵீభ ‫ீܵ ת‬ర ് ‫׎‬ܶ ି௞ ܵுయ ‫ீܵ ת‬భ ് ‫׎‬G2ܶ ି଼௞ାଶ ൌ ‫ܩ‬ଶ ܶ ଶ௞ ‫ܩ ל‬ଶ ܶ ିଶ௞ ‫ܪ‬ସ ‫ܩ ל‬ଷܶ ି௞ ܵீమ ‫ீܵ ת‬య ് ‫׎‬ܶ ି௞ ܵுర ‫ீܵ ת‬మ ് ‫׎‬

H2ܶ ି଼௞ାଶ ൌ ‫ܩ‬ଵ ܶ ଶ௞ ‫ܩ ל‬ଵ ܶ ିଶ௞ ‫ܪ‬ସ ‫ܩ ל‬ଷܶ ି௞ ܵீభ ‫ீܵ ת‬య ് ‫׎‬ܶ ି௞ ܵுర ‫ீܵ ת‬భ ് ‫׎‬‫ܪ‬ଵ ‫ܩ ל‬ଶ‫ܪ‬ଵ ‫ܩ ל‬ଶ

ܶ

ି௞

ܶ

ି௞

ଚࡑড়݇ 2ܶ

A2ଶ௞ଚࡑড়݇ 1ି଼௞ିଶܵீర ‫ீܵ ת‬మ ് ‫׎‬ܵீయ ‫ீܵ ת‬మ ് ‫׎‬

ܶ

ି௞ିଵ

ܵுభ ‫ீܵ ת‬ర ് ‫׎‬

ܶ

ି௞ିଵ

ܵுభ ‫ீܵ ת‬య ് ‫׎‬

A1 ͹ড়݇ͳ͢ͱ, ‫ܪ‬ଷ ‫ݖ‬௞ ൌ ‫ݖ‬௞ ͳ‫ܩ‬ଶ ‫ݖ‬ଶ௞ ൌ ‫ݖ‬ଶ௞ ͍͗Ζ. ͞ΗΝཤ༽͢ͱࡠ༽ ‫ܩ‬ଶ ܶ ଶ௞ ‫ܪ‬ଷ ͗߅౵ࣺ଀Ͳ
͍Ζ͞ͳ͗ࣖͦΖ.
‫ܩ‬ଶ ܶ ଶ௞ ‫ܪ‬ଷ ‫ݖ‬௜ ൌ ‫ܩ‬ଶ ܶ ଶ௞ ‫ݖ‬ଶ௞ି௜ ൌ ‫ܩ‬ଶ ‫ݖ‬ସ௞ି௜ ൌ ‫ݖ‬௜ Ǥ

(43)

௞

࣏Ͷଲসત͹‫ܐؖ‬Ν಍͑͞. ࠪଲ߻ͳӊଲ߻͖Δ, ܶ ܵுభ ‫ܵ ת‬ுయ ് ‫ ׎‬Νຮ఼ͪͤ͗ A1 Νຮͪͤ࢝॑ଲসब‫ٌغ‬ಕͲ͍Ζ . ࣏Ͷ߅౵ࣺ଀Ν࣏͹Γ͑Ͷॽ͘௜ͤ .
ܶ ଶ௞ ‫ܪ‬ଷ ‫ݖ‬௜ ൌ ‫ܩ‬ଶ ‫ݖ‬௜ Ǥሺ44ሻ

ଲ ߻ ܶ ଶ௞ ‫ܪ‬ଷ Ͷ ͯ ͏ ͱ ෈ ร ͵ ॄ ߻ ͺ ܶ ௞ ܵுయ Ͳ ͍ Ε , ‫ܩ‬ଶ Ͷ ͯ ͏ ͱ ෈ ร ͵ ॄ ߻ ͺ ܵீమ Ͳ ͍ Ζ . Γ ͮ ͱ ,ܶ ௞ ܵுయ ‫ீܵ ת‬మ ് ‫͗׎‬ಚΔΗΖ.
࣏Ͷ A2 ͹ଲ߻න‫ݳ‬Ν಍ड़ͤΖ. Ήͥ, ܶ ି଼௞ିଶ Ν࣏͹Γ͑Ͷ෾մͤΖ.
ܶ ି଼௞ିଶ ൌ ܶ ିଶ௞ ܶ ିଵ ܶ ି଺௞ିଵ ൌ ܶ ିଶ௞ ‫ܩ‬ସ ‫ܪ ל‬ସ ܶ ି଺௞ିଵ ‫ܩ‬ଶ ‫ܩ ל‬ଶ ൌ ‫ܩ‬ସ ܶ ଶ௞ ‫ܪ ל‬ସ ܶ ି଺௞ିଵ ‫ܩ‬ଶ ‫ܩ ל‬ଶ Ǥ࣏Ͷ୊್͹ࡠ༽Νॽ͘௜ͤ .
‫ܪ‬ସ ܶ ି଺௞ିଵ ‫ܩ‬ଶ ൌ ‫ܪ‬ସ ܶ ି଺௞ିଵ ‫ܩ‬ଶ ‫ܩ‬ଵ ‫ܩ ל‬ଵ ‫ܪ‬ଵ ‫ܪ ל‬ଵ ൌ ‫ܪ‬ସ ܶ ି଺௞ିଵ ܴܶ ିଵ ‫ܪ‬ଵt 19 tሺ45ሻൌ ‫ܪ‬ସ ܶ ି଺௞ିଵ ܶ ସ௞ାଵ ܶ ିଵ ‫ܪ‬ଵ ൌ ‫ܩ‬ସ ‫ܩ ל‬ସ ‫ܪ‬ସ ܶ ିଶ௞ିଵ ‫ܪ‬ଵ ൌ ‫ܩ‬ସ ‫ି ܶ ל‬ଵ ܶ ିଶ௞ିଵ ‫ܪ‬ଵ ൌ ‫ܩ‬ସ ܶ ିଶ௞ିଶ ‫ܪ‬ଵ Ǥሺ46ሻҐ৏ΓΕ, Ծ‫ى‬͹ଲ߻න‫͗ݳ‬ಚΔΗΖ .
ሺ47ሻܶ ି଼௞ିଶ ൌ ‫ܩ‬ସ ܶ ଶ௞ ‫ܩ ל‬ସ ܶ ିଶ௞ିଶ ‫ܪ‬ଵ ‫ܩ ל‬ଶ Ǥ
A2 ͹ ড় ݇ Ͳ ͍ Ζ ‫ܪ‬ଵ ‫ݖ‬ଶ௞ାଵ ൌ ‫ݖ‬ଶ௞ାଵ ͳ ‫ܩ‬ସ ‫ݖ‬௞ ൌ ‫ݖ‬௞ Ν ཤ ༽ ͤ Ζ ͳ , ‫ܩ‬ସ ܶିଶ௞ିଶ‫ܪ‬ଵ ͗ ߅ ౵ ࣺ ଀ Ͳ ͍ Ζ ͞ͳ͗಍͖ΗΖ.
‫ܩ‬ସ ܶ ିଶ௞ିଶ ‫ܪ‬ଵ ‫ݖ‬௜ ൌ ‫ܩ‬ସ ܶ ିଶ௞ିଶ ‫ݖ‬ସ௞ାଶି௜ ൌ ‫ܩ‬ସ ‫ݖ‬ଶ௞ି௜ ൌ ‫ݖ‬௜ Ǥࠪଲ߻ͳӊଲ߻͖Δ, ܶ

ି௞

ሺ48ሻ

ܵீర ‫ீܵ ת‬మ ് ‫ ׎‬Νຮ఼ͪͤ͗ A2 Νຮͪͤ࢝॑ଲসब‫ٌغ‬ಕͲ͍Ζ .

߅౵ࣺ଀ΝԾ‫ى‬͹Γ͑Ͷॽ͘௜ͤ .
ܶ ିଶ௞ିଶ ‫ܪ‬ଵ ‫ݖ‬௜ ൌ ‫ܩ‬ସ ‫ݖ‬௜ Ǥሺ49ሻ

ଲ ߻ ܶ ିଶ௞ିଶ ‫ܪ‬ଵ Ͷ ͯ ͏ ͱ ෈ ร ͵ ॄ ߻ ͺ ܶ ି௞ିଵ ܵுభ Ͳ ͍ Ε , ‫ܩ‬ସ Ͷ ͯ ͏ ͱ ෈ ร ͵ ॄ ߻ ͺ ܵீర Ͳ ͍ Ζ . Γͮͱ, ܶ ି௞ିଵ ܵுభ ‫ீܵ ת‬ర ് ‫͗׎‬ಚΔΗͪ. ಚΔΗͪ݃ՎΝΉͳΌͱໍୌ 4.2 ͳͤΖ.
ໍୌ 4.2. ߅౵ࣺ଀ΝȞͳॽ͚. ‫ܐؖ‬Ȟ‫ ݖ‬ൌ ‫ݖ‬Νຮ఼ͪͤ‫ݖ‬ͺ, ࢝॑ଲসब‫ٌغ‬ಕ͹ٌಕ఼Ͳ͍Ζ .
ͪͫ͢, ఼‫ݖ‬ͺ෈ಊ఼Ͳ͵͏ͳͤΖ.

ਦ 7ʁC1 ͹৖߻. ߅౵ࣺ଀ ‫ܩ‬ଵ ܶ ଶ ‫ܪ‬ଷ Ͷ͕͜Ζٌಕ఼͹ಊ͘: ‫ݖ‬ଽ ՜ ‫ݖ‬ଷ ՜ ‫ݖ‬ହ ՜ ‫ݖ‬ଽ . ‫ ݍ‬ൌ ͳͲ, ܽ ൌ ͳǤͷ.
C1 ͹ब‫ غ‬10 ͹࢝॑ଲসब‫ٌغ‬ಕΝྭͳ͢ͱ , ߅౵ࣺ଀ͶΓΖٌಕ఼Νಊ͘Νਦ 7 Ͷࣖͪ͢. ͞ ͞ Ͳ ‫ ଺ݖ‬ൌ ‫ܪ‬ଷ ‫ ͍ Ͳ ଺ݖ‬Ζ. ͞ Η ͺ ‫ݖ‬ଽ ൌ ‫ܪ‬ଷ ‫ݖ‬ଷ ͳ ॽ ͘ ௜ ͦ Ζ . ٌ ಕ ఼ ‫ݖ‬ଽ Ͷ ‫ܪ‬ଷ Ν ࡠ ༽ ͤ Ζ ͳ , ٌಕ ఼ ͺ ‫ݖ‬ଷ Ͷ Ң ಊ ͤ Ζ . ࣏ Ͷ , ܶ ଶ Ͷ Γ ͮ ͱ ‫ݖ‬ହ Ͷ Ң Ζ . ‫ܩ‬ଵ ‫ ଻ݖ‬ൌ ‫ ଻ݖ‬Γ Ε , ‫ܩ‬ଵ ‫ݖ‬ହ ൌ ‫ݖ‬ଽ ͗ ಍ ͖ Η Ζ . Γ ͮͱ, ߅౵ࣺ଀Ͳٌಕ఼ͺड़൅఼ ‫ݖ‬ଽ Ͷ໯Ζ.
࠹‫ޛ‬Ͷ࢝॑ଲসब‫ٌغ‬ಕ͹੓࣯ΝΉͳΌͱఈཀྵ 4.3 ͳͤΖ.
ఈཀྵ 4.3. ࢝॑ଲসब‫ٌغ‬ಕͶͯ͏ͱ (1)͖Δ(5)ΉͲ͗੔Εཱིͯ.
(1) ब‫ݍغ‬ͺ, 2 Ґ৏͹۰਼ . ͪͫ͢, 4 ͹ഔ਼Νঈ͚ . ‫ ݍ‬ൌ Ͷ݊ ൅ ʹሺ݊ ൒ Ͳሻ.
(2) ࣺ଀ܶ ௤ ͺࠪଲ߻, ߅౵ࣺ଀ (Ȟ), ӊଲ߻͹ࢀͯ͹ࡠ༽Ͷ෾մ͠ΗΖ .

t 20 t(3) ࠪ ଲ߻ Ͷ ͯ ͏ͱ ෈ ร ͵ॄ ߻ Ν ࣭௟ ͳ͢, ӊ ଲ߻ Ͷ ͯ͏ ͱ ෈ ร͵ ॄ ߻ ࣭௥ ͳͤ Ζ . ‫ ݖ ܐؖ‬ൌ ࣭௟ ‫ת‬࣭௥ Νຮ఼ͪͤ ‫ݖ‬ͺ࢝॑ଲসब‫ٌغ‬ಕ͹ٌಕ఼Ͳ͍Ζ . ఼‫ݖ‬ͺ෈ಊ఼Ͳ͵͏ͳͤΖ.
(4) ড় ݇ Ȟ‫ ݓ‬ൌ ‫ݓ‬Ν ຮ ͪ ͤ ఼ ‫ݓ‬ͺ ࢝ ॑ ଲ স ब ‫ ٌ غ‬ಕ ͹ ٌ ಕ ఼ Ͳ ͍ Ζ . ఼ ‫ݓ‬ͺ ෈ ಊ ఼ Ͳ ͵ ͏ ͳ ͤΖ.
(5) ࢝॑ଲসब‫ٌغ‬ಕͺ 180 ౕյ఺ଲস੓Ν΍ͯ.

5. ࢝॑ଲসब‫ٌغ‬ಕ͹෾‫ݳز‬ে
5.1. ྡֆ஍ͳ : ࣊‫ܙ‬෾‫ز‬ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ͹஦Ͳ࠹΍Κ͖ΕΏͤ͏ब‫ غ‬2 Νཤ༽͢ͱ෾‫ݳز‬ে͹ઈ໎Νߨ͑ .
࠹ॵͶब‫ غ‬2 ͗ਫ਼ͣΖྡֆ஍͹಍ड़Νߨ͑ .

ਦ 8ʁब‫ غ‬2 ͹ٌಕ఼ ‫ Ͳݖ‬, ‫ ͳݖ‬ͺଲসતܵ‫ ͳܪ‬, ܵ‫ܪ‬Ͷ , ܵ‫ ʹܩ‬, ܵ‫ ͵ܩ‬͹ި఼Ͷ͍Ζ. ܽ ൌ ʹǤͷ.
න 4 ͹ B1 ΓΕ, ब‫ غ‬2 ͹ٌಕ఼‫ݖ‬଴ ͳ‫ݖ‬ଵ ͺ࢝ͯ͹ଲসત (ܵுభ , ܵுర , ܵீమ , ܵீయ )͹ި఼Ͷ͍Ζ͞
ͳ͗಍͖ΗΖ(ਦ 8). ͞ΗΔ͹஦͖Δ, ܵுర ሺ‫ ݕ‬ൌ ‫ ݔ‬െ ߨሻͳܵீమ ሺ‫ ݕ‬ൌ െሺܽȀʹሻ‫ݔ‬ሻΝમ΁. ܽ ൐ ʹͲͺ,͞ Η Δ ͹ ި ఼ ͺ ෈ ಊ ఼ ܳଵ଴ Ґ ֐ Ͷ ್ ͯ ਫ਼ ͣ Ζ . ਫ਼ ͣ ͪ ި ఼ Ν ‫ݖ‬଴ ൌ ሺߨ ൅ ‫ݐ‬ǡ ‫ݐ‬ሻሺͲ ൏ ‫ ݐ‬൏ ߨሻͳ ‫ ͳݖ‬ൌሺߨ െ ‫ݐ‬ǡ െ‫ݐ‬ሻͳͤΖ. ‫ Ͳݖ‬Νܵ‫ ʹܩ‬͹ࣞͶେ೘ͤΖͳ, ‫ݐ‬Ν݀ΌΖࣞ(50)͗ಚΔΗΖ.
‫ ݐ‬ൌ ሺܽȀʹሻ‫ݐ‬Ǥሺ50ሻત‫ܙ‬Կߨ྽ ‫ܯ‬͹ Trace‫͍Ͳܯ‬Ζ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻͺҐԾ͹Γ͑ͶಚΔΗΖ.
‫ݎ‬ሺܽሻ ൌ ʹ െ Ͷܽଶ  ଶ ‫ ݐ‬൅ ܽସ  ସ ‫ݐ‬Ǥ

ሺ51ሻ

࠹ॵ Ͷ ್॑ ಋ ब‫ غ‬෾ ‫( ز‬ಋब ‫ غ‬෾‫ ز‬΍ ‫؜‬΋ )͹ ྡ ֆ஍ Ν ௒΄ Ζ . ͞͹ ৖ ߻ , ‫ݎ‬ሺܽሻ ൌ ʹͳͤΖͳ, ࣞ(51)ΓΕ࣏ࣞ͗಍͖ΗΖ.
ܽʹ ʹ ‫ݐ‬ሺܽʹ ʹ ‫ ݐ‬െ Ͷሻ ൌ ͲǤ‫ ݐ‬ൌ ͲΓΕ, ‫ ݐ‬ൌ ߨȀʹ͗‫ٽ‬ΉΖ.

͞ΗΝࣞ(50)Ͷେ೘͢ͱྡֆ஍͗ܽሺʹሻܿ ൌ ߨͳಚΔΗΖ.

t 21 tሺ52ሻሺଶሻ઎Ͷ਒΋઴Ͷ ܽ ൌ ܽ௖ Ͷ͕͜Ζ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻ͹ܑ͘Ν‫ٽ‬ΌΖ.
݀‫ݎ‬ሺܽሻ݀‫ݐ‬݀‫ݐ‬ൌ െͺܽ ʹ ‫ ݐ‬൅ ͺܽʹ ‫ ݐݐ‬൬ ൰ ൅ Ͷܽ͵ Ͷ ‫ ݐ‬െ ͶܽͶ ͵ ‫ ݐݐ‬൬ ൰Ǥ
݀ܽ
݀ܽ
݀ܽ

ሺ53ሻ

͞͞Ͳ ݀‫ݐ‬Ȁ݀ܽ ൌ ‫ݐ‬Ȁሺʹ െ ܽ ‫ݐ‬ሻ. ͞͹ࣞͶ ‫ ݐ‬ൌ ߨȀʹΝେ೘ ͤΖͳ , ͪͫͬ Ͷ҈ఈ ੓ ‫ݎ ਼ܐ‬ሺܽሻ͹ܑሺଶሻ͗͘κϫͲ͍Ζ͞ͳ͗ಚΔΗΖ . ͞Ηͺܽ ൌ ܽ௖ Ͳ‫ݎ‬ሺܽሻ͗‫ ݎ‬ൌ ʹͳં͢ͱ͏Ζ͞ͳΝқັ͢ͱ͏Ζ.
࣏Ͷܽʹ ʹ ‫ ݐ‬ൌ Ͷͳࣞ(50)ΓΕ,‫ ݐ‬ൌ ඥܽଶ ȀͶ െ ͳ(54)͗ಚΔΗΖ. ͞ΗΝࣞ(50)Ͷେ೘͢ͱྡֆ஍Ν݀ΌΖࣞඥܽʹ ȀͶ െ ͳ ൌ ሺܽȀʹሻඥܽʹ ȀͶ െ ͳ

ሺ55ሻ

ሺʹሻ
͗‫ٽ‬ΉΖ. ਼஍‫ࢋܯ‬ΓΕ, ྡֆ஍ͺ ܽܿሺͶሻ ൌ ͶǤͷʹ͵͸ͷͳ݀ΉΖ. ܽሺͶሻܿ ൐ ܽܿ Ͳ͍Ζ. ͞͹ࣆࣰΓΕ,ሺͶሻ್॑ಋ ब‫غ‬෾ ‫ز‬͹ ྡֆ஍ ͗ ܽሺʹሻܿ ൌ ߨͲ, ಋब‫ غ‬෾‫ ز‬͹ྡֆ ஍͗ ܽܿ ൌ ͶǤͷʹ͵͸ͷ Ͳ͍Ζ ͞ͳ͗ Κ͖Ζ.
࣏Ͷ್ ॑ब ‫غ‬ഔ ෾‫ ز‬͹ྡ ֆ஍Ν ݀Ό Ζࣞ Ν಍ ड़ͤ Ζ . ͞͹ ৖߻ , ‫ݎ‬ሺܽሻ ൌ െʹ͗੔Εཱི ͯ . ‫ؖ‬‫ ݐ ʹ ʹܽࣞܐ‬ൌ ʹͳ ࣞ(50)ΓΕ, ࣏ࣞ͗ಚΔΗΖ.
‫ܨ‬ሺܽሻ ‫ܽ ؠ‬ሺඥܽଶ െ ʹȀʹሻ െ ඥܽଶ െ ʹ ൌ ͲǤሺ56ሻ

ࣞ (56)͹ ή ϧ ϓ Ν ਦ 9 Ͷ ඵ ͏ ͪ . ‫ܨ‬ሺܽሻ ൌ ͲΝ ຮ ͪ ͤ մ ͺ ್ ͯ ͍ Ζ . ܽܿሺͳሻ ൌ ʹǤ͵Ͳͷͷͺͳ ܽሺ͵ሻ
ܿ ൌ
ሺଵሻ
ሺଶሻ
ͶǤͲͻͺͲ͵Ͳ ͍ Ζ . ͞ Η Δ ͺ ‫ ڠ‬Ͷ ್ ॑ ब ‫ غ‬ഔ ෾ ‫ ز‬͹ ྡ ֆ ஍ Ͳ ͍ Ζ . Ґ ৏ Γ Ε ʹ ൏ ܽ௖ ൏ ܽ௖ ൏ሺଷሻሺସሻܽ௖ ൏ ܽ௖ ͗੔Εཱིͯ. ͯΉΕ, ࠹ॵͶ್॑ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣ , ࣏Ͷ್॑ಋब‫غ‬෾‫͗ز‬ਫ਼ͣΖ .
2 յ໪͹್॑ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣͪ‫ޛ‬Ͷಋब‫غ‬෾‫͗ز‬ਫ਼ͣΖ .
ሺଵሻ

ሺଷሻ

࠹‫ޛ‬Ͷ, ܽ ൌ ܽ௖ ͳ ܽ ൌ ܽ௖ Ͳ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻ͗‫ ݎ‬ൌ െʹͳં͢ ͱ͏Ζ͞ ͳΝ ࣖ ͢ͱ͕͚ . ҈ ఈ੓ ‫ ਼ ܐ‬ͺ ‫ݎ‬ሺܽሻ ൌ ሺܽଶ  ଶ ‫ ݐ‬െ ʹሻଶ െ ʹͳ ॽ ͘ ௜ ͦ Ζ . ఈ ੓ ‫ ਼ ܐ‬͹ ࠹ ঘ ஍ െʹͲ ͍ Ζ . ͞ Η Γ Ε ҈ ఈ੓‫ ݎ਼͗ܐ‬ൌ െʹ ͳં͢ͱ͏Ζ͞ͳ͗಍͖ΗΖǤਦ 9ʁ‫ܨ‬ሺܽሻ ൌ ͲΝຮ್ͪͤͯ͹մ͹‫ ܐؖ‬. 1 յ໪ͺ࠹ॵͶਫ਼ͣΖ್॑ब‫غ‬ഔ෾‫ز‬Νන͢ , 2 յ໪ͺ್॑ಋब‫غ‬෾‫ز‬͹‫ޛ‬Ͷਫ਼ͣΖ್॑ब‫غ‬ഔ෾‫ز‬Νන͢ͱ͏Ζ .

t 22 tਦ 10ʁ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻ͹ W ࣊‫ܽܙ‬ғଚ੓ (W ࣊‫ܙ‬෾‫)ز‬.

5.2. ෾‫Ͳز‬ਫ਼ͣͪଲসब‫ٌغ‬ಕ͹੓࣯ࠕ‫ޛ‬͹٠࿨Ͳචགྷͳ͵ΖໍୌΝ়ղͤΖ .
ໍୌ 5.1. ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ್͗॑ब‫غ‬ഔ ෾‫ز‬Νً͢͞ͱਫ਼໊ͣͪ ब‫ٌغ‬ಕͺҲ॑ଲসब‫ٌغ‬ಕͲ͍Ζ.
ৄ໎. ਫ਼໊ͣͪब‫ٌغ‬ಕ͗Ҳ॑ଲসब‫ٌغ‬ಕҐ֐͹ब‫ٌغ‬ಕͲ͵͏͞ͳΝৄ໎ͤΖ .
(1) ໊ब‫ٌغ‬ಕ͗࢝॑ଲসब‫ٌغ‬ಕͲ͵͏͞ͳ .
ఈཀྵ 4.3(1)ΓΕ, ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ͹ब‫ݍ غ‬ͺ‫ ݍ‬ൌ Ͷ݊ ൅ ʹͲ͍Ζ. ໊ब‫ٌغ‬ಕ͹ब‫غ‬ͺʹ‫͍Ͳݍ‬Ζ. ໊ब‫ٌغ‬ಕ͹ब‫غ‬ͺ 4 ͹ഔ਼(Ͷሺʹ݊ ൅ ͳሻ)Ͷ͵Ζ. ͞Ηͺఈཀྵ 4.3(1)ͶൕͤΖ.
(2) ໊ब‫ٌغ‬ಕ್͗॑ଲসब‫ٌغ‬ಕͲ͵͏͞ͳ .
฾࢝॑ଲসब‫ٌغ‬ಕ͖Δ෾‫Ͳز‬ਫ਼໊ͣͪଲসब‫ٌغ‬ಕͺ࢝॑ଲসब‫ٌغ‬ಕ͗࢘༽͢ͱ͏Ζଲসત৏Ͷٌಕ఼Ν΍ͯ . ໊ब‫ٌغ‬ಕ͹ब‫ ͗غ‬4 ͹ഔ਼Ͳ͍Ζ͞ͳΝཤ༽ͤΖ . න 2 ΓΕ, ͞͹ब‫غ‬Ͷ‫ͤؖ‬Ζড়݇Νຮ್ͪͤ॑ଲসब‫ٌغ‬ಕͺ෾ྪ (a), (d), (e), (h)Ͷ‫ݸ‬ΔΗΖ. න 4 Ν‫ݡ‬Ζͳ, ෾ྪ(a)Ͷ͕͜Ζ ଲ সત ܵுభ ͳܵுమ Ͷٌ ಕ఼Ν ΍ͯ ฾ ࢝॑ଲ সब ‫ٌغ‬ಕ ͺଚ ࡑ ͢͵͏ . ࢔Ε͹৖߻Ͷ͕͏ͱ΍ಋ༹Ͳ͍Ζ .
(3) ໊ब‫ٌغ‬ಕ͗ඉଲসब‫ٌغ‬ಕͲ͵͏͞ͳ .
ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ͹ٌಕ఼ ‫ݖ‬଴ ͗ଲসત ܵுమ ৏Ͷ͍ΖͳͤΖ. ͞Ηͺන 4 ͹ C1 Ήͪͺ
D1 ͹৖߻Ͷ૮౲ͤΖ. ໊ब‫ٌغ‬ಕ఼ ‫ݑ‬଴ ͳ‫ݑ‬௤ ͺଲসત৏Ͷ͵͏ͳͤΖ. ‫ݑ‬௤ ൌ ‫ܪ‬ଶ ‫ݑ‬଴ ͺ ੔Εཱིͯ
͖ Δ , ྈ ว Ͷ ܶ ି௤Ȁଶ Ν ࡠ ༽ ͤ Ζ . ࠪ ว ͺ ‫ݑ‬௤Ȁଶ Ͳ , ӊ ว ͺ ܶ ି௤Ȁଶ ‫ܪ‬ଶ ‫ݑ‬଴ ൌ ‫ܪ‬ଶ ܶ ௤Ȁଶ ‫ݑ‬଴ ൌ ‫ܪ‬ଶ ‫ݑ‬௤Ȁଶ . ͯ Ή Ε ,‫ݑ‬௤Ȁଶ ൌ ‫ܪ‬ଶ ‫ݑ‬௤Ȁଶ ͗ ಚ Δ Η Ζ . ͞ Η ͺ , ٌ ಕ ఼ ‫ݑ‬௤Ȁଶ ͗ ଲ স ત ܵுమ ৏ Ͷ ͍ Ζ ͞ ͳ Ν қ ັ ͤ Ζ . Γ ͮ ͱ ,໅८͗ ಍͖Ηͪ.
ໍୌ 5.2. ବԃ‫ܗ‬ଲসब‫ٌغ‬ಕ͗ಋब‫غ‬෾‫ز‬Ήͪ ͺൕಋब‫غ‬෾‫ز‬Νً͢͞ ͱਫ਼໊ͣͪब‫ٌغ‬ಕͶͯ͏ͱ, (i), (ii)ͳ(iii)͗੔Εཱིͯ.
(i) ବԃ‫ܗ‬ଲসब‫ٌغ‬ಕ͗Ҳ॑ଲসٌಕ͵Δͻ , ໊ब‫ٌغ‬ಕͺඉଲসٌಕͲ͍Ζ .
(ii) ବԃ‫ܗ‬ଲসब‫ٌغ‬ಕ್͗॑ଲসٌಕ͵Δͻ , ໊ब‫ٌغ‬ಕͺҲ॑ଲসٌಕͲ͍Ζ .
(iii) ବԃ‫ܗ‬ଲসब‫ٌغ‬ಕ͗࢝॑ଲসٌಕ͵Δͻ , ໊ब‫ٌغ‬ಕͺ್॑ଲসٌಕͲ͍Ζ .
ৄ໎. (i)ͳ(ii)ͺ‫ع‬Ͷࢂߡช‫[ݛ‬1]Ͳৄ໎͠Ηͱ͏Ζ. ͞͞Ͳͺ(iii)͹ৄ໎Νߨ͑.
t 23 t(1) ໊ब‫ٌغ‬ಕͺ್॑ଲসब‫ٌغ‬ಕΉͪͺ࢝॑ଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳ .
ਦ 7 Ͳ࢘༽ͪ͢ C1 Νཤ༽ͤΖ. ͞͞Ͳब‫غ‬ͺ ‫ ݍ‬ൌ ͺ݇ ൅ ʹͲ͍Ζ. ٌಕ఼‫ݖ‬଴ ͺଲসતܵுమ ৏Ͷ͍Ζ. ٌಕ ఼ ‫ݖ‬଴ ΓΕ ଲস ત ৏Ͷ ‫ݑ‬଴ ͳ ‫ݒ‬଴ ሺ് ‫ݑ‬଴ ሻ͗ ਫ਼ͣ ͪͳ ͤ Ζ . ٌಕ఼ ‫ݖ‬଴ ͺ ࢝॑ ଲস ब ‫ٌ غ‬ಕ͹ ٌ ಕ ఼ Ͳ ͍ Ζ ͖ Δ , ٌ ಕ ఼ ‫ݖ‬௤Ȁଶ ΍ ଲ স ત ܵுమ ৏ Ͷ ͍ Ζ . ݃ Վ ͳ ͢ ͱ ٌ ಕ ఼ ‫ݖ‬௤Ȁଶ Γ Ε ‫ݑ‬௤Ȁଶ ͳ‫ݒ‬௤Ȁଶ͗ਫ਼ͣΖ.
‫ݖ‬௤Ȁଶ ൌ ‫ܪ‬ଷ ‫ܩ‬ଶ ‫ܪ‬ଷ ‫ݖ‬଴ ͗ ੔ Ε ཱི ͯ ͞ ͳ Ν ࣖ ͤ . ଲ স ત ܵுయ ৏ Ͷ ٌ ಕ ఼ ‫ݖ‬௞ ͗ ଚ ࡑ ͤ Ζ ͖ Δ , ‫ܪ‬ଷ ‫ݖ‬௞ ൌ ‫ݖ‬௞͗ ੔ Ε ཱི ͯ . ྈ ว Ͷ ܶ ௞ Ν ࡠ ༽ ͤ Ζ ͳ , ࠪ ว ͺ ܶ ௞ ‫ܪ‬ଷ ‫ݖ‬௞ ൌ ‫ܪ‬ଷ ܶ ି௞ ‫ݖ‬௞ ൌ ‫ܪ‬ଷ ‫ݖ‬଴ Ͳ , ӊ ว ͺ ‫ݖ‬ଶ௞ Ͳ ͍ Ζ .
‫ܪ‬ଷ ‫ݖ‬଴ ൌ ‫ݖ‬ଶ௞ ͗ ಚ Δ Η Ζ . ଲ স ત ܵுమ ৏ Ͷ ٌ ಕ ఼ ‫ݖ‬଴ ͗ ଚ ࡑ ͤ Ζ ͖ Δ , ‫ܪ‬ଶ ‫ݖ‬଴ ൌ ‫ݖ‬଴ ͗ ੔ Ε ཱི ͯ .ྈ ว Ͷ‫ܩ‬ଶ Ν ࡠ ༽ ͤ Ζ ͳ , ࠪ ว ͺ ‫ܩ‬ଶ ‫ܪ‬ଶ ‫ݖ‬଴ ൌ ܶ ିଵ ‫ݖ‬଴ ൌ ‫ିݖ‬ଵ Ͳ ͍ Ζ ͖ Δ ‫ିݖ‬ଵ ൌ ‫ܩ‬ଶ ‫ݖ‬଴ ͗ ಚ Δ Η Ζ . ྈ ว Ͷ ܶ ିଶ௞Ν ࡠ ༽ ͤ Ζ ͳ , ࠪ ว ͺ ‫ିݖ‬ଶ௞ିଵ ͳ ͵ Ζ . ब ‫ غ‬੓ Γ Ε ‫ିݖ‬ଶ௞ିଵ Ν ‫଺ݖ‬௞ାଵ ͳ ॽ ͘ ௜ ͤ . ӊ ว ͺ ܶ ିଶ௞ ‫ܩ‬ଶ ‫ݖ‬଴ ൌ‫ܩ‬ଶ ܶ ଶ௞ ‫ݖ‬଴ ൌ ‫ܩ‬ଶ ‫ݖ‬ଶ௞ Ͳ ͍ Ζ . Ή ͳ Ό ͱ , ‫ܩ‬ଶ ‫ݖ‬ଶ௞ ൌ ‫଺ݖ‬௞ାଵ ͗ ಚ Δ Η Ζ . ‫ܪ‬ଷ ‫ݖ‬௞ ൌ ‫ݖ‬௞ ͹ ྈ ว Ͷ ܶ ିହ௞ିଵ Ν ࡠ ༽
ͤ Ζ . ࠪ ว ͺ ܶ ିହ௞ିଵ ‫ܪ‬ଷ ‫ݖ‬௞ ൌ ‫ܪ‬ଷ ܶ ହ௞ାଵ ‫ݖ‬௞ ൌ ‫ܪ‬ଷ ‫଺ݖ‬௞ାଵ ͳ ͵ Ζ . ӊ ว ͺ ܶ ିହ௞ିଵ ‫ݖ‬௞ ൌ ‫ିݖ‬ସ௞ିଵ Ͳ ͍ Ζ ͗ , ब‫غ‬੓ΓΕ‫ିݖ‬ସ௞ିଵ Ν‫ݖ‬ସ௞ାଵ ൌ ‫ݖ‬௤Ȁଶ ͳॽ͘௜ͤ. Ґ৏ΝΉͳΌΖͳ ‫ݖ‬௤Ȁଶ ൌ ‫ܪ‬ଷ ‫ܩ‬ଶ ‫ܪ‬ଷ ‫ݖ‬଴ ͗ಚΔΗΖ.
͞ ͞ Ͳ ‫ܩ‬ଶ ‫ܪ‬ଷ ൌ ‫ܪ‬ଷ ‫ܪ‬ଵ (ࣞ (30)Ν ‫ ݡ‬Γ )Ν ཤ ༽ ͤ Ζ ͳ , ‫ݖ‬௤Ȁଶ ൌ ‫ܪ‬ଷ ‫ܩ‬ଶ ‫ܪ‬ଷ ‫ݖ‬଴ ൌ ‫ܪ‬ଷ ‫ܪ‬ଷ ‫ܪ‬ଵ ൌ ‫ܪ‬ଵ ‫ݖ‬଴ ͗ ੔ Ε
ཱིͯ. ͞͞Ͳ‫ݖ‬଴ ൌ ‫ܪ‬ଶ ‫ݖ‬଴ Νཤ༽ͤΖͳ, ‫ݖ‬௤Ȁଶ ൌ ‫ܪ‬ଵ ‫ܪ‬ଶ ‫ݖ‬଴ ൌ ܴ‫ݖ‬଴ ಚΔΗΖ. ٌಕ఼‫ݖ‬௤Ȁଶ ͺ‫ݖ‬଴ Νൕ఺ͪ͢Ғ ஖ Ͷ ͍ Ζ ͖ Δ, ‫ݖ‬௤Ȁଶ ͺ ଲ স ત ܵுమ ৏ Ͷ ͍ Ζ. ٌ ಕ ఼ ‫ݑ‬଴ ͳ ‫ݒ‬଴ Ͷ ‫ܪ‬ଵ Ν ࡠ ༽ ͢ ͱ ΍ ಋ ༹ ͹ ൕ ఺ ͗ ਫ਼
ͣ Ζ. ٌ ಕ ఼ ‫ݖ‬௤Ȁଶ ͖ Δ࢝ ఼ ‫ݑ‬௤Ȁଶ , ‫ݒ‬௤Ȁଶ , ‫ܪ‬ଵ ‫ݑ‬଴ , ‫ܪ‬ଵ ‫ݒ‬଴ ͗ ਫ਼ ͣ Ζ͞ ͳ ͺ ໅ ८ Ͳ ͍ Ζ ͖ Δ , ‫ܪ‬ଵ ‫ݒ‬଴ ൌ ‫ݒ‬௤Ȁଶ͖ͯ‫ܪ‬ଵ ‫ݑ‬଴ ൌ ‫ݑ‬௤Ȁଶ ͗੔Εཱིͯ৖߻ͳ, ‫ܪ‬ଵ ‫ݒ‬଴ ൌ ‫ݑ‬௤Ȁଶ ͖ͯ‫ܪ‬ଵ ‫ݑ‬଴ ൌ ‫ݒ‬௤Ȁଶ ͗੔Εཱིͯ৖߻͍͗Ζ. ͞͞Ͳ
ͺ, ઴ ं͹ ৖ ߻ Ͷ ͯ ͏ ͱ ௒ ΄ Ζ . ࠪ ว ͺ ‫ܪ‬ଵ ‫ݒ‬଴ ൌ ‫ܪ‬ଵ ‫ܪ‬ଶ ‫ݒ‬଴ ൌ ܴ‫ݒ‬଴ Ͳ ͍ Ζ ͖ Δ, ܴ‫ݒ‬଴ ൌ ‫ݒ‬௤Ȁଶ ͗ ಚΔ Η
Ζ. ͞ Ηͺ ٌ ಕ ఼ ‫ݒ‬଴ ͖ Δ ड़ ൅ ͢ ͪ ब‫ ٌ غ‬ಕ ܱሺ‫ݒ‬଴ ሻ͹ ब ‫ ͍ Ͳݍ ͗ غ‬Ζ ͞ͳ Ν қ ັ ͢ ͱ ͏ Ζ . ब ‫غ‬ٌಕܱሺ‫ݑ‬଴ ሻ͹ब‫غ‬΍ ‫͍Ͳݍ‬Ζ. ͯΉΕ, ब‫ٌغ‬ಕ ܱሺ‫ݑ‬଴ ሻͳܱሺ‫ݒ‬଴ ሻͺٌಕ఼ ‫ݖ‬଴ ΓΕ(ൕ)ಋब‫غ‬෾‫Ͳز‬ਫ਼ͣ ͪ ब‫ٌ غ‬ಕ Ͳ ͍Ζ ͞ ͳ͗ ࣖ ͠Η ͪ . ब‫ ٌغ‬ಕ ܱሺ‫ݑ‬଴ ሻͳ ܱሺ‫ݒ‬଴ ሻ͗ଲ সब ‫ ٌغ‬ಕ Ͳ͍ Ζ ͞ͳ ͺ
໎Δ͖Ͳ, ߍͶ͞ΗΔͺ 180 ౕյ఺ଲস੓Ν࣍ͯ. ໍୌ 4.1 ΓΕ, ब‫ٌغ‬ಕ ܱሺ‫ݑ‬଴ ሻͳܱሺ‫ݒ‬଴ ሻͺ್॑ଲসब‫ٌغ‬ಕΉͪͺ ࢝॑ଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳ͗಍͖Ηͪ .
(2) ໊ब‫ٌغ‬ಕͺ࢝॑ଲসब‫ٌغ‬ಕͲ͵͏͞ͳ .
ਫ਼್ͣͪͯ͹໊ब‫ٌغ‬ಕ͗࢝॑ଲসब‫ٌغ‬ಕͳͤΖͳ , ͞ΗΔͺ฾ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕͳಋͣড়݇Νຮͪͤ͞ͳͶ͵Ζ. ͯΉΕ್ͯ͹໊ब‫ٌغ‬ಕͳ฾ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ͗Ҳ ஗ ͤ Ζ͞ ͳ Ͷ ͵Ζ . Γͮ ͱ, ࢝ ॑ଲ স ब‫ ٌ غ‬ಕ͹ ໊ ब ‫ ٌغ‬ಕ ͺ ਫ਼ͣ ͵ ͏ . Ґ৏ Ͳ ৄ໎ Νश͓Ζ.
৏‫ى‬͹ৄ໎Ͳཤ༽್ͪ॑͢ଲসब‫ٌغ‬ಕ͗௪գͤΖ΍͑Ҳͯ͹ଲসતΝ݀ఈͤΖ . ͞͞Ͳͺ ‫ݑ‬଴ ൌ ‫ܪ‬ଶ ‫ݑ‬଴ Ν ཤ ༽ ͤ Ζ . ྈ ว Ͷ ‫ܩ‬ଶ Ν ࡠ ༽ ͤ Ζ ͳ , ࠪ ว ͺ ‫ܩ‬ଶ ‫ݑ‬଴ Ͳ , ӊ ว ͺ ‫ܩ‬ଶ ‫ܪ‬ଶ ‫ݑ‬଴ ൌ ܶ ିଵ ‫ݑ‬଴ ൌ‫ିݑ‬ଵ . Γ ͮ ͱ , ‫ܩ‬ଶ ‫ݑ‬଴ ൌ ‫ିݑ‬ଵ ͗ ಚ Δ Η Ζ . ྈ ว Ͷ ‫ܩ‬ଵ Ν ࡠ ༽ ͤ Ζ . ࠪ ว ͺ ‫ܩ‬ଵ ‫ܩ‬ଶ ‫ݑ‬଴ ൌ ܴ‫ݑ‬଴ ൌ ܶ ௤Ȁଶ ‫ݑ‬଴ ൌ‫ݑ‬௤Ȁଶ Ͳ, ӊวͺ‫ܩ‬ଵ ‫ିݑ‬ଵ . ‫ݑࣞܐؖ‬௤Ȁଶ ൌ ‫ܩ‬ଵ ‫ିݑ‬ଵ ͗੔Εཱིͯ. ࣏Ͷ‫ ݍ‬ൌ ͺ݇ ൅ ʹΝཤ༽ͤΖͳ, ͞͹‫ܐؖ‬ͺ ‫ݑ‬ସ௞ାଵ ൌ ‫ܩ‬ଵ ‫ିݑ‬ଵ ͳ ॽ ͘ ௜ ͦ Ζ . ྈ ว Ͷ ܶ ିଶ௞ିଵ Ν ࡠ ༽ ͤ Ζ . ࠪ ว ͺ ‫ݑ‬ଶ௞ Ͳ , ӊ ว ͺ ܶ ିଶ௞ିଵ ‫ܩ‬ଵ ‫ିݑ‬ଵ ൌ‫ܩ‬ଵ ܶ ଶ௞ାଵ ‫ିݑ‬ଵ ൌ ‫ܩ‬ଵ ‫ݑ‬ଶ௞ . Γͮͱ, ‫ݑ‬ଶ௞ ൌ ‫ܩ‬ଵ ‫ݑ‬ଶ௞ ͗੔Εཱིͯ. ͞Ηͺٌಕ఼ ‫ݑ‬ଶ௞ ͗ଲসત ܵீభ ৏Ͷ͍Ζ͞ͳΝқັ͢ͱ ͏Ζ . ͞͞Ͳߡ͓ͱ͏Ζ ್॑ଲসब ‫ٌغ‬ಕͺ , ଲসત ܵுమ ৏ ͳܵீభ ৏Ͷٌಕ఼Ν΍ͯ.
ໍୌ 5.3. ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ್͗॑ಋब‫ غ‬෾‫ز‬Νً͢͞ͱਫ਼໊ͣͪ ब‫ٌغ‬ಕͺҲ॑ଲসٌಕͲ͍Ζ.
t 24 tৄ໎. ್॑ಋब‫ غ‬෾‫ز‬ͺ ಋ࣎Ͷಋब ‫غ‬෾‫ ͳز‬ൕಋ ब‫غ‬෾‫ ͗ز‬ਫ਼ͣΖ‫ ݳ‬েͲ ͍Ζ . ͞͹‫ݳ‬েΝ್ͯͶ෾͜ ͱߡ͓Ζ . ࠹ॵͺବԃ‫॑࢝ ܗ‬ଲস ब‫ٌ غ‬ಕ͗ಋब ‫غ‬෾‫ز‬Ν ً͞ ͤͳߡ͓Ζ . ໍୌ
5.2(iii)ΓΕ, ਫ਼໊ͣͪब‫ٌغ‬ಕͺ್॑ଲসब‫ٌغ‬ಕͲ͍Ζ. ࣏Ͷਫ਼್໊ͣͪ॑ଲসब‫ٌغ‬ಕ͗ൕ ಋ ब‫ غ‬෾ ‫ز‬Ν ً ͤ͞ ͳ ߡ ͓Ζ. ໍୌ 5.2(ii)ΓΕ, ਫ਼ͣ ͪ ໊ ब‫ ٌ غ‬ಕ ͺҲ ॑ ଲস ब ‫ٌغ‬
ಕͲ͍Ζ.

5.3. ࠹ॵ͹್॑ब‫غ‬ഔ෾‫ز‬ሺଵሻܽ௖ ൌ ʹǤ͵ͲͷͷͺͲ࠹ॵ͹್॑ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣΖ . ໍୌ 5.1 ΓΕ, ͞͹෾‫Ͳز‬ਫ਼ͣΖ໊ब‫غ‬ٌಕͺસͱҲ॑ଲসब‫ٌغ‬ಕͲ͍Ζ . ଲসતܵுభ ͳܵீమ Ͷਫ਼ͣͪब‫ غ‬4 ͹ٌಕ఼ͺବԃ‫͍Ͳܗ‬Ζ(ਦ 11 (a-1),(a-2),(b-1),(b-2)). ਦͶ͕͏ͱٌಕ఼ ‫ݖ‬௞ ͺ੖਼݇Ͳන‫͠ݳ‬Ηͱ͏Ζ. ब‫ غ‬2 ͹ٌಕ఼‫ݖ‬଴ ͳ‫ݖ‬ଵ ͹बΕͶͨΗͩΗ 4 ‫ݺ‬͹ٌಕ఼͗ਫ਼ͣ, ߻‫ ܯ‬8 ‫ݺ‬͹ٌಕ఼͗ଚࡑͤΖ͞ͳͶ͵
Ζ. ܶ ସ ΝୱҒͳͤΖͳϛΠϱΩϪࢨ਼͹࿪ͺ (+8)Ͳ͍Ζ. ଲসતܵுర ͳܵீయ Ͷਫ਼ͣͪब‫ غ‬4 ͹γχϩ‫ٌܗ‬ಕ఼͗ਫ਼ͣΖ (ਦ 11 (c-1),(c-2),(d-1),(d-2)). ݃Վͳ͢ͱ, ब‫ غ‬2 ͹ٌಕ఼‫ݖ‬଴ ͳ‫ݖ‬ଵ͹बΕͶ 8 ‫ݺ‬͹γχϩ‫ٌܗ‬ಕ఼͗‫ݳ‬ΗΖ . ܶ ସ ΝୱҒͳͤΖͳϛΠϱΩϪࢨ਼͹࿪ͺ (-8)Ͳ͍
Ζ. ͯΉΕ, ሺ൅ͺሻ ൅ ሺെͺሻ ൌ Ͳͳ‫͏ޕ‬Ͷ଩ͬভ͍͑͢. ݃Վͳ͢ͱ, ब‫ غ‬2 ͹ٌಕ఼͹ϛΠϱΩϪࢨ਼ͺ(+1)͹ΉΉรԿ͢͵͏. ͯΉΕ, ್॑ब‫غ‬ഔ෾‫ز‬͹઴‫Ͳޛ‬ब‫ غ‬2 ͹฾ٌಕ఼͹҈ఈ੓ͺବԃ‫Ͳܗ‬รԿ͢͵͏ .

t 25 tਦ 11ʁ࠹ॵ͹್॑ब‫غ‬ഔ෾‫Ͳز‬ਫ਼ͣͪବԃ‫ܗ‬Ҳ॑ଲসब‫ٌغ‬ಕ (a-1),(a-2),(b-1),(b-2). ࠹ॵ ͹ ್ ॑ ब ‫ غ‬ഔ ෾ ‫ Ͳ ز‬ਫ਼ ͣ ͪ γ χ ϩ ‫ ܗ‬Ҳ ॑ ब ‫ ٌ غ‬ಕ (c-1),(c-2),(d-1),(d-2). ब ‫ غ‬4. ܽ ൌʹǤͷ.
t 26 t5.4. ್॑ಋब‫غ‬෾‫ز‬್॑ಋब‫غ‬෾‫͗ز‬ਫ਼ͣΖ͞ͳ͗ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ͹ಝ௅Ͳ͍Ζ . ໍୌ 5.3 ΓΕ, ್॑ಋब‫غ‬෾‫Ͳز‬ਫ਼ͣΖ໊ब‫ٌغ‬ಕͺҲ॑ଲসब‫ٌغ‬ಕͲ͍Ζ .
ବ ԃ ‫ ܗ‬ब ‫ ٌ غ‬ಕ ͺ ଲ স ત ܵுభ ৏ ͳ ܵீమ ৏ Ͷ ਫ਼ ͣ Ζ ͞ ͳ ͗ , ਼ ஍ ‫ ݃ ࢋ ܯ‬Վ Ͳ ಚ Δ Η ͪ (ਦ 12(a-1),(a-2),(b-1),(b-2)). ͞Η Δ ͹ ੓ ࣯ ͺ ್ ॑ ब ‫ غ‬ഔ ෾ ‫ز‬͹ ৖ ߻ ͳ ಋ ͣ Ͳ ͍ Ζ . ଲস તܵுభ ৏ͳܵீమ ৏ͨΗͩΗ 2 ‫ݺ‬͹ٌಕ఼͗ਫ਼ͣΖ. ब‫ غ‬2 ͹ٌಕ఼ ‫ݖ‬଴ ͳ‫ݖ‬ଵ ͹बΕͶͨΗͩΗ 4 ‫ݺ‬͹ٌಕ఼͗ਫ਼ͣ, ߻‫ ܯ‬8 ‫ݺ‬͹ٌಕ఼͗ଚࡑͤΖ͞ͳͶ͵Ζ . ܶ ଶ ΝୱҒͳͤΖͳϛΠϱΩϪࢨ਼͹࿪ͺ (+8) Ͳ͍Ζ.
γχϩ‫ܗ‬ब‫ٌغ‬ಕͺଲসત ܵுర ৏ͳܵீయ ৏Ͷਫ਼ͣΖ(ਦ 12(c-1),(d-2),(d-1),(d-2)). ब‫ غ‬2 ͹ٌಕ఼‫ݖ‬଴ ͳ‫ݖ‬ଵ ͹बΕͶͨΗͩΗ 4 ‫ݺ‬͹ٌಕ఼͗ਫ਼ͣ, ߻‫ ܯ‬8 ‫ݺ‬͹ٌಕ఼͗ଚࡑͤΖ͞ͳͶ͵
Ζ. ܶ ଶ ΝୱҒͳͤΖͳϛΠϱΩϪࢨ਼͹࿪ͺ (-8)Ͳ͍Ζ.
Ґ ৏ ͹ ݃ Վ Ν Ή ͳ Ό Ζ . ϛ Π ϱ Ω Ϫ ࢨ ਼ ͹ ࿪ Ͷ ͕ ͏ ͱ , ሺ൅ͺሻ ൅ ሺെͺሻ ൌ Ͳ͗ ੔ Ε ཱི ͯ ͖ Δ ,ब‫ غ‬2 ͹฾ٌಕ఼͹҈ఈ੓ͺ್॑ಋब‫غ‬෾‫ز‬͹઴‫Ͳޛ‬รԿ͢͵͏ . ͯΉΕ, ฾ٌಕ఼͹҈ఈ੓ͺ್॑ಋब‫غ‬෾‫ز‬͹௜઴ͺବԃ‫͖͍ͪͮͲܗ‬Δ , ್॑ಋब‫غ‬෾‫ز‬͹௜‫ޛ‬΍ବԃ‫͍Ͳܗ‬Ζ .

t 27 tਦ 12ʁ್॑ಋब‫غ‬෾‫Ͳز‬ਫ਼ͣͪବԃ‫ܗ‬Ҳ॑ଲসब‫ٌغ‬ಕ (a-1),(a-2),(b-1),(b-2). ್॑ಋब‫غ‬෾‫Ͳز‬ਫ਼ͣͪγχϩ‫ܗ‬Ҳ॑ब‫ٌغ‬ಕ (c-1),(c-2),(d-1),(d-2). ब‫ غ‬2. ܽ ൌ ߨ ൅ ͲǤͳ.

t 28 t5.5. ୊  յ໪͹್॑ब‫غ‬ഔ෾‫ز‬
୊ 2 յ໪͹್॑ब‫غ‬ഔ෾‫Ͳز‬ਫ਼ͣΖ໊ब‫ٌغ‬ಕ΍ , સͱҲ॑ଲসब‫ٌغ‬ಕͲ͍Ζ (ਦ 13).
ଲসતܵுభ ৏ͳܵீమ ৏Ͷବԃ‫ܗ‬ब‫ غ‬4 ब‫ٌغ‬ಕ఼͗ਫ਼ͣ, ଲসતܵுర ৏ͳܵீయ ৏Ͷγχϩ‫ܗ‬ब‫ غ‬4ब‫ٌغ‬ಕ఼͗ਫ਼ͣΖ. ͞ΗΔ͹੓࣯ͺ ୊ 1 յ໪͹್॑ब‫غ‬ഔ෾‫ͳز‬ಋͣͲ͍Ζ .

t 29 tਦ 13ʁ್॑ब‫غ‬ഔ෾‫Ͳز‬ਫ਼ͣͪବԃ‫ܗ‬Ҳ॑ଲসब‫ٌغ‬ಕ (a-1),(a-2),(b-1),(b-2). ್॑ब‫غ‬ഔ෾‫Ͳز‬ਫ਼ͣͪγχϩ‫ܗ‬Ҳ॑ब‫ٌغ‬ಕ (c-1),(c-2),(d-1),(d-2). ब‫ غ‬4. ܽ ൌ ͶǤʹ.

t 30 t5.6. ಋब‫غ‬෾‫ز‬࠹‫ޛ‬͹ಋब‫غ‬෾‫Ͳز‬ਫ਼ͣΖ໊ब‫ٌغ‬ಕͺ್॑ଲসब‫ٌغ‬ಕͲ͍Ζ (ໍୌ 5.2(iii)). ब‫ غ‬2 ͺ
න 2(b)͹৖߻Ͷ૮౲͢, ਫ਼್ͣͪ॑ଲসब‫ٌغ‬ಕͺଲসત ܵுభ ͳܵீమ ͹ި఼Ͷٌಕ఼Ν΍ͯ(ਦ
14). ಋब‫غ‬෾‫ز‬͹‫ޛ‬, ฾ٌಕ఼ͺγχϩ‫ܗ‬ब‫ٌغ‬ಕ఼ͳ͵Ζ . ϏϧϟʖνΝଁՅ͢ͱ΍฾ٌಕ఼͹҈ఈ੓ͺรΚΔ͵͏ͪΌ W ࣊‫ܙ‬෾‫ز‬ͺಋब‫غ‬෾‫׮Ͳز‬྅ͤΖ .

ਦ 14ʁಋब‫غ‬෾‫Ͳز‬ਫ਼ͣͪବԃ‫್॑ܗ‬ଲসब‫ٌغ‬ಕ . ब‫ غ‬2. ܽ ൌ ͶǤ͹.

6. : ࣊‫ܙ‬෾‫ز‬Ͷ‫ͤؖ‬ΖմएW ࣊‫ܙ‬෾‫҈Ͳز‬ఈ੓‫ݎ਼ܐ‬ሺܽሻ͹รԿΝ‫ݡ‬Ζͳ, ್॑ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣΖΉͲͺ฾ٌಕ఼͹बΕ͹յ఺͹࢕๏͗ଐ͚͵Ε , ್॑ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣͪ‫ޛ‬ͺங͚͵Ζ͹Ͳͺ͵͏͖ͳࢧΚΗΖ. Ή ͪ , ್ ॑ ಋ ब ‫ غ‬෾ ‫ ͗ ز‬ਫ਼ ͣ ͪ ‫ ޛ‬ͺ յ ఺ ͹ ࢕ ๏ ͗ ࠸ ; ଐ ͚ ͵ Ζ ͳ ߡ ͓ Δ Η Ζ . ͢ ͖ ͢ ,͞͹Γ͑͵ߡ͓ͺ‫ޣ‬ΕͲ͍Ζ . ฾ٌಕ఼͹बΕ͹յ఺͹࢕๏ͺϏϧϟʖν͹ଁ୉ͳͳ΍Ͷଁ
Յ͢ͱ͏Ζ͹Ͳ͍Ζ. ྭΝ༽͏ͱ͞͹ࣆࣰΝઈ໎ͤΖ .
฾ٌಕ఼ͳ͢ͱब‫ غ‬2 ͹ٌಕ఼Νߡ͓Ζ. ฾ٌಕ఼‫ݖ‬଴ ͺ྘ҮͲ ൏ ‫ ݔ‬൏ ߨͶ͍Ε, ‫ݖ‬ଵ ൌ ሺ‫ݔ‬ଵ ǡ ‫ݕ‬ଵ ሻͺ ྘ Ү ߨ ൏ ‫ ݔ‬൏ ʹߨͶ ͍ Ζ. ͞ ͞ Ͳ, ܵመீమ ͺ ଲ স ત ܵீమ ͹ ‫ ݔ‬൒ ‫ݔ‬ଵ Ν ຮ ͪ ͤ ෨ ෾ ͳ ͤ Ζ. ਦ 15 Ͷ ͺ ,ଲ স ત ܵீమ , ܵுర ͵ Δ ; Ͷ ଀ ܶ ଶ ܵመீమ Ν ඵ ͏ ͪ . ฾ ٌ ಕ ఼ ‫ݖ‬ଵ Ͷ ඏ ঘ ϗ έ φ ϩ ‫ݒ‬Ν ͕ ͚. ଲ স ત ܵீమ͹ંત๏޴Ͳ ‫࠴ݔ‬ඬ͹ଁՅͤΖ๏޴Νඏঘϗέφϩ ‫ݒ‬͹๏޴ͳͤΖ. ਦ 15 Ͳͺ, ඏঘϗέφϩ‫ݒ‬Νଢ͏໾һͲඵ͘, ଀ܶ ଶ ‫ݒ‬ͺࡋ͏໾һͲඵ͏ͪ.
ବԃ ‫ ෈ܗ‬ಊ఼ ܳଵ଴ ͹ब ‫غ‬ഔ ෾ ‫ز‬͹ ྡֆ ஍ͺ ܽ௖ ሺͳȀʹሻ ൌ ʹͲ͍ Ζ. ͞ ͹ྡ ֆ஍ ΓΕ ܽ͹஍ ΝଁՅͤΖ. ਦ 15(a)͖Δਦ 15(l)ΉͲΝॳͶ௧ͮͱ, ଀ܶ ଶ ‫ݒ‬͹յ఺͹࢕๏Ν؏ࡱ͢ͱΆ͢͏ . ඏঘϗέφϩ‫ ܶ଀ͳݒ‬ଶ ‫ݒ‬͹͵ֱౕͤΝߠͳͤΖ. յ఺͹๏޴ͺൕ࣎‫ܯ‬յΕͲ͍Ζ͖Δ , ֱౕߠ΍ൕ࣎‫ܯ‬յΕͶ଎Ζ. ਦ(a)͖Δਦ(d)ΉͲյ఺ֱౕ ߠͺଁՅͤΖ. ਦ(a)͖Δ(d)ΉͲ͹գఖͺ, W ࣊‫ܙ‬෾ ‫ ز‬Ͷ ͕ ͜Ζ ҈ ఈ ੓ ‫ݎ ਼ ܐ‬ሺܽሻ͹ ࠹ ॵ ͹ ‫ݰ‬ঙ ෨ ෾ Ͷ ଲ Ԣ ͢ ͱ ͏ Ζ . ਦ(d)(ܽ ൌ ʹǤ͵)Ͳ͹ յ ఺ ֱ ߠ
ͺ 180 ౕͶঙ͢ଏΕ͵͏ . ਦ(e)(ܽ ൌ ʹǤͷ)Ͳ͹յ఺ֱߠͺ 180 ౕΝӿ͢ͱ͏Ζ. ͞Ηͺ, ब‫ غ‬2್͗॑ब‫غ‬ഔ෾‫ز‬Νًͪ͢͞‫ޛ‬͹য়‫͍Ͳڱ‬Ζ . ਦ(f)(ܽ ൌ ʹǤͺ)Ͳͺ, յ఺ֱߠͺ 270 ౕΝӿ͢
ͱ͏Ζ. ͞ΗΔ͹ য়‫ ڱ‬ͺ , ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻ͹࠹ ॵ͹ଁ Յ෨෾ ͶଲԢ ͢ͱ ͏Ζ . ਦ(g)Ͳͺյ఺ֱߠͺΆ· 360 ౕͲ, ਦ(h)Ͳͺյ఺ֱ ߠͺ 360 ౕҐ৏Ͳ͍Ζ. ਦ(g)ͳਦ(h)͹ؔͲ್॑ಋब‫غ‬෾‫͗ز‬ਫ਼ͣͱ͏Ζ.
t 31 tਦ 15ʁඏঘϗέφϩ ‫ ܶ଀ͳݒ‬ଶ ‫ݒ‬͹‫ܐؖ‬.

್॑ಋब‫غ‬෾‫͗ز‬ਫ਼ͣͪ‫Ͳޛ‬ͺ, ҈ఈ੓‫ݎ਼ܐ‬ሺܽሻͺ࠸ ;‫ݰ‬ঙͤΖ. ଀ܶ ଶ ‫ݒ‬ͺٌಕ఼ ‫ݖ‬ଵ ͹बΕΝ 360 ౕҐ৏յ఺͢ͱ͏Ζ. ਦ(i)ͳਦ(j)͹ؔͲ 2 յ໪͹್॑ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣͱ͏Ζ . 2 յ໪͹್॑ब‫غ‬ഔ෾‫͗ز‬ਫ਼ͣͪ‫ ޛ‬, ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻͺଁՅͤΖ. ਦ(k)ͳਦ(l)͹ؔͲಋब‫غ‬෾‫ز‬͗ਫ਼ͣͱ͏Ζ. ਦ(l)Ͳͺ଀ ܶ ଶ ‫ݒ‬ͺ‫ݖ‬ଵ ͹बΕΝ್յ఺Ґ৏͢ͱ͏Ζ . Ґ৏͹݃ՎΓΕ, ್॑ब‫غ‬ഔ෾‫್॑ͳز‬ಋब‫غ‬෾‫ز‬Ν੓࣯ 6.1 ͹Γ͑Ͷ ࠸մएͲ͘Ζ. ͞ΗΓΕ, ఈཀྵ 6.2 ͗ಚΔΗΖ.
੓࣯ 6.1. յ఺਼ͳब‫غ‬ͺࣺ଀ ܶ ଶ ΝୱҒͳͤΖ.
(1) ࠹ॵ͹್॑ब‫غ‬ഔ෾‫ز‬ͺ , յ఺਼ͳȀʹ͹յ఺෾‫Ͳز‬ब‫ غ‬4 ͗ਫ਼ͣͪ͞ͳͳಋ౵Ͳ͍Ζ .
(2) ࠹ॵ͹್॑ಋब‫غ‬෾‫ز‬ͺ , յ఺਼ͳȀͳ͹յ఺෾‫Ͳز‬ब‫ غ‬2 ͗ਫ਼ͣͪ͞ͳͳಋ౵Ͳ͍Ζ .
(3) ್ౕ໪͹್॑ब‫غ‬ഔ ෾‫ز‬ͺ, յ఺਼͵Ȁʹ͹յ఺෾‫Ͳز‬ब‫ غ‬4 ͗ਫ਼ͣͪ͞ͳͳಋ౵Ͳ͍
Ζ.
ఈཀྵ 6.2. ବԃ‫ܗ‬ଲসब‫ٌغ‬ಕ͹ब‫غ‬Ν ‫͢ͳݍ‬, յ఺਼ͺ ܶ ௤ ΝୱҒͳͤΖ. ବԃ‫॑࢝ܗ‬ଲসबt 32 t‫ٌغ‬ಕͶ͕͜Ζ W ࣊‫ܙ‬෾‫ز‬͹৖߻, յ఺෾‫ز‬͹յ఺਼ۢؔͺ ሺͲǡʹሻͲ͍Ζ.

7. ݃‫ޢ‬࢝॑Ն‫ٱ‬໚੷ฯଚύʖϏʖࣺ଀Ͷ͕͏ͱຌ࿨ชͲಚΔΗͪ݃ՎΝ؈ୱͶΉͳΌΖ .
(1) ࢝ ॑ ଲ স ब ‫ ٌ غ‬ಕ Ν ಝ ௅ ෉ ͜ Ζ ͪ Ό Ͷ, ଲ ߻ න ‫ ݳ‬Ν ಍ ೘ ͢ ͪ. Ҳ ब ‫ غ‬෾ ͹ ࣺ ଀ ͺ, ࠪ ଲ߻, ߅౵ࣺ଀, ͨ͢ͱӊଲ߻Ͷ෾մ͠ΗΖ .
(2) ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ͹҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻͺ W ࣊‫ܙ‬Νࣖͤ͞ͳ͗ಚΔΗͪ . W ࣊‫ܙ‬ͶΓͮͱବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ͹෾‫݀͗ز‬ఈ͠ΗΖ .
(3) ବԃ‫॑࢝ܗ‬ଲসब‫ٌغ‬ಕ͹յ఺෾‫ز‬͹յ఺਼ۢؔͺۢؔ ሺͲǡʹሻͲ͍Ζ.

ःࣛຌ࿨ชͶଲ͢ͱࠬಣंΓΕଡ͚͹‫͵॑ي‬αϟϱφΝ͏ͪͫ͘Ήͪ͢ . ৼΓΕ‫״‬ः͏ͪ͢Ήͤ.

ࢂߡช‫ݛ‬[1] ࢃ޳‫ج‬ദ, ್॑Ն‫ ٱ‬໚੷ฯଚࣺ଀Ͷ͕͜Ζ್ ॑ଲসब‫ٌغ‬ಕ , ‫ܙ‬͹Ռ ָճࢿ , ୊ 34 ‫ת‬,
୊ 1 ߺ (2019), 13-42. https://katachi-jp.com/gakkaishi[2] Harper, P. G., Single band motion of conduction electrons in a uniform magnetic field,
Proc. Soc. London A8 (1955), 874-878.
https://iopscience.iop.org/article/10.1088/0370 -1298/68/10/304/meta[3] Yamaguchi, Y. and Tanikawa, K., Periodic orbits in the quadruply reversible non-twist
Harper map, Prog. Theor. Phys. 111 (2004), 757-762.
https://academic.oup.com/ptp/article/111/5/757/1875581[4] Birkhoff, G. D.,

Dynamical systems . American Mathematical Society, Colloquium

Publications, Vol.9. (AMS, 1966). Revised edition.
https://archive.org/details/dynamicalsystems00birk/page/n7[5] ࢃ޳‫ج‬ദ, ୫ઔੜཾ, ഇఛ΃͹ಕ (‫ཱིڠ‬ड़൝, 2016).

t 33 tt 34 t࿨ช‫ܙ‬͹Ռָճࢿ ୊ 35 ‫ ת‬୊ 1 ߺඬ६ࣺ଀Ͷ͕͜Ζҡ৙յ఺෾‫ز‬ࢃ޳‫ج‬ദ
ˡ-

એཁ‫ࠅࢤݬࢤݟ‬෾࣋ୈ஦ԟ --

FKDRVIUDFWDO#L&ORXGFRP$QRPDORXVURWDWLRQELIXUFDWLRQLQWKHVWDQGDUGPDS<RVKLKLUR<DPDJXFKL--.RNXEXQMLGDL-&KXR,FKLKDUD&KLED--DSDQ(2020 ೧ 7 ݆ 23 ೖण෉ɾ2020 ೧ 9 ݆ 15 ೖणཀྵ)Abstract: The anomalous rotation bifurcation in the standard map ܶ on cylinder is discussed.
The elliptic fixed point ܳ of ܶ undergoes the ordinary rotation bifurcation and theperiod-doubling bifurcation. As a result, the elliptic daughter periodic orbit with the rotation
number ‫݌‬Ȁ‫ Ͳ( ݍ‬൏ ‫݌‬Ȁ‫ ݍ‬൑ ͳȀʹ) appears. This orbit is represented as ‫݌‬Ȁ‫ݍ‬-BE. We study whether
the anomalous bifurcation of ‫݌‬Ȁ‫ݍ‬-BE occurs. Let ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-BE (BS) be the elliptic (saddle)
periodic orbit appearing through the ordinary or anomalous rotation bifurcation of ‫݌‬Ȁ‫ݍ‬-BE.
The period of ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-BE (BS) is ‫ݎݍ‬. We prove that the doubly symmetric periodic orbitͳȀʹ:ʹȀ͵-BE (BS) appears through the anomalous rotation bifurcation of ͳȀʹ-BE and thesingly symmetric periodic orbit ͳȀʹ:ͳȀ͵-BE (BS) appears through the ordinary rotationbifurcation of ͳȀʹ-BE.
Keywords: Standard map, Symmetric periodic orbit, Ordinary and anomalous rotationbifurcations

1. ຌ࿨ช͹໪ద໚੷ฯଚࣺ଀Ͷ͕͜Ζବԃ‫ܗ‬ब‫ٌغ‬ಕ఼ͺϏϧϟʖνΝଁՅͤΖͳյ఺෾‫͵ز‬Δ;Ͷब‫غ‬ഔ෾ ‫ز‬Ν ً ͤ͞ ͞ͳ ͺΓ ͚எ ΔΗ ͱ ͏Ζ [1]. յ ఺෾ ‫ز‬Ͷ ͺ ਜ਼৙ յ఺ ෾‫ͳ ز‬ҡ ৙յ ఺ ෾‫[ ز‬1,2, 3]͍͗Ζ͞ͳΝ, ਦ 1 ͳ 2 Ν༽͏ͱઈ໎ͤΖ. Ϗϧϟʖνʖ͹ଁՅͶͯΗͱߡ͓ͱ͏Ζବԃ ‫ ෈ ܗ‬ಊ ఼ ܳ͹ ۛ ๥ ͹ յ ఺ ͹ ࢕ ๏ ͺ ଐ ͚ ͵ Ζ ͳ ͤ Ζ . ਜ਼ ৙ յ ఺ ෾ ‫ ͗ ز‬ਫ਼ ͣ Ζ ৖ ߻ , ବ ԃ ఼͹ۛ๥͹յ఺͹࢕๏ͺବԃ఼͖Δ཯ΗΖͶͯΗͱ ங͚͵Ζ. ͞Ηͺବԃ఼͹ۛ๥͹յ఺͹࢕๏͹ୱ௒͸ͣΗ੓Νқັ͢ͱ͏Ζ . ਦ 1(a)Ͳյ఺਼ͳ͢ͱ 1/4 ͳॽ͏ͱ͍Ζ͍ͪΕͶ, յ఺਼
1/4 ͹ब‫ٌغ‬ಕ఼͗ଚࡑͤΖ . ͞͹৖߻, յ఺਼ 1/4 ͹ब‫ٌغ‬ಕ఼ͺବԃ‫෈ܗ‬ಊ఼ ͖ܳΔਫ਼ͣͱ, Ϗϧϟʖνʖ͹ଁՅͶͯΗͱବԃ఼͖Δ཯Ηͱ͏͚ . յ఺਼ 1/4 ͹ब‫ٌغ‬ಕ͗ਫ਼ͣΖྡֆ஍Νܽ௖ ሺͳȀͶሻΝͤΖͳ, ܽ ൐ ܽ௖ ሺͳȀͶሻͲյ఺਼ 1/4 ͹ब‫ٌغ‬ಕ఼͗ଚࡑͤΖ(ਦ 2(a)).

t 35 tਦ 1ʁ(a) ਜ਼৙յ఺෾‫ز‬Ͷ͕͜Ζବԃ‫෈ܗ‬ಊ఼ ܳ͹ۛ๥͹ӣಊ. (b) ҡ৙յ఺෾‫ز‬Ͷ͕͜Ζବԃ‫෈ܗ‬ಊ఼ ܳ͹ۛ๥͹ӣಊ.

ਦ 2ʁ(a) ਜ਼৙յ఺෾‫ز‬. ܽ ൐ ܽ௖ ሺ‫݌‬Ȁ‫ݍ‬ሻͶ͕͏ͱ, ‫݌‬Ȁ‫ݍ‬-B ͗ଚࡑͤΖ. (b) ҡ৙յ఺෾‫ز‬Ͷ͕͜Ζҡ৙յ఺ۢؔ.
࣏ Ͷ ҡ ৙ յ ఺ ෾ ‫ ز‬Ͷ ͯ ͏ ͱ ઈ ໎ ͤ Ζ . ҡ ৙ յ ఺ ෾ ‫ ͗ ز‬ਫ਼ ͣ Ζ ৖ ߻ , ବ ԃ ‫ ෈ ܗ‬ಊ ఼ ܳ͹ ۛ ๥͹յ఺͹࢕๏ͺବ ԃ఼͖ Δ཯ΗΖͶͯΗͱ ଐ͚͵ Ε, ౏஦͖Δங͚ ͵Ζ . յ఺਼ͳ͢ͱ 1/3ͳॽ͏ͱ͍Ζ͍ͪΕͶյ఺਼ 1/3 ͹ब‫ٌغ‬ಕ఼͗ଚࡑͤΖ (ਦ 1(b)). ͞͹৖߻, յ఺਼ 1/3
͹ ब ‫ ٌ غ‬ಕ ఼ ͺ ବ ԃ ‫ ෈ ܗ‬ಊ ఼ ͖ܳ Δ ཯ Η ͪ ྘ Ү Ͳ ਫ਼ ͣ Ζ. ਫ਼ ͣ ๏ ͺ γ χ ϩ ό ʖ χ ෾ ‫ Ͳ ز‬, ବԃ ‫ ܗ‬ब ‫ ٌ غ‬ಕ ఼ ͳ γ χ ϩ ‫ ܗ‬ब ‫ ٌ غ‬ಕ ఼ ͗ ਫ਼ ͣ Ζ . ྡ ֆ ஍ Ν ܽ௖ୱ௡ ሺ‫݌‬Ȁ‫ݍ‬ሻͳ ͤ Ζ . Ϗ ϧ ϟ ʖ ν ʖ Νܽ௖ୱ௡ ሺ‫݌‬Ȁ‫ݍ‬ሻΓΕଁՅͤΖͳ, γχϩ‫ܗ‬ब‫ٌغ‬ಕ఼͗ବԃ‫෈ܗ‬ಊ఼ ܳͶ޴͜ͱۛͰ͘, ବԃ‫ܗ‬ब‫غ‬ٌ ಕ ఼ ͺ ਫ਼ ͣ ͪ ஏ ఼ ͖ Δ ԗ ͡ ͖ ͮ ͱ ͏ ͚ . ܽ ൌ ܽ௖ ሺ‫݌‬Ȁ‫ݍ‬ሻͲ , γ χ ϩ ‫ ܗ‬ब ‫ ٌ غ‬ಕ ఼ ͳ ܳ͗ Ҳ ஗ ͤ
Ζ . ܽ ൐ ܽ௖ ሺ‫݌‬Ȁ‫ݍ‬ሻͲ γ χ ϩ ‫ ܗ‬ब ‫ ٌ غ‬ಕ ఼ ͺ ବ ԃ ‫ ෈ ܗ‬ಊ ఼ ͖ܳ Δ ड़ ͱ , ͖ܳ Δ ཯ Η Ζ . ͞ ͹ ৖ ߻ ,ҡ ৙ յ ఺ Ϗ ϧ ϟ ʖ ν ۢ ؔ ሾܽ௖ୱ௡ ሺ‫݌‬Ȁ‫ݍ‬ሻǡ ܽ௖ ሺ‫݌‬Ȁ‫ݍ‬ሻሻ͗ ఈ ٝ Ͳ ͘ Ζ(ਦ 2(b)). ҡ ৙ յ ఺ Ϗ ϧ ϟ ʖ ν ۢؔͲͺବԃ ఼͹बΕ ͹ୱ ௒͸ͣΗ੓ ͗ഃΗͱ ͏Ζ . ࣰࡏ, ବԃ఼͖ Δ཯Η Ζͳյ఺͹ ࢕๏͗ବԃ఼ΓΕ ଐ͚͵Ζ . γχϩ‫ܗ‬ब‫ٌ غ‬ಕ఼Ν գ͙ Ζͳյ఺͹ ࢕๏ͺங ͚͵ Ζ . ͞͹Γ͑͵ବԃ఼͹बΕ ͹ୱ௒͸ ͣΗ ੓͹ഃΗͶ Γͮͱ , ҡ৙ յ఺෾‫ ͗ز‬ਫ਼ͣΖ͞ ͳ͗ Κ͖Ζ . ٠࿨ͤ΄͘໲ ୌ ͺ , ବԃ ఼ ͹ बΕ ͹ୱ ௒ ͸ ͣΗ ੓ ͹ ഃΗ ͗ ਫ਼ͣ Ζ ‫ ݬ‬ҾͲ ͍ Ζ . ҐԾ ͹٠ ࿨ Ͳ ͺ , յ఺
෾ ‫( ز‬Rotation bifurcation) Ν ਜ਼ ৙ յ ఺ ෾ ‫( ز‬Ordinary rotation bifurcation) ͳ ҡ ৙ յ ఺ ෾ ‫ز‬(Anomalous rotation bifurcation)͹૱সͳ͢ͱཤ༽ͤΖ.
t 36 tԃ౷໚ ሺͲ ൑ ‫ ݔ‬൏ ʹߨǡ െλ ൏ ‫ ݕ‬൏ λሻͲఈٝ͠Ηͪඬ६ࣺ଀Ν়ղͤΖ [4].
ܶ ‫ݕ  ׷‬௡ାଵ ൌ ‫ݕ‬௡ ൅ ݂ሺ‫ݔ‬௡ ሻǡ‫ݔ‬௡ାଵ ൌ ‫ݔ‬௡ ൅ ‫ݕ‬௡ାଵ ሺ‫ߨʹ݀݋‬ሻǤሺ1ሻ͞͞Ͳ, ݂ሺ‫ݔ‬ሻ ൌ ܽ‫ݔ‬ሺܽ ൒ Ͳሻ.
ࣺ ଀ ܶሺܽ ൐ ͲሻͲ ͺ , ݂ሺ‫ݔ‬ሻ ൌ Ͳͳ ‫ ݕ‬ൌ ͲΝ ຮ ͪ ͤ ब ‫ ٌ غ‬ಕ ఼ ͗ ෈ ಊ ఼ Ͳ ͍ Ζ . ඬ ६ ࣺ ଀ Ͷ ͕ ͏ͱ, ෈ಊ఼ͺ ܲ ൌ ሺͲǡͲሻͳܳ ൌ ሺߨǡ ͲሻͲ͍Ζ. ෈ಊ఼ܲͺγχϩ‫෈ܗ‬ಊ ఼ Ͳ͍Ζ . ෈ಊ఼ܳͺ, Ͳ ൏ܽ ൏ ͶͶ͕͏ͱବԃ‫෈ܗ‬ಊ఼Ͳ , ܽ ൐ ͶͲͺൕ఺Ν൒͑γχϩ‫෈ܗ‬ಊ఼Ͳ͍Ζ .
ଲ স ब ‫ ٌ غ‬ಕ Ν ఈ ٝ ͤ Ζ ͪ Ό Ͷ ࣺ ଀ ܶ͹ ଲ ߻ න ‫ ݳ‬Ν ಍ ೘ ͤ Ζ . ୊ Ҳ ͹ ଲ ߻ න ‫ ݳ‬Ν ࣖ ͤ . ࣞ(1)͹ න ‫ Ͳ ݳ‬ͺ , ୊ Ҳ ͹ ଲ ߻ න ‫ ܶ ݳ‬ൌ ݄ ‫݃ ל‬ͺ ৙ Ͷ Ն ೵ Ͳ ͍ Ζ. ࣺ ଀ ܶͺ ್ ͯ ͹ ଲ ߻ ͳ ‫ ͻ ݼ‬Η Ζࣺ଀͹߻੔Ͳ‫ى‬फ़͠ΗΖ . ‫לߺى‬͹ӊଈͶ͍Ζଲ߻ ݃Ν઎Ͷࡠ༽͢, ࣏Ͷଲ߻ ݄Νࡠ༽ͤΖ. ҐԾͲଲ߻͹නࣞΝࣖͤ.
െ‫ݕ‬‫ݕ‬‫ݕ‬െ‫ ݕ‬െ ݂ሺ‫ݔ‬ሻ݃ቀ ቁ ൌ ቀቁ ǡ݄ ቀ ቁ ൌ ቀ‫ ݔ‬െ ‫ݕ‬ቁǤ‫ݔ‬‫ݔ‬‫ݔ‬

ሺ2ሻࣺ଀‫਼݂ؖ‬ሺ‫ݔ‬ሻ͗‫͍Ͳ਼ؖد‬Ζ͞ͳͶΓΕ , ࣏͹୊್͹ଲ߻න‫ ܶ ݳ‬ൌ ‫ ܩ ל ܪ‬΍Ն೵Ͳ͍Ζ.
‫ݕ‬‫ ݕ‬൅ ݂ሺ‫ݔ‬ሻ‫ݕ‬‫ݕ‬‫ܩ‬ቀ ቁ ൌ൬൰ ǡ‫ ܪ‬ቀ ቁ ൌ ቀ‫ ݕ‬െ ‫ݔ‬ሺ‫ߨʹ݀݋‬ሻቁǤ
ሺ3ሻ
‫ݔ‬
‫ݔ‬
െ‫ݔ‬ሺ‫ߨʹ݀݋‬ሻ͞͞Ͳ, ݃ ‫ ݃ ל‬ൌ ݄ ‫ ݄ ל‬ൌ ‫ ܩ ל ܩ‬ൌ ‫ ܪ ל ܪ‬ൌ ݅݀Ǥ.
ଲ߻ͺ໚੷ฯଚͲ͍Ζ͗๏޴ൕ఺ࣺ଀Ͳ͍Ζ . ଲ߻͹෈ಊ఼͹ॄ߻͗ଲসતͲ͍Ζ .
ܵ௚ ǣ‫ ݕ‬ൌ െ݂ሺ‫ݔ‬ሻȀʹǡ

ሺ4ሻ

ܵ௛ ǣ

‫ ݕ‬ൌ Ͳǡ

ሺ5ሻ

ܵீ ǣ

‫ ݔ‬ൌ Ͳǡ‫ ݔ‬ൌ ߨǡ

ሺ6ሻ

ܵு ǣ

‫ ݕ‬ൌ ʹ‫ݔ‬ǡ‫ ݕ‬ൌ ʹሺ‫ ݔ‬െ ߨሻǤሺ7ሻԃ౷໚Ͳఈٝ͠Ηͪඬ६ࣺ଀Ͷ͕͜Ζ࢝ͯ͹ଲসતΝਦ 3 Ͷඵ͏ͪ͹ͲࢂߡͶ͢ͱཋ͢͏ .

ਦ 3 ʁ ԃ ౷ ໚ Ͳ ఈ ٝ ͠ Η ͪ ඬ ६ ࣺ ଀ Ͷ ͕ ͜ Ζ ଲ স ત . ܲ ൌ ሺͲǡͲሻͳ ܳ ൌ ሺߨǡ Ͳሻͺ ෈ ಊ ఼ . ܲԢ ൌሺʹߨǡ Ͳሻ. ԃ౷໚Ͳͺܲ ൌ ܲԢ. ܵ௚ ͳܵ௛ ͺ৏Ծ͹ଲসતͲ, ܵீ ͳܵு ͺࠪӊ͹ଲসત. ܽ ൌ ͷ.
ଲ߻ͶΓΖब‫ٌغ‬ಕ఼͹ಊ͘ͺଲসતͶ‫ͤؖ‬Ζൕ఺Ͳ͍Ζ . Γͮͱଲ߻ͶΓΖब‫ٌغ‬ಕ఼͹ ಊ ͘ Γ Ε , ܵ௚ ͳ ܵ௛ Ν ৏ Ծ ͹ ଲ স ત ͳ ‫ ; ݼ‬, ܵீ ͳ ܵு Ν ࠪ ӊ ͹ ଲ স ત ͳ ‫ ΁ ݼ‬. ࣺ ଀ ͗ ଲ ߻ ͹ ੷Ͳන‫͠ݳ‬ΗΖ৖߻, ‫ܧ‬ͺώʖαϓ͹қັͲ͹Ն‫ٱ‬੓Ν΍ͯͳ͏͑ [1]. ৏‫ى‬͹Γ͑Ͷ್झྪ͹ҡ͵ͮͪଲ ߻͹੷Ͳ න‫͠ ݳ‬ΗΖࣺ଀ ͺ , ್॑Ն‫ ͻݼͳ଀ࣺ ٱ‬ΗΖ . ್॑Ն ‫଀ࣺٱ‬Ͷͺ Ҳ॑ଲt 37 tস ब ‫ ٌ غ‬ಕ ͳ ್ ॑ ଲ স ब ‫ ٌ غ‬ಕ ͗ ଚ ࡑ ͤ Ζ [5]. ଲ স ત Ν ཤ ༽ ͢ ͱ ଲ স ब ‫ ٌ غ‬ಕ ͳ ඉ ଲ স ब‫ٌغ‬ಕΝఈٝ͢, ࣏ͶҲ॑ଲসब‫ٌغ‬ಕͳ್॑ଲসब‫ٌغ‬ಕΝ ఈٝͤΖ.
ఈٝ 1.1 (ଲসब‫ٌغ‬ಕͳඉଲসब‫ٌغ‬ಕ͹ఈٝ ). ଲসત৏Ͷٌಕ఼Ν΍ٌͯಕΝଲসٌಕ
ͳ͏͑. ଲসٌಕ͗ ଲস ત৏Ͷ۰਼ ‫ݺ‬͹ٌಕ ఼Ν ΍ͯ͵Δͻ , ͞͹ٌಕͺ ଲসब‫ ٌغ‬ಕͳ‫ݼ‬ͻΗ Ζ. ຌ ࿨ช Ͳ ͺ ෈ಊ ఼ͺ ଲ স ब‫ ٌ غ‬ಕ Ͷ‫ ؜‬Ό ͥ , ෈ಊ ఼ ͳ͢ ͱ ѽ͑ . ଲস ત ৏ Ͷٌ ಕ ఼Ν΍ͪ͵͏ब‫ٌغ‬ಕΝඉଲসब‫ٌغ‬ಕͳఈٝͤΖ .
ఈٝ 1.2 (Ҳ॑ଲসब‫ٌغ‬ಕ͹ఈٝ ). ৏Ծ͹ଲসત͹ΊͶٌಕ఼Ν΍ͯब‫ٌغ‬ಕΝҲ॑৏Ծଲসब‫ ٌغ‬ಕͳఈٝ ͤΖ . Ήͪ, ࠪӊ͹ଲ সત͹ ΊͶٌಕ఼ Ν΍ͯब ‫ ٌغ‬ಕΝҲ॑ࠪ ӊଲসब‫ٌغ‬ಕͳఈٝͤΖ. ͞ΗΔΝΉͳΌͱҲ॑ଲসब‫ٌغ‬ಕͳ‫ ΁ݼ‬.
ब‫ٌغ‬ಕ͗ଲসત৏Ͷٌಕ఼Νͽͳͯ΍ͯͳ , พ͹ٌಕ఼͗ଲসત৏Ͷ͍Ζͳ͏͑੓࣯͗
͍Ζ. ͨ͹ͪΌͶҲ ॑ଲ সब‫ٌغ‬ಕ ͺଲসત ৏Ͷ ್ٌͯಕ఼ Ν΍ͯ . ್ͯ ͹ٌಕ఼͗ Ҳ఼Ͷ͵ͮͪ৖߻͗෈ಊ఼Ͳ͍Ζ .
ఈٝ 1.3 (್॑ଲসब‫ٌغ‬ಕ͹ఈٝ ). ৏Ծͳࠪӊ͹ଲসત৏Ͷٌಕ఼Ν΍ͯब‫ٌغ‬ಕΝ್॑ଲসब‫ٌغ‬ಕͳఈٝͤΖ .
್॑ଲসब‫ٌغ‬ಕͺ৏Ծ͹ଲসત৏Ͷ್ٌͯಕ఼Ν΍͖ͬͯࠪӊ͹ଲসત৏Ͷ್ٌͯಕ఼
Ν΍ͯ. ͤ͵Κ್ͬ ॑ଲ সब‫ٌغ‬ಕ ͺଲসત ৏Ͷ ٌ࢝ͯ ಕ఼ Ν΍ͯ . ࢝ͯ ͹ٌಕ఼͗ ఼್Ͷ͵ͮͪ৖߻ͺब‫ غ‬2 ͹ٌಕ఼Ͳ͍Ε, ࢝ͯ͹ٌಕ఼͗Ҳ఼Ͷ͵ͮͪ৖߻͗෈ಊ఼Ͳ͍Ζ .
ବ ԃ ‫ ෈ ܗ‬ಊ ఼ ܳ͗ յ ఺ ෾ ‫ ز‬Ν ً ͞ ͢ ͱ ਫ਼ ͣ ͪ ໊ ब ‫ ٌ غ‬ಕ Ν ‫݌‬Ȁ‫ݍ‬-B (Ͳ ൏ ‫݌‬Ȁ‫ ݍ‬൏ ͳȀʹ) ͳ ॽ ͚(B ͺ Birkhoff ΝқັͤΖ). ‫݌‬Ȁ‫ݍ‬-B ͺବԃ‫໊ܗ‬ब‫ٌغ‬ಕ ‫݌‬Ȁ‫ݍ‬-BE ͳ, γχϩ‫໊ܗ‬ब‫ٌغ‬ಕ ‫݌‬Ȁ‫ݍ‬-BS ͹૱সͲ͍Ζ. ͞͞Ͳ, E ͺ Elliptic Νқັ͢, S ͺ Saddle ΝқັͤΖ. ବԃ‫෈ܗ‬ಊ఼ ܳ͗ब‫ غ‬ഔ෾ ‫ز‬Ν ً ͢͞ ͱਫ਼ ͣͪ ବԃ ‫ ໊ܗ‬ब ‫ ٌغ‬ಕΝ ͳȀʹ-BE ͳ ॽ͚. ͞Η Δ͹ ब ‫ ٌغ‬ಕͶ ͯ͏
ͱ੓࣯ 1.4 ͗੔Εཱིͯ. ৆ࡋͺࢂߡช‫[ݛ‬5]Ν‫ݡ‬ΔΗͪ͏.
੓࣯ 1.4.
(1) ब‫͗ݍغ‬۰਼͹৖߻, ‫݌‬Ȁ‫ݍ‬-B ͳͳȀʹ-BE ͺ್॑ଲসब‫ٌغ‬ಕͲ͍Ζ .
(2) ब‫਼د͗ݍغ‬͹৖߻, ‫݌‬Ȁ‫ݍ‬-B ͺҲ॑ଲসब‫ٌغ‬ಕͲ͍Ζ .
͞͞Ͳຌ࿨ช͹໪దΝΉͳΌ , ࣏ͶҐԾ͹ઇ͹಼༲Ν؈ୱͶ়ղͤΖ .
໪ ద 1.5. ବԃ ‫ ෈ ܗ‬ಊ ఼ ܳ͗ յ ఺ ෾ ‫ ز‬Ή ͪ ͺ ब ‫ غ‬ഔ ෾ ‫ ز‬Ν ً ͞ ͢ ͱ ਫ਼ ͣ ͪ ବ ԃ ‫ ໊ ܗ‬ब ‫ٌ غ‬ಕ‫݌‬Ȁ‫ݍ‬-BE ͹յ఺෾‫ز‬Ν௒΄Ζ͞ͳ .
୊ 2 ઇ. ຌ࿨ชͲ࢘༽ͤΖ਼ָదಕ۫Ͷͯ͏ͱઈ໎ͤΖ .
୊ 3 ઇ. ବԃ‫໊ܗ‬ब‫ٌغ‬ಕ‫݌‬Ȁ‫ݍ‬-BE ͹յ఺෾‫ز‬Ͷͯ͏ͱ਼஍‫Ͳࢋܯ‬ಚΔΗͪ݃ՎΝ়ղͤΖ .
୊ 4 ઇ. ਼஍‫Ͳࢋܯ‬ಚΔΗͪ݃Վ͹Ҳ෨Ͷৄ໎Ν༫͓ , ఈཀྵ 4.1 ͳໍୌ 4.5 ͳͤΖ.
୊ 5 ઇ. ຌ࿨ชͲಚΔΗͪ݃ՎΝΉͳ ΌΖ.

t 38 t2. ਼ָదಕ۫
2.1. ब‫ٌغ‬ಕ͹҈ఈ੓‫਼ܐ‬ब‫ ٌغ‬ಕ͹ ҈ఈ ੓Ν ൓இ ͤΖ๏ ๑Ν ়ղ ͤΖ . ͞͞Ͳ ‫ݍ‬ሺ൒ ͳሻΝब‫ٌ غ‬ಕ ͹ब ‫ͤ ͳغ‬Ζ . ब‫ݍغ‬͹ब‫ٌغ‬ಕ఼Ν ‫ݖ‬௞ ൌ ሺ‫ݔ‬௞ ǡ ‫ݕ‬௞ ሻሺͲ ൑ ݇ ൑ ‫ ݍ‬െ ͳሻͳͤΖ. ٌಕ఼‫ݖ‬௞ Ͷ͕͜Ζત‫ܙ‬Կߨ྽ ‫ܯ‬ሺ‫ݖ‬௞ ሻͺͳ ݂Ԣሺ‫ݔ‬௞ ሻ‫ܯ‬ሺ‫ݖ‬௞ ሻ ൌ ൬൰ሺ8ሻͳ ͳ ൅ ݂Ԣሺ‫ݔ‬௞ ሻ
ͳಚΔΗΖ. ͞͞Ͳ, ݂Ԣሺ‫ݔ‬௞ ሻ ൌ ܽ ‫ݔ‬௞ . ໚੷ฯଚࣺ଀Ͳ͍Ζ͖Δ, ‫ܯ‬ሺ‫ݖ‬௞ ሻ͹ߨ྽ࣞͺ 1 Ͳ͍Ζ.
ࣺ଀ܶ ௤ ͹ત‫ܙ‬Կߨ྽Ν ‫ܯ‬௤ ൌ ‫ܯ‬ሺ‫ݖ‬௤ିଵ ሻ ‫ܯ ڮ‬ሺ‫ݖ‬ଵ ሻ‫ܯ‬ሺ‫ݖ‬଴ ሻͳͤΖͳ, ‫༙ݽ‬஍ߣΝ݀ఈͤΖ๏ఖࣞߣଶ െ ‫ݎ‬ሺܽሻߣ ൅ ͳ ൌ Ͳሺ9ሻ͗ ಚ Δ Η Ζ . ‫ݎ ਼ ܐ‬ሺܽሻ ൌTrace‫ܯ‬௤ Ν ҈ ఈ ੓ ‫ ΁ ݼ ͳ ਼ ܐ‬. ࣞ (9)͹ մ Ͳ ͍ Ζ ‫ ༙ ݽ‬஍ Ν ߣേ ͳ ॽ ͚ .
յ఺਼‫݌‬Ȁ‫ݍ‬͹ब‫ٌغ‬ಕ͗ਜ਼৙յ఺෾‫͗ز‬ਫ਼ͣΖ৖߻ , Ծ‫ى‬͹‫͗ܐؖ‬੔ཱིͤΖ.
ߣേ ൌ‫ݎ‬ሺܽሻ േ ݅ඥͶ െ ‫ ݎ‬ଶ ሺܽሻൌ ሺʹߨ‫݌‬Ȁ‫ݍ‬ሻ േ ሺʹߨ‫݌‬Ȁ‫ݍ‬ሻǤʹሺ10ሻ͞͹ࣞΓΕ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻͳյ఺਼ ‫݌‬Ȁ‫ݍ‬͹‫͗ܐؖ‬ಚΔΗΖ.
‫ݎ‬ሺܽሻ ൌ ʹ ሺʹߨ‫݌‬Ȁ‫ݍ‬ሻǤ

ሺ11ሻ

2.2. ܑ͘౵͹ࣺ଀
ࣞ(1)͹୊ 1 ࣞͳ୊ 2 ࣞ͹ྈวΝ‫ݔ‬௡ Ͳඏ෾ͤΖͳ, Ծ‫ى‬͹್ࣞ͗ಚΔΗΖ.
݀‫ݕ‬௡ାଵ ݀‫ݔ‬௡ାଵ ݀‫ݕ‬௡ାଵ ݀‫ݕ‬௡ൌൌ൅ ݂Ԣሺ‫ݔ‬௡ ሻǡ݀‫ݔ‬௡݀‫ݔ‬௡ ݀‫ݔ‬௡ାଵ ݀‫ݔ‬௡݀‫ݔ‬௡ାଵ ݀‫ݕ‬௡ൌ൅ ݂Ԣሺ‫ݔ‬௡ ሻ ൅ ͳǤ
݀‫ݔ‬௡
݀‫ݔ‬௡

ሺ12ሻ
ሺ13ሻ

఼ ‫ݖ‬௡ ൌ ሺ‫ݔ‬௡ ǡ ‫ݕ‬௡ ሻ͗ ‫ ۄ‬ત ‫ ݕ‬ൌ ‫ܨ‬ሺ‫ݔ‬ሻ৏ Ͷ ͍ Ζ ͳ ͤ Ζ . ఼ ‫ݖ‬௡ Ͷ ͕ ͜ Ζ ‫ ۄ‬ત ͹ ܑ ͘ Ν න ͤ ร ਼ ߦ௡ ൌ݀‫ܨ‬ሺ‫ݔ‬௡ ሻȀ݀‫ݔ‬௡ ൌ ݀‫ݕ‬௡ Ȁ݀‫ݔ‬௡ Ν಍೘ͤΖ. ࣞ(13)ΓΕ, ܑ͘ߦ௡ ͗ॊࣺ͑଀͗ಚΔΗΖ.
ߦ௡ ൅ ݂Ԣሺ‫ݔ‬௡ ሻߦ௡ାଵ ൌ
Ǥߦ௡ ൅ ݂Ԣሺ‫ݔ‬௡ ሻ ൅ ͳሺ14ሻܑ ͘ ͹ ࣺ ଀ ͹ ྈ ว Ν ‫ݔ‬௡ Ͳ ඏ ෾ ͤ Ζ ͳ , ‫ ۄ‬ત ‫ ݕ‬ൌ ‫ܨ‬ሺ‫ݔ‬ሻ ͹ ఼ ‫ݖ‬௡ Ͷ ͕ ͜ Ζ ್ ֌ ඏ ෾ ‫ߟ ਼ ܐ‬௡ ൌ݀ߦ௡ Ȁ݀‫ݔ‬௡ ͗ॊࣺ͑଀͗ಚΔΗΖ.
ߟ௡ାଵ ൌߟ௡ ൅ ݂ԢԢሺ‫ݔ‬௡ ሻǤሺߦ௡ ൅ ݂Ԣሺ‫ݔ‬௡ ሻ ൅ ͳሻଷሺ15ሻ͞͞Ͳ݀ߦ௡ାଵ Ȁ݀‫ݔ‬௡ ൌ ሺ݀‫ݔ‬௡ାଵ Ȁ݀‫ݔ‬௡ ሻሺ݀ߦ௡ାଵ Ȁ݀‫ݔ‬௡ାଵ ሻ ൌ ߟ௡ାଵ ሺ݀‫ݔ‬௡ାଵ Ȁ݀‫ݔ‬௡ ሻΝཤ༽ͪ͢.
࣏ Ͷ ್ ֌ ඏ ෾ ‫ ਼ ܐ‬͹ ࣺ ଀ ͹ ྈ ว Ν ‫ݔ‬௡ Ͳ ඏ ෾ ͤ Ζ ͳ , ‫ ۄ‬ત ‫ ݕ‬ൌ ‫ܨ‬ሺ‫ݔ‬ሻ͹ ఼ ‫ݖ‬௡ Ͷ ͕ ͜ Ζ ࢀ ֌ ඏ ෾‫ߞ਼ܐ‬௡ ൌ ݀ߟ௡ Ȁ݀‫ݔ‬௡ ͗ॊࣺ͑଀͗ಚΔΗΖ.
ߞ௡ାଵ ൌߞ௡ ൅ ݂ԢԢԢሺ‫ݔ‬௡ ሻ͵ሺߟ௡ ൅ ݂ԢԢሺ‫ݔ‬௡ ሻሻଶെǤሺߦ௡ ൅ ݂Ԣሺ‫ݔ‬௡ ሻ ൅ ͳሻସ ሺߦ௡ ൅ ݂Ԣሺ‫ݔ‬௡ ሻ ൅ ͳሻହሺ16ሻ͞͞Ͳ݀ߟ௡ାଵ Ȁ݀‫ݔ‬௡ ൌ ሺ݀‫ݔ‬௡ାଵ Ȁ݀‫ݔ‬௡ ሻሺ݀ߟ௡ାଵ Ȁ݀‫ݔ‬௡ାଵ ሻ ൌ ߞ௡ାଵ ሺ݀‫ݔ‬௡ାଵ Ȁ݀‫ݔ‬௡ ሻΝཤ༽ͪ͢.
Ґ ৏ ͹ ࣞ Ͳ , ݂Ԣሺ‫ݔ‬௡ ሻ ൌ ܽ ‫ݔ‬௡ , ݂ԢԢሺ‫ݔ‬௡ ሻ ൌ െܽ‫ݔ‬௡ ൌ െ݂ሺ‫ݔ‬௡ ሻ, ݂ԢԢԢሺ‫ݔ‬௡ ሻ ൌ െ݂Ԣሺ‫ݔ‬௡ ሻ. ͞ ͞ Ͳ ಚ Δ Ηࣺͪ଀Νཤ༽ͤΖͪΌͶͺٌಕ఼ ‫ݖ‬௡ ͹৚ๅͳܑ͘౵͹ॵ‫غ‬஍͹৚ๅ͗චགྷͲ͍Ζ .

t 39 t2.3. ҡ৙յ఺෾‫͗ز‬ਫ਼ͣΖͪΌ͹චགྷড়݇ବԃ‫෈ܗ‬ಊ఼ ܳ͗ҡ৙յ఺෾‫͗ز‬ਫ਼ͣΖͪΌ͹චགྷ ড়݇Ν఑ࣖ͢Γ͑ . ࣺ଀ܶͲܳ͹बΕΝյ఺ͤΖब‫ٌغ‬ಕ͹յ఺๏޴ͺ , ਦ 4(a)Ͳͺ࣎‫ܯ‬յΕͲ, ਦ 4(b)Ͳͺൕ࣎‫ܯ‬յΕͲ͍Ζ.
ਦ 4(a)Ͷඵ͖Ηͪଲসત͹଀ͺଲসત͹ࠪଈͶ͍Ζ . Ϗϧϟʖν͹ଁ୉Ͷͳ΍͵͏ ܳͶ͕͜Ζଲসત͹଀͹ܑ͗͘ଲসત͹ܑ͘ͶۛͰ͚ . ଲসતͳଲসત͹଀͹‫ིۄ‬ͺ‫ڠ‬Ͷਜ਼Ͳ͍Ζ .
ਜ਼ ৙ յ ఺ ෾ ‫ Ͳ ز‬ͺ ͖ܳ Δ ब ‫ ٌ غ‬ಕ ఼ ͗ ਫ਼ ͣ Ζ ͞ ͳ Ͷ ஭ қ ͢ Γ ͑ . ΍ ͢ , ଲ স ત ͹ ‫ ͗ ི ۄ‬κϫͲ͍ͮͪΕෝͲ͍Ζͳଲসત͹଀͹ܑ͗͘ଲসત͹ܑ͗͘Ҳ஗ͪ͢‫ ޛ‬, ଲসતͳଲসત͹଀͹ ި ࠫ ఼ ͺ ܳΓ Ε ਫ਼ ͣ ܳΓ Ε ཯ Η ͱ ͏ ͚ . ଲ স ત ͳ ଲ স ત ͹ ଀ ͹ ި ࠫ ఼ ͗ յ ఺ ෾ ‫ Ͳ ز‬ਫ਼ ͣ ͪब‫ٌغ‬ಕ͹ ٌಕ఼͹ Ҳͯ Ͳ͍Ζ . ͞ΗΓΕҡ ৙յ ఺෾‫͗ز‬ਫ਼ ͣΖͪΌ Ͷͺ , ଲসત͹‫ིۄ‬΍
ਜ਼Ͳ͍Ζ͞ͳ͗චགྷͲ͍Ζ͞ͳ͗಍͖ΗΖ . ͯΉΕਦ 4(a)͗ड़‫ͤݳ‬Ζ͵Δͻ, ҡ৙յ఺෾‫ز‬͗ ਫ਼ ͣ Ζ Ն ೵ ੓ ͗ ͍ Ζ . ҡ ৙ յ ఺ ෾ ‫ Ͳ ز‬ͺ , ͖ܳ Δ ཯ Η ͪ Ғ ஖ Ͳ γ χ ϩ ό ʖ χ ෾ ‫ Ͳ ز‬γ χϩ‫ܗ‬ब‫ٌغ‬ಕͳବԃ‫ܗ‬ब‫ٌغ‬ಕ͗ड़‫ͤݳ‬Ζ͞ͳΝ࠸ౕ஭қ͢ͱ͕͚ .
ਦ 4(b)Ͷඵ͖Ηͪଲসત͹଀ͺଲসત͹ӊଈͶ͍Ζ . Ϗϧϟʖν͹ଁ୉Ͷͳ΍͵͏ ܳͶ͕͜Ζଲসત͹଀͹ܑ͗͘ଲসત͹ܑ͘ͶۛͰ͘յ఺෾‫͗ز‬ਫ਼ͣΖ . ͞͹৖߻Ͳͺଲসતͳଲ
সત͹଀͹‫ིۄ‬ͺ‫ڠ‬ͶෝͲ͍Ζ . ͞͹Γ͑͵৖߻Ͷ΍ҡ৙յ఺෾‫͗ز‬ਫ਼ͣΖՆ೵੓͍͗Ζ .

ਦ 4ʁҡ৙յ఺෾‫͗ز‬ਫ਼ͣΖՆ೵੓͍͗Ζഓ஖ . (a) ࣺ଀ΓΖܳ͹बΕ͹յ఺๏޴͗࣎‫ܯ‬յΕ͹৖߻. ବԃ‫෈ܗ‬ಊ఼ ܳͶ͕͜Ζଲসતͳଲসત͹଀͹‫ڠི͗ۄ‬Ͷਜ਼ . (b) ࣺ଀ΓΖ ܳ͹बΕ͹յ ఺๏ ޴ ͗ൕ ࣎‫ ܯ‬յΕ ͹৖ ߻ . ବ ԃ‫෈ ܗ‬ಊ ఼ ܳͶ͕͜ Ζଲ সત ͳ ଲস ત͹ ଀͹ ‫ ڠ͗ ིۄ‬Ͷෝ.
ବ ԃ ‫ ෈ ܗ‬ಊ ఼ ܳΝ Ҳ ൢ ͹ ବ ԃ ‫ ٌ ܗ‬ಕ ఼ ‫ ͪ ͢ ͳݖ‬৖ ߻ Ͷ ͕ ͜ Ζ ҡ ৙ յ ఺ ෾ ‫ ز‬Ν ً ͞ ͤ ͪ Ό Ͷචགྷ͵ড়݇Νড়݇ 2.1 ͳ͢ͱΉͳΌͱ͕͚[3].
ড়݇ 2.1. ҡ৙յ఺෾‫ز‬Νًͤͪ͞ΌͶචགྷ͵ড়݇ͺ , ବԃ‫ٌܗ‬ಕ఼ ‫ݖ‬͹बΕ͹յ఺͹࢕๏͗(ൕ) ࣎‫ܯ‬յΕ͹৖߻͹৖߻, ‫ི͗ۄ‬ਜ਼ (ෝ) ͹ଲসતͳ‫ི͗ۄ‬ਜ਼ (ෝ) ͹ଲসત͹଀ٌ͗ಕ఼‫ݖ‬Ν௪գ͢ͱ͏Ζ͞ͳͲ͍Ζ .
Φόϱࣺ଀ଔͶ‫؜‬ΉΗΖࣺ଀Ͳͺ , ବԃ‫෈ܗ‬ಊ఼ ܳ͹बΕ͹յ఺͹࢕๏ͺ࣎‫ܯ‬յΕͲ͍Ζ.
ਦ 4(a)͹৖߻֚͗౲͢, յ఺਼ͳȀ͵͹ଲসब‫ٌغ‬ಕ͗ҡ৙յ఺෾‫ܳ Ͳز‬ΓΕਫ਼ͣΖ͞ͳ͗ࣖ͠Ηͱ ͏ Ζ [3]. ඬ६ ࣺ଀ Ͳ΍ , ବԃ ‫෈ ܗ‬ಊ ఼ ܳ͹ब Ε͹ յ ఺͹ ࢕๏ ͺ ࣎‫ ܯ‬յ ΕͲ ͍ Ζ . ඬ६ ࣺt 40 t଀ͲͺܳΝ௪գͤΖଲসતͲ ܳͶ͕͜Ζ‫ི͗ۄ‬ਜ਼͹ଲসતͺଚࡑ͢͵͏ . Γͮͱ, ඬ६ࣺ଀Ͳ
ͺড়݇ 2.1 ͗ຮͪ͠Η͵͏ͪΌ, ܳͺҡ৙յ఺෾‫ز‬Νً͞͠͵͏ . ͞Ηͺ॑གྷ͵݃ՎͲ͍Ζ͹Ͳໍୌ 2.2 ͳͤΖ.
ໍୌ 2.2. ඬ६ࣺ଀ܶͶ͕͏ͱ, ବԃ‫෈ܗ‬ಊ఼ ܳͺҡ৙յ఺෾‫ز‬Νً͞͠͵͏ .

3. ਼஍‫݃ࢋܯ‬Վ್॑ଲসब‫ٌغ‬ಕ ͳȀʹ-BE ͺବԃ‫෈ܗ‬ಊ఼ ܳ͹ब‫غ‬ഔ෾‫Ͳز‬ਫ਼ͣΖ. ͞͞Ͳͺ ͳȀʹ-BE ͹յ఺ ෾ ‫ ز‬Ν ௒ ΄ Ζ . घ ॳ Ν ઈ ໎ ͤ Ζ . Ϗ ϧ ϟ ʖ ν ܽΝ ݀ Ό , ࠹ ॵ Ͷ ͳȀʹ-BE ͹ ٌ ಕ ఼ Ν ‫ ٽ‬Ό Ζ .
࣏Ͷٌಕ఼͹৚ๅΓΕ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻΝ݀ఈͤΖ. ͞ΗΔͺસͱ਼஍‫ͤࢬࣰͲࢋܯ‬Ζ .
‫ ع‬Ͷ फ़ ΄ ͪ Γ ͑ Ͷ , ବ ԃ ‫ ෈ ܗ‬ಊ ఼ ܳ͹ (Ҳ ࣏ ͹ )յ ఺ ෾ ‫ ز‬Ͷ Γ Ε ‫݌‬Ȁ‫ݍ‬-BE ͗ ਫ਼ ͣ Ζ . ͨ ͢ ͱ‫݌‬Ȁ‫ݍ‬-BE ͹յ఺෾‫ز‬ͺ್࣏͹յ఺෾‫ͻݼͳز‬ΗΖ . ್࣏͹յ఺෾‫Ͳز‬յ఺਼ ‫ݎ‬Ȁ‫͗ݏ‬ਫ਼ͣͪ৖߻,͞͹ब‫ٌغ‬ಕΝյ఺਼ ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬͹ब‫ٌغ‬ಕͳ‫;ݼ‬, ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-B ͳॽ͚. ͞͞Ͳյ఺਼‫ݎ‬Ȁ‫ݏ‬ͺܶ ௤ ΝୱҒ ͳ ͢ͱ ଎ ͮͪ յ ఺਼ Ͳ͍ Ζ . յ఺ ਼ ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬͹ब ‫ ٌغ‬ಕ͹ ࣰ ࡏ͹ ब ‫غ‬ͺ ‫͍ Ͳݏݍ‬Ζ . ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ
‫ݏ‬-B ͺ, ବԃ‫ܗ‬ब‫ٌغ‬ಕ ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-BE ͳγχϩ‫ܗ‬ब‫ٌغ‬ಕ ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-BS ͹૱সͲ͍Ζ. ବԃ‫෈ܗ‬ಊ఼ܳΝଢཇͶᄽ͓, ͳȀʹ-BE Νஏ‫ځ‬Ͷᄽ͓Ζ͵Δͻ, ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-B ͺஏ‫ځ‬͹Ӷ੗Ͳ͍Ζ݆Ͷᄽ͓ΔΗΖ.
ͳȀʹ-BE ͹҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻ͹ܽғଚ੓Νਦ 5 Ͷࣖͪ͢. ͳȀʹ-BE ͺܽ ൐ ͶͶ͕͏ͱଚࡑͤΖ. Γͮ ͱ , ܽΝ 4 Γ Ε ଁ Յ ͤ Ζ ͳ ҈ ఈ ੓ ‫ݎ ਼ ܐ‬ሺܽሻͺ ‫ݎ‬ሺܽሻ ൌ െʹΉ Ͳ ‫ ݰ‬ঙ ͤ Ζ . ͨ ͹ ‫ ޛ‬, ଁ Յ ͤ Ζ .
ͳȀʹ-BE ͺ್॑ଲসब‫ٌغ‬ಕͲ͍ΖͪΌ , ‫ݎ‬ሺܽሻ ൌ െʹΝຮͪͤܽ͹஍Ͳब‫غ‬ഔ෾‫ز‬Νً͞͠͵͏[5].
ͳȀʹ-BE ͹յ఺෾‫ز‬Ν௒΄ΖͪΌ ܶ ଶ ΝୱҒͳ͢ͱब‫ͳغ‬յ఺਼Ν଎Ζ . յ఺਼ͳȀ͵ͳʹȀ͵͹ब‫ٌغ‬ಕ͗յ఺෾‫ͳ Ͳز‬Ȁʹ-BE ΓΕਫ਼ͣΖ. ͳȀʹ:ͳȀ͵-B (ਦ 5 Ͷ͕͜ΖͳȀʹǣ ͳȀ͵) ͺਜ਼৙յ఺෾
‫Ͳز‬ਫ਼ͣ, ྡֆ஍ͺ ܽ௖ ሺͳȀʹǣ ͳȀ͵ሻ ൌ ͶǤͷͳͺͷͲ͍Ζ͞ͳ਼͗ ஍‫Ͳ ࢋܯ‬ಚΔ Ηͪ . Ҳ๏, ͳȀʹ:ʹȀ͵-B
ͺҡ৙յ఺෾‫Ͳز‬ਫ਼ͣΖ . ྡֆ஍ͺܽ௖ ሺͳȀʹǣ ʹȀ͵ሻ ൌ ͷǤ͸͸ʹͲͲ͍Ζ(ਦ 5 Ͷ͕͜Ζ ͳȀʹǣ ʹȀ͵). ͞͹ब‫ ٌغ‬ಕͺ ܽ௖ ሺͳȀʹǣ ʹȀ͵ሻΓΕ ঙ͢ ঘ͠ ͵ ܽ͹஍ Ͳγ χϩ όʖ χ෾ ‫Ͳز‬ਫ਼ ͣΖ ͞ͳ Ν஭ қ͢ ͱ͕͚.

ਦ 5ʁͳȀʹ-BE ͹҈ఈ੓‫ݎ਼ܐ‬ሺܽሻ͹ܽғଚ੓. ͳȀʹ:ͳȀ͵ͳͳȀʹǣ ʹȀ͵͹қັͺຌชΝ‫ݡ‬Γ.

t 41 t࠹ॵͶ ͳȀʹ:ͳȀ͵-B Ͷͯ͏ͱ௒΄ ΖͪΌͶ , Ҳ॑ଲসब‫ٌ غ‬ಕͶଲͤΖ ܶ ଺ ͹Ծ‫ى‬͹ଲ ߻න‫ݳ‬
Νཤ༽ͤΖ.
ܶ ଺ ൌ ‫ ܶܩ ל ܪ‬ହ Ǥହӊଲ߻‫ ܶܩ‬Ͷ‫෈ͱؖ͢‬ร͵ॄ߻ͺ ܶହ‫ ܶܩ‬ሺܶ

ିଷ

ିଷ

ሺ17ሻ

ܵு Ͳ͍Ζ.

ܵு ሻ ൌ ‫ ܶܩ‬ଶ ܵு ൌ ܶ ିଶ ‫ܵܩ‬ு ൌ ܶ ିଶ ‫ܵܪܩ‬ு ൌ ܶ ିଷ ܵு Ǥሺ18ሻ͞ΗΓΕ, ͳȀʹ:ʹȀ͵-B ͺܵு ‫ି ܶ ת‬ଷ ܵு ് ‫׎‬Ͷٌಕ఼Ν࣍ͯ. ͞Ηͺܶ ଷ ܵு ‫ܵ ת‬ு ് ‫ͳ׎‬΍ॽ͜Ζ.
ଲ߻ͶΓΖ෾մͳ͢ͱܶ ଺ ൌ ܶ ହ݄ ‫݃ ל‬ሺ19ሻ΍Ն೵Ͳ͍Ζ. ଲ߻ܶ ହ ݄Ͷͯ͏ͱ෈ร͵ॄ߻ͺ ܶ ଷ ܵ௚ Ͳ͍Ζ. Γͮͱ, ܶ ଷ ܵ௚ ‫ܵ ת‬௚ ് ‫͗׎‬ಚΔΗΖ.
ब‫ٌغ‬ಕͳȀʹ:ͳȀ͵-BS ͗ਜ਼৙յ఺෾‫Ͳز‬ਫ਼ͣΖ༹ࢢΝਦ 6 Ͷࣖͪ͢. ଲসત͹଀ܶ ଷ ܵுି ͳଲসતܵுା ͹ި఼Ͷ ͳȀʹ-BE ͳͳȀʹ:ͳȀ͵-BS ͹ٌಕ఼ͺ͍Ζ. ͞͞Ͳ, ܵுା ͺ‫ ݕ‬ൌ ʹሺ‫ ݔ‬െ ߨሻሺ‫ ݕ‬൐ ͲሻͲ͍Ε,ܵுି ͺ‫ ݕ‬ൌ ʹሺ‫ ݔ‬െ ߨሻሺ‫ ݕ‬൏ ͲሻͲ͍Ζ. ਦ 6(a)Ͳͺ, ଀ܶ ଷ ܵுି ͹ͳȀʹ-BE ͹ٌಕ఼Ͷ͕͜Ζܑ͘ͺଲসતܵுା ͹ܑ͘ ΓΕঘ͠ ͏ . ܽΝଁՅͤΖ ͳ଀ ܶ ଷ ܵுି ͹ܑ͘ͺଁ Յ͢ (ਦ 6(b)), ଲসત ܵுା ͳંৰͤΖΓ͑Ͷ͵Ζ(ਦ 6(c)). ંৰ͹࢕๏ͺࢀ࣏‫਼ؖ‬ద Ͳ͍Ζ. ࢀ࣏‫਼ؖ‬దંৰͳͺ, ଀ ܶ ଷ ܵுି ͳଲস ત ܵுା ͹ ྈ ं ͹ ܑ ͘ ͳ ್ ֌ ඏ ෾ ‫ ͗ ਼ ܐ‬౵ ͢ ͚ , ࢀ ֌ ඏ ෾ ‫ ͗ ਼ ܐ‬ҡ ͵ Ζ ં ৰ Ν қ ັ ͤ Ζ . ਦ
6(c) Ͷ ͕ ͏ ͱ , ଀ ܶ ଷ ܵுି ͹ ࢀ ֌ ඏ ෾ ‫ ਼ ܐ‬ͺ ෝ Ͳ , ଲ স ત ܵுା ͹ ࢀ ֌ ඏ ෾ ‫ ਼ ܐ‬ͺ κ ϫ Ͳ ͍ Ζ . ͳȀʹ-BE ͹ ٌ ಕ ఼ Ͷ ͕ ͜ Ζ ଲ স ત ͹ ଀ ܶ ଷ ܵுି ͹ ܑ ͘ ͳ , ଲ স ત ܵுା ͹ ܑ ͘ ͗ ౵ ͢ ͚ ͵ Ζ ৖ ߻ ( ਦ
6(c))͗ਜ਼৙յ఺෾‫ز‬͹ྡֆ஍Ν༫͓Ζ . ਦ 6(d)Ͳͺ, ଀ܶ ଷ ܵுି ͳଲসત ܵுା ͳࢀ఼ͲިࠫͤΖ.
ͳȀʹ-BE ͹ ٌಕ ఼͖ Δ ਫ਼ͣ ್ͪ ఼͹ ҈ఈ ੓ Ν௒ ΄ͱ , ͞ΗΔ ͗ ͳ΍ Ͷγ χϩ ‫͍ Ͳܗ‬Ζ ͞ͳ ͗෾͖ͮͪ͹Ͳ, ͞ΗΔΝ ͳȀʹ:ͳȀ͵-BS ͳॽ͚.
ͳȀʹ-BE ͹ٌಕ఼͖Δଲসત ܵ௚ା ৏Ͷͺବԃ‫ٌܗ‬ಕ఼್͗ͯਫ਼ͣΖ͞ͳΝ஭қ͢ͱ͕͚ (ਦ 7).
͞ΗΔΝͳȀʹ:ͳȀ͵-BE ͳॽ͚.

ਦ 6ʁͳȀʹ:ͳȀ͵-BS ͺਜ਼৙յ఺෾‫Ͳز‬ਫ਼ͣΖ . (a) ܽ ൌ ͶǤ͵, (b) ܽ ൌ ͶǤͶͷ, (c) ܽ ൌ ͶǤͷͳͺͷ,(d) ܽ ൌ ͶǤ͸ͷ.
t 42 tਦ 7ʁͳȀʹ:ͳȀ͵-BE ͺਜ਼৙յ఺෾‫Ͳز‬ਫ਼ͣΖ. (a) ܽ ൌ ͶǤͷͳͺͷ, (b) ܽ ൌ ͶǤ͸ͷ.
ਜ਼ ৙ յ ఺ ෾ ‫ ز‬͹ ઴ ‫ ޛ‬Ͷ ͕ ͜ Ζ ϛ Π ϱ Ω Ϫ ࢨ ਼ [1]Ν ௒ ΄ Ζ . ͞ ͞ Ͳ ͺ ࣺ ଀ ͳ ͢ ͱ ܶ ସ Ν ཤ ༽
ͤΖ. ܶ ଶ Ͷ‫ͱؖ͢‬ବԃ‫͍Ͳܗ‬Ζ ͳȀʹ-BE ͺ, ܶ ସ Ͷ‫ͱؖ͢‬΍ବԃ‫͍Ͳܗ‬Ζ. ෾‫ز‬઴Ͳͺ, ͳȀʹ-BE͹ ٌ ಕ ఼ ͹ Ί Ͳ ͍ Ζ ͖ Δ ϛ Π ϱ Ω Ϫ ࢨ ਼ ͺ ሺ൅ͳሻ Ͳ ͍ Ζ . ෾ ‫ Ͳ ޛ ز‬ͺ , γ χ ϩ ‫ ٌ ܗ‬ಕ ఼(ͳȀʹ:ͳȀ͵-BS)ͳବԃ‫ٌܗ‬ಕ఼(ͳȀʹ:ͳȀ͵-BE)͗ͳ΍Ͷ 2 ‫ݺ‬ਫ਼ͣΖ . ͞ΗΔ͹ϛΠϱΩϪࢨ਼ͺܶ ସ Ͷ‫ͱؖ͢‬, ʹ ൈ ሺെͳሻͳʹ ൈ ሺ൅ͳሻͲ͍Ζ. ͞ΗΔ͹࿪ͺሺͲሻͲ͍Ζ. ϛΠϱΩϪࢨ਼ͺ෾‫ز‬͹઴‫Ͳޛ‬ฯଚ͠ΗΖ͖Δ, ͳȀʹ-BE ͹ٌಕ఼͹ϛΠϱΩϪࢨ਼ͺ ሺ൅ͳሻͲ͍Ζ. ਜ਼৙յ఺෾‫͗ز‬ਫ਼ͣͱ΍, ͳȀʹ-BE ͹ٌಕ఼͹҈ఈ੓ͺବԃ‫Ͳܗ‬รԿ͢͵͏͞ͳ͗಍͖Ηͪ .
͞͞Ͳߨ͵ͮͪ٠࿨Ν ͳȀʹ:ͳȀʹ-BS ͳͳȀʹ:ͳȀʹ-BE ͗ਫ਼ͣΖ৖߻Ͷͯ͏ͱ΍ߨ͓Ζ . ݃Վͳ͢ͱͳȀʹ:ͳȀʹ-B ͗ਫ਼ͣͱ΍ͳȀʹ-BE ͹ٌಕ఼͹҈ఈ੓ͺବԃ‫Ͳܗ‬รԿ͢͵͏͞ͳ͗ࣖͦΖ . ͞ΗΓΕ, ਦ 5 Ͳ҈ఈ੓‫ݎ ਼ܐ‬ሺܽሻ͗ሺെʹሻΓΕঘ͚͠͵Δ͵͏͞ͳ͗಍͖ΗΖ .
ਦ 6(c)ΓΕ, ਜ਼৙յ఺෾‫͗ز‬ਫ਼ͣΖͪΌ͹ড়݇͗ಚΔΗΖ . 1/2-BE ͹ٌಕ఼‫ݖ‬଴ Ͷ͕͜Ζଲস ત ܵு ͹ܑ ͘ Ν ߦ௚ ሺൌ ʹሻͳ͢, ್ ֌ ඏ ෾‫ ਼ ܐ‬Ν ߟ௚ ሺൌ Ͳሻͳ ͢, ࢀ֌ ඏ ෾ ‫ ਼ܐ‬Ν ߞ௚ ሺൌ Ͳሻͳ ͤ Ζ.
ಋ༹Ͷଲসત͹଀͹ܑ͘Ν ߦͳ͢, ್֌ඏ෾‫਼ܐ‬Ν ߟͳ͢, ࢀ֌ඏ෾‫਼ܐ‬Ν ߞͳͤΖ. ਦ 6(c)͖
Δߦ ൌ ߦ௚ , ߟ ൌ ߟ௚ ͗ಚΔΗΖ . ࢀ֌ඏ෾‫਼ܐ‬Ͷͯ͏ͱͺߞ ൏ Ͳሺൌ ߞ௚ ሻ͗੔Εཱིͯ. ͞͹Γ͑͵ড়݇͗੔Εཱི ͯ৖߻ , ਜ਼৙ յ఺෾‫ ͗ز‬ਫ਼ͣΖ . ‫͓׷ ͏ݶ‬Ζͳࢀ ࣏‫਼ؖ‬ద ંৰ য়‫ڱ‬͹ड़‫͗ ݳ‬ਜ਼৙յ఺෾‫͗ز‬ਫ਼ͣΖͪΌ͹ড়݇Ͳ͍Ζ .
࣏Ͷब‫ٌغ‬ಕ ͳȀʹ:ʹȀ͵-B ͺҡ৙յ఺෾‫Ͳز‬ਫ਼ͣΖ͞ͳΝઈ໎ͤΖ . ਦ 8 Ͷඵ͏ͪब‫ٌغ‬ಕͺͳȀʹ:ʹȀ͵-BE Ͳ್॑ଲসब‫ٌغ‬ಕͲ͍Ζ . ब‫ ݍغ‬ൌ ʹ ൈ ͵್͗॑ଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳΝߡྂ͢ͱଲ߻න‫ݳ‬Ν༫͓Ζ [5].
ܶ ଺ ൌ ܶ ଶ ‫ ܶܩ ל ܪ‬ଶ ݄ ‫݃ ל‬Ǥࠪଲ߻ܶ ଶ ‫ܪ‬Ͷ‫෈ͱؖ͢‬ร͵ॄ߻͗ ܶܵு Ͳ͍Ζ( ܶ ଶ ‫ܪ‬ሺܶܵு ሻ ൌ ܶ ଶ ܶ ିଵ ‫ܵܪ‬ு ൌ ܶܵு ).

t 43 tሺ20ሻਦ 8ʁͳȀʹ:ʹȀ͵-B ͺҡ৙յ఺෾‫Ͳز‬ਫ਼ͣΖ . (a) ܽ ൌ ͷǤͷͶ, (b)ܽ ൌ ͷǤͷͷͶ͹, (c) ܽ ൌ ͷǤͷ͸,(d) ܽ ൌ ͷǤ͸͸ʹͲ.
ਦ 8(a)Ͳͺ, ଲসત͹଀ ܶܵுି ͺଲসતܵ௚ା ͳިࠫ͢ͱ͏Ζ. ͞͹఼ͺͳȀʹ-BE ͹ٌಕ఼Ͳ͍Ζ.
଀ܶܵுି ͹ި఼Ͳ͹ܑ͘ͺଲসત ܵ௚ା ͹ܑ͘ΓΕঘ͠͏. ਦ 8(b)Ͳͺ, ଲসત͹଀ ܶܵுି ͗ଲসતܵ௚ା
ͳંৰ͢ͱ͏Ζ. ંৰ఼ͺͳȀʹ-BE ͹ٌಕ఼ΓΕ཯Ηͱ͏Ζ͞ͳͶ஭қ͢Γ͑ . ͞͹য়‫͗ڱ‬ҡ৙ յ ఺ ෾ ‫ ز‬͹ ྡ ֆ ఼ Ͳ ਫ਼ ͣ Ζ . ྡ ֆ ஍ Ν ܽ௖௦௡ ሺͳȀʹǣ ʹȀ͵ሻͳ ͤ Ζ . ં ৰ ఼ Γ Ε ବ ԃ ఼ ͳ γ χ ϩ ఼͗ਫ਼ͣΖ. Ϗϧϟʖνܽ͹ଁՅͶ൒͏γχϩ఼ͺ ͳȀʹ-BE ͹ٌಕ఼Ͷ޴͜ͱҢಊͤΖ (ਦ 8(c)).
ବԃ ఼ͺ ӊଈ Ͷ ཯Η ͱ͏ ͚ . γχ ϩ ఼ͳ ͳȀʹ-BE ͹ ٌ ಕ఼ ͳҲ ஗͢ ͪ য়‫͗ ڱ‬ਜ਼ ৙յ ఺෾ ‫ز‬͹ྡֆ఼Ͳ͍Ζ(ਦ 8(d)). ྡֆ஍Νܽ௖ ሺͳȀʹǣ ʹȀ͵ሻͳͤΖ. ਦ 8(c)͖Δਦ 8(c)ΉͲ͹ۢؔΝҡ৙յ఺ Ϗ ϧ ϟ ʖ ν ۢ ؔ ሾܽ௖௦௡ ሺͳȀʹǣ ʹȀ͵ሻǡ ܽ௖ ሺͳȀʹǣ ʹȀ͵ሻሻͳ ͤ Ζ . ϛ Π ϱ Ω Ϫ ࢨ ਼ Ν ཤ ༽ ͤ Ζ ͳ , ҡ ৙ յ఺෾ ‫ ͗ز‬ਫ਼ͣ ͱ ΍ ͳȀʹ-BE ͹ٌ ಕ఼ ͹҈ ఈ ੓ͺ ବԃ ‫ Ͳ ܗ‬รԿ ͢͵ ͏ ͞ͳ ͗಍ ͜Ζ . ৆ࡋ ͺ঴͚. ͳȀʹ-BE ͹ٌಕ఼(‫ݖ‬଴ ͳ‫ݖ‬ଵ )ͳ, ͳȀʹ:ʹȀ͵-BE ͹ٌಕ఼(‫ݓ‬௞ ǡ Ͳ ൑ ݇ ൑ ͷ)Νਦ 9 Ͷඵ͏ͪ͹ͲࢂߡͶ͢ͱཋ͢͏.

ਦ 9ʁͳȀʹ-BE ͹ٌಕ఼(‫ݖ‬଴ , ‫ݖ‬ଵ )ͳ, ͳȀʹ:ʹȀ͵-BE ͹ٌಕ఼(‫ݓ‬௞ ,Ͳ ൑ ݇ ൑ ͷ). ܽ ൌ ͷǤ͹.
t 44 tਦ 8(d)ΓΕ, ҡ৙յ఺෾‫͗ز‬ਫ਼ͣΖͪΌ͹ড়݇͗ಚΔΗΖ . ਜ਼৙յ఺෾‫͗ز‬ਫ਼ͣΖͪΌ͹ড়݇ Ν ಍͏ ͪ ͠͏ Ͷ࢘ ༽ ͪ͢ ‫ߺ ى‬Ν ࠸ ౕཤ ༽ͤ Ζ . ਦ 8(d)͖Δ ߦ ൌ ߦ௚ ͗ಚ ΔΗ Ζ . ٌ ಕ ఼ ‫ݖ‬଴ΓΕӊ৏๏Ͷ್ͯ͹‫ۄ‬તΝ௧͑ͳ , ଲসત͗ଲসત͹଀ΓΕଐ͚ӊͶ‫͗ۄ‬Ζ͖Δ , ߟ௚ ൏ ߟ ൏ Ͳ͗ಚ ΔΗ Ζ . ͞͹Γ ͑͵ ড়݇ Νຮ ͪͤ ͳ ͳȀʹ-BE ͹ ٌಕ ఼Γ Ε཯ Η ͪҒ ஖Ͳ ್࣏ ‫ ਼ؖ‬దં ৰ
͗ਫ਼ͣΖ. ॶҨ, γχϩόʖχ෾‫͗ز‬ਫ਼ͣΖ .
͞͞ΉͲͲಚΔΗͪ൓ఈড়݇Ν൓ఈড়݇ 3.1 ͳͤΖ.
൓ఈড়݇ 3.1. ଲসતͳଲসત͹଀͹ި఼ ‫ݖ‬௞ ͺବԃ‫ܗ‬ब‫ٌغ‬ಕ͹ٌಕ఼Ͳ͍ΖͳͤΖ .
(a) ਜ਼৙յ఺෾‫͗ز‬ਫ਼ͣΖড়݇ : ٌಕ఼‫ݖ‬௞ Ͳ, ߦ ൌ ߦ௚ , ߟ ൌ ߟ௚ ൌ Ͳͳߞ ൏ ߞ௚ ൌ Ͳ ͗੔Εཱིͯ.
(b) ҡ৙յ఺෾‫͗ز‬ਫ਼ͣΖড়݇ : ٌಕ఼‫ݖ‬௞ Ͳ, ߦ ൌ ߦ௚ ͳߟ௚ ൏ ߟ ൏ Ͳ͗੔Εཱིͯ.
ଲসતͳଲসત͹଀͹‫ش‬Ձָద͵ഓ஖Ν΍ͳͶҡ৙յ఺෾‫͗ز‬ਫ਼ͣΖড়݇Νড়݇ 2.1 ͳ͢
ͪ. ড়݇ 2.1 Ͳଲসતͳଲসત͹଀͹ ͳȀʹ-BE ͹ٌಕ఼Ͷ͕͜Ζ‫ི͗ۄ‬ෝ͹৖߻Ͷ͕͏ͱ , ড়
݇ 2.1 Ν͠ΔͶ੠ືԿͪ͢ড়݇͗൓ఈড়݇ 3.1 Ͳ͍Ζ.
൓ఈড়݇ 3.1 Νཤ༽͢ͱ, ‫݌‬Ȁ‫ݍ‬ǣ ͳȀ͵-B ͳ‫݌‬Ȁ‫ݍ‬ǣ ʹȀ͵-B ͹ਫ਼ͣ๏Ν਼஍‫Ͳࢋܯ‬௒΄ͪ . ͏͚͖ͯ͹‫݌‬Ȁ‫ݍ‬Ͷͯ͏ͱಚΔΗͪ݃ՎΝන 1 ͳන 2 Ͷࣖͤ. න 1 Ͷͺਜ਼৙յ఺෾‫͗ز‬ਫ਼ͣΖ৖߻Νࣖ͢ ,න 2 Ͷͺҡ৙յ఺෾‫͗ز‬ਫ਼ͣΖ৖߻Νࣖͪ͢ . නͶ͕͜ΖҲ॑ͺ, ਫ਼ͣͪब‫ٌغ‬ಕ͗Ҳ॑ଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳΝқັ͢ͱ͏Ζ . ್॑Ͷͯ͏ͱ΍ಋ༹͹қັͲ͍Ζ .
න 1ʁਜ਼৙յ఺෾‫ز‬͹ྭͳȀʹ:ͳȀ͵-B (Ҳ॑) ͳȀͺ:ͳȀ͵-B (Ҳ॑)͵ȀͳͲ:ͳȀ͵-B (Ҳ॑)ͳȀͶ:ͳȀ͵-B (Ҳ॑) ͵Ȁͺ:ͳȀ͵-B (Ҳ॑)ͳȀͳʹ:ͳȀ͵-B (Ҳ॑)ͳȀ͸:ͳȀ͵-B (Ҳ॑) ͳȀͳͲ:ͳȀ͵-B (Ҳ॑) ͷȀͳʹ:ͳȀ͵-B (Ҳ॑)න 2ʁҡ৙յ఺෾‫ز‬͹ྭͳȀʹ:ʹȀ͵-B (್॑) ͳȀ͸:ʹȀ͵-B (್॑) ͵Ȁͺ:ʹȀ͵-B (್॑)ͳȀ͵:ͳȀ͵-B (Ҳ॑) ͳȀ͹:ͳȀ͵-B (Ҳ॑) ͳȀͻ:ͳȀ͵-B (Ҳ॑)ͳȀͶ:ʹȀ͵-B (್॑) ʹȀ͹:ͳȀ͵-B (Ҳ॑) ͶȀͻ:ͳȀ͵-B (Ҳ॑)ͳȀͷ:ͳȀ͵-B (Ҳ॑) ͵Ȁ͹:ͳȀ͵-B (Ҳ॑) ͳȀͳͲ:ʹȀ͵-B (್॑)ʹȀͷ:ͳȀ͵-B (Ҳ॑) ͳȀͺ:ʹȀ͵-B (್॑) ͵ȀͳͲ:ʹȀ͵-B (್॑)
න 1 ͳන 2 Ν΍ͳͶ༩૟ 3.2 ͗ಚΔΗΖ.
༩૟ 3.2.
[i] ब‫͗ݍغ‬۰਼͹৖߻: Ҳ॑ଲসब‫ٌغ‬ಕ ‫݌‬Ȁ‫ݍ‬:ͳȀ͵-B ͺ ਜ਼৙յ఺෾‫Ͳز‬ਫ਼ͣ , ್॑ଲসब‫غ‬ٌಕ‫݌‬Ȁ‫ݍ‬:ʹȀ͵-B ͺҡ৙յ఺෾‫Ͳز‬ਫ਼ͣΖ .
[ii] ब‫਼د͗ݍغ‬͹৖߻: Ҳ॑ଲসब‫ٌغ‬ಕ ‫݌‬Ȁ‫ݍ‬:ͳȀ͵-B ͺҡ৙յ఺෾‫Ͳز‬ਫ਼ͣΖ .

t 45 t4. ఈཀྵ͹ৄ໎
༩૟ 3.2 Ͷ͕͜Ζ‫݌‬Ȁ‫ ݍ‬ൌ ͳȀʹ͹৖߻Ͷ‫ͤؖ‬Ζఈཀྵ 4.1 Νৄ໎ͤΖ͞ͳ͗ຌઇ͹क໪దͲ͍Ζ .
ఈཀྵ 4.1.
[1] ್॑ଲসब‫ٌغ‬ಕ ͳȀʹ:ʹȀ͵-B ͺ, ͳȀʹ-BE ͹ҡ৙յ఺෾‫Ͳز‬ਫ਼ͣΖ.
[2] Ҳ॑ଲসब‫ٌغ‬ಕ ͳȀʹ:ͳȀ͵-B ͺ, ͳȀʹ-BE ͹ਜ਼৙յ఺෾‫Ͳز‬ਫ਼ͣΖ.
࠹ॵͶ, ఈཀྵ 4.1 ͹ৄ໎Ͳචགྷͳ͵Ζ੓࣯ 4.2 ͳ 4.3 Ν়ղͤΖ. ࣏Ͷໍୌ 4.4 ͳໍୌ 4.5
Ν়ղ͢ৄ໎Ν༫͓Ζ. ੓࣯ 4.2 Ͷͯ͏ͱͺࢂߡช‫[ݛ‬1, 2]Ν, ੓࣯ 4.3 Ͷͯ͏ͱͺࢂߡช‫ݛ‬[5]Ν‫ݡ‬ΔΗͪ͏.
੓࣯ 4.2. ବԃ‫෈ܗ‬ಊ఼ܳ͗յ఺෾‫ز‬Νً͢͞ͱਫ਼ͣͪ ‫݌‬Ȁ‫ݍ‬-B ͗ܳ͹बΕΝ࣎‫ܯ‬յΕͶյ఺ͤΖ͵Δͻ, ‫݌‬Ȁ‫ݍ‬-B ͖Δյ఺෾‫Ͳز‬ਫ਼ͣͪ ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-B ͺ‫݌‬Ȁ‫ݍ‬-BE ͹ٌಕ఼͹बΕΝൕ࣎‫ܯ‬յΕͶյ఺ͤΖ.
ඬ६ࣺ ଀Ͷ͕ ͏ͱ , ‫݌‬Ȁ‫ݍ‬-B ͺବԃ ‫෈ܗ‬ಊ ఼ ܳ͹बΕΝ ࣎‫ܯ‬յ ΕͶյ ఺ͤΖ ͹Ͳ , ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-Bͺ‫݌‬Ȁ‫ݍ‬-BE ͹ٌಕ఼͹बΕΝൕ࣎‫ܯ‬յΕͶյ఺ͤΖ .
੓࣯ 4.3. ඬ६ࣺ଀Ͷ͕͏ͱ, ବԃ‫෈ܗ‬ಊ఼ ܳ͹յ఺෾‫Ͳز‬ਫ਼ͣͪ ‫݌‬Ȁ‫ݍ‬-B ͺԾ‫ى‬͹(i)ͳ(ii)Ν
ຮͪͤ.
(i) ‫਼د͗ݍ‬. ‫݌‬Ȁ‫ݍ‬-B ͺҲ॑ଲসब‫ٌغ‬ಕ. ‫݌‬Ȁ‫ݍ‬-BE ͹յ఺෾‫ز‬͹յ఺਼ۢؔͺ ሺͲǡͳȀʹሻͲ͍Ζ.
յ఺෾‫Ͳز‬ਫ਼ͣͪब‫ٌغ‬ಕΝ ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-B Νॽ͚ͳ, Ͳ ൏ ‫ݎ‬Ȁ‫ ݏ‬൏ ͳȀʹ͗੔Εཱིͯ. ‫݌‬Ȁ‫ݍ‬-BE ͺब‫غ‬ഔ෾‫ز‬Νًͤ͞(ii) ‫͗ݍ‬۰਼. ‫݌‬Ȁ‫ݍ‬-B ͺ್॑ଲসब‫ٌغ‬ಕ. ‫݌‬Ȁ‫ݍ‬-BE ͹յ఺෾‫ز‬͹յ఺਼ۢؔͺ ሺͲǡͳሻͲ͍Ζ. յ఺෾‫Ͳز‬ਫ਼ͣͪब‫ٌغ‬ಕΝ ‫݌‬Ȁ‫ݍ‬:‫ݎ‬Ȁ‫ݏ‬-B Νॽ͚ͳ, Ͳ ൏ ‫ݎ‬Ȁ‫ ݏ‬൏ ͳ͗੔Εཱིͯ. ‫݌‬Ȁ‫ݍ‬-BE ͺಋब‫غ‬෾‫ز‬Νًͤ͞.
ໍୌ 4.4. ‫ݍ‬Ν‫͢ͳ਼د‬, ‫݌‬Ȁ‫ݍ‬-BE ͗յ఺෾‫ز‬Νً͢͞ͱब‫ ݍ غ‬ൈ ͵͹ब‫ٌغ‬ಕ͗ਫ਼ͣͪͳͤΖ .
ब‫ ݍغ‬ൈ ͵͹ब‫ٌغ‬ಕͺ, Ҳ॑ଲসब‫ٌغ‬ಕͲ ‫݌‬Ȁ‫ݍ‬:ͳȀ͵-B Ͳ͍Ζ.
ৄ໎. ब‫਼د͗ݍغ‬͹৖߻, ‫݌‬Ȁ‫ݍ‬-BE ͺҲ॑ଲসब‫ٌغ‬ಕͲ͍Ζ͖Δյ఺෾‫ز‬͹յ఺਼ۢؔͺ
੓࣯ 4.3(i)ΓΕሺͲǡͳȀʹሻͲ͍Ζ. Γͮͱ, յ఺਼ͺචͥ ‫݌‬Ȁ‫ݍ‬:ͳȀ͵Ͷ͵Ζ.
࣏Ͷ, Ҳ॑ଲসब‫ٌغ‬ಕ͖Δյ఺෾‫Ͳز‬ਫ਼ͣΖब‫ٌغ‬ಕͺҲ॑ଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳΝৄ ໎ ͤ Ζ. ٌ ಕ ఼ ‫ݖ‬଴ Ν Ҳ ॑ ଲ স ब ‫ ٌ غ‬ಕ ͹ ٌ ಕ ఼ ͳ ͢ , ٌ ಕ ఼ ‫ݖ‬଴ Ν ॵ ‫ ͤ ͳ ఼ غ‬Ζ ब ‫ ٌ غ‬ಕ Νܱሺ‫ݖ‬଴ ሻͳॽ͚.
͞͞Ͳͺ ܶ ௤ ൌ ‫ ܶܩ ל ܪ‬௤ିଵ ͳ෾ׄͤΖ. ‫ܩ‬෠ ൌ ‫ ܶܩ‬௤ିଵ ͳͤΖ. ଲ߻‫ܪ‬Ͷͯ͏ͱ෈ร͵ॄ߻Ͳ͍Ζଲসતͳ͢ͱ‫ ݕ‬ൌ ʹሺ‫ ݔ‬െ ߨሻΝમ΁. ͞ΗΝܵு ሺߨሻͳॽ͚. ଲ߻‫ܩ‬෠ Ͷͯ͏ͱ ෈ร͵ॄ߻Ͳ͍ΖଲসતΝܵீ෠ ͳॽ͚. ٌಕ఼‫ݖ‬଴ ͺ͞ΗΔ͹ި఼Ͳ͍Ζ. ͞͹ި఼Ͷͺ৏Ծ͹ଲসતͺ௪գ͢ͱ͏͵ ͏ .
Γͮͱٌಕ఼ ‫ݖ‬଴ ͖Δյ఺෾‫Ͳز‬ਫ਼ͣͪब‫ٌغ‬ಕ್͗॑ଲসब‫ٌغ‬ಕͶ͵Ζ͞ͳͺ͵͏ .
࣏Ͷଲসત৏Ͷ͍Ζब‫ٌغ‬ಕ఼ ‫ݖ‬଴ ͖Δյ఺෾‫Ͳز‬ਫ਼ͣͪब‫ٌغ‬ಕ͗ඉଲসब‫ٌغ‬ಕͲ͍Ζͳ͢ͱ ໅८ Ν಍ ͚ . ٌಕ ఼ ‫ݖ‬଴ ͖Δ ਦ 10(a)Ͳࣖ͢ ͪ ༹Ͷब ‫ غ‬3(ܶ ௤ Νୱ Ғͳ ͢ ͱ଎ͮ ͪब ‫) غ‬͹ඉଲসγ χϩ‫ܗ‬ब ‫ ٌغ‬ಕ఼ ሼ‫ݏ‬଴ ǡ ‫ݏ‬ଵ ǡ ‫ݏ‬ଶ ሽ͗ਫ਼ͣͪͳͤΖ. ଲস੓ΓΕඉ ଲসγχ ϩ‫ ܗ‬ब‫ٌغ‬ಕ
t 46 t఼ሼ‫ݐ‬଴ ǡ ‫ݐ‬ଵ ǡ ‫ݐ‬ଶ ሽ΍ਫ਼ͣΖ͗, ܱሺ‫ݏ‬଴ ሻ ് ܱሺ‫ݐ‬଴ ሻͲ͍Ζ. ਦ 10(a)ͲͺҐԾ͹‫͗ܐؖ‬੔Εཱིͯ .
‫ݐ‬଴ ൌ ‫ݏܪ‬଴ ǡ ‫ݐ‬ଶ ൌ ‫ݏܪ‬ଵ ǡ‫ݐ‬ଵ ൌ ‫ݏܪ‬ଶ ǡሺ21ሻ‫ݐ‬ଵ ൌ ‫ܩ‬෠ ‫ݏ‬଴ ǡ‫ݐ‬଴ ൌ ‫ܩ‬෠ ‫ݏ‬ଵ ǡ‫ݐ‬ଶ ൌ ‫ܩ‬෠ ‫ݏ‬ଶ Ǥሺ22ሻ

ਦ 10(a)Ͷ͕͏ͱ, ࣰત͹໾һ‫ߺى‬ͺଲ߻ ‫ܪ‬ͶΓΖ఼͹ҢಊΝන͢, ఼ત͹໾һ‫ߺى‬ͺଲ߻ ‫ܩ‬෠Ͷ఼͹ಊ͘Νන͢ͱ͏Ζ . ਦ 10(a)Ͳ఼તͲҕΞͫวΕΝ֨୉͢ , ඉଲসγχϩ‫ٌܗ‬ಕ఼ ‫ݏ‬଴ ͳ‫ݐ‬଴ ͹෈҈ ఈଡ༹ ରͳ҈ ఈଡ ༹ରΝ ਦ 10(b)Ͷඵ͏ͪ . ෈҈ఈଡ༹ ର͹‫ ҈ͳ ނ‬ఈଡ༹ ର͹‫ͳ ނ‬Νཤ༽͢ͱ྘Ү ܼሺ‫ݏ‬଴ ǡ ‫ݐ‬଴ ሻ (ਦ 10(b)Ͷ͕͜Ζօ৯྘ Ү )Νߑ੔Ͳ͘Ζ. ‫ܼߺى‬ሺ‫ݏ‬଴ ǡ ‫ݐ‬଴ ሻͺྣΕ ߻ͮͪඉଲসγχϩ‫ٌܗ‬ಕ఼ ‫ݏ‬଴ ͳ‫ݐ‬଴ ͹෈҈ఈଡ༹ରͳ҈ఈଡ༹ରΝཤ༽͢ͱߑ੔ͪ͢‫ڠ‬໒྘Ү [1]Νන͢ͱ͏Ζ. ಋ༹Ͷ͢ͱྣΕ߻ͮͪඉଲসγχϩ‫ٌܗ‬ಕ఼͹෈҈ఈଡ༹ରͳ҈ఈଡ༹ରΝཤ༽͢ͱ߻‫ݺ̔ܯ‬͹‫ڠ‬໒྘Ү͗ߑ੔͠ΗΖ . ܼሺ‫ݏ‬଴ ǡ ‫ݐ‬଴ ሻ͹஦Ͷͺ ‫ݖ‬଴ ΓΕਫ਼ͣͪඉଲসବԃ‫ܗ‬ब‫غ‬ٌ ಕ ఼ ߙ଴ ͗ ଚ ࡑ ͢ ͱ ͏ Ζ ͺ ͥ Ͳ ͍ Ζ . ଲ স ੓ ͖ Δ ߚ଴ ൌ ‫ߙܪ‬଴ ΍ ଚ ࡑ ͢ ͱ ͏ Ζ . ٌ ಕ ఼ ߙ଴ ͳ ߚ଴ ͗ͳ΍Ͷඉଲসब‫ٌغ‬ಕ఼ Ͳ͍Ζ͖Δ ߙ଴ ് ߚ଴ ͗੔Εཱིͯ. ݃Վͳ͢ͱ, Ҳͯ͹‫ڠ‬໒྘ҮͶ್ͯ͹ବԃ‫ܗ‬ब‫ٌغ‬ಕ͗‫؜‬ΉΗΖ . ବԃ‫ܗ‬ब‫ٌغ‬ಕ͹‫਼͗ݺ‬γχϩ‫ܗ‬ब ‫ٌغ‬ಕ͹‫਼ݺ‬͹್ഔͳ͵Ε, ϛΠϱΩϪࢨ਼͹ฯଚଉͶൕ͢໅८Ͳ͍Ζ . Ґ৏Ͳໍୌ 4.4 ͹क௃͗ࣖ͠Ηͪ.

ਦ 10ʁ(a) ଲসતܵு ሺߨሻ৏Ͷ͍Ζब‫ٌغ‬ಕ఼ ‫ݖ‬଴ ͖Δਫ਼ͣͪඉଲসγχϩ‫ܗ‬ब‫ٌغ‬ಕ఼ . (b) ‫ڠ‬໒྘Үܼሺ‫ݏ‬଴ ǡ ‫ݐ‬଴ ሻ͹ߑ੔.
ໍୌ 4.5. ‫ݍ‬Ν۰਼ͳ͢, ‫݌‬Ȁ‫ݍ‬-BE ͗յ఺෾‫ز‬Νً͢͞ͱब‫ ݍ غ‬ൈ ͵͹ब‫ٌغ‬ಕ͗ਫ਼ͣͪͳͤΖ .
ब‫ ݍغ‬ൈ ͵͹ब‫ٌغ‬ಕͺ, ್॑ଲসब‫ٌغ‬ಕͲ ‫݌‬Ȁ‫ݍ‬:ʹȀ͵-B Ͳ͍Ζ.
ৄ໎͹๏ਓ. ‫͗ݍ‬۰਼Ͳ͍Ζ͖Δ, ‫݌‬Ȁ‫ݍ‬-BE ͺ್॑ଲসब‫ٌغ‬ಕͲ͍Ζ . ‫݌‬Ȁ‫ݍ‬-BE ͗, ଲসતܵ௚
৏Ͷٌಕ఼Ν࣍ͯ৖߻(ίʖη 1)ͳ, ଲসતܵ௛ ৏Ͷٌಕ఼Ν࣍ͯ৖߻(ίʖη 2)Ͷ෾͜ͱৄ໎
ͤΖ. ͨΗͩΗ͹৖߻ΝߍͶ ‫ ݍ‬ൌ Ͷ݊͹৖߻([i])ͳ‫ ݍ‬ൌ Ͷ݊ െ ʹ͹৖߻([ii])Ͷ෾͜Ζ.
ίʖη 1 ͹ৄ໎. ‫݌‬Ȁ‫ݍ‬-BE ͗ଲসતܵ௚ ৏Ͷٌಕ఼Ν࣍ͯ৖߻.
[i] ‫ ݍ‬ൌ Ͷ݊ሺ݊ ൒ ͳሻ͹৖߻(ଲসતܵீ (‫ ݔ‬ൌ ߨ)৏Ͷٌಕ఼Ν࣍ͯ৖߻).
ܶ ௤ ͹ଲ߻න‫ݳ‬Νࣖͤ[5].
ܶ ସ௡ ൌ ܶ ଶ௡ିଵ ‫ ܶܩ ל ܪ‬ଶ௡ିଵ ݄ ‫݃ ל‬Ǥ
ࠪଲ߻ ܶ

ଶ௡ିଵ

௡

ሺ23ሻ
௡

‫ܪ‬Ͷͯ͏ͱ෈ร͵ॄ ߻ͺ ܶ ܵீ Ͳ͍Ζ. ‫݌‬Ȁ‫ݍ‬-BE ͹ٌಕ ఼ͺ ܶ ܵீ ͳ ܵ௚ ͹ި఼Ͷ͍Ζ.

͞ΗΝ‫ݖ‬௡ ͳͤΖ. ॵ‫ݖ఼غ‬଴ ͺܵீ ৏Ͷ͍Ζ. ‫ݖ‬௡ ΓΕܶ ௤ ΝୱҒͳ͢ͱब‫ غ‬3 ͗ਫ਼ͣΖ. ଲসતܵ௚ ৏Ͷ ਫ਼ ͣ ͪ ٌ ಕ ఼ Ν ‫ݓ‬௡ ͳ ͤ Ζ . ٌ ಕ ఼ ‫ݖ‬௡ Γ Ε , ܵ௚ ৏ Ͷ ͵ ͏ ٌ ಕ ఼ ߙ ൌ ܶ ଶ௡ିଵ ‫ݓܪ‬௡ ͳ ߚ ൌ ݃ߙ͗ ਫ਼ ͣ
Ζ. ٌಕ఼‫ݓ‬௡ , ߙ, ߚͲब‫ غ‬3 ͗ߑ੔͠ΗΖ͖Δ, ਦ 11(a)͹ഓ஖Ͳͺ ߚ͗ܶ ௡ ܵீ ৏Ͷ͍Ζ.
t 47 tߙ ൌ ܶ ଼௡ ‫ݓ‬௡ ͳߚ ൌ ܶ ସ௡ ‫ݓ‬௡ ͗੔Εཱིͯ͞ͳΝ಍͚. ͞͞Ͳ, ߙ ൌ ܶ ସ௡ ‫ݓ‬௡ ͳՀఈ͢໅८Ν಍͚.
ߙ ൌ ܶ ସ௡ ‫ݓ‬௡ ൌ ܶ ସ௡ିଵ ݄ ‫ି ܶܪ ל‬ሺଶ௡ିଵሻ ‫ ܶ ל‬ሺଶ௡ିଵሻ ‫ݓ݃ ל ܪ‬௡ Ǥ࠹ॵ Ͷ ‫ݓ݃ ࣞܐ ؖ‬௡ ൌ ‫ݓ‬௡ ͳ ߙ ൌ ܶଶ௡ିଵ‫ݓܪ‬௡ Ν ཤ༽ ͢ , ࣏Ͷ ܴ ൌ ݄‫ܴܶ ͳܪ‬ିሺଶ௡ିଵሻ

ሺ24ሻൌܶିሺଶ௡ିଵሻ

ܴΝཤ ༽ ͤΖͳߙ ൌ ܶ ସ௡ିଵ ݄ ‫ି ܶܪ ל‬ሺଶ௡ିଵሻ ‫ ߙ ל‬ൌ ܶ ଶ௡ ܴߙǤሺ25ሻ

ܴͺ 180 ౕյ఺ࡠ༽Ͳ͍Ζ . ͞͞Ͳߡ͓ͱ͏Ζٌಕ఼‫ݓ‬௡ ͹ब‫ٌغ‬ಕ͹ब‫غ‬ͺ ͳʹ݊Ͳ͍Ζ. Γͮͱ, ܴ ൌ ܶ േ଺௡ . ܴ ൌ ܶ ି଺௡ Νࡀ༽ͤΖͳ, ߙ ൌ ܶ ିସ௡ ߙ͗ಚΔΗΖ. ब‫ٌغ‬ಕܱሺߙሻ͹ब‫غ‬ͺ, ܱሺ‫ݖ‬௡ ሻ͹ब‫ͳغ‬ಋͣͶ͵Ε, ໅८Ͳ͍Ζ. Γͮͱ, ߚ ൌ ܶ ସ௡ ‫ݓ‬௡ ͳߙ ൌ ܶ ଼௡ ‫ݓ‬௡ ͗੔Εཱིͯ.

ਦ 11 ʁ (a) ‫ ݍ‬ൌ Ͷ݊ሺ݊ ൒ ͳሻ ͹ ৖ ߻ ͹ ഓ ஖ ( ܵீ : ‫ ݔ‬ൌ ߨ ). (b) ‫ ݍ‬ൌ Ͷ݊ െ ʹሺ݊ ൒ ͳሻ ͹ ৖ ߻ ͹ ഓ ஖(ܵு ǣ ‫ ݕ‬ൌ ʹሺ‫ ݔ‬െ ߨሻ).
͞͞Ͳ੓࣯ 4.2 Νཤ༽ͤΖ. ٌಕ఼‫ݓ‬௡ ͖Δड़൅͢‫ݖ‬௡ ͹बΕΝൕ࣎‫ܯ‬յΕͶյ఺͢ , ࠹ॵͶߚͶ ୣ ͢, ࣏ Ͷ ߙͶ౺ ஥ ͢ , ࠹ ‫ ޛ‬Ͷ ‫ݓ‬௡ Ͷ ໯ Ζ. ͞ ͹ؔ Ͷ , ͞ ͹ ब ‫ ٌغ‬ಕ ͺ ٌ ಕ ఼ ‫ݖ‬௡ ͹ ब Ε Ν 2յ఺͢ͱ͏Ζ͞ͳ͗಍͖ΗΖ . ͯΉΕ, ͞͹ब‫ٌغ‬ಕ͹յ఺਼ͺ ʹȀ͵Ͳ͍Ζ.
࣏Ͷյ఺෾‫Ͳز‬ਫ਼ͣ ͪब ‫ٌغ‬ಕ͗ ‫݌‬Ȁ‫ݍ‬:ʹȀ͵-B ͳ‫ى‬फ़͠Η Ζ͵Δͻ , ͞͹ब‫ٌغ‬ಕ ್͗॑ଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳ Ν ࣖͤ . յ఺਼͗ʹȀ͵Ͳ͍Ζ͞ ͳ͖Δ ߚ ൌ ܶ ସ௡ ‫ݓ‬௡ ͳ ߙ ൌ ܶ ସ௡ ߚ͗੔Εཱིͯ.
Ή ͪ , ߚ ൌ ݃ܶ ଶ௡ିଵ ‫ݓܪ‬௡ (ߚ ൌ ݃ߙ, ߙ ൌ ܶ ଶ௡ିଵ ‫ݓܪ‬௡ )΍ ੔ Ε ཱི ͯ . ߚ ൌ ܶ ସ௡ ‫ݓ‬௡ ͹ ྈ ว Ͷ ܶ ଶ௡ିଵ ‫ܪ‬Ν ࡠ ༽
ͤΖ. ӊวΝԾ‫ى‬͹Γ͑Ͷॽ͘‫͓׷‬Ζ .
ܶ ଶ௡ିଵ ‫ ܶܪ‬ସ௡ ‫ݓ‬௡ ൌ ܶ ଶ௡ିଵ ܶ ିସ௡ ‫ݓܪ‬௡ ൌ ܶ ିଶ௡ିଵ ‫ି ܶ ל‬ሺଶ௡ିଵሻ ݃ ‫ ܶ݃ ל‬ଶ௡ିଵ ‫ݓܪ‬௡ൌ ܶ ିସ௡ ݃ߚ ൌ ܶ ିସ௡ ߙ ൌ ߚǤ(26)Ґ৏ Ͳ ߚ ൌ ܶ ଶ௡ିଵ ‫͠ ࣖ͗ߚܪ‬Η ͪ . ͞ Η ͺ ߚ͗ଲস ત ܶ ௡ ܵீ ৏Ͷ ͍Ζ ͞ ͳ Νқ ັͤ Ζ . ͯΉΕ ್॑ଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳ͗ࣖ͠Ηͪ .
[ii] ‫ ݍ‬ൌ Ͷ݊ െ ʹሺ݊ ൒ ͳሻ͹৖߻(ଲসતܵு (‫ ݕ‬ൌ ʹሺ‫ ݔ‬െ ߨሻ)৏Ͷٌಕ఼Ν࣍ͯ৖߻).
ܶ ௤ ͹ଲ߻න‫ݳ‬Νࣖͤ.
ܶ ସ௡ିଶ ൌ ܶ ଶ௡ିଶ ‫ ܶܩ ל ܪ‬ଶ௡ିଶ ݄ ‫݃ ל‬Ǥ
ࠪଲ߻ܶ

ଶ௡ିଶ

‫ܪ‬Ͷͯ͏ͱ෈ร͵ॄ߻ͺ ܶ

௡ିଵ

(27)

ܵு Ͳ͍Ζ. ‫݌‬Ȁ‫ݍ‬-BE ͹ٌಕ఼ͺ ܶ௡ିଵܵு ͳܵ௚ ͹ި఼Ͷ͍௤Ζ. ͞ΗΝ‫ݖ‬௡ିଵ ͳͤΖ. ॵ‫ݖ఼غ‬଴ ͺܵு ৏Ͷ͍Ζ. ٌಕ఼‫ݖ‬௡ିଵ ΓΕܶ ΝୱҒͳ͢ͱब‫ غ‬3 ͗ਫ਼ͣ
Ζ . ଲ স ત ܵ௚ ৏ Ͷ ਫ਼ ͣ ͪ ٌ ಕ ఼ Ν ‫ݓ‬௡ିଵ ͳ ͤ Ζ . ٌ ಕ ఼ ‫ݖ‬௡ିଵ Γ Ε , ܵ௚ ৏ Ͷ ͵ ͏ ٌ ಕ ఼ ߙ ൌܶ ଶ௡ିଵ ‫ݓܪ‬௡ିଵ ͳߚ ൌ ݃ߙ͗ਫ਼ͣΖ. ٌಕ఼ߚͺ, ਦ 11(b)͹ഓ஖Ͳͺ ܶ ௡ିଵ ܵு ৏Ͷ͍Ζ.
͞͞Ͳ, ߙ ൌ ܶ ସ௡ିଶ ‫ݓ‬௡ ͳՀఈ͢໅८Ν಍͚.
ߙ ൌ ܶ ସ௡ିଶ ‫ݓ‬௡ିଵ ൌ ܶ ସ௡ିଷ ݄ ‫ି ܶܪ ל‬ሺଶ௡ିଶሻ ‫ ܶ ל‬ሺଶ௡ିଶሻ ‫ݓ݃ ל ܪ‬௡ Ǥ‫ݓ݃ࣞܐؖ‬௡ ൌ ‫ݓ‬௡ , ߙ ൌ ܶଶ௡ିଶ‫ݓܪ‬௡ , ܴ ൌ ݄‫ ܪ‬Νཤ༽ͤΖͳ
t 48 t(28)ߙ ൌ ܶ ସ௡ିଷ ݄ ‫ି ܶܪ ל‬ሺଶ௡ିଶሻ ‫ ߙ ל‬ൌ ܶ ଶ௡ିଵ ܴߙǤ(29)͞ ͞ Ͳ ߡ ͓ ͱ ͏ Ζ ‫ݓ‬௡ ͹ ब ‫ ٌ غ‬ಕ ͹ ब ‫ غ‬ͺ ͳʹ݊ െ ͸Ͳ ͍ Ζ . Γ ͮ ͱ , ܴ ൌ ܶࡀ༽ͤΖͳ, ߙ ൌ ܶିସ௡ାଶേሺ଺௡ିଷሻ

. ܴൌܶ

ି଺௡ାଷ

Ν

ߙ͗ ಚ Δ Η Ζ . ब ‫ ٌ غ‬ಕ ܱሺߙሻ͹ ब ‫ غ‬ͺ , ܱሺ‫ݖ‬௡ିଵ ሻ͹ ब ‫ ͳ غ‬ಋ ͣ Ͷ ͵ Ε ,໅८Ͳ͍Ζ. Γͮͱ, ߚ ൌ ܶ ସ௡ିଶ ‫ݓ‬௡ ͳߙ ൌ ܶ ଼௡ିସ ‫ݓ‬௡ ͗੔Εཱིͯ.
ٌಕ఼‫ݓ‬௡ିଵ ͖Δड़൅͢ ‫ݓ‬௡ିଵ Ͷ໯ΖؔͶ͞͹ब‫ٌغ‬ಕͺٌಕ ఼‫ݖ‬௡ିଵ ͹बΕΝൕ࣎‫ܯ‬յΕ Ͷ 2
յ ఺ ͢ ͱ ͏ Ζ . ͞ ͹ ब ‫ ٌ غ‬ಕ ͹ յ ఺ ਼ ͺ ʹȀ͵Ͳ ͍ Ζ . Ґ ৏ Ͳ ҡ ৙ յ ఺ ෾ ‫ Ͳ ز‬ਫ਼ ͣ ͪ ब ‫ ٌ غ‬ಕ್͗॑ଲসब‫ٌغ‬ಕͲ ‫݌‬Ȁ‫ݍ‬:ʹȀ͵-B ͳ‫ى‬फ़͠ΗΖ͞ͳ͗ࣖ͠Ηͪ .
࣏Ͷյ఺෾‫Ͳز‬ਫ਼ͣ ͪब ‫ٌغ‬ಕ͗ ‫݌‬Ȁ‫ݍ‬:ʹȀ͵-B ͳ‫ى‬फ़͠Η Ζ͵Δͻ , ͞͹ब‫ٌغ‬ಕ ್͗॑ଲস ब ‫ ٌ غ‬ಕ Ͳ ͍ Ζ ͞ ͳ Ν ࣖ ͤ . ͞ ͞ Ͳ ߚ ൌ ܶ ସ௡ିଶ ‫ݓ‬௡ିଵ ͳ ߙ ൌ ܶ ସ௡ିଶ ߚ ͗ ੔ Ε ཱི ͯ . Ή ͪ , ߚ ൌ݃ܶ ଶ௡ିଶ ‫ݓܪ‬௡ିଵ ΍ ੔ Ε ཱི ͯ . ߚ ൌ ܶ ସ௡ିଶ ‫ݓ‬௡ିଵ ͹ ྈ ว Ͷ ܶ ଶ௡ିଶ ‫ܪ‬Ν ࡠ ༽ ͤ Ζ . ӊ ว Ν Ծ ‫ ى‬͹ Γ ͑ Ͷ ॽ͘‫͓׷‬Ζ.
ܶ ଶ௡ିଶ ‫ ܶܪ‬ସ௡ିଶ ‫ݓ‬௡ିଵ ൌ ܶ ଶ௡ିଶ ܶ ିସ௡ାଶ ‫ݓܪ‬௡ିଵ ൌ ܶ ିሺସ௡ିଶሻ ݃ ‫ ܶ݃ ל‬ଶ௡ିଶ ‫ݓܪ‬௡ିଵൌ ܶ ିሺସ௡ିଶሻ ݃ߚ ൌ ܶ ିሺସ௡ିଶሻ ߙ ൌ ߚǤҐ ৏Ͳ ߚ ൌ ܶଶ௡ିଶ‫ ͠ ࣖ ͗ߚܪ‬Ηͪ . ͞ Η ͺ ߚ͗ଲ স ત ܶ

௡ିଵ

(30)

ܵு ৏ Ͷ͍ Ζ ͞ ͳ Ν қັ ͤ Ζ . ί ʖη 1͹ৄ໎Νश͓Ζ.
ҾΊͶ৏‫ ى‬͹ৄ໎ Ͷ࢘༽ ͪ͢ ‫ݓ‬௡ିଵ (ਦ 11(b))ͺ, ਦ 9 Ͳͺ ‫ݓ‬଴ Ͷ૮౲ͤΖ. Ήͪ, ߙ ൌ ‫ݓ‬ସ Ͳߚ ൌ ‫ݓ‬ଶ Ͳ͍Ζ.
ίʖη 2 ͹ৄ໎. ‫݌‬Ȁ‫ݍ‬-BE ͗ଲসતܵ௛ ৏Ͷٌಕ఼Ν࣍ͯ৖߻.
[i] ‫ ݍ‬ൌ Ͷ݊ሺ݊ ൒ ͳሻ͹৖߻(ଲসતܵு (‫ ݕ‬ൌ ʹሺ‫ ݔ‬െ ߨሻ)৏Ͷٌಕ఼Ν࣍ͯ৖߻).
ܶ ௤ ͹ଲ߻න‫ݳ‬Νࣖͤ.
ܶ ସ௡ ൌ ݄ ‫ ܶ݃ ל‬ଶ௡ିଵ ‫ ܶܩ ל ܪ‬ଶ௡ିଵ Ǥࠪଲ߻‫ܶܩ‬ଶ௡ିଵͶͯ͏ͱ෈ร͵ॄ߻ͺ ܶି௡ܵு Ͳ͍Ζ. ‫݌‬Ȁ‫ݍ‬-BE ͹ٌಕ఼ͺ ܶ

(31)
ି௡

ܵு ͳܵ௛ ͹ި఼Ͷ͍Ζ.

௤͞ΗΝ‫ݖ‬଴ ͳͤΖ. ॵ‫ݖ఼غ‬଴ ͺܵ௛ ৏Ͷ͍Ζ. ‫ݖ‬଴ ΓΕܶ ΝୱҒͳ͢ͱब‫ غ‬3 ͗ਫ਼ͣΖ. ଲসતܵ௛ ৏Ͷ ਫ਼ ͣ ͪ ٌ ಕ ఼ Ν ‫ݓ‬଴ ͳ ͤ Ζ . ٌ ಕ ఼ ‫ݖ‬଴ Γ Ε , ܵ௛ ৏ Ͷ ͵ ͏ ٌ ಕ ఼ ߙ ൌ ‫ ܶܩ‬ଶ௡ିଵ ‫ݓ‬଴ ͳ ߚ ൌ ݄ߙ͗ ਫ਼ ͣ
Ζ. ٌಕ఼ߚͺ, ਦ 12(a)͹ഓ஖Ͳͺܶ ି௡ ܵு ৏Ͷ͍Ζ.
ߚ ൌ ܶ ସ௡ ‫ݓ‬଴ ΝՀఈ͢໅८Ν಍͚.
ߚ ൌ ܶ ସ௡ ‫ݓ‬଴ ൌ ܶ ସ௡ ‫ି ܶ ל‬ሺଶ௡ିଵሻ ‫ ܶܩ݄ ל ݄ܩ‬ሺଶ௡ିଵሻ ‫ݓ‬଴ ൌ ܶ ଶ௡ାଵ ‫ߚ ל ݄ܩ‬Ǥ(32)

‫ ݄ܩࣞܐؖ‬ൌ ‫ ݄ܪ ל ܪܩ‬ൌ ܶ ିଵ ܴͳܴ ൌ ܶ ି଺௡ Νཤ༽ͤΖͳ, ࣏ࣞ͗ಚΔΗΖ.
ߚ ൌ ܶ ଶ௡ ܴߚ ൌ ܶ ିସ௡ ߚǤ

(33)

ਦ 12ʁ(a) ‫ ݍ‬ൌ Ͷ݊ሺ݊ ൒ ͳሻ͹৖߻͹ഓ஖ (ܵு ǣ ‫ ݕ‬ൌ ʹሺ‫ ݔ‬െ ߨሻ). (b) ‫ ݍ‬ൌ Ͷ݊ െ ʹሺ݊ ൒ ͳሻ͹৖߻͹ഓ஖(ܵீ ǣ ‫ ݔ‬ൌ ߨ).
t 49 tब‫ٌغ‬ಕ ܱሺߚሻ͹ब‫غ‬ͺ ܱሺ‫ݖ‬଴ ሻ͹ब‫ͳغ‬ಋͣͶ͵Ζ. ͞Ηͺ໅८Ͳ͍Ζ. ߙ ൌ ܶ ସ௡ ‫ݓ‬଴ ͳߚ ൌ ܶ ସ௡ ߙ
͗ಚΔΗ, ٌಕ఼‫ݓ‬଴ ͖Δड़൅͢‫ݓ‬଴ Ͷ໯ΖؔͶ͞͹ब‫ٌغ‬ಕͺٌಕ఼ ‫ݖ‬଴ ͹बΕΝൕ࣎‫ܯ‬յΕͶ
2 յ఺͢ͱ͏Ζ͞ͳ͗ࣖ͠Ηͪ .
յ఺෾‫Ͳز‬ਫ਼ͣͪब ‫ ٌغ‬ಕ͗ ‫݌‬Ȁ‫ݍ‬:ʹȀ͵-B ͳ‫ى‬फ़͠ΗΖ͵ Δͻ , ͞͹ब‫ٌغ‬ಕ್͗ ॑ଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳ͹ৄ໎ͺ঴͚ . ࣏͹[ii]͹৖߻΍ಋ༹Ͳ͍Ζ.
[ii] ‫ ݍ‬ൌ Ͷ݊ െ ʹሺ݊ ൒ ͳሻ͹৖߻(ଲসતܵீ (‫ ݔ‬ൌ ߨ)৏Ͷٌಕ఼Ν࣍ͯ৖߻).
ܶ ௤ ͹ଲ߻න‫ݳ‬Νࣖͤ.
ܶ ସ௡ିଶ ൌ ݄ ‫ ܶ݃ ל‬ଶ௡ିଶ ‫ ܶܩ ל ܪ‬ଶ௡ିଶ Ǥ(34)ࠪଲ߻‫ ܶܩ‬ଶ௡ିଶ Ͷͯ͏ͱ෈ร͵ॄ߻ͺ ܶ ିሺ௡ିଵሻ ܵீ Ͳ͍Ζ. ‫݌‬Ȁ‫ݍ‬-BE ͹ٌಕ఼ͺ ܶ ିሺ௡ିଵሻ ܵீ ͳܵ௛ ͹ި఼Ͷ͍Ζ. ͞ΗΝ‫ݖ‬଴ ͳͤΖ. ॵ‫ݖ఼غ‬଴ ͺܵ௛ ৏Ͷ͍Ζ. ‫ݖ‬଴ ΓΕܶ ௤ ΝୱҒͳ͢ͱब‫ غ‬3 ͗ਫ਼ͣΖ. ଲসતܵ௛ ৏Ͷਫ਼ٌͣͪಕ఼Ν ‫ݓ‬଴ ͳͤΖ. ٌಕ఼‫ݖ‬଴ ΓΕ, ܵ௛ ৏Ͷ͵͏ٌಕ఼ ߙ ൌ ‫ ܶܩ‬ଶ௡ିଶ ‫ݓ‬଴ ͳߚ ൌ ݄ߙ
͗ਫ਼ͣΖ. ٌಕ఼ߚͺ, ਦ 12(b)͹ഓ஖Ͳͺ ܶ ିሺ௡ିଵሻ ܵீ ৏Ͷ͍Ζ.
ߚ ൌ ܶ ସ௡ିଶ ‫ݓ‬଴ ͳՀఈ͢໅८Ν಍͚.
ߚ ൌ ܶ ସ௡ିଶ ‫ݓ‬଴ ൌ ܶ ସ௡ିଶ ‫ି ܶ ל‬ሺଶ௡ିଶሻ ‫ ܶܩ݄ ל ݄ܩ‬ሺଶ௡ିଶሻ ‫ݓ‬଴ ൌ ܶ ଶ௡ ‫ߚ ל ݄ܩ‬Ǥ‫ ݄ܩࣞܐؖ‬ൌ ‫ ݄ܪ ל ܪܩ‬ൌ ܶିଵܴͳܴ ൌ ܶ

ି଺௡ାଷ

(35)

Νཤ༽ͤΖͳ࣏ࣞ͗ಚΔΗΖ .

ߚ ൌ ܶ ଶ௡ିଵ ܴߚ ൌ ܶ ିሺସ௡ିଶሻ ߚǤ(36)ब ‫ ٌ غ‬ಕ ܱሺߚሻ͹ ब ‫ غ‬ͺ , ܱሺ‫ݖ‬଴ ሻ͹ ब ‫ ͳ غ‬ಋ ͣ Ͷ ͵ Ζ . ͞ Η ͺ ໅ ८ Ͳ ͍ Ζ . ߙ ൌ ܶ ସ௡ିଶ ‫ݓ‬଴ ͳ ߚ ൌܶ ସ௡ିଶ ߙ͗ಚΔΗͪ. ͞ΗΓΕ ٌ ಕ఼ ‫ݓ‬଴ ͖Δड़൅ ͢ ‫ݓ‬଴ Ͷ໯Ζؔ Ͷ͞͹ब‫ ٌغ‬ಕͺٌಕ ఼ ‫ݖ‬଴ ͹बΕΝൕ࣎‫ܯ‬յΕͶ 2 յ఺͢ͱ͏Ζ͞ͳ͗ࣖ͠Ηͪ . ίʖη 2 ͹ৄ໎Νश͓Ζ.
ఈཀྵ 4.1 ͹ৄ໎Νߨ͑઴Ͷ६ඍΝߨ͑ . ब‫ غ‬2 ͹ٌಕ఼Ν ‫ݖ‬଴ ൌ ሺ‫ݔ‬଴ ǡ ‫ݕ‬଴ ሻ ൌ ሺߨ െ ‫ݐ‬ǡ െʹ‫ݐ‬ሻͳ‫ݖ‬ଵ ൌሺ‫ݔ‬ଵ ǡ ‫ݕ‬ଵ ሻ ൌ ሺߨ ൅ ‫ݐ‬ǡ ʹ‫ݐ‬ሻͳͤΖ. ͞͞Ͳ, ‫ ݐ‬൐ Ͳ.
ब‫ غ‬2 ͹ٌಕ఼ͺʹሺ‫ ݔ‬െ ߨሻ ൌ െሺܽȀʹሻ‫ݔ‬(37)

͹մͲ͍Ζ(‫)ߨ ് ݔ‬. ‫ ݔ‬ൌ ߨ ൅ ‫ ݐ( ݐ‬൐ Ͳ)Νཤ༽ͤΖͳࣞ(37)ͺʹ‫ ݐ‬ൌ ሺܽȀʹሻ‫ݐ‬(38)

ͳ͵Ζ. ܽ ൐ ͶͲ, ࣞ(38)ͺ‫ ݐ‬൐ Ͳ͹մΝ΍ͯ͹ͲҐԾ͹٠࿨Ͳͺ ܽ ൐ ͶͳͤΖ.
͞ΗΓΕ୊ 2.2 ઇͲ಍ड़ܑͪ͘͢ ߦ௡ Ͷ‫ͤؖ‬Ζࣺ଀, ್֌ඏ෾‫ߟ਼ܐ‬௡ Ͷ‫ͤؖ‬Ζࣺ଀ͳࢀ֌ඏ෾ ‫ߞ ਼ ܐ‬௡ Ͷ ‫ ͤ ؖ‬Ζ ࣺ ଀ Ν ཤ ༽ ͤ Ζ . න ‫ ݳ‬Ν ؈ ݁ Ͷ ͤ Ζ ͪ Ό Ͷ ร ਼ ‫ݏ‬ሺ൐ Ͳሻͳ ݀ሺ൐ ͲሻΝ ಍ ೘ ͤΖ.
݂Ԣሺ‫ݔ‬଴ ሻ ൌ ݂Ԣሺ‫ݔ‬ଵ ሻ ൌ െ݂ԢԢԢሺ‫ݔ‬଴ ሻ ൌ െ݂ԢԢԢሺ‫ݔ‬ଵ ሻ ‫ ؠ‬െ‫ݏ‬ǡ(39)݂ԢԢሺ‫ݔ‬଴ ሻ ൌ െ݂ԢԢሺ‫ݔ‬ଵ ሻ ‫ ؠ‬െ݀Ǥ

(40)

ఈཀྵ 4.1[1]͹ৄ໎. ໍୌ 4.5 Ͳ‫݌‬Ȁ‫ ݍ‬ൌ ͳȀʹͳͤΖͳ್॑ଲসब‫ٌغ‬ಕ ͳȀʹ:ʹȀ͵-B ͗ਫ਼ͣΖ͞ͳ͗಍͖ΗΖ. ͞͞Ͳͺਫ਼ͣ๏͗ҡ৙յ఺෾‫͍Ͳز‬Ζ͞ͳΝৄ໎ͤΖ .
ब‫ غ‬6 ͹ٌಕ఼Ν‫ݑ‬௞ ൌ ሺ‫ݔ‬௞ ǡ ‫ݕ‬௞ ሻሺͲ ൑ ݇ ൑ ͷሻͳॽ͚. ॵ‫ݑ఼غ‬଴ ͺଲসત ܵு ৏Ͷ͍ΖͳͤΖ. ॵ‫ݑ ఼ غ‬଴ ͗ ଲ স ત ܵு ৏ Ͷ ͍ Ζ ͞ ͳ Γ Ε , ܑ ͘ ߦ௡ ͳ ್ ֌ ඏ ෾ ‫ߟ ਼ ܐ‬௡ Ν ‫ ٽ‬Ό Ζ ͪ Ό ͹ ॵ ‫ غ‬஍ ͺ ,ߦ଴ ൌ ʹ, ߟ଴ ൌ ͲͲ͍Ζ. ब‫ غ‬6 ͹್॑ଲসब‫ٌغ‬ಕ͗ब‫ غ‬2 ͹ٌಕ఼‫ݑ‬ଵ ͖Δյ఺෾‫Ͳز‬෾‫ͤز‬Ζড়݇ͺ, ߦଵ ͗‫ݑ‬ଵ Ͷ͕͜Ζܵ௚ ͹ܑ͘Ͷ౵͢͏͞ͳͲ͍Ζ . ͞ΗΓΕ,ߦଵ ൌ െሺͳȀʹሻ݂Ԣሺ‫ݔ‬ଵ ሻ
͗ಚΔΗΖ. ࣞ(14)ΓΕ, ͞͹ড়݇ͺԾ‫ى‬͹Γ͑Ͷॽ͜Ζ .

t 50 t

(41)

ߦଵ ൌ

ߦ଴ ൅ ݂Ԣሺ‫ݔ‬଴ ሻ݂Ԣሺ‫ݔ‬ଵ ሻൌെǤߦ଴ ൅ ݂Ԣሺ‫ݔ‬଴ ሻ ൅ ͳʹሺ42ሻ͞͞Ͳร਼ ‫ݏ‬Νཤ༽͢, ߍͶߦ଴ ൌ ʹΝେ೘ͤΖͳ࣏ࣞ͗ಚΔΗΖ .
ʹെ‫ݏ ݏ‬ൌ Ǥ͵െ‫ʹ ݏ‬ሺ43ሻ͞ΗΓΕԾ‫ى‬͹್࣏๏ఖࣞ͗ಚΔΗΖ .
(44)ሺ‫ ݏ‬െ ͳሻሺ‫ ݏ‬െ Ͷሻ ൌ ͲǤ್ ࣏ ๏ ఖ ࣞ ͹մ ͳ ͢ ͱ ‫ ݏ‬ൌ ͳͳ ‫ ݏ‬ൌ Ͷ͗ ಚ Δ Η Ζ. ‫ ݏ‬ൌ Ͷͳͤ Ζ ͳ , ܽ ‫ݔ‬଴ ൌ െͶ͗ ੔ Ε ཱི ͯ. ܽ ൒ͶͲ, ͞͹ড়݇Νຮͪͤ ‫ݔ‬଴ ͗ଚࡑͤΖ. ܽ ൌ Ͷͺ, ବԃ‫෈ܗ‬ಊ఼ܳ͗ब‫غ‬ഔ෾‫ز‬Νً͞ ͤྡֆ஍Ͳ͍Ζ. Γͮͱ, ͞͞Ͳͺ‫ ݏ‬ൌ ͳΝࡀ༽ͤΖ.
ܽ ‫ ݐ‬ൌ ͳǤ(45)ሺͶ‫ݐ‬ሻଶ ൅ ͳ ൌ ܽଶ(46)Ͷ‫ ݐ‬ൌ ඥܽଶ െ ͳǤ(47)

ࣞ(38)ͳࣞ(45)ΓΕ,͗ಚΔΗΖ. ‫ ݐ‬൐ ͲΓΕ,
ࣞ(38)ͳࣞ(47)ΓΕԾ‫ى‬͹‫͗ܐؖ‬ಚΔΗΖ .
(48)

ܽ‫ ݐ‬ൌ ඥܽଶ െ ͳǤ࣏Ͷ್֌ඏ෾‫ߟ ਼ܐ‬ଵ Ν௒΄Ζ. ࣞ(15)Ͷ͕͏ͱߟ଴ ൌ Ͳͳ͢ͱߟଵ ͗ಚΔΗΖ.
ߟଵ ൌ݂ԢԢሺ‫ݔ‬଴ ሻെܽሺ‫ݔ‬଴ ሻ െܽ‫ݐ‬ξܽଶ െ ͳ
ൌ
ൌ
ൌെ
Ǥ
ଷ
ଷ
ሺ͵ ൅ ݂Ԣሺ‫ݔ‬଴ ሻሻሺ͵ െ ‫ݏ‬ሻ
ͺ
ͺ

(49)

ٌಕ఼‫ݑ‬ଵ Ͷ͕͜Ζଲসત ܵ௚ ͹್֌ඏ෾‫ߟ ਼ܐ‬௚ ሺ‫ݔ‬ଵ ሻͺߟ௚ ሺ‫ݔ‬ଵ ሻ ൌ െሺͳȀʹሻ݂ԢԢሺ‫ݔ‬ଵ ሻ ൌ ሺܽȀʹሻ‫ݔ‬ଵ ൌ െሺܽȀʹሻ‫ ݐ‬ൌ െξܽଶ െ ͳ
Ǥ
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ߟ௚ ሺ‫ݔ‬ଵ ሻ ൏ ߟଵ ൏ Ͳሺ51ሻ͗ࣖ͠Ηͪ. ൓ఈড়݇ 3.1(b)ΓΕ, յ఺਼ʹȀ͵͹್॑ଲসब‫ٌغ‬ಕ͗ब‫ غ‬2 ͹್॑ଲসब‫ٌغ‬ಕΓΕ ҡ৙յ఺෾‫Ͳز‬ਫ਼ͣͪ͞ͳΝқັͤΖ . Ґ৏Ͳఈཀྵ 4.1[1]͗ৄ໎͠Ηͪ.
ఈཀྵ 4.1[2]͹ৄ໎. ͞͞Ͳͺଲসત ܵு ৏Ͷਫ਼ͣΖब‫ غ‬6 Ͷͯ͏ͱ௒΄Ζ. ब‫ غ‬6 ͹ٌಕ఼Ν‫ݒ‬௞ ሺͲ ൑ ݇ ൑ ͷሻͳॽ͚. ॵ‫ݒ఼غ‬଴ ͗ଲসતܵு ৏Ͷ͍ΖͳͤΖͳ, ॵ‫ݒ఼غ‬ଷ ΍ଲসતܵு ৏Ͷ͍Ζ.
ٌಕ఼‫ݒ‬଴ ͳ‫ݒ‬ଷ ͗ଲসત ܵு ৏Ͷ͍Ζড়݇ΓΕ, ͳȀʹǣ ͳȀ͵-B ͺҲ॑ଲসब‫ٌغ‬ಕͲ͍Ζ .
ٌಕ఼‫ݒ‬଴ ‫ܵ א‬ு Ͷ͕͜Ζଲসત͹ܑ͘ͺ ߦ଴ ൌ ʹͲ͍Ζ. ॳ࣏ߦଵ , ߦଶ , ߦଷ Ν݀Ό, ߦଷ ൌ ʹͳͤΖ.
ߦଷ ൌ‫ ݏ‬ଷ െ ͸‫ ݏ‬ଶ ൅ ͻ‫ ݏ‬െ ʹൌ ʹǤ‫ ݏ‬ଷ െ ͹‫ ݏ‬ଶ ൅ ͳͶ‫ ݏ‬െ ͹

ሺ52ሻ

͞ΗΓΕ
ሺ‫ ݏ‬െ ͳሻሺ‫ ݏ‬െ ͵ሻሺ‫ ݏ‬െ Ͷሻ ൌ Ͳሺ53ሻ͗ಚΔΗΖ. ‫ ݏ‬ൌ Ͷͺବԃ‫෈ܗ‬ಊ఼ ܳ͗ब‫غ‬ഔ෾‫ز‬Νًͤ͞৖߻ͶଲԢ͢ , ‫ ݏ‬ൌ ͳͺ[1]͹ৄ໎͹৖߻ͶଲԢͤΖ. Γͮͱ, ͞͞Ͳͺ‫ ݏ‬ൌ ͵Νཤ༽ͤΖ.
್֌ඏ෾‫਼ܐ‬Ν௒΄Ζ. ߟ଴ ൌ Ͳͳ͢ͱ, ॳ࣏ߟଵ , ߟଶ , ߟଷ Ν݀ΌΖͳ ߟଷ ͗ ಚΔΗΖ.
ߟଷ ൌ݀ሺ‫ ݏ‬െ ͵ሻሺ‫ ݏ‬ହ െ ͳʹ‫ ݏ‬ସ ൅ ͷͶ‫ ݏ‬ଷ െ ͳͳʹ‫ ݏ‬ଶ ൅ ͳͲͷ‫ ݏ‬െ ͵͵ሻǤሺ‫ ݏ‬ଷ െ ͹‫ ݏ‬ଶ ൅ ͳͶ‫ ݏ‬െ ͹ሻଷ͞͹ࣞͶ‫ ݏ‬ൌ ͵Νେ೘͢ͱ, ߟଷ ൌ Ͳ͗ಚΔΗΖ.
t 51 tሺ54ሻ࣏Ͷࢀ֌ඏ෾‫਼ܐ‬Ν௒΄Ζ . ߞ଴ ൌ Ͳͳ͢ͱ, ॳ࣏ߞଵ , ߞଶ , ߞଷ Ν݀Ό, ࠹‫ޛ‬Ͷ‫ ݏ‬ൌ ͵Νେ೘ͤΖͳԾ‫ى‬͹ߞଷ ͗ಚΔΗΖ.
ߞଷ ൌ ͸ െ ͵݀ ଶ ൌ ͸ െ ͵ሺܽଶ െ ͻሻ ൌ ͵͵ െ ͵ܽଶ Ǥ
ଶ

ଶ

ଶ

ଶ

ଶ

ሺ55ሻ

ଶ

͞͞Ͳ݀ ൌ ሺܽ‫ݐ‬ሻ ͳ͵ ൌ ሺܽ ‫ݐ‬ሻ ΓΕ, ݀ ൌ ܽ െ ͻ͗੔Εཱི ͯ͞ͳΝཤ༽ͪ͢ . ͞͹ࣞ͹ܽͳ͢ͱ, ͳȀʹǣ ͳȀ͵-B ͗ਫ਼ͣΖྡֆ஍Νେ೘ͤΖචགྷ͍͗Ζ . ྡֆ஍ͺ 4 ΓΕ୉͘͏͹Ͳߞଷ ൏ Ͳ͗
ಚΔΗΖ. ൓ఈড়݇ 3.1(a)ΓΕ, ଲসત͹଀ͳଲসત ܵு ሺߨሻ͹ંৰ͹࢕๏ͺࢀ࣏‫਼ؖ‬దંৰͲ
͍Ζ͞ͳ͗಍͖Ηͪ. Ґ৏Ͳఈཀྵ 4.1[2]͗ৄ໎͠Ηͪ.
ࣞ(48)͹ࠪวͶࣞ(47)Νେ೘͢ร਼ ‫ݐ‬Νভ‫ͤڊ‬Ζͳྡֆ஍Ν݀ΌΖࣞ͗ಚΔΗΖ .
ܽሺඥܽଶ െ ͳȀͶሻ ൌ ඥܽଶ െ ͳǤሺ56ሻ͞ΗΝ਼஍‫Ͳࢋܯ‬մ͚ͳյ఺෾‫͗ز‬ਫ਼ͣΖྡֆ஍ ܽ௖ ሺͳȀʹǣ ʹȀ͵ሻ ൌ ͷǤ͸͸ʹͲ͗ಚΔΗΖ(ਦ 5 ͲͺͳȀʹǣ ʹȀ͵Ͳ ࣖ ͠ Η ͱ ͏ Ζ ). ҡ ৙ յ ఺ ෾ ‫ ͗ ز‬ਫ਼ ͣ Ζ ྡ ֆ ஍ ͹ ਼ ஍ ‫ ݃ ࢋ ܯ‬Վ ͺ ͷǤͷͷͶ͹Ͳ ͍ Ζ . ͯΉΕ, ͳȀʹǣ ʹȀ͵-B ͹ҡ৙յ఺Ϗϧϟʖνۢؔͺ ሾͷǤͷͷͶ͹ǡͷǤ͸͸ʹͲሻͲ͍Ζ.
࠹‫ޛ‬ͶͳȀʹǣ ͳȀ͵-B ͗ਫ਼ͣΖྡֆ஍Ν‫ٽ‬ΌΖ . ࣞ(56)͹ξܽଶ െ ͳΝξܽଶ െ ͻͶ஖͘‫͓׷‬Ζ͞ͳͲྡֆ஍Ν݀ΌΖࣞ͗ಚΔΗΖ .
ܽሺඥܽଶ െ ͻȀͶሻ ൌ ඥܽଶ െ ͻǤሺ57ሻ͞ Η Ν ਼ ஍ ‫ Ͳ ࢋܯ‬մ ͚ ͳ ਜ਼ ৙ յ ఺ ෾ ‫ ͗ز‬ਫ਼ ͣ Ζ ྡ ֆ ஍ ܽ௖ ሺͳȀʹǣ ͳȀ͵ሻ ൌ ͶǤͷͳͺͷ͗ ಚ Δ ΗΖ (ਦ 5ͲͺͳȀʹǣ ͳȀ͵Ͳࣖ͠Ηͱ͏Ζ).

5. ݃‫ޢ‬ຌ࿨ชͲಚΔΗͪ݃ՎΝΉͳΌΖ .
(1) ͳȀʹ-BE ͗ҡ৙յ఺෾‫ز‬Νً͢͞ͱ ͳȀʹ:ʹȀ͵-B ͗ਫ਼ͣΖ͞ͳͳ, ͳȀʹ-BE ͗ਜ਼৙յ఺෾‫ز‬Νً͢͞ͱͳȀʹ:ͳȀ͵-B ͗ਫ਼ͣΖ͞ͳΝৄ໎ͪ͢ (ఈཀྵ 4.1).
(2) ‫݌‬Ȁ‫ݍ‬:ʹȀ͵-B Ν, ‫݌‬Ȁ‫ݍ‬-BE (‫ݍ‬ͺ ۰ ਼)͗ յ ఺ ෾ ‫ ز‬Ν ً ͞ ͢ ͱ ਫ਼ ͣ ͪ ब ‫ ٌ غ‬ಕ ͳ ͤ Ζ . ‫݌‬Ȁ‫ݍ‬:ʹȀ
͵-B ್͗॑ଲসब‫ٌغ‬ಕͲ͍Ζ͞ͳΝৄ໎ͪ͢ (ໍୌ 4.5).
࠹‫ޛ‬Ͷජं͹ߡ͓Νफ़΄ͪ͏ . ࣺ଀յ਼ΝଁΏͦͻब‫ٌغ‬ಕ఼Ͳড়݇ 2.1 ͗੔ཱིͤΖΓ͑Ͷ͵Ζͳߡ͓ΔΗΖ. ͞͹͞ͳͺҡ৙յ఺෾‫͗ز‬ଡ͚͹ࣺ଀Ͳઞࡑ͢ͱ͏Ζ͞ͳΝࣖࠨ͢ͱ͏Ζ. ҡ৙յ఺෾‫ز‬͹‫ݬ‬ҾͲ͍Ζୱ௒͸ͣΗ੓͹ഃΗ͗ΩΨη‫ݳ‬েͶʹ͹Γ͑͵‫ر‬༫Ν͢ͱ͏Ζ͹͖. ͞͹໲ୌͶͯ͏ͱͺࠕ‫ޛ‬͹‫ݗ‬౾՟ୌͳͪ͢͏ .

ःࣛຌ࿨ชͶଲ͢ͱࠬಣंΓΕ‫͵॑ي‬αϟϱφΝ͏ͪͫ͘Ήͪ͢ . ৼΓΕ‫״‬ःਅ͢৏͝Ήͤ.

ࢂߡช‫ݛ‬[1] ࢃ޳‫ج‬ദ, ୫ઔੜཾ, ഇఛ΃͹ಕ (‫ཱིڠ‬ड़൝, 2016).
[2] Dullin, H. R., Meiss, J. D. and Sterling, D., Symbolic codes for rotational orbits.
SIAM J. Appl. Dyn. Sys. 4 (2005), 515-562. https://doi.org/10.1137/040612877[3] ࢃ޳‫ج‬ദ, ҡ৙յ఺෾‫΃ز‬͹ௗ͏ಕ , ‫ܙ‬͹Ռָճࢿ, ୊ 34 ‫ת‬, ୊ 2 ߺ (2019), 56-77.
https://katachi-jp.com/paper/34_2_web.pdf
t 52 t[4] ඬ६ࣺ଀Ͷͯ͏ͱͺ WolframMathWorld ͹մઈ͗ࢂߡͶ͵Ζ.
http://mathworld.wolfram.com/StandardMap.html[5] ࢃ޳‫ج‬ദ, ್॑Ն‫ ٱ‬໚੷ฯଚࣺ଀Ͷ͕͜Ζ್ ॑ଲসब‫ٌغ‬ಕ , ‫ܙ‬͹Ռ ָճࢿ , ୊ 34 ‫ת‬,
୊ 1 ߺ (2019), 13-42. https://katachi-jp.com/paper/34_1_web.pdf

t 53 tt 54 t‫ݬ‬ஸ‫ܙ‬͹Ռָճࢿ ୊ 35 ‫ ת‬୊ 1 ߺ

Python Ν࢟Όͱ̏0 ೖͲΰʖϞΝࡠΗΖΓ͑Ͷ͵Ζ͹͖ʃt ̔̏ࡂ͖Δ͹ேકʁ͖̎Δ࢟ΌΖ Python Programming t
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(2020 ೧ 8 ݆ 3 ೖण෉ɾ2020 ೧ 10 ݆ 7 ೖणཀྵ)Abstract: Python is one of the most popular programming languages and is employed invarious fields. In particular, applications associated with machine learning and deeplearning are often developed by the Python programming langu age. Additionally, a lot of
Python libraries are available for studying artificial intelligence. Based on this context, it
would be interesting to survey whether or not an elderly person can learn a newprogramming language such as Python within a short period. In this paper, the proposedcase study was conducted for analyzing whether a beginner, who is older than sixty years,
can develop a game after self-studying for ten days. Furthermore, a questionnaire survey
regarding programming languages was also conducted.
Keywords: Python, Elderly Person, Beginner, Ten Days Learning, Game Programming,
Self-Study.

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ճ٠ʰWWDC2017ʱͲɼ82 ࡂ͹ΠϕϨ֋൅ं͗ࣘ໪Ν͍ͯΌͪ͹ͺɼ߶ྺंͶ༒‫ـ‬Ν༫͓Ζ΍͹Ͳ͍ͮͪ[3]ɽҲൢదͶɼؒྻΝӿ͓ͱ΍৿͢͏‫ޢݶ‬Νָ΄Ζ͹ͫΘ͖͑ʃ 60 ࡂΝӿ͓ͱɼ୻‫͏͢৿Ͳؔغ‬ϕϫήϧϝϱή‫ޢݶ‬ΝसಚͲ͘Ζ͹ͫΘ͖͑ʃΉͪɼ৿͚͢सಚͤΖϕϫήϧϝϱή‫ͱ͢ͳޢݶ‬ͺɼՁ͗૮Ԣ͢͏ͫΘ͖͑ɽ 4 ͯ͹؏఼͖Δߡ͓Γ͑ɽ1)ϟζϡʖ͵‫͍Ͳޢݶ‬Ζɼ2)ॵৼंͲ΍ֺ͢ΌΖ‫͍Ͳޢݶ‬Ζ(ΰʖϞΝ͚ͯΗΖ͵ʹ͹ಊ‫͘Ͳ͗͜෉ؽ‬Ζ)ɼ3)ॵৼंͶ༑͢͏‫͍Ͳޢݶ‬Ζɼ4)αηφ͖͖͗Δ͵͏͞
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Python Ͳ *RRJOHbV 3\WKRQ &ODVV>@Ν६ඍ͢ͱ͕ΕɼYouTube Ͳ‫ݡ‬Ζ͞ͳ͗Ͳ͘Ζɽ͠ΔͶɼAmazon ͹ྖࢢαϱϒϣʖνʤAWSʥ͹ Amazon Braket ͹ϕϫήϧϝϱή‫͢ͳޢݶ‬ͱɼPython ͗ࡀ༽͠Ηͱ͏Ζ͵ʹɼࠕ‫ޛ‬͹൅ఴ੓΍‫غ‬ଶͲ͘Ζɽ2)ॵৼंͲ΍ֺ͢ΌΖ‫ݶ‬‫͖ޢ‬ʃPython ͲΰʖϞΝࡠΘ͑ͳ͏͑ॽ੸͵ʹ͗ଡ͚ड़൝͠Ηͱ͏Ζ[6]ɽ3)ॵৼंͶΏ͢͠͏‫͍Ͳޢݶ‬Ζɽଡ͚͹୉ָͲ‫گ‬ү༽͹ϕϫήϧϝϱή‫ޢݶ‬Ͷࡀ༽͠Ηͱ͏Ζɽ 4)αηφ͖͖͗Δ͵͏ɽPython ͺໃྋͲ͍ΕɼFree ͹֋൅‫͗ڧ؂‬६ඍ͠Ηͱ͏ΖɽΉͪɼWeb৏Ͷɼໃྋ͹ָसαʖη͍͗Ζɽྭ͓ͻɼKrishna ͗અཱིͪ͢ Guru99[7]ͺɼ‫گ‬ү͹‫ؽ‬ճ‫ۋ‬౵Ν໪ࢨͤ΍͹Ͳ͍Εɼ෾͖ΕΏͤ͏ Python ͹οϣʖφϨΠϩ͗६ඍ͠Ηͱ͏Ζɽຌ࿨ชͲͺɼ60 ࡂΝ௔͓ͱ Python ͶॵΌͱৰΗͱɼ10 ೖؔ͹ಢसͲΰʖϞΝࡠΗΖΓ
͑Ͷ͵Ζ͹͖͹ࣆྭ‫ڂݜ‬Ν͕͞͵ͮͪ͹ͲๅࠄͤΖɽಋ࣎Ͷɼॵৼं͗ 10 ೖؔ͹ಣॽͲʹ͹Γ͑Ͷ Python ͹‫گ‬ຌΝཀྵմ͢ͱ͏Ζ͖͹ࣆྭΝࣖͤɽΉͪɼϕϫήϧϝϱή‫ޢݶ‬͹ָसͶ͕͏ͱɼऑ೧ंͳ߶ྺं͹ؔͶ͍ࠫ͗Ζ͖ʹ͖͑ΝஎΖͪΌɼΠϱίʖφ௒ࠬΝ͕͞͵ͮͪ͹ͲๅࠄͤΖɽ

2. 3\WKRQ ϕϫήϧϝϱήͶேક
2.1. 3\WKRQ ͳͺPython ͺɼ1991 ೧ͶΨϧϱξਕ͹ή΢χʀϲΟϱϫργϞͶΓͮͱ֋൅͠ΗͪϕϫήϧϞϝϱή‫͍Ͳޢݶ‬Ζʤή΢χʀϲΟϱϫργϞ͗ Python 0.90 ͹λʖηαʖχΝޮ֋ͪ͢ ʥɽ Google ͗ क གྷ ‫ ͏ ͱ ͢ ͳ ޢ ݶ‬Ζ ͵ ʹ ɼ ‫ ࡑ ݳ‬΍ ͮ ͳ ΍ ஭ ໪ ͠ Η ͱ ͏ Ζ ‫ ͍ Ͳ ޢ ݶ‬Ζ[8]-[12][IEEE 2015-2019]ʤਦ 1 ࢂলʥɽΉͪɼೖ‫ܨ‬ͶΓΖͳɼೖຌͶ͕͜Ζϕϫήϧϝϱ
t 56 tή‫ޢݶ‬͹ਕ‫Ͳـ‬΍ Python ͺ৏ҒͲ͍Ζ[13]ɽ͞͞ͲͺɼC ‫ ͳޢݶ‬C++‫ޢݶ‬ΝΉͳΌͱɼखΕѽͮͱ͏Ζ͹ͲɼPython ͗ 2 ҒͶ͵ͮͱ͏Ζ͗ɼୱରͲͺ IEEE ͹݃Վͳಋ͚ͣɼ
Python ͗ 1 ҒͶ͵ΖͳࢧΚΗΖɽ
Python Ͷͯ͏ͱ৆͚͢Ίͱ͏͑͞ɽPython ͺɼ΢ϱνϕϨν‫͍Ͳޢݶܗ‬ΖɽҲൢͶ΢ϱνϕϨν‫ޢݶܗ‬ͺɼC ‫ʹ͵ޢݶ‬͹αϱϏ΢ϧ‫ͳޢݶܗ‬ർֳ͢ͱɼଐౕ͹఼Ͳ྾Ζɽ͖͢͢ɼαϱϒϣʖν͹߶ଐԿͳͳ΍ͶɼPython ͹Γ͑͵΢ϱνϕϨν‫Ͳޢݶܗ‬΍൜༽ϕϫήϧϞ‫ޢݶ‬Ͷ͵ΗΖΓ͑Ͷ͵ͮͱͪ͘ɽ͠ͱɼPython Ͷͺɼ΢ϱϕϪϟϱτʖεϥϱͶΓͮ
ͱɼCPythonɼJythonɼPyPy ͵ʹ͍͗Ζɽ‫Ͳ఼࣎ݳ‬ɼ΍ͮͳ΍ଡ͚࢘ΚΗͱ͕Εɼϟϱτψϱη͠Ηͱ͏Ζ͹͗ C ‫ىͲޢݶ‬फ़͠Ηͪ CPython Ͳ͍ΖɽCPython ͺɼC ‫ͳޢݶ‬͹਎࿪੓΍Γ͚ɼC ֨௃͹΢ϱνʖϓΥ΢ηΝ༙͢ͱ͏Ζɽ߶ଐͲॴཀྵͪ͢͏෨෾Ν C ‫Ͳޢݶ‬‫ى‬फ़ͪ͢γϔϩʖοϱͶࣰߨͦ͠Ζ͞ͳͲɼ΢ϱνϕϨν‫ޢݶܗ‬͹ओ఼Νึ͏ɼ߶ଐॴཀྵ
͗Ն೵ͳ͵ͮͱ͏Ζɽ
0

Ranking of Programming Languages

1
2
3
4
Python

Java

C

C++

Year 5

2015

2016

2017

2018

2019

Python

4

3

1

1

1

Java

1

2

3

3

2

C

2

1

2

4

3

C++

3

4

4

2

4

ਦ̏ʁϕϫήϧϝϱή‫ޢݶ‬͹ਕ‫ـ‬ϧϱΫϱήɽ‫ݫ‬υʖν [8]-[12]͖Δ͹Ӏ༽ɽ
160

Number of Persons

140
120
100
80
60

40
20
0

Languageਦ̐ʁೖຌͶ͕͜Ζϕϫήϧϝϱή‫ޢݶ‬ਕ‫ـ‬ϧϱΫϱήɽ‫ݫ‬υʖν [13]͖Δ৏Ғ 10 ҒΉͲΝϕϫρφͪ͢ʤ440 ਕյ౶ʥɽ

t 57 t͠ΔͶɼPython ͹ಝ௅͹Ҳͯͳ͢ͱɼΨʖϕϱλʖη͹λϓφΤΥΠͲ͍Ζ͞ͳ΍‫͝ڏ‬ΔΗΖɽϤʖδʗΝ‫؜‬ΌͪևͲүͱͱͪ͘ϕϫήϧϝϱή‫͍Ͳޢݶ‬Εɼ Python Software
Foundation [14]͗؇ཀྵ͢ͱ͕ΕɼλϓφΤΥΠΝໃྋͲξΤϱϫʖχͲ͘Ζɽ͞͞Ͳɼ
Python Ͷ‫ͤؖ‬ΖชॽԿ΍͕͞͵ͮͱ͏Ζ Python Software Foundation ͹ PEP Ͷͯ͏ͱ΍΀Ηͱ͕͚ɽ͞͞Ͷɼpython.org ͖Δ PEP Ͷͯ͏ͱ๱ಆ͹෨෾ΝӀ༽͢Γ͑ɽ
What is a PEP?
PEP 1 PEP Purpose and Guidelines(PEP stands for Python Enhancement Proposal.

A PEP is a design document providing

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Python Ͷ‫خͱؖ͢‬ຌద͵͞ͳͺɼpython.org ͲΆͳΞʹմ݀ͤΖɽ
Google ͺ Python Νकགྷ‫͕ͱ͢༽ࡀͱ͢ͳޢݶ‬ΕɼGoogle for Education Ͳ *RRJOHbV
Python Class Ν఑‫͏ͱ͢ڛ‬Ζ[5]ɽGoogle Python Class ͺ YouTube ͶΠρϕ͠Ηͱ͕Εɼ
Google ͶΓΖߪसΝϑυΨͲ‫ͱݡ‬ΊΓ͑ɽGoogle Python Class Day 1 YouTubeʤPart̏༁ 52 ෾ 2,867,000 Ґ৏͹ࢻ௎ɼPart2 ༁ 35 ෾ 644,000 Ґ৏͹ࢻ௎ɼPart3 ༁ 29 ෾ 467,000Ґ৏͹ࢻ௎ʥͳ Google Python Class Day 2 YouTubeʤ Part̏༁ 42 ෾ 470,000 Ґ৏͹ࢻ௎ɼPart2 ༁ 20 ෾ 284,000 Ґ৏͹ࢻ௎ɼPart̑༁ 26 ෾ 270,000 Ґ৏͹ࢻ௎ʥͲ͍Ζʤ2020
೧ 2 ݆ 14 ೖ‫ࡑݳ‬ʥɽ͞ΗΔͺɼGoogle Developers ͶΓͮͱ੏ࡠ͠Ηɼ2010 ೧ 3 ݆ 1 ೖΠρϕϫʖχ͠Ηͪ΍͹Ͳ͍Ζɽ͵͕ɼώʖζϥϱ͗ Python 2 Ͳ͍Ζ͞ͳΝ෉‫͚͕ͱ͢ى‬ɽ
͞͞ͲɼPython ͳଠ͹ϕϫήϧϝϱή‫ͳޢݶ‬͹ർֳΝࢾΊΖɽ۫ରదͶͺɼPython ͹ϧ΢ώϩ‫ͱ͢ͳޢݶ‬ɼPython ͳஂਫ਼ͪ࣎͢‫͏ۛ͗غ‬ϕϫήϧϞ‫͍Ͳޢݶ‬Ζ Ruby [15]ͳർ
ֳͤΖɽRuby ͺɼ1995 ೧Ͷޮ֋͠ΗͪΨϔζΥέφࢨ޴ηέϨϕφ‫͍ Ͳޢݶ‬Ζɽೖຌਕ͹Ήͯ΍ͳͽΘΑ͘ࢱ͗ɼPython ͶຮଏͲ͘͵͚ͱ֋൅ͪ͢΍͹Ͳ͍Ζɽ
͠ͱɼλϓφΤΥΠ͹ָसγ΢φ Guru99[7]Ͷͺɼϕϫήϧϝϱή‫ޢݶ‬͹ർֳ͗‫͠ࡎܟ‬Ηͱ͏ΖɽҐԾͶɼPython ͳ Ruby Νർֳͪ͢‫ࣆى‬Ν࿪༃͢ͱࡎͦͱ͕͚ɽ
ʴRuby ͳർ΄ͱɼPython ͺυʖνߑଆɼ಼෨‫਼ؖ‬ɼΓΕྒྷ͏ namespace ͹खΕѽ͏ɼ
module Ώ iterator ͹࢘͏๏͹๝͖͵ιρφΝ࣍ͮͱ͏ΖɽRuby ͶͺͲ͘͵͏͗ɼPythonͺଡ॑‫ܩ‬টΝγϛʖφ͢ͱ͏ΖɽRuby ͺϔϫρέ͹शΚΕͶɼend ͍Ζ͏ͺ } Νචགྷͳ
ͤΖ͗ɼPython ͺ‫ۯ‬പ͹ΊͲྒྷ͏ɽRuby ͺ Web ͹֋൅Ώ functional programming Ͷ޴͏ͱ͕ΕɼPython ͺ Academic and scientific programming Ͷ޴͏ͱ͏ΖɽRuby Ͷͺɼ|ঙ ͵ ͏ ͞ ͳ Ͳ ଡ ͚ ͹ ͞ ͳ Ν ͵ ͤ |ɼ Python Ͷ ͺ | Ҳ ͯ ͹ ਜ਼ ͢ ͏ Ώ Ε ๏ Ͳ ΍ ͹ ͟ ͳ Ν ͵ ͤn ͳ͏͑ఫָ͍͗ΖɽʵҐ৏ɼPython Ͷͯ͏ͱΊͱͪ͘ɽ֕གྷΝཀྵմͲͪ͘͹Ͳͺ͵͏ͫΘ͖͑ɽ࣏Ͷɼʹ͹Γ͑Ͷָ΄ͻΓ͏͖ΝΊͱ͏͑͞ɽ

2.2. Ώ͢͠͏೘໵ॽ
ؒྻΝ௔͓ͱॵΌͱ Python ͳ͏͑ϕϫήϧϝϱή‫ޢݶ‬Ͷड़ճ͏ɼ‫͢͵ࢥگ‬ͶಢसͤΖͪΌɼ΍ͮͳ΍Ώ͢͠͏೘໵ॽΝ୵ͪ͢ɽ͞͞ͲͺɼघͶ೘ΗΏͤ͏ 3 ࡯͹ຌΝ೘໵ॽͳ͢ͱમΞͫʤ2020 ೧ 2 ݆ 10 ೖʤ݆ʥෳҬ‫ݟ‬ෳҬࢤ಼͹ॽవʤঋ໨ॽవ ৿್͹‫ٸ‬వʥͶͱ߬೘ʥɽΉͪɼ೘໵ॽ͗ཀྵմ͢Ώͤ͏͖ʹ͖͑Ͷ୉͘͵Ӫ‫ڻ‬Ν༫͓ΖͫΘ͑ɼ೘໵ॽ͗ਬ
ও͢ͱ͏Ζ֋൅‫ڧ؂‬Ν๱ಆͶࣖͤ͞ͳͳͤΖɽ͵͕ɼ͞͹হ͹क͵໪దͺɼPython ͹ॵָ
t 58 tं͗ʹ͹Γ͑Ͷ೘໵ॽΝཀྵմ͖ͪ͢͹ࣆྭΝࣖͤ͞ͳͲ͍Εɼຌ͹մઈ͗क໪దͲ͵͚ɼΉͪմઈͶยΕ͍͗Ζ͖΍͢Η͵͏͞ͳΝ෉‫͚͕ͱ͢ى‬ɽ

2.2.1.  ෉ࡢᩍᮏ‫͢͵ࢥگ‬͹ಢस͹ͪΌͶમΞͫ‫گ‬ຌͺɼ̏ʥPython ௔೘໵[16]ɼ̐ʥ͏ͬͻΞΏ͢͠͏
Python ͹‫گ‬ຌ[17]ɼ̑ʥκϫ͖ΔΏ͚͢͠ͺͣΌΖ Python ೘໵[6]͹ 3 ࡯Ͳ͍Ζɽ3 ࡯͹‫گ‬ຌ͹಼༲ΝॳͶΊͱ͏͑͞ɽ
̏ʥPython ௔೘໵ ʷϠϱτΡͳָ΁ͺͣΌͱ͹ϕϫήϧϝϱήʷ˔֋൅‫ڧ؂‬ͺɼIDLE Ν࢘༽͢ͱ͏Ζɽ
Python ͶΓΖඵժΝυϠϱηφϪʖεϥϱͤΖͪΌͶɼTurtle GraphicsʤPython ͹ඬ६ϧ΢ϔϧϨʥΝ࢘ͮͱ͏Ζ͹͗ಝ௅Ͳ͍Ζɽ஦߶ਫ਼Ͷ΍෾͖ΕΏͤ͏Γ͑ͶɼඵժΝ‫؜‬ΌͪϕϫήϧϝϱήΝઈ໎͢ͱ͏Ζɽಣंͳ͢ͱ஦߶ਫ਼Νϟ΢ϱνʖΰρφͶ͢ͱ͕Εɼαϱϒϣʖν͹ಚқ͵߶ߏਫ਼ͳॵৼं͹஦ָਫ਼͹ճ࿫ͶΓͮͱɼ೘໵ॽͳ͢ͱ͹ην΢ϩ
͗੔ཱི͢ͱ͏ΖɽͪͫɼखΕૌΊΏͤ͠Ν॑གྷࢻ͢ͱ͏ΖͪΌɼ৓ௗ੓ͺ൳Ό͵͏һেΝणͪ͜ɽ͞ ͹ ຌ ͹ ಝ ௅ ͹ Ҳ ͯ Ͳ ͍ Ζ ॵ ৼ ं Ͷ ΍ ख Ε ૌ Ί Ώ ͤ ͏ ή ϧ ϓ Ρ ρ έ η Ͳ ͍ Ζ Turtle
Graphics Ͷͯ͏ͱफ़΄ΖɽTurtle Graphics ͺɼ1960 ೧େͶ‫گ‬ү޴͜ϕϫήϧϝϱή‫ޢݶ‬ͳ͢ͱ֋൅͠Ηͪ΍͹ͲɼॵৼंͶͺखΕૌΊΏͤ͏Ͳ΍͹͍ΖɽPython Ν Python.org͖Δ΢ϱηφʖϩͤΖͳɼඬ६ϧ΢ϔϧϨ͹ Turtle Graphics ΝϠζϣʖϩͳ͢ͱ͓࢘ΖΓ͑Ͷ͵Ζɽ͵͕ɼTurtle Graphics ͺɼήϧϓΡρέη͹‫ ͱ͢ͳેخ‬tkinter Ν࢘ͮͱ͏ΖɽͨΗΑ͓ɼtkinter Ν௜ં࢘͑Ά͑͗࢘͏Ώͤ͏ͳ͏͑ਕ΍ଡ͏͖΍͢Η͵͏͗ɼॵৼंͶͺɼtkinter Ν࢘͑ΓΕ΍ Turtle Graphics Ν࢘ͮͪඵժ͹๏͗ྒྷ͏ͳࢧ͓ΖɽTurtle
Graphics ͺɼ‫ͳ਼ؖ‬ϟλρχͳ͏͑ଈ໚Ν࣍ͮͱ͏Ζ͗ɼ͞͞Ͳͺɼϟλρχͳ͢ͱ࢘༽
͢ͱ͏Ζɽ‫گ‬ຌͶͪ͗ͮ͢ͱָसͤΗͻɼϕϫήϧϞͺಊࡠͤΖ͗ɼ಼༲Νཀྵմ͢Γ͑ͳ
ͤΖͳೋ͢͏ɽΉͪɼητʖζ 8 ΉͲͺ෾͖ΕΏͤ͏͗ɼήϧϓΡρέ͹ητʖζ 9 ͖Δ
12 ͺɼঙ͢ઈ໎෈ଏͲ͍ΖΓ͑Ͷࢧ͓Ζɽྭ͓ͻɼt = turtle.Pen() ͺϟλρχͳ͢ͱ࢘
༽͢ͱ͏Ζ͗ɼt = turtle Ͳ΍ಊ͚ɽWXUWOH˲W ͳ঴ྲྀͪ͢͏ͫ͜Ͳ͍Ηͻɼtype Ν୻͚Ͳ͘ΖϟϨρφΓΕɼॵৼंͶͺ෾͖ΕͶ͚͚͵ΖυϟϨρφ͹๏͗୉͘͏Γ͑Ͷࢧ͓Ζɽ
͞͞Ͳɼྒྷ͏఼Ν‫͝ڏ‬ΖɽಣΊศ͢ͱΊΖͳɼᠵ(஦ 1)ͳϠϱτΡ͹ճ࿫Ͳͤͤ΋Γ͑޽෋͠Ηͱ͏Ζ͹͗෾͖ΖɽΉͪɼҐԾͶ‫͝ڏ‬Ζɼ
1ʥౌ৖ਕ෼͹ༀׄ෾୴ͶΓΖճ࿫Ͳ͹ઈ໎
2ʥึ͑Γ͑Ͷ͍Ζઈ໎(αϧϞ)
3ʥহ຦͹໲ୌ

check ͳ challenge͹ߑ੔ͺɼण‫ॽߡࢂݩ‬Ͷࣇͪην΢ϩͲ͍Εɼ஦߶ਫ਼Ͷͺ͵ͣΊΏͤ͏ͫΘ͑ɽ͖͢͢ɼ
‫ݺ‬ʓͶͺ͞͵Ηͱ͏͵͏෨෾΍ଡ͏Γ͑Ͷࢧ͓Ζɽஸं͹γʖϑη੠ਈ͖͓͗ͮͱɼಣं
Νࠠཛྷͦ͠ΖՆ೵੓͍͗Ζ͖΍͢Η͵͏ɽ͵͕ɼ֋൅‫ͱ͢ͳڧ؂‬ɼIDLE Ͷ‫׵‬ΗͪΔɼ࣏ͺΓΕสཤ͵ PyCharm Νਬ͢ͱ͏Ζɽ
PyCharm ͺɼGuru99.com ͗఑‫ͤڛ‬Ζαʖηʁʰ7 ೖͲɼPython Νָ΁ʱͲ΍࢘ΚΗͱ͏Ζɽt 59 t̐ʥ͏ͬͻΞΏ͢͠͏ Python ͹‫گ‬ຌ ʷਕ‫͓گ͗ࢥߪـ‬Ζ‫͖ેخ‬Δγʖώʖγ΢χ֋൅ΉͲʷ˔֋൅‫ڧ؂‬ͺɼAtom Ν࢘༽͢ͱ͏Ζɽຌ͹ν΢φϩͺ͏ͬͻΞΏ͢͠͏ Python ͹‫گ‬ຌͳ͵ͮͱ͏Ζ͗ɼࣰͺϕϫήϧϞ‫ݩܨ‬ΝՀఈ͢ͱ͏Ζ෨෾͗݃ߑ͍ΖɽॵָंͶͺɼ݀͢ͱΏ͚͢͠͵͏ Python ͹‫گ‬ຌͲ͍ΖɽΉͪɼPython.org ͖Δඬ६దͶ Python Ν΢ϱηφʖϩͤΖͳಋ࣎Ͷ΢ϱηφʖϩ͠ΗΖ
IDLE Ͳͺ͵͚ɼτΫηφΦυΡνͳ͢ͱ Atom(https://atom.io/͖ΔໃྋͲξΤϱϫʖχͲ͘Ζ)Νਬও͢ͱ͏Ζ͹΍ಝ௅Ͳ͍Ζɽ࣏Ͷɼ֦হ͟ͳ͹නୌͳͳ΍Ͷɼ಼༲ΝΊͱ͏͑͞ɽ
Chap. 1 ʰPython Νָ΁६ඍΝ͢Γ͑ʱͲͺɼ΢ϱηφʖϩ͹๏๑ɼͨ͢ͱଲ࿫ϠʖχɼϓΟ΢ϩͲ͹ࣰߨ๏๑͗फ़΄ΔΗͱ͏Ζɽ
Chap. 2 ʰαϜϱχϕϫϱϕφͶ͵ΗΓ͑ʱͲͺɼPython ͨ͹΍͹Ͳͺ͵͚ɼPython Νָ΁৏ͲචགྷͶ͵Ζ OS ͹αϜϱχ͹ઈ໎͗͵͠Ηͱ͏Ζɽ
Chap. 3 ʰ‫ેخ‬Νָ;͵͗ΔϕϫήϧϞΝࡠ੔͢Γ͑ʱͲͺɼ؈ୱ͵ྭୌΝ΍ͳͶॵָंͶɼPython ͶΓΖϕϫήϧϞ͹಍೘Ν͕͞͵ͮͱ͏ΖɽΉͪɼ‫ͳ਼ؖ‬ϟλρχༀׄ͹ҩ͏Ν‫݃׮‬Ͷઈ໎͢ͱ͏Ζɽ‫ͳ਼ؖ‬ϟλρχͺɼPython Νָ΁ͳ͘Ͷɼ᪶͘Ώͤ͏ͳࢧΚΗΖགྷોͲ͍Εɼಢཱིͪ͢হΝઅ͜ͱ͏Ζ͹ͺ਎઀Ͳ͍Ζɽ
Chap. 4ʰ‫܃‬Εศ͢ͳড়݇෾‫ز‬ʱͲͺɼPython Ͷ‫ݸ‬ΔͥɼϕϫήϧϞͲঙ͢෵ࡸ͵ॴཀྵ

Νͦ͠Ζ৖߻Ͷචགྷ͵‫܃‬Εศ͢ͳড়݇෾‫ز‬Νմઈ͢ͱ͏Ζɽ
Chap. 5 ʰࣛॽͳϓΟ΢ϩ͹ѽ͏๏ʱͲͺɼPython Ν࢘ͮͪԢ༽ͶಁΊࠒΞͲ͏ΖɽPythonͺชহ͵ʹ text ॴཀྵͶ΍Γ͚༽͏ΔΗͱ͕ΕɼPython ͹ಚқ͵෾໼Ͳ͍ΖɽC ‫ʹ͵ޢݶ‬͗‫ۦ‬घͳͤΖ෾໼͗ɼPython Ν࢘͑ͳ؈ୱͶͲ͘Ζ͞ͳࣰ͗‫͘Ͳ״‬ΖΓ͑Ͷ͵ͮͱ͏Ζɽ
Chap. 6 ʰճ࿫ bot ΝࡠΘ͑ʱͲͺɼ‫ݳ‬େదͲ‫ັڷ‬਄͏τʖϜΝୌࡒͶ઴হΉͲͶָस಼ͪ͢༲Ν਄ΌΖΓ͑Ͷ͵ͮͱ͏Ζɽͪͫɼ‫͙ٺ‬ଏͶ͵ͮͱ͕Εɼॵָंͺͯ͏ͱ͏͜͵͏Ն೵੓͍͗Ζ͞ͳͺ൳Ό͵͏ɽྭ͓ͻɼϕϫήϧϞΝࣖ͢͵͗Δࣰߨͦ͠ͱ͏͵͏͹ͺɼॵָंͶͺ෈਎઀ͳࢧΚΗΖɽ
Chap. 7 ʰϧ΢ϔϧϨΝ࢘͏͞͵ͨ͑ʱͲͺɼϧ΢ϔϧϨΝ࢘ͮͪϕϫήϧϞͶಁΊࠒΞ
Ͳ͏ΖɽPython ͹‫ک‬Ίͺɼ๝෍͵ϧ΢ϔϧϨͶ͍Ζɽͨ͹қັͲɼಢཱིͪ͢হΝઅͪ͜͹ͺɼྒྷ͏఼Ͳ͍ΖɽΉͪɼlesson 44ʛ47 ͲɼϕϫήϧϞ͹ΦϧʖͶଲԢͤΖ๏๑ΝͳΕ͍͝ɼߑชΦϧʖͳྭ֐͹ 2 झྪΝͳΕ৏͝ͱ͏Ζ͹ͺࣰ༽ద͵ϕϫήϧϞΝߡྂ͢ͱ͏Ζɽ͖͢͢ɼॵৼंͶͺೋ͢͏͖΍͢Η͵͏ɽ
Chap. 8ʰγʖχϏʖτΡ੣ϏρίʖζΝ࢘͏͞͵ͨ͑ʱͲͺɼ Python ͹‫ک‬ΊͲ͍Ζ๝෍ͶଚࡑͤΖ͹γʖχϏʖτΡ੣ϏρίʖζΝͳΕ͍͝ͱ͏ΖɽΉͪɼLesson 50 ͲͺɼՀ૟‫ڧ؂‬ΝͳΕ͍͝ͱ͕Εɼ͞ΗͶΓΕ෵਼͹γʖχϏʖτΡ੣ϏρίʖζΝαϱϓϨέ
φ͵͢Ͷ҈સͶ͓࢘ΖΓ͑Ͷ͵Ζ͞ͳΝઈ໎͢ͱ͏Ζ͹ͺ਎઀Ͷࢧ͓Ζɽ
Chap. 9ʰWeb ΠϕϨίʖεϥϱΝࡠ੔͢Γ͑ʱͺɼΠϕϨίʖεϥϱΝࡠͮͱΊͪ͏ͳࢧ͑ಣंͶͺ‫ັڷ‬Νऔ͖ΗΖহͲ͍Ζɽ͖͢͢ɼࡋ෨ͺ‫؛‬౦͢͝ͱ͏Ζܑ޴͍͗Εʤྭɼ৆͚͢ͺʞΝ‫ͱݡ‬Ծ͠͏ʨϭϱϛ΢ϱφ͹αϧϞͲಝͶʩʥɼ෈ॉ෾͵಼༲Ͳ͍ΖһেΝ༫͓Ζɽ
Chap. 10ʰ͠ΔͶஎࣟΝਐͶ෉͜ΖͪΌ͹ָ;๏ʱͺɼ͕Ή͜͹Γ͑͵হͲ͍Ζɽ͠Δ

͵Ζָस͹ͪΌ͹ Web γ΢φɼॽ੸ɼαϝϣωτΡͶͯ͏ͱ‫͠ٶݶ‬Ηͱ͏Ζɽ
t 60 tસରΝ௪͢ͱɼC ‫ʹ͵ޢݶ‬͹ຌͳҡ͵ΕɼWeb ΠϕϨ͵ʹͶ॑͘ɼཀྵՌ‫ܧ‬͹਼஍‫͵ࢋܯ‬ʹͺ͍ΉΕखΕѽͮͱ͏͵͏͹͗ಝ௅Ͳ͍Ζɽ̑ʥκϫ͖ΔΏ͚͢͠ͺͣΌΖ Python ೘໵ ʷ‫͖ેخ‬Δηνʖφ͢ͱɼΰʖϞͰ͚Εɼ‫ؽ‬փָसΉͲָ·͑ʄʷ˔֋൅‫ڧ؂‬ͺɼIDLE Ν࢘༽͢ͱ͏Ζɽ͞͞ͲͳΕ৏ͪ͝ 3 ࡯͹‫گ‬ຌ͹஦Ͳͺɼ࠹΍ϚϨϣʖϞ͍͗Εɼ͖ͯώϧϱη͗ͳΗͪ‫گ‬ຌͲ͍Ζɽ෯ୌ͗ɼ
ʰ‫خ‬ຌ͖Δηνʖφ͢ͱɼΰʖϞͰ͚Εɼ‫ؽ‬փָसΉͲָ·͑ʄʱͳ͍ΖΓ͑Ͷɼॵิ͖Δ Python ͗Γ͚͖ͯΚΗͱ͏Ζ‫ؽ‬փָसΉͲखΕѽͮͱ͏Ζɽ࣏Ͷɼ֦হ͟ͳ͹නୌͳͳ΍Ͷɼ಼༲ΝΊͱ͏͑͞ɽ
Chap. 1 ʰϕϫήϧϝϱήͮͱ͵ΞͫΘ͑ʱͲͺɼαϱϒϣʖν΍ Python Ν࢘ΚͥͶϕϫήϧϝϱή͹ର‫ݩ‬Νͦ͠Ζ޽෋͗͵͠Ηͱ͏ΖɽΉͪɼPython ͹΢ϱηφʖϩ͹࢕๏Νઈ໎͢ͱ͏Ζ͹ͺɼ਎઀Ͷࢧ͓Ζɽ
Chap. 2 ʰϕϫήϧϝϱήΝͺͣΌΓ͑ʱͲͺɼIDLE ͹࢘͏๏͗ϟ΢ϱͲ͍ΖɽIDLE ͹τΫηφΦυΡν࢘ͮͱɼร਼Ώช࣊྽ΝཀྵմͲ͘ΖΓ͑͵޽෋͗͵͠Ηͱ͏Ζɽ
Chap. 3 ʰ੏‫ߑޜ‬ช࢘ͮͱΊΓ͑ʱͲͺɼ੏‫ߑޜ‬ช͹‫خ‬ຌͲ͍Ζʰড়݇෾‫ز‬ʱʰ‫܃‬Εศ͢ॴ
ཀྵʱ͗खΕѽΚΗͱ͏Ζɽif ชͳ while ช͹ཀྵմ͗͞͹হ͹ϟ΢ϱͲ͍Ζɽ͠ΔͶɼΰʖϞϕϫήϧϞ͹ॵิͳ͢ͱɼཛྷ਼Ν༽͏ͪήϧϓΡρέηΝ࢘Κ͵͏ text ͶΓΖͣΎΞ͜ΞΰʖϞΝͳΕ͍͝ͱ͏Ζɽ؈ୱ͵ΰʖϞΝ಍೘Ͷ࢘ͮͱɼॵৼं͹ΏΖ‫ـ‬Ν߶Όͱ͏Ζ
ͳࢧ͑ɽChap. 4 ʰϨηφͳࣛॽ‫ܗ‬Ν࢘ͮͱΊΓ͑ʱͲͺɼ1 ࣏‫͖ݫ‬ΔͺͣΌͱ 2 ࣏‫ݫ‬͹Ϩηφɼ͠ΔͶࣛॽ‫ܗ‬͹υʖν͹ѽ͏๏Νઈ໎͢ͱ͏ΖɽϨηφΝ΍ͬ͏ͱɼช࣊྽Ν͍͖ͯͮͪ
3 ୔έ΢θ͹ϕϫήϧϞ͗ͳΕ৏͝ΔΗͱ͏ΖɽΉͪɼΫʖͶΓͮͱυʖνͶΠέιηͤΖࣛॽ‫ܗ‬ΝखΕѽͮͱ͏ΖɽϨηφΏࣛॽ‫ܗ‬ͺɼชহॴཀྵͶ͕͜Ζ Python ͹ௗॶ͹ͽͳ
ͯͲ͍Ζ͞ͳ͗෾͖ͮͪɽ
Chap. 5 ʰ‫਼ؖ‬Ν࢘ͮͱΊΓ͑ʱͲͺɼఈٝ๏๑Ώ‫਼ؖ‬͹࢘͏๏Νָ΁ɽPython ͲͺɼγϔϩʖοϱͲͺ͵͚ɼ‫਼ؖ‬Νఈٝ͢ɼ‫਼ؖ‬Ν‫;ݼ‬ड़ͤͳ͏͑Γ͑ͶϕϫήϧϞΝࡠ੔ͤΖɽ͠ΔͶɼϠζϣʖϩʤϧ΢ϔϧϨʥ͹ཤ༽๑ɼήϫʖώϩร਼ʀϫʖΩϩร਼͹ઈ໎͗͵͠Ηͱ͕ΕɼಣंͶ Python ͗࢘͏Ώͤ͏‫͍Ͳޢݶ‬Ζͳ͏͑һেΝ༫͓ͱ͚ΗΖ ɽ
Chap. 6 ʰΰʖϞΝࡠͮͱΊΓ͑ʱͲͺɼ઴হΉͲΝ‫ͱ͢ͳેخ‬ɼԢ༽Ͷ͍ͪΖຌ֪ద͵ΰʖϞϕϫήϧϞͶேક͢ͱ͏Ζɽtext ϗʖη͹ΰʖϞͲͺ͵͚ɼຌ֪ద͵ήϧϓΡρέ
ΰʖϞͶखΕૌΞͲ͏Ζɽ͵͕ɼήϧϓΡρέͶͺɼtkinter Ν࢘ͮͱ͏ΖɽChap. 1ʛ5
ͳͺҡ͵ΕɼϪϗϩ͗‫ٺ‬Ͷ৏͗ͮͱ͏Ζɽָसं͗ಢྙͲΰʖϞΝࡠΖͳ͏͑ΓΕɼΰʖϞ͗ർֳద؈ୱͶࡠΗΖ͞ͳΝ়ղ͢ͱ͏Ζɽ͖͢͢ɼॵৼंͶͺͯ͏ͱ͏͜͵͏͹Ͳͺͳ͏͑෈҈Ν༫͓Ζ͖΍͢Η͵͏ɽ
Chap. 7 ʰ‫ؽ‬փָसͶேક͢Γ͑ʱͲͺɼPython ͹ਕ‫ـ‬͹གྷҾͲ͍Ζ‫ؽ‬փָस͗ͳΕ৏͝
ΔΗͱ͏ΖɽStep 1 ‫ؽ‬փָस͹֕གྷɼStep 2‫ؽ‬փָसϧ΢ϔϧϨΝಋࠟͪ͢ Anaconda ͹΢ϱηφʖϩɼStep 3ʛ5 ͺɼຌ֪ద͵ժ଀υʖν͹෾ྪ໲ୌ͗ͳΕ৏͝ΔΗͱ͏Ζɽϧ΢ϔϧϨͳ͢ͱͺɼNumPyɼmatplotlib ͗࢘ΚΗͱ͏ΖɽસରదͶͺɼॵิ͖ΔԢ༽ΉͲखΕѽͮͪώϧϱη͹ͳΗͪ‫گ‬ຌͲ͍Ζɽ͖͢͢ɼஸंͶͳͮͱͺʰκϫ͖ΔʱͲ͍Ζ͗ɼಣंͶͳͮͱͺʰκϫ͖ΔʱͶࢧ͓͵͏ͳ͞Θʤ༩ඍt 61 tஎ͍ࣟ͗Ζ΍͹ͳՀఈ͠Ηͱ͏ΖʥͶϝηϜροϱή͍͗ΖΓ͑͵‫ͤ͗ـ‬ΖɽΑ͓ͶɼͬΒͮͳͪ͢ͳ͞ΘͲͯΉͥ͏ͱ͢Ή͖͑΍͢Η͵͏ɽҐ৏ɼ‫ݺ‬ʓ͹‫گ‬ຌ͹಼༲ͳಝ௅͕Γ;ॵৼं͗‫ͪͣ״‬һেΝΉͳΌͪɽ࣏Ͷɼ 3 ࡯͹‫گ‬ຌ͹સରద͵ಝ௅ΝΊͱ͏͚ɽ

2.2.2.  ෉࡟ඹ㏻ࡍࡿⰋ࠸࡜ᝏ࠸Ⅼ઴হͲͺɼ3 ࡯͹‫گ‬ຌΝ‫ݺ‬ʓͶ‫͗ͪ͘ͱݡ‬ɼ͞͞Ͳͺॵָं༽͹‫گ‬ຌͳ͢ͱ͹ 3 ࡯Ͷ‫ڠ‬
௪ͤΖಝ௅Νྒྷ͏ͳ͞Θʀѳ͏ͳ͞Θͳ͏͑؏఼͖ΔΊͱ͏͚ɽྒྷ͏ͳ͞Θʁ
̏ʥָस಼༲͗হͳ͢ͱΉͳΌΔΗͱɼ਒Ί۫߻͗෾͖ΕΏͤ͏Γ͑Ͷ޽෋͠Ηͱ͏Ζɽ
̐ʥ΢ϱηφʖϩ͹ஊ֌͖Δմઈ͠Ηͱ͏Ζɽಢཱིͪ͢হ͗ΚΕ΀ ΔΗͱ͏Ζɽ
̑ʥໃັ‫ף‬૫Ͷ͵Δ͵͏Γ͑ͶΰʖϞ͗Ԣ༽ͳ͢ͱɼͳΕ͍͝ΔΗͱ͏Ζɽ̒ʥγϱϕϩϕϫήϧϞ͗ Web ͖ΔξΤϱϫʖχͲ͘ΖΓ͑Ͷ͵ͮͱ͏Ζɽѳ͏ͳ͞Θʁ
̏ʥઈ໎෈ଏ͵ͳ͞Θ͍͗Ζɽ͵ͧ͗ൊ͜ͱ͕ΕɼళԾΕదͶॽ͖Ηͱ͏Ζɽ̐ʥஸंͺɼॵৼंͲͺ͵͏͹Ͳɼ෾͖ͮͱ͏Ζ΍͹ͳ͢ͱॽ͏ͱ͏Ζɽ̑ʥ࠸‫ؾ‬ఈٝ͵ʹͶ΀ΗΔΗͱ͏͵͏ʤϓϧένϩ͵ʹɼ‫ܙ‬͹ՌָճదͶͺ෼ଏΕ͵͏ʥɽ
̒ʥClass ͵ʹ͹ߴ໪͗͵͏ɽঙ͢਒ΞͫϕϫήϧϝϱήͲͺࠖΖͫΘ͑ɽ͞͹Γ͑ͶΊͱ͏͚ͳɼॵΌͱ Python Ͷ΀Ηͪॵৼं͗ϕϫήϧϝϱήΝ޹͘Ͷ͵ΕɼΰʖϞ͵ʹ͹ϕϫήϧϞΝࣙ༟ͶࡠΗΖΓ͑Ͷ͵ΖͶͺɼΉͫΉͫଡ͚͹՟ୌ͗࢔ͮͱ͏
ΖΓ͑Ͷࢧ͓Ζɽ

2.3.  ೖͲ  ࡯͹೘໵ॽΝಣΊश͓Ζ઴হͲ়ղͪ͢ 3 ࡯͹೘໵ॽΝ 10 ೖͲಣΊश͓ͪɽ͵ͧɼ10 ೖͳ͏͑੏‫ݸ‬Νઅఈ͖ͪ͢Ͷͯ͏ͱͺɼ3 ೖ͖Δ 1 ुؔͲಣΊश͓Ζ͹ͺɼ୻͙ͤͱන໚Ν͵ͩΖͫ͜Ͷ͵ΖՆ೵
੓͍͗ΖɽΉͪɼ1 ͖݆΍͖͜ͱಣ΋͹Ͳͺɼ౏஦Ͳ‫ݑ‬Ͷ͵ΖՆ೵੓͍͗Ζͳߡ͓͖ͪΔ
Ͳ͍ΖɽҐԾͶɼ10 ೖͲ 3 ࡯͹೘໵ॽΝಣΞͫ‫ى‬࿧Νɼ६ඍ‫ؔغ‬Ν‫؜‬Όͱɼන͕̏Γ;න̐ͶΉͳΌͪɽ
න̏ʁ६ඍ‫ؔغ‬ೖ ෉ߨಊ‫ى‬࿧

2020 ೧ 2 ݆ 7 ೖ (ۜ)ঋ໨ॽవ(৿್͹‫ٸ‬వ)Ͷͱ 3 ࡯͹ຌ߬೘

2020 ೧ 2 ݆ 8 ೖ (ౖ)

Pythonista3 Ν apple store Ͳ߬೘2020 ೧ 2 ݆ 10 ೖ(ೖ)

Python3.8.1 Ν PC Ͷ΢ϱηφʖϩ઴হͲ়ղͪ͢ 3 ࡯͹೘໵ॽΝ 10 ೖͲಣΊश͓ͪɽ͵ͧɼ10 ೖͳ͏͑੏‫ݸ‬Νઅఈ͖ͪ͢Ͷͯ͏ͱͺɼ3 ೖ͖Δ 1 ुؔͲಣΊश͓Ζ͹ͺɼ୻͙ͤͱන໚Ν͵ͩΖͫ͜Ͷ͵ΖՆ೵
੓͍͗ΖɽΉͪɼ1 ͖݆΍͖͜ͱಣ΋͹Ͳͺɼ౏஦Ͳ‫ݑ‬Ͷ͵ΖՆ೵੓͍͗Ζͳߡ͓͖ͪΔ
Ͳ͍ΖɽҐԾͶɼ10 ೖͲ 3 ࡯͹೘໵ॽΝಣΞͫ‫ى‬࿧Νɼ६ඍ‫ؔغ‬Ν‫؜‬Όͱɼන͕̏Γ;න̐ͶΉͳΌͪɽ
t 62 tන̐ʁಣॽ‫ى‬࿧*ೖ ఖ௔೘໵ [1]͏ͬͻΞΏ͢͠͏ [2]κϫ͖ΔΏ͚͢͠ [3]1 ೖ໪ητʖζ 1ʛητʖζ 15

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2 ೖ໪

Chap. 1ʛChap. 4

3 ೖ໪

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Chap. 5

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4 ೖ໪

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Chap. 6

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5 ೖ໪

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Chap. 7ʛChap. 8

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6 ೖ໪

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Chap. 9ʛChap. 10

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7 ೖ໪

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Chap. 1ʛChap. 2

8 ೖ໪

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Chap. 3ʛChap. 4

9 ೖ໪

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Chap. 5ʛChap. 6

10 ೖ໪

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Chap. 7*3 ࡯͹ຌΝɼ1 ೖ໪Ͷ 1 ࡯໪ɼ2 ೖ໪͖Δ 6 ೖ໪Ͳ 2 ࡯໪ɼ7 ೖ໪͖Δ 10 ೖ໪Ͳ 3 ࡯໪ΝಣΊ͕͓ͪɽ͵͕ɼ[1]ͺ Python ௔೘໵ɼ[2]ͺκϫ͖ΔΏ͚͢͠ͺͣΌΖ Python ೘໵ɼ[3]ͺκϫ͖ΔΏ͚͢͠ͺͣΌΖ Python ೘໵Νྲྀ͢ͱ͏Ζɽ1 ೖ໪ʁ2 ݆ 12 ೖɼ2 ೖ໪ʁ
2 ݆ 14 ೖɼ3 ೖ໪ʁ2 ݆ 15 ೖɼ4 ೖ໪ʁ2 ݆ 17 ೖɼ5 ೖ໪ʁ2 ݆ 19 ೖɼ6 ೖ໪ʁ2 ݆ 20
ೖɼ7 ೖ໪ʁ2 ݆ 22 ೖɼ8 ೖ໪ʁ2 ݆ 23 ೖɼ9 ೖ໪ʁ2 ݆ 24 ೖɼ10 ೖ໪ʁ2 ݆ 25 ೖͲ͍Ζɽ‫گ‬ຌΝಢसͤΖͳͳ΍ͶɼΨϨζψϩ͵ΰʖϞϕϫήϧϝϱήͶேકͪ͢ɽ PC ͳ‫ܠ‬ଵ୼຦ʤiPhoneɼiPadʥͲ͕͞͵ͮͪΰʖϞϕϫήϧϝϱήࣰભ͹գఖΝන 3 ͶΉͳΌͪɽ͵͕ɼॵΌ͹ஊ֌ͺɼ‫گ‬ຌͶͪ͗ͮͪ͢ϕϫήϧϝϱή͹ָस಼༲ͳΨϨζψϩ͵ΰʖϞϕϫήϧϝϱή͗Ψʖώʖϧρϕ͢ͱ͏ͪɽࣰભͪ͢ϕϫήϧϝϱήͶͯ͏ͱɼࡠ੔ͪ͢
ϕϫήϧϞΝࣖ͢͵͗Δɼන̑͹‫ى‬࿧Νึͮͱ͏͚ɽ͞͹ஊ֌Ͳͺɼϕϫήϧϝϱήٗढ़ͳ෾͖ΕΏͤ͠ʤॵৼं͚͗ͯΕΏͤ͠Ͷ఼॑Ν͕͏ͱ͏Ζ͹Ͳɼ_main()_ͳ͏͑Γ͑͵ࡠΕ๏Ͷ͢ͱ͏͵͏ɽ৏͖ΔॳͶࣰߨ͠ΗΖϕϫήϧϞͶ͵ͮͱ͏ΖɽʥͶ఼॑Ν͕͏ͱ͏
Ζɽ۫ରద͵ೖఖͳ಼༲ͺɼҐԾ͹௪ΕͲ͍Ζɽ2 ݆ 19 ೖʁ͕Ί͚ͣϕϫήϧϞΝ͚ͯΖɽ
Python ௔೘໵ʤDay5 Stage17 ͶɼʗϏ΢λϱͲ͕Ί͚ͣΝ͚ͯΘ͑ʗ1 ೖ໪(2 ݆ 12 ೖ)ʥ
Ͷࣇͪ΍͹͍ͮͪʤ2 ݆ 27 ೖͶ‫͚෉͗ـ‬ʥɽཛྷ਼Ν͖͕ͯͮͪΊ͚ͣ͹ϕϫήϧϞ͹Ϩη
φΝ෉࿧̏ϕϫήϧϞ̏ʛ̑Ͷࡎͦͱ͕͚ɽϕϫήϧϞ͖̏Δ̑ͺɼϕϫήϧϞ͹ߨ਼Ν୻͚؈݁Ͷ͵ΖΓ͑Ͷͪͫ͢͜ͲɼಋͣಊࡠΝͤΖϕϫήϧϞͲ͍Ζɽ͵͕ɼॵৼंͺɼߨ਼͗ଡ͚͵ͮͱ΍ɼ෾͖ΕΏ͚ͤϕϫήϧϝϱήͪ͢๏͗ྒྷ͏ͳͫΘ͑ͳ͏͑‫״‬૟Ͳ͍
Ζɽસରͳ͢ͱɼʰΏ͢͠͏೘໵ॽͺɼॵৼंͶ༑͚͢͵͏ɽʱͳ͏͑һেͲ͍ΖɽΉͪɼ͞͞Ͳ͍ͪ͝ 3 ࡯͹‫گ‬ຌΝಣΊश͓ͪ‫ޛ‬Ͷɼࢂߡͳ͵ΖຌͶͯ͏ͱफ़΄ͱ͕͚ɽ೘໵
Python3[4]ͺɼC ‫ޢݶ‬Ͷ͕͜ΖηνϱξʖχͲ͍ΖΩʖωύϱˏϨροʖ͹ຌ [18]ͶଲԢ͢ɼࣛॽదͶ΍ϜωϣΠϩదͶ΍͓࢘ΖɽΉͪɼঙ͢਒Ξͫ೘໵ॽͳ͢ͱɼಅࢃྰஸ͹ຌ[19] ͗ྒྷ͏ͫΘ͑ɽࠖͮͪͳ͘΢ϱνʖϋρφͲ‫͢ࡩݗ‬ɼඉ৙Ͷༀཱིͮͪ HomePage ͹಼༲ΝΉͳΌͱຌͶͪ͢΍͹Ͳ͍Ζɽ

t 63 tන̑ʁϕϫήϧϝϱήࣰભฦೖ෉

PC

Python 3.8.1 (Anaconda)

iPad & iPhone

Pythonista 3

2 ݆ 19 ೖ͕Ί͚ͣϕϫήϧϞ

͕Ί͚ͣϕϫήϧϞ

2 ݆ 20 ೖήϧϓΡρέ͹࿇सήϧϓΡρέ͹࿇सʛ

TKinter Ν࢘ͮͱΊΖ

Examples(Game Tutorial)ΝΊΖ

2 ݆ 28 ೖTurtle Graphics Ν࢘͑Turtle Graphics Ν࢘͑

3݆1ೖ

έϧη

‫ܩ‬ট͹࿇स

3݆2ೖ

࠸‫ؾ‬ఈٝ΃ேક

3݆4ೖ

Ωϱφʖϩɼαρϙॄ߻ϓϧένϩਦ‫ܙ‬

3݆5ೖϓϧένϩਦ‫ܙ‬3݆3ೖηΫʖϕϫήϧϞ֋࢟3݆6ೖ

ηΫʖΰʖϞ

֌৒
‫׮‬੔

‫׮‬੔
ηΫʖϕϫήϧϞ֋࢟(ΫʖϚʖχ೘ྙαϱφϫʖϩ) ‫׮‬੔3݆9ೖηΫʖΰʖϞ
(Յଐౕιϱγʖαϱφϫʖϩ) ‫׮‬੔

3. 3\WKRQ ϕϫήϧϝϱήͶேક
10 ೖͲ‫گ‬ຌ͹ָसΝश͓ͪ͹ͬɼ‫گ‬ຌ͖ΔҲ୶཯ΗɼಢࣙͶඵժϕϫήϧϞͶேકͪ͢ɽ
‫ܙ‬͹Ռָճ͹୉͘͵τʖϜͲ͍Ζʰժ଀ʱͶ‫ؖ‬࿊͢ͱɼPython Ͷ಼ใ͠Ηͱ͏Ζ Turtle
Graphics ‫ ;ٶ‬tkinter Ν࢘ͮͱ͕ֈඵ͘ΝͤΖɽ

3.1. ࠸‫ؾ‬దΠϩβϨθϞ
Python Ν࢘ͮͱɼࣙΔΝ‫;ݼ‬ड़ͤߑଆΝ΍ͮͪ࠸‫ؾ‬దΠϩβϨθϞΝߡ͓Γ͑ɽ࠸‫ؾ‬దΠϩβϨθϞͺɼ3 ࡯͹‫گ‬ຌͲͺखΕѽΚΗͱ͕ΔͥɼॵΌͱซ͚ͳೋͨ͑͢Ͷซ͓͞Ζ͖΍͢Η͵͏ɽ͖͢͢ɼ࠸‫ؾ‬దΠϩβϨθϞͺɼ֌৒͹‫ࢋܯ‬Ώϓϧένϩਦ‫ܙ‬͹ඵժͶ࢘ΚΗΖɽ͵͕ɼϓϧένϩਦ‫ܙ‬ͺɼαρϙ‫ۄ‬તΏϘϱϫʖθν΢ϩ͗‫ܙ‬͹Ռָճࢿ͹නࢶͳ͢ͱ΍࢘ΚΗͱ͏Ζɽ࠹ॵͶɼ࠸‫ؾ‬ఈٝΝ࢘ͮͪ؈ୱͲࣰ͖ͯ༽ద͵ྭͳ͢ͱɼ֌৒͹‫ࢋܯ‬Νߡ͓Γ͑ɽ̿͹֌৒ͺɼnʄ Qʹ Qʷ1)ʄͲఈٝ͠ΗΖΓ͑Ͷɼnʄͺ࠸‫ؾ‬దͶ‫͠ࢋܯ‬ΗΖɽ͞ΗΝ PythonͲॽ͏ͱΊΖͳɼ1

def kaijyou(n) :2

total = 13

if 1 > n :4
5

return total
return n * kaijyou( n t 1 )ͪͮͪ̓ߨ͹‫͜ॽͱ͢ͳ਼ؖ‬Ζɽ‫ ਼ؖ‬kaijyou() ͹஦Ͳɼ5 ߨ໪Ͳɼkaijyou()Ν‫;ݼ‬ड़͢ͱ͏Ζ͹͗෾͖Ζɽ‫਼ؖ‬͹஦Ͳࣙ෾ࣙਐΝ‫;ݼ‬ड़͢ͱ͕Εɼ࠸‫ؾ‬దͶॽ͚͞ͳ͗Ͳͪ͘ɽ
࣏Ͷɼ࠸‫(਼ؖؾ‬recursive function)Ν࢘ͮͱɼࣙ‫ހ‬૮ࣇ(self-similar)͵ਦ‫ܙ‬Νඵ͏ͱΊt 64 tΓ͑ɽਦ̑ͺɼ࠸‫਼ؖؾ‬Ν࢘ͮͱඵ͏ͪɼΩϱφʖϩॄ߻ (Cantor Set)ͳαρϙ‫ۄ‬ત(KochCurve)Ͳ͍Ζɽϓϧένϩ‫ش‬ՁָͲɼΓ͚எΔΗͪਦ‫͗ܙ‬ඵͪ͜ɽΩϱφʖϩॄ߻(Cantor
Set)ͳαρϙ‫ۄ‬ત(Koch Curve)Νඵ͚ϕϫήϧϞΝ෉࿧ 1 ϕϫήϧϞ 4 Ͷࣖͤɽ͵͕ɼIDLE
Ͳ Python Shell Νཱིͬ৏͝ͱ Help ͹ Turtle Demo Νમ΁ͳɼChaosʤLogistic ࣺ଀Ν΍ͬ͏ͪ ChaosʥɼFractalɼPenrose tiling ͵ʹ͹υϠΝֺ͢΋͞ͳ͗Ͳ͘Ζɽ

DEਦ̑ʁϓϧένϩਦ‫ܙ‬ʁ a) αρϙ‫ۄ‬ત(Koch curve)ɼb) Ωϱφʖϩॄ߻(Cantor Set)ɽ

4. 3\WKRQ ϕϫήϧϝϱήͶேક
4.1. ࠸‫ؾ‬దΠϩβϨθϞ
Pythonista3 Νཤ༽͢ͱɼiPhoneˏ iPad Ͳ Python ͹ϕϫήϧϝϱήΝ͕͞͵ͮͪɽ͵
͕ɼPythonista3 ͺɼPython̐ͳ Python̑Νγϛʖφ͢ͱ͏Ζ͗ɼ͞͞ͲͺɼPython̑(ver,3.6)Ν࢘ͮͪɽΉͪɼPythonista3 ͹ಝ௅͹ҲͯͺɼiPhone (iPad)͹֦झιϱγʖΝ͓࢘Ζϧ΢ϔϧϨΝγϛʖφ͢ͱ͏Ζ͞ͳͲ͍Ζɽ͞͹ཤ఼Νਫ਼͖ͤͳɼଡ༹͵ϕϫήϧϞ͗
੣ࡠͲ͘Ζɽ͞ ͞ Ͳ ͺ ɼ iPhone (iPad)͹ Յ ଐ ౕ ι ϱ γ ʖ Ν ࢘ ͮ ͪ ΰ ʖ Ϟ Ν ࡠ ͮ ͪ ɽ Ή ͥ ɼ importmotion Ͳϧ΢ϔϧϨΝಣΊࠒ΋͞ͳͶΓΕɼmotion.get_gravity()Ͳ(x, y, z)͹॑ྙΝ‫ܯ‬଎Ͳ͘ΖΓ͑Ͷ͵Ζɽ஭қ͢͵͜Ηͻ͵Δ͵͏͹ͺɼmotion.start_updates()ͳ͢ͱ͕͖͵͏ͳɼ‫ܯ‬଎Ν࢟ΌΔΗ͵͏͞ͳͲ͍Ζɽਦ 4 ͶɼηΫʖΰʖϞ͗ಊࡠ͢ͱ͏ΖྭΝࣖͤɽΉͥɼϪϗϩΝ͖̏̐ͶΓͮͱમ΁ͳΰʖϞ͗ηνʖφͤΖɽͨ͢ͱɼiPhone Νܑ͜Ζ͞ͳͶΓͮͱɼηΫʖϢʖΝࠪӊͶҢಊͦ͠Ζɽηνʖφஏ఼͖Δβʖϩϧ΢ϱͶࢺΖΉͲ͹ؔͶɼϜʖέΝ௪Ζͳɼϛ΢ϱφΝΰρφͲ͘Ζͳ͏͑ΰʖϞͲ͍Ζɽ

t 65 tਦ̒ʁηΫʖΰʖϞ͹ࣰߨྭΝ 2 ྭࣖͤɽa)ɼc)Ͷ Turtle Graphics ͹ड़ྙΝɼb)ɼd)Ͷ Console͹ड़ྙΝࣖͤɽ

5. ͕ΚΕͶ
60 ࡂΝ௔͓ͱ৿͢͏ϕϫήϧϝϱή‫ޢݶ‬Νָ΁͞ͳ͗Ͳ͘Ζ͖͹ࣆྭ‫ڂݜ‬Ν͕͞͵ͮ
ͪɽPython ͶॵΌͱৰΗͱɼʹ͹͚Δ͏͹‫Ͳؔغ‬ΰʖϞϕϫήϧϞ͚͗ͯΗΖ͖Ͷேકͪ͢ɽ͠ͱɼऑ೧ंͳ߶ྺंͶ୉͘͵ࠫͺ͍Ζ͹ͫΘ͖͑ʃऑ͏ָਫ਼͗ Python Νʹ͹Γ͑ͶͳΔ͓ͱ͏Ζ͖ΝஎΖͪΌɼ୉ָਫ਼͕Γ;୉ָӅਫ਼ΝଲেͶ Python Ͷ‫ͤؖ‬ΖΠϱίʖφ௒ࠬΝ͕͞͵ͮͪɽָਫ਼͹ Python ͶଲͤΖָसয়‫ڱ‬ΝஎΖ͞ͳ͗Ͳ͘ɼ͠ΔͶ৿͢͏ϕϫήϧϞ‫ޢݶ‬ͶखΕૌ΋ͳ͘Ͷɼ߶ྺंͳऑ೧ं͹ؔͶࠫҡ͍͗Ζ͖ʹ͖͑Ν໎Δ͖Ͷ
Ͳ͘ΖՆ೵੓͍ΖɽෳҬ୉ָ޽ָ෨‫ؽ‬փʀεητϞ޽ָՌ͹ඉત‫ܙ‬Ռָ‫࣪ڂݜ‬͹ ָ෨ָਫ਼ͳ୉ָӅਫ਼ΝଲেͶ 2020 ೧ 5 ݆ 11 ೖͶϕϫήϧϝϱήͶ‫ͤؖ‬ΖΠϱίʖφ௒ࠬΝ͕͞͵ͮͪɽΠϱίʖφ͹಼༲ͺɼPython Ν஦ৼͶ޹͘͵ϕϫήϧϝϱή‫ޢݶ‬ΏՁΝͪ͢͏͖Ͷ‫ͤؖ‬Ζ΍͹Ͳ͍Ζɽ
ࣰࡏ͹Πϱίʖφͺɼ෉࿧̐ͳ͢ͱ‫͏ͱ͢ࡎܟ‬Ζɽ͵͕ɼ৿‫ ܗ‬αϫψΤΡϩη‫״‬ઝ঳ʤCOVID-19ʥ͹ͪΌɼGoogle Classroom ղ͢ͱΠϱίʖφ௒ࠬΝࣰࢬͪ͢ɽΠϱίʖφ݃ՎͺɼϋρφΝཤ༽͢ͱ PDF ϓΟ΢ϩͶ͢ͱశ෉ϓΟ΢ϩͳ͢ͱɼϟʖϩͲૻͮͱ΍Δ͏ɼյफͪ͢(5 ݆ 11 ೖ 6 ਕɼ12 ೖ 1 ਕɼ13 ೖ 6 ਕɼ14 ೖ 2 ਕɼ15 ೖ 1 ਕ)ɽ‫ؽ‬փʀεητϞ޽ָՌ͹ 4 ೧ਫ਼ 8 ਕɼஎ೵εητϞ޽ָ઒ߊ͹୉ָӅਫ਼ 1 ೧ਫ਼ 3 ਕɼ2 ೧ਫ਼ 5 ਕ
͹ 16 ਕ͹஋ࢢָਫ਼͖Δյ౶ΝಚΖ͞ͳ͗Ͳͪ͘ɽΠϱίʖφ͹ॄ‫݃ܯ‬ՎΝන̒ͶࣖͤɽΠϱίʖφ͖Δ Python Ͷ‫ͤؖ‬Ζ‫͗ັڷ‬ඉ৙Ͷ߶͏͞ͳ͗෾͖ΖɽΠϱίʖφͶ‫͢ྙڢ‬ͱ͚Ηָͪਫ਼સҽ͗ Python Νห‫ͳ͏ͪ͢ک‬౶͓ͱ͏ΖɽΉͪɼΆͳΞʹ͹ָਫ਼͗ PythonͲϕϫήϧϞͪ͢‫ݩܨ‬Ν΍ͮͱ͏ͪɽ͵͕ɼෳҬ୉ָ޽ָ෨͹‫ؽ‬փʀεητϞָՌͲͺࣰ‫ݩ‬ΏԍसͲ Python Ν࢘ͮͱ͏ΖɽҲൢదͶɼસࠅ͹޽ָ෨ͲɼC ‫͕ޢݶ‬Γ; Python ͺद‫ۂ‬ͶૌΊࠒΉΗͱ͏Ζɽָਫ਼͹‫ັڷ‬Ͷ‫ͱؖ͢‬ͺɼͽͳͯͶยͮͱ͏͵͖ͮͪ͗ʰਕ޽எ೵ʱ
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60 Python ͺɼඉ৙Ͷ‫ॴޢݶ͵ྙک‬ཀྵͶ‫ͤؖ‬Ζ‫͏ͱ༙͢ʹ͵਼ؖ‬Ζ[4][19]ɽͨΗΑ͓ɼͨ͹ઞࡑ೵ྙΝӀ͘ड़ͨ͑ͳͤΖͳʤ༹ʓ͵‫ʹ͵਼ؖ‬Ν͕࢘͑ͳͤΖͳʥɼ݀͢ͱ༑͢͏ϕϫήϧϝϱή‫͏͵͓ݶͳ ޢݶ‬Ն೵੓͍͗ΖɽC ‫ʹ͵ޢݶ‬͹Άְ͓͑͗Ζ͞ͳ͗ঙ͵͏ɽ2ͯ͹‫ޢݶ‬͹ηνϱξʖχ͵ຌɼΩʖωύϱˏϨροʖ͹ C ‫ޢݶ‬͹ຌ[18]ͳ Python3[4]͹
ຌ͹޲͠ΝർֳͤΖͳ෾͖Ζɽ‫ٱ‬ͶɼPython ͺ‫਼ؖ‬Ώϧ΢ϔϧϨ͗ैࣰ͢ͱ͏Ζ͹ͲɼҲ୶ɼཀྵմͤΗͻֺ͖΍͢Η͵͏ɽ͞Ηͺɼ‫਼ؖ‬Ώϧ΢ϔϧϨΝ࢘͏͞͵ͤ͹Ͷ͖͖࣎ؔ͗
Ζ͞ͳΝқັͤΖɽ‫ܙ‬͹ՌָճͶ͕͏ͱɼժ଀ॴཀྵɼϏνʖϱ೟ࣟͺɼָճઅཱི౲ॵ͖Δ͹୉͘͵‫ڂݜ‬τʖϜͲ͍ΖɽPython ͺɼժ଀ॴཀྵΏϏνʖϱ೟ࣟͶɼ๴୉͵ϧ΢ϔϧϨΝ༙͢ͱ͏ΖɽໃྋͲ͍Ζ͹΍୉͘͵ಝ௅Ͳ͍Εɼॵ‫غ‬αηφΝචགྷͳָͦͥसΝ࢟ΌΔΗΖɽॵิϪϗϩ͹೘໵ॽͲ͓͠ɼժ଀ॴཀྵΝର‫͘Ͳݩ‬Ζ[6]ɽ࣏ͶɼPython Ͷঙ͢‫׵‬ΗͪΔಣ΋ͳ໚പ͏ຌ
Ͷͯ͏ͱफ़΄Ζɽ‫ࡑݳ‬ɼPython ͗஭໪͠Ηͱ͏Ζ͹ͺɼAI ͳ‫ؖ‬࿊͗਄͏ͪΌͲ΍͍Ζɽ
AIɼಝͶɼDeep Learning ͹෾໼Ͷ͕͏ͱͺ Python ͗ηνϱξʖχͶ͵ͮͱ͏Ζ[20][21]ɽ
Python ͳ Deep Learning Νห‫ͱ͢ک‬Ίͪ͏ͳࢧͮͪ๏Ͷͺɼࡊ౽ஸʁκϫ͖ΔࡠΖ Deep
Learning[20]ΝપΌΖɽ࠹ఁ‫ݸ‬͹ Python ͹෰स͖Δ࢟ΉΕɼ‫ܙ‬͹Ռָճ͹कགྷ͵τʖϜ͹ҲͯͲ͍Ζժ଀೟ࣟͶ‫ͱؖ͢‬ɼϏʖιϕφϫϱ͖Δωϣʖϧϩϋρφϭʖέ΃ͳմઈ͢
ͱ͏ΖɽͯΉͥ͘Ώͤ͏ͳ͞ΘΝ஺ೣͶึ͏͵͗Δɼ෾͖Εҝ͚ཀྵմͲ͘ΖΓ͑Ͷ޽෋͠
Ηͱ͏Ζɽ‫ࡑݳ‬ɼ‫ܠ‬ଵ୼຦͓͍͠ΗͻɼPC ͗͵͚ͱ΍ Python ͹ϕϫήϧϝϱήͶேકͲ͘Ζ͹Ί͵Δͥɼ਄૜ָस͵ʹΝର‫͘Ͳݩ‬ΖɽGoogle Colaboratory [22]Ώ Sony ͹ Neural
Network Console[23]ͺɼPython Ͳϧ΢ϔϧϨΝ࢘͏ Web ৏Ͳ਄૜ָसΝࣰߨͲ͘ΖΠϕϨίʖεϥϱͲ͍Ζɽ͞ΗΔͶΓͮͱɼPython ͕Γ;‫ؽ‬փָसΏ Deep Learning Νָ΁͘͢͏͗ఁ͚͵ͮͱ͘ͱ͏Ζ͞ͳͺ‫ັڷ‬਄͏͞ͳͲ͍ΖɽҐ৏ɼ60 ࡂΝӿ͓ͱ 10 ೖͲ Python ͲΰʖϞΝࡠΗΖΓ͑Ͷ͵Ζ͖ɼ͕Γ;‫گ‬ຌ͹ಢ
t 67 tसͶΓΖָसգఖ͹ࣆྭΝࣖͤ͞ͳ͗Ͳͪ͘ɽΉͪɼ20 ೖ༪ΕͲɼιϱγʖ͵ʹΝ࢘ͮͪຌ֪ద͵ΰʖϞΝ੣ࡠͲ͘ΖΓ͑Ͷ͵ͮͪࣆྭΝࣖͦͪɽ‫ࡑݳ‬ɼ৿‫ܗ‬αϫψΤΡϩη‫״‬ઝ঳Ք͹ॶҟͲɼࣙ୒‫ۊ‬ແ͗ਬও͠Ηͱ͏ΖɽҲൢదͶࡂΝͳΖͳͳ΍Ͷ৿͢͏͞ͳͶேક
ͤΖ͞ͳ͗‫ۦ‬घͶ͵ͮͱ͚Ζ͗ɼࣙ୒‫ۊ‬ແ͹ཤ఼Νਫ਼͖͢ɼ৿͢͏͞ͳͶேકͤΖοϡϱη͖΍͢Η͵͏ɽ࠹‫ޛ‬Ͷɼຌ‫͗ڂݜ‬ɼ৿͚͢ϕϫήϧϝϱή‫ޢݶ‬Ͷேક͢Γ͑ͳͤΖ๏ʓ͹ϐϱφͶ͵Ζ͞ͳΝ‫غ‬ଶͤΖɽःࣛຌ‫ڂݜ‬Ν͕͞͵͑৏Ͳ༙ӻ͵٠࿨Ν͢ͱ͚ͫͮͪ͠∃ీभध‫گ‬द͕Γ;ΠϱίʖφͶ‫͚ͱ͢ྙڢ‬ΗͪෳҬ୉ָ޽ָ෨ඉત‫ܙ‬Ռָ‫࣪ڂݜ‬͹ϟϱώʖͶ‫״‬ः͏ͪ͢Ήͤɽ࠹‫ޛ‬Ͷɼ࿨ชΝրఙͤΖ৏Ͳ༙ӻ͵αϟϱφΝ͚ͫͮͪࠬ͠ಣं͕Γ;ฦॄं͹ੁӞ໎߄ࢱͶ‫״‬ः͏ͪ͢Ήͤɽ

ࢂߡช‫ݛ‬[1]

Laura T. Germine, Bradley Duchaine, Ken Nakayama: Where cognitive development andaging meet: Face learning ability peaks after age 30, Cognition, Volume 118, Issue 2,
201-210 (2011)[2]ಛౣ‫ݟ‬ҫࢥճ HomePage:

https://tokushima.med.or.jp/[3]ন ଞ ଢ ࿢ : iPhone Ŋ 82 ࡂ ೖ ຌ ਕ ֋ ൅ ं ŋ ͺ Ձ ͗ η β ΢ ͹ ͖ . ౨ ༺ ‫ ࡃ ܨ‬ONLINE,
2017/06/05 https://toyokeizai.net/articles/-/174799[4]

Bill Lubanovic ஸɼࡊ౽߃‫ ؿ‬؄༃ɼௗඎ߶߄ ༃ʁ೘໵ Python3ɼΨϧ΢ϨʖʀζϡϏϱɼpp. 567 (2015)[5]

Google for Education ͹ *RRJOHbV3\WKRQ&ODVV:
https://developers.google.com/edu/python/[6]έζϧඊߨ‫ط‬ʁκϫ͖ΔΏ͚͢͠ͺͣΌΖ Python ೘໵ ʗ‫͖ેخ‬Δηνʖφ͢ͱɼΰʖϞͰ͚Εɼ‫ؽ‬փָसΉͲָ·͑ʄʗɼϜ΢ψϑड़൝ɼ pp. 255 (2018)[7]

Krishna : Guru99

www.guru99.com[8]

The 2015 Top Ten Programming Languages
https://spectrum.ieee.org/computing/software/the-2015-top-tenprogramming-languages[9]

The 2016 Top Programming Languages
https://spectrum.ieee.org/computing/software/the-2016-top-programming-languages[10] The 2017 Top Programming Languages t IEEE Spectrum Ranking
https://spectrum.ieee.org/static/interactive-the-top-programming-languages-2017[11] The 2018 Top Programming Languages t IEEE Spectrum Ranking
https://spectrum.ieee.org/static/interactive-the-top-programming-languages-2018[12] The Top Programming Languages 2019 t IEEE Spectrum Ranking
https://spectrum.ieee.org/computing/software/the-top-programming-languages-2019[13] ҈౽ਜ਼๗ʁೖ‫ ܨ‬Xtech (2019 ೧ 11 ݆ 18 ೖ)

t 68 thttps://xtech.nikkei.com/atcl/nxt/column/18/01068/111100001/[14] Python Software Fundation: https://www.python.org/[15] Ruby 2.7.0 ϨϓΟϪϱηϜωϣΠϩ :https://docs.ruby-lang.org/ja/latest/doc/glossary.html[16] ‫ٶ‬ઔ͓ΕࢢʁPython ௔೘໵ ʷϠϱτΡͳָ΁ͺͣΌͱ͹ϕϫήϧϝϱήʷɼΨʖϞऀɼpp.179 (2020)[17] ྵ໨͖ͪ͹Ε ˏ ੁ୫໽݆ʁ͏ͬͻΞΏ͢͠͏ Python ͹‫گ‬ຌʷਕ‫͓گ͗ࢥߪـ‬Ζ‫ેخ‬͖Δγʖώʖγ΢χ֋൅ΉͲʷɼ΢ϱϕϪηɼ pp. 271 (2017)[18] ΩʖωύϱɼϨροʖ ੶ీ੘‫ٳ‬༃:ϕϫήϧϝϱή‫ ޢݶ‬C ୊ 2 ൝

ANSI ‫֪و‬६‫ڎ‬ɼ‫ڠ‬ཱིड़൝ɼpp.360 (1989)[19] ಅࢃྰʁPython ࣰભ೘໵ɼٗढ़඲࿨ऀɼ pp. 349 (2020)[20] ࡊ౽ ߃‫ؿ‬ʁκϫ͖ΔࡠΖ Deep Learning ʗPython Ͳָ΁υΡʖϕϧʖωϱή͹ཀྵ࿨ͳࣰૹʗɼΨϧ΢ϨʖʀζϡϏϱɼ pp.298 (2016)[21] Antonio Gulli (ΠϱφωΨʀήρϨ) ˏ Sujit Pal (γζρφʀϏϩ)ஸɼ୉۴ਜ਼໾ɼ‫ٳ‬ฯཾ޼ɼ஦ࢃޭध༃ʁ௜‫ ״‬Deep Learning tPythonʹKeras ͲΠ΢υΠΝ‫ܙ‬ͶͤΖϪεϒ -ɼΨϧ΢ϨʖʀζϡϏϱɼ pp.309 (2018)[22] Google Colaboratory: https://colab.research.google.com/notebooks/intro.ipynb#[23] Sony Neural Network Console: https://dl.sony.com/console/#/dashboard

t 69 t෉࿧̏ Python ϕϫήϧϞ͹Ϩηφ
ʴ஭ʵՁ࣎ϕϫήϧϞ͗‫׮‬੔ͪ͢͹͖͗෾͖ΖΓ͑ͶɼϕϫήϧϞ໌͹ӊวͶ‫ ׮‬੔ೖ࣎͗‫͠ى‬Ηͱ͏Ζɽ͵͕ɼϕϫήϧϞ 5 ͹ηΫʖΰʖϞͺ iPhone(iPad)Pythonista 3 ༽Ͳ͍ΖɽϕϫήϧϞ̏# ͕Ί͚ͣ

2020 ೧ 2 ݆ 19 ೖfrom random import *ofuda = ['୉٤', '٤', 'ঘ٤', '຦٤', '‫'ڡ‬, '୉‫]'ڡ‬
print('ࠕೖ͹ӣ੐ͺ', end=' ')dice = 6 * random()
print(ofuda[int(dice)], end=' ')print('Ͳͤɽ')ϕϫήϧϞ 2ϕϫήϧϞ 1 Ν؈݁Ͷॽ͏ͪ΍͹from random import *ofuda = ['୉٤', '٤', 'ঘ٤', '຦٤', '‫'ڡ‬, '୉‫]'ڡ‬
print('ࠕೖ͹ӣ੐ͺ', + choice(ofuda) +'Ͳͤɽ')ϕϫήϧϞ̑

ϕϫήϧϞ 1 Ν 2 ߨͲ؈݁Ͷॽ͏ͪ΍͹from random import *print('ࠕೖ͹ӣ੐ͺ{}Ͳͤɽ'.format(choice(['୉٤', '٤', 'ঘ٤', '຦٤', '‫'ڡ‬, '୉‫)))]'ڡ‬ϕϫήϧϞ̒# Ωϱφʖϩॄ߻αρϙ‫ۄ‬ત2020 ೧ 3 ݆ 4 ೖ

import turtle
def moveturtle(x, y):turtle.penup()
turtle.goto(x,y)
turtle.pendown()
def koch(length1, n):
if n == 0 :turtle.forward(length1)else:
koch(length1/3, n - 1)
turtle.left(60)koch(length1/3, n - 1)
turtle.right(120)koch(length1/3, n - 1)
turtle.left(60)koch(length1/3, n - 1)
t 70 tdef cantor(length2, n):
if n < 1 :turtle.forward(length2)
else:
cantor(length2/3, n - 1)
turtle.penup()
turtle.forward(length2/3)
turtle.pendown()
cantor(length2/3, n - 1)
turtle.setup(400, 300)
turtle.speed(0)turtle.hideturtle()
turtle.color('blue')
moveturtle(-150, 0)koch(300, 5)
turtle.width(2)turtle.color('black')
moveturtle(-150, -75)
cantor(300, 5)ϕϫήϧϞ̓ʤiPad ༽ηΫʖ# ηΫʖηϧϫʖϞιϱγʖ॑ྙՅଐౕ࢘༽ʥ

2020 ೧ 3 ݆ 3 ೖ(րྒྷ

3 ݆ 9 ೖ)import motion
import soundimport turtle
import time
def moveP(x, y):turtle.penup()
turtle.goto(x, y)turtle.pendown()
turtle.setup(600, 500) # ήϧϓΡρέ͹γ΢θΝ͘ΌΖ
turtle.speed(500)t 71 t# Drawing Goal Lineturtle.color('green')
moveP(-200, -150)
turtle.goto(200, -150)# Define the position and color of targets
target ={1:(0, 100), 2:(50, 10), 3:(-30, -50), 4:(-70,-120)}
turtle.color('red')for i in range(4):x = target[i+1][0]y = target[i+1][1]moveP(x,y)
turtle.begin_fill()
turtle.circle(5)turtle.end_fill()
turtle.color('blue')turtle.speed(8) # speed(6)͖ speed(9)͚Δ͏͗న౲ͨΗҐ৏ͫͳඵժ͗ؔͶ߻Κ͵͏

turtle.right(90) # turtle (໾һ) Ν઴޴͘ͶͤΖ# Starting position (0, 150)
x=0
y = 150
moveP(x, y)turtle.dot(1)# Select game level
gamelevel = 1
print('Select the game level. 1 = easy or 2 = hard')
gamelevelstring = input()gamelevel = int(gamelevelstring)
if gamelevel != 2: gamelevel = 1
dx = 0score = 0
motion.start_updates() # ͞͹αϜϱχ͗͵͏ͳɼmotion.get_gravity()ً͗ಊ͢͵͏sound.play_effect('arcade:Coin_4')
while y > - 150:
t 72 ttime.sleep(0.01) # ֶ࣎ؔؔභୱҒ

0.1 භ΍ಊࡠͤΖΓ͑ͫ

3݆7ೖ

command = 's' # ͞͞Ͳɼร਼ command Νఈٝ͢ͱ͕͖͵͏ͳɼNameError ͗൅ਫ਼gravity = 10 * motion.get_gravity() # gravity = (x, y, z) ͳ͵ͮͱ͏Ζ gravity[1]Ͳ x αϱϛʖφΝ࢘༽
if gravity[1] > 0.2: command = 'l'

if gravity[1] < - 0.2 : command = 'r'
if 's' in command : dx = 0
if 'l' in command : dx = - 2
if 'r' in command : dx = 2
y -= 1x += dx
turtle.goto(x, y)# Player ͹Ңಊfor i in range(4):# Checking touch the targets

xtmp = target[i+1][0]ytmp = target[i+1][1]rr = (x - xtmp)**2 + (y - ytmp)**2
if rr < (10 / gamelevel) :sound.play_effect('arcade:Coin_4')
score += 100 * gamelevelsound.play_effect('arcade:Explosion_3') # End of game.
print('Game Level =', gamelevel,'Score =', score)t 73 t෉࿧̐ϕϫήϧϝϱή‫ޢݶ‬Ͷ‫ͱؖ͢‬೧݆ೖ੓พʤ ஋ ʀ ঃ ʥ೧ྺΠϱίʖφͶ‫͢͏ب͕ྙڢ‬Ήͤɽࡂʤ˚ͶοΥρέ͢ͱ͚ͫ͠͏ʥ

1. Python Ν࢘ͮͱϕϫήϧϝϱήͪ͢͞ͳ͍͗ΕΉ͖ͤʃ
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6. ҐԾͶࣖ͠Ηͪ෾໼͹஦Ͳ‫͍͗ັڷ‬Ζ΍͹Ν‫͏͚ͫ͠ͱ͓گ‬ʤ෵਼Նʥɽ
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৚ๅ͹खΕѽ͏Ͷͯ͏ͱʁຌΠϱίʖφͲಚΔΗͪ‫ݺ‬ਕ৚ๅͶ‫ͱؖ͢‬ͺɾ‫׈ڂݜ‬ಊ͹໪ద͹ΊͲ࢘༽͢ɾ͟ຌਕ͹ಋқ͵͚୊ࢀंͶ֋ࣖʀ఑‫ͤڛ‬Ζ͞ͳͺ͏ͪ͢ΉͦΞɿ

t 74 t࿊ࡎߪ࠴‫ܙ‬͹Ռָճࢿ ୊ 35 ‫ ת‬୊ 1 ߺColaboratory Ͳ‫ܙ‬͹ՌָΝֺ͢΍͑t ୊ 1 յ αρϙ‫ۄ‬તΝඵ͏ͱΊΓ͑ t
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Python ͺɼ‫ࡑݳ‬΍ͮͳ΍ਕ‫͍͗ـ‬Ζϕϫήϧϝϱή‫͍Ͳޢݶ‬ΖɽΉͪɼϕϫήϧϝϱή
ॵৼंͶ΍ָस͗༲ҝ͵ϕϫήϧϝϱή‫ͱ͢ͳޢݶ‬எΔΗͱ͏Ζɽ͖͢͢ɼαϱϒϣʖνͶ෈‫׵‬Η͵ॵৼंͶͳͮͱͺɼָस‫ڧ؂‬Ν੖͓Ζஊ֌ͲͯΉͥ͏ͱ͢Ή͖͑΍͢Η͵͏ɽͨ͹ͪΌɼଡ͚͹ Python ͹‫گ‬ຌͺɼ΢ϱηφʖϩͶ̏ͯ͹হΝׄ͏ͱ͏Ζɽ΍͢ɼ΢ϱ
ηφʖϩͤΖ͞ͳ͵͚ɼͤ͛Ͷ Python Ν͓࢘ͪΔɼϕϫήϧϝϱή͹ύʖχϩ͗ఁ͚͵ΖͫΘ͑ɽ
Google ͗఑‫ͤڛ‬Ζ Colaboratory[1]ͺɼPC ͵ʹΝචགྷͳͦͥɼηϜʖφϓΧϱ͵ʹ͹‫ܠ‬ଵ୼຦͍͗Ε Web ϘʖζͶΠέιη͓͠Ͳ͘Ηͻཤ༽Ͳ͘ΖɽColaboratory ͺɼJupyter
Notebook ΝϗʖηͶ͢ͱ͕Εɼ‫ؽ‬փָसɼAIʤArtificial Intelligenceʥ͵ʹͶ΍༲ҝͶखΕૌΌΖΓ͑Ͷ͵ͮͱ͏Ζɽ͞͞ͲͺɼColaboratory Ν࢘ͮͱɼ‫ܙ‬͹Ռָ͹໚പ͠Ν়ղͤΖɽ

2. &RODERUDWRU\ ͳͺColaboratory ͺ ɼ Web ৏ Ͳ Python Ν ࢘ ͓ Ζ ‫ ͍ Ͳ ڧ ؂‬Ζ ɽ Google Π Ω Τ ϱ φ
ʤxxx@gmail.comʥΝ΍ͮͱ͕Ε Colaboratory ͹ HomePage ͶΠέιη͓͠Ͳ͘Ηͻɼͤ͛Ͷ͓࢘ΖΓ͑Ͷ͵Ζɽ͵͕ɼGoogle ΠΩΤϱφͺ؈ୱͶ೘घͲ͘Ζ͹ͲɼͤͲͶ࣍ͮͱ
͕ΔΗΖ๏΍ଡ͏ͫΘ͑ɽΉͫɼGoogle ΠΩΤϱφΝ΍ͮͱ͏͵͏ಣंͺɼΉͥΠΩΤϱφΝ೘घ͢Γ͑ɽ࣏ͶɼColaboratory ͹ HomePageʤhttps://colab.research.google.com/ʥͶΠέιη͢Γ͑ɽਦ 1 ͶࣖͤΓ͑͵ʰColaboratory ΃Γ͑ͨ͞ʱ͹Ϙʖζ͗ड़ͱ͚ΖͫΘ͑ɽ

t 75 tਦ̏ʁʰColaboratory ΃Γ͑ͨ͞ʱ͹ Web ϘʖζɽColaboratory ͹ͺͣΌͶͪ͗ͮ͢ͱɼColab όʖφϔρέΝ࢘ͮͱΊΖɽColab όʖφϔρέͺɼPython ͹αʖχΝ‫ى‬फ़͢ͱࣰߨͲ͘Ζ‫͍Ͳڧ؂‬Ζɽਦ 1 ͹αʖχιϩͲͺɼు୐ͳ͢ͱ Python Ν࢘ͮͱ͏ΖɽҐԾͶͨ͹ྭΝ‫ͱݡ‬ΊΓ͑ɽ
seconds_in_a_day = 24 * 60 * 60
seconds_in_a_day

1 ߨ໪Ͳร਼ seconds_in_a_day Ͷ਼஍Νେ೘͢ɼ2 ߨ໪Ͳร਼ seconds_in_a_day ͹஍Νड़ྙͦ͠ͱ͏Ζɽਦ̏͹Ծ෨Ͷ͍Ζ[Ηͱ]͹Ғ஖Ͷϛ΢ϱνʖΝҢಊͦ͠ΖͳɼࣰߨϚνϱ⾥͗ࠪ৏୼Ͷ‫ݳ‬ͳ͵Ζ͹ͲɼέϨρέͤΖ͞ͳͲࣰߨ͠ΗΖɽαʖχιϩͳ‫݃ࢋܯ‬Վ͹ड़ྙΝࣖͤɽਦ̐ʁ࠹ॵͶड़ͱ͚Ζαʖχιϩͳड़ྙྭɽ

t 76 tPython ͺϧ΢ϔϧϨΝ import ͤΖ͞ͳͶΓΕɼήϧϓΡΩϩͶήϧϓΝඵ͚͞ͳ͗Ͳ
͘Ζɽ΍ͬΘΞɼColaboratory ΍ժ଀΍खΕѽ͓ΖɽPython ͹਼஍‫ࢋܯ‬ϧ΢ϔϧϨͲ͍Ζ
numpy Ν࢘ͮͱ‫݃ͪ͢ࢋܯ‬ՎΝɼmatplotlib Ν࢘ͮͱඵժͪ͢ήϧϓ͹ྭ΍ࣖ͠Ηͱ͏Ζɽਦ̑ͶɼPython ͹αʖχιϩͳड़ྙ͹෨෾Νൊ͘ड़ͪ͢΍͹Νࣖͤɽਦ̑ʁPython ͹αʖχιϩͳήϧϓड़ྙɽਦ 3 ͹Ծ෨͹ήϧϓͺཛྷ਼Νϕϫρφͪ͢΍͹Ͳ͍Ζɽnumpy ϧ΢ϔϧϨ͹ random ‫ؖ‬਼ͶΓΕ 0 ͖Δ 100 ͹ཛྷ਼Νਫ਼੔͢ɼʶ200 ͹ώ΢ΠηΝՅ͓ͪ΍͹Ν 100 ‫͚ͯݺ‬Εϕϫ
ρφ͢ͱ͍Ζɽαʖχιϩ͹஦͹਼஍Νฦॄ͢ͱɼ͏Θ͏Θͳร͓ͱΊΓ͑ɽΉͥͺ͍ΉΕ͵ΏΉͥɼΏͮͱΊΖ͞ͳͶΓΕ‫׵‬ΗΓ͑ɽϕϫήϧϝϱή͹ָसͶ͕͏ͱͺɼ‫ݩܨ‬৏ʰस͑ΓΕ‫׵‬ΗΓʱ͹੠ਈͲேકͤΖ͹͗ۛಕ͹Γ͑͵‫ͤ͗ـ‬Ζɽ

t 77 t3. αρϙ‫ۄ‬તΝඵ͚
͠ͱɼColaboratory Ν࢘ͮͱʰ‫ܙ‬͹ՌָʱΝֺ͢ΞͲΊΓ͑ɽΉͥɼ৿͢͏αʖχιϩΝ६ඍͤΖʤਦ̒ࢂলʥɽ

ԾͶαʖχιϩΝૢ೘ ʤέϨρέͤΖʥ৿͢͏αʖχιϩ͗ࡠΔΗͪਦ̒ʁ਩͘ड़͢(ᄽᄾ )ͲαʖχιϩΝૢ೘ͤΖ༹ࢢΝઈ໎͢ͱ͏ΖɽતͲҕΉΗͱ͏Ζ෨෾͗૤ࡠͳ Colaboratory ͹ಊࡠͲ͍ΖɽαʖχιϩͶɼ࣏͹ϕϫήϧϞΝେ೘͢Γ͑ɽ!pip install ColabTurtle
import ColabTurtle.Turtle as turtle
turtle.initializeTurtle()
turtle.penup()
turtle.goto(100,400)
turtle.pendown()
turtle.speed(10) # 1(slow) ± 10(fast)
def koch(length1, n):
if n == 0 :turtle.forward(int(length1))# integer ࡟ࡋ࡞࠸࡜࢚࣮ࣛ࡟࡞ࡿ
else:koch(length1/3, n - 1)
turtle.left(60)koch(length1/3, n - 1)

t 78 tturtle.right(120)koch(length1/3, n - 1)
turtle.left(60)koch(length1/3, n - 1)
turtle.right(90)koch(600,4)ਦ̓ʁαρϙ‫ۄ‬તɽਦ̓ͺɼTurtle Graphics Ν࢘ͮͱɼαρϙ‫ۄ‬તΝඵժͪ͢΍͹Ͳ͍Ζɽ͵͕ɼColaboratory
Ͳ Turtle Graphics Ν͓࢘ΖΓ͑ͶͤΖͪΌͶɼColabTurtle Ν import ͪ͢ɽ!pip install ColabTurtle
import ColabTurtle.Turtle as turtle
turtle.initializeTurtle()

pip αϜϱχΝ࢘͏৏‫ى‬͹ 3 ߨͲɼColabTurtle[2]ϧ΢ϔϧϨΝ͓࢘ΖΓ͑Ͷ͢ͱ͏ΖɽColabTurtle ͺɼ੏‫ݸ‬ͺ͍Ζ΍͹͹ Colaboratory Ͳ Turtle Graphics Ν͓࢘ΖΓ͑ͪ͢΍͹Ͳ͍Ζ[2]ɽʰ͕Ήͣ͵͏ʱͳߡ͓ͱ΍ΔͮͱΓ͏ͫΘ͑ɽt 79 t‫ܙ‬͹ՌָճͲ͕͵ͣΊ͹ϓϧένϩਦ‫ܙ‬ʤਦ̓ࢂলʥ͗ඵͪ͜ɽϕϫήϧϞΝঙͫ͢͜
մઈͤΖͳɼ࠸‫ؾ‬ఈٝΝ࢘ͮͱ‫ ਼ؖ‬koch(length1, n)Νఈٝ͢ͱ͏Ζɽ۫ରదͶͺɼ୊̒੊େ(೘Ηࢢߑଆ͗̒͹΍͹)͹αρϙ‫ۄ‬તΝ koch(600,4)ͳ͢ͱඵ͏ͱ͏Ζɽ͵͕ɼQˀ̓
ͶͤΖͳࡋ͖͚͵Ε͙ͤͱ͢Ή͑ɽ͏Θ͏Θͳࢾ͢ͱΊΓ͑ɽ
Python ͺɼָ;Ώͤ͏ϕϫήϧϝϱή‫͍Ͳޢݶ‬Εʤྭ͓ͻɼ[3]ΝࢂলʥɼColaboratoryΝ࢘͑ͳͤ͛Ͷ࢟ΌΔΗΖɽͧͽɼηϜʖφϓΧϱ͵ʹͲֺ͢ΞͲ௘ͪ͜Δࢧ͑ɽ࣏յ͖
Δ Ω Ψ η ʀ ϓ ϧ έ ν ϩ ͵ ʹ ༹ ʓ ͵ ΍ ͹ Ν ় ղ ͢ ͱ ͏ ͚ ɽ Ή ͪ ɼ ϕ ϫ ή ϧ Ϟ ͺ ‫ ܙ‬͹ Ռ ָճ
HP ͖Δ download Ͳ͘ΖΓ͑ͶͤΖ༩ఈͲ͍Ζɽ
ʴʤ஭ʥColaboratory ͹ HomePage ͖ΔΫϡϕοϡʖͪ͢ਦ̏ʛ̓ͺɼຌཔΩϧʖͲ͍ΕപࠉͲһࡰͤΖͳ‫ݡ‬Ͷ͚͏৖߻͍͗ΕΉͤɽΩϧʖͲળ໎Ͷ‫͏ͪݡ‬৖߻ͺɼ Colaboratory
͹ HomePage ͶΠέιη͢ͱ͚ͫ͠͏ɽʵ

ःࣛ‫ߚݬ‬ΝಣΞͲ༙ӻ͵αϟϱφΝ͢ͱ͚ͫͮͪ͠∃ీभध‫گ‬दɼ୉ָӅਫ਼͹߶௣࿪‫܇ن‬ΝͺͣΌͳͤΖෳҬ୉ָ୉ָӅ޽ָ‫ڂݜ‬Ռஎ೵εητϞ޽ָ઒ߊඉત‫ܙ‬Ռָ‫࣪ڂݜ‬͹ϟϱώʖͶ‫״‬ः͏ͪ͢Ήͤɽࢂߡช‫ݛ‬[1]

Google Colaboratory

https://colab.research.google.com/notebooks/intro.ipynb#[2]

ColabTurtle[3]ฑీཾ߀ʁPython Ν࢟Όͱ̏0 ೖͲΰʖϞΝࡠΗΖΓ͑Ͷ͵Ζ͹͖ʷ̔̏ࡂ͖Δ͹https://github.com/tolgaatam/ColabTurtleேકʁ͖̎Δ࢟ΌΖ Python Programmingʷɾ‫ܙ‬͹Ռָճࢿʤ౦ߚ஦ʥ

t 80 t