小千谷二荒神社の紛失算額
長岡工業高等専門学校研究紀要 第51巻（2015）⺰ᢥዊජ⼱࠾⨹␹␠ߩ⚗ᄬ▚㗵ᶗ↰ ๺⧐1࡮ᄖᎹ ৻ੳ21৻⥸ᢎ⢒⑼㧙ᢙቇ㧔Liberal Arts-Mathematics, National Institute of Technology, Nagaoka College㧕2㔚ሶ೙ᓮᎿቇ⑼㧔Department of Electronic Control Engineering, National Institute of Technology, Nagaoka College㧕

The Sangaku Lost from the Nikkǀ Shrine in Ojiya
Kazuyoshi WAKUTA1 and Kazuhito TOGAWA 2Abstract
A mathematical tablet called sangaku was dedicated to the Ojiya Nikkǀ Jinjyaʊthe Shinto shrine in1833. Although the sangaku was lost, fortunately it was recorded in a book “Sanpǀkaishǌ”. The sangaku
is a historical work that shows the wasanʊthe traditional Japanese mathematics in the Edo period. Then,we try to restore the sangaku through the drawing by computer. Also, we show the solution to theproblems described in the sangaku.
Key Words : wasan, sangaku, restoration by computer, solution㧝㧚ߪߓ߼ߦ૒⮮⸃⸥⪺‫▚ޡ‬ᴺ⸃㓸‫ߦ ޢ‬㧘‫ޟ‬ᚲឝዊජ⼱ᣣశ␠৻੐‫ߡߒߣޠ‬㧘ᄤ଻ 㧔㧕ᐕ㧘ዊජ⼱ੑ⨹␹␠ߦᄺ⚊ߐࠇߚ▚㗵ߩ໧㗴߇⸃ᴺߣߣ߽ߦឝタߐࠇߡ޿ࠆ㧚૒⮮⸃⸥߇㧘㐷ੱߩખ▚ᱜ㧘ᷰㇱศ⍱ߣᄺ⚊ߒߚ߽ߩߢ޽ࠆ㧚ߥ߅㧘‫▚ޡ‬ᴺ⸃㓸‫  ߪޢ‬ౠࠃࠅߥࠅ㧘᭽‫▚ߥޘ‬ᴺᦠ߆ࠄ㓸߼ߚ  㗴ߩ໧㗴ࠍ⸃޿ߚ߽ߩߢ޽ࠆ㧚⷏⣁ᷣਃ㇢⪺‫ޡ‬૒⮮㔐ጊ⇛વ෸▚ᴺ౞ℂਃบ⪺⠪⠨‫߫ࠇࠃߦ  ޢ‬㧘૒⮮⸃⸥ߪᢥൻ 㧔㧕ᐕߦዊජ⼱ߦ↢߹ࠇ㧘❗Ꮣ໡㧘ᓟᐕ㧘⮎⒳໡ࠍ༡ߺ㧘ߘߩறࠄ㧘ᢙቇ㧘ᥲቇࠍቇࠎߛ㧚㔐ጊߪ㧘૒⮮⸃⸥ߩภߢ޽ࠆ㧚ೋ߼⁛ቇߢቇ߮㧘ੑචᱦߩߣ߈㧘ዊජ⼱ੑ⨹␹␠ߦ਄ㅀߩ▚㗵ࠍᄺ⚊ߒߡ޿ࠆ㧚ᄤ଻  ᐕߦ㧘ጊญ๺ߦᏧ੐ߒߚߣ޿߁㧚ጊญ๺ߪ᳓ේߩੱߢ㧘ᒰᤨߩ⪺ฬߥ▚ቇ⠪㐳⼱Ꮉኡߩ㜞ᒉߢ޽ࠆ㧚ᢙቇ㆏႐ߣ๭߫ࠇߡ޿ߚᳯᚭߩ㐳⼱ᎹኡߩႶߩᜰዉ⠪ߢ޽ߞߚ㧚ㆆᱧ▚ኅߣߒߡ߽⍮ࠄࠇߡ޿ࠆ㧚૒⮮⸃⸥߽㐳⼱Ꮉᵷߩ᦭ജ⠪ߢ޽ࠅ㧘‫▚ޡ‬ᴺ౞ℂਃบ‫⪺ߩ ޢ‬⠪ߣߒߡ⍮ࠄࠇࠆ㧚‫▚ޡ‬ᴺ౞ℂਃบ‫ߪޢ‬㧘㤥ὐ゠✢㧔゠〔㧕㧘㊒ု㧔㊀ᔃ㧕㧘౞ℂᭂᢙ㧔ᭂ୯㧕ߩ  ⒳㘃ߩ໧㗴ࠍ⸃޿ߡ޿ࠆ㧚ਃบߪฎઍਛ࿖ߩᤊߩฬ೨ߢ޽ࠅ㧘ਃ㗴ߦ߆ߌߡ޿ࠆ  㧚ᄙߊߩᒉሶࠍ⢒ߡ㧘቟᡽ 㧔㧕ᐕ㧘྾ච౐ᱦߢᴚߒߚ‫ޕ‬੖ච፲⑲ᄥ㇢⪺‫⹏ޡ‬વ૒⮮㔐ጊ‫ߪߦޢ‬㧘⹦ߒ޿વ⸥߇޽ࠆ  㧚ዊජ⼱ੑ⨹␹␠ߩ▚㗵ߪ㧘ᳯᚭᤨઍߩᢙቇ̆๺▚ࠍવ߃ࠆ⾆㊀ߥ⾗ᢱߢ޽ࠆ㧚⃻࿷ᄬࠊࠇߡߒ߹ߞߚ߇㧘‫▚ޡ‬ᴺ⸃㓸‫ࠅࠃޢ‬ឬ↹࠰ࡈ࠻ࠍ↪޿ߡᓳరߔࠆ㧚࿑ᒻߩ໧㗴߇  㗴ߢ޽ࠆ㧚⸃ᴺ߽⸥タߐࠇߡ޿ࠆߩߢ㧘ߎࠇࠍ⚫੺ߔࠆ㧚๺▚ߪ㧘᣿ᴦએᓟ㧘⑼ቇผߩ┙႐߆ࠄ⎇ⓥ߇ⴕࠊࠇ㧘ᶏᄖߦ߽⚫੺ߐࠇ෻㗀ࠍ๭ࠎߛ  㨪 㧚ㄭᐕߪ㧘ቇⴚ⊛ߥ⎇ⓥߛߌߢߥߊ㧘৻⥸ߩੱ‫ߩ߳ޘ‬๺▚ߩ⚫
੺߿ᢙቇᢎ⢒ߩ┙႐߆ࠄߩ⎇ⓥ߽⋓ࠎߢ޽ࠆ  㨪 㧚−35−涌田和芳・外川㧞㧚▚㗵ᓳర࿑‫▚ޡ‬ᴺ⸃㓸‫ߦޢ‬ၮߠ޿ߡ㧘▚㗵ࠍᓳరߔࠆ㧚ߎࠇߪⓂᧄߢ޽ࠅ㧘ߘߩ౮ᧄߦ▚㗵ߩ⺑᣿ᢥߣ࿑߇޽ࠆ㧚▚㗵ߩᄢ߈ߐ㧘߅ࠃ߮⺑᣿ᢥߣ࿑ߩ㈩⟎╬ߪਇ᣿ߢ޽ࠆ㧚ṽሼߪ㧘ේೣߣߒߡᣥሼ૕ࠍ↪޿ߚ㧚࿑ߩᓀ⦡ߦߟ޿ߡߪ㧘⃻ሽߔࠆઁߩ▚㗵ࠍෳ⠨ߦߒߚ㧚࠰ࡈ࠻ߪ㧘ឬ↹࠷࡯࡞㧔#FQDG+NNWUVTCVQTEU㧕ࠍ↪޿ߚ㧚૞ᚑߒߚᓳర࿑ࠍ␜ߔ㧔࿑㧙㧕㧚㧟㧚㗵ᢥߩ⸃⺑㧟㧚㧝㗵ᢥߩᦠਅߒᢥ 㗵ᢥߩᦠਅߒᢥࠍ৻᩺ߣߒߡᦠ߈㧘ߘߩ⃻ઍ⺆⸶ࠍ␜ߔ㧚๺▚ߩ↪⺆ߦߟ޿ߡߪ⺑᣿ࠍߒߥ޿߇㧘⃻ઍ⺆⸶ߣᲧセߔࠇ߫㧘ߘߩᗧ๧߇ಽ߆ࠆ㧚╙৻໧੹㧘࿑ߩᅤߊ㧘⋥ߩౝߦᒐ⢛ࠍ㓒ߡ╬౞߅ࠃ߮ㅙ౞ᢙ୘㧔઒ߦ౐୘ࠍ↹ߊ㧕ࠍኈࠆࠆ᦭ࠅ㧚ด੔߰㧘㐳ੑච౎ኸ㧚෶੔߰㧘╬౞ᓘ྾ኸ㧘ᧃ౞ᓘ৻ኸ㧚Ꮐฝ౞ᢙᐞ૗ߣ໧߰㧚╵߳ߡᦚߊ㧘Ꮐฝ౞ᢙ౐୘㧚ⴚߦᦚߊ㧘ᧃ౞ᓘ㧔એਅ౞ᓘੑሼߎࠇࠍ⇛ߔ㧕ࠍએߡ╬ࠍ㒰ߒ㧘ᐔᣇߦ㐿ߊ㧚ౝ৻୘ࠍᷫߓ㧘㐳╬Ꮕࠍਸ਼ߓ㧘╬ࠍએߡߎࠇࠍ㒰ߒ㧘Ꮐฝ౞ᢙࠍᓧߡ໧߭ߦวߔ㧚╙ੑ໧੹㧘࿑ߩᅤߊ㧘╬౞ߩ੤➱ߦ྾౞ࠍ↹ߊ᦭ࠅ㧚ด੔߰㧘↲౞ᓘච౎ኸੑಽ㧘ਸ౞ᓘੑච৾ኸਃಽ㧘਑౞ᓘ਻ኸ৻ಽ㧚ৼ౞ᓘᐞ૗ߣ໧߰㧚╵߳ߡᦚߊ㧘ৼ౞ᓘ৻ኸ྾ಽ㧚ⴚߦᦚߊ㧘ਸ౞ᓘ㧔એਅ౞ᓘੑሼߎࠇࠍ⇛ߔ㧕ࠍ⟎߈਑ࠍട߳㧘ᄤߣฬߠߊ㧚↲ࠍട߳㧘ਸ౹ࠍਸ਼ߕ㧚↲ࠍ࿃ߒߚࠆ਑౹ࠍએߡ㧘ߎࠇࠍ㒰ߒ㧘ౝ৻୘ࠍᷫߕ㧚એߡᄤࠍ㒰ߒ㧘ৼࠍᓧߡ໧߭ߦวߔ㧚╙ਃ໧ ੹㧘࿑ߩᅤߊ㧘ᒐߩౝߦਃ╬౞᦭ࠅ㧚ด੔߰㧘ᒏ྾ኸ౎ಽ㧘⍫৻ኸੑಽ㧚╬౞ᓘᐞ૗ߣ໧߰㧚╵߳ߡᦚߊ㧘╬౞ᓘ৻ኸ㧚ⴚߦᦚߊ㧘⍫౹྾Ბࠍ⟎߈㧘ⷺߣฬߠߊ㧚ᒏ౹ࠍട߳㧘એߡⷺࠍ㒰ߒ㧘৻୘ࠍട߰㧚એߡ⍫ࠍ㒰ߒ㧘╬౞ᓘࠍᓧߡ໧߭ߦวߔ㧚㧟㧚㧞⃻ઍ⺆⸶一仁╙৻໧࿑ߩࠃ߁ߦ㧘㐳ᣇᒻߩਛߦᒐࠍ㓒ߡ╬౞ߣߘࠇࠍㅙ߁ᐞߟ߆ߩ౞㧔઒ߦ  ୘↹ߊ㧕߇޽ࠆ㧚ߚߛߒ㧘㐳ᣇᒻߩ৻ㄝߪ  ኸ㧚߹ߚ㧘╬౞ߩ⋥ᓘߪ  ኸ㧘ᧃ౞ߩ⋥ᓘߪ  ኸߣߔࠆ㧚Ꮐฝߩ㧔ㅙ㧕౞ߩ୘ᢙߪ޿ߊߟ߆㧚  
−36−࿑ᓳర࿑

小千谷二荒神社の紛失算額╵㧚Ꮐฝߩ㧔ㅙ㧕౞ߩ୘ᢙߪ  ୘ߢ޽ࠆ㧚ⴚ㧚ᧃ౞ߩ⋥ᓘߢ╬౞ߩ⋥ᓘࠍഀࠅ㧘ߘߩᐔᣇᩮࠍขࠆ㧚ߎࠇ߆ࠄ  ࠍᒁ߈㧘㐳ߣ╬౞ߩ⋥ᓘߩᏅࠍដߌ㧘╬౞ߩ⋥ᓘߢഀߞߡ㧘Ꮐฝߩ㧔ㅙ㧕౞ߩ୘ᢙࠍᓧࠆ㧚╙ੑ໧࿑ߩࠃ߁ߦ㧘╬౞ߩ㑆ߦ  ߟߩ౞߇޽ࠆ㧚ߚߛߒ㧘↲౞ߩ⋥ᓘߪ  ኸ  ಽ㧘ਸ౞ߩ⋥ᓘߪ  ኸ  ಽ㧘਑౞ߩ⋥ᓘߪ  ኸ  ಽߣߔࠆ㧚ৼ౞ߩ⋥ᓘߪ޿ߊࠄ߆㧚╵㧚ৼ౞ߩ⋥ᓘߪ  ኸ  ಽߢ޽ࠆ㧚ⴚ㧚ਸ౞ߩ⋥ᓘߦ਑౞ߩ⋥ᓘࠍട߃㧘ᄤߣߔࠆ㧚↲౞ߩ⋥ᓘട߃㧘ਸ౞ߩ⋥ᓘߩ౹ࠍដߌࠆ㧚↲౞ߩ⋥ᓘߣਸ౞ߩ⋥ᓘߩ౹ࠍដߌߚ߽ߩߢ㧘ߎࠇࠍഀߞߡ ࠍᒁߊ㧚ߎࠇߢᄤࠍഀߞߡ㧘ৼ౞ߩ⋥ᓘࠍᓧࠆ㧚╙ਃ໧࿑ߩࠃ߁ߦ㧘ᒐߩౝߦ  ߟߩ╬౞߇޽ࠆ㧚ߚߛߒ㧘ᒏߪ  ኸ  ಽ㧘⍫ߪ  ኸ  ಽߣߔࠆ㧚╬౞ߩ⋥ᓘߪ޿ߊࠄ߆㧚╵㧚╬౞ߩ⋥ᓘߪ  ኸߢ޽ࠆ㧚ⴚ㧚⍫ߩ౹ࠍ  ୚ߒⷺߣߒ㧘ᒏߩ౹ࠍട߃ࠆ㧚ߎࠇߢⷺࠍഀߞߡ  ࠍട߃ࠆ㧚ߎࠇߢ⍫ࠍഀߞߡ㧘╬౞ߩ⋥ᓘࠍᓧࠆ㧚 ࿑ ߩࠃ߁ߦ㧘౞㧔ᒐ㧕ࠍ O 㧘╬౞ࠍ O1 㧘ㅙ౞ࠍ Ok 㧘ᧃ౞ࠍ On ߣߒa : 㐳ᣇᒻߩᮮߩ㐳ߐ㧘 r : ౞㧔ᒐ㧕 O ߩ⋥ᓘ

r1 : ╬౞ O1 ߩ⋥ᓘ㧘 rk : ㅙ౞ Ok ߩ⋥ᓘߣ߅ߊ㧚࿑ ࠃࠅ
T1T2ߎࠇࠃࠅ 
r1r2



O1O2

TT2

  ߒߚ߇ߞߡ
r1r r2  r  r2

r r3

ห᭽ߦr2r  r3   ࠍ  ߦઍ౉ߒߡr1‫▚ޡ‬ᴺ⸃㓸‫⸃ࠆ޽ߦޢ‬ᴺࠍዋߒ⵬޿ߥ߇ࠄ⚫੺ߔࠆ㧚ߚߛߒ㧘⴫⸥ߪ㧘⃻ઍᢙቇߦᓥ߁㧚㧠㧚㧝 ╙৻໧ߦߟ޿ߡ
O


rr1  rr2

㧠㧚ⴚߩ⸃⺑ 

TT1  TT2    r r3r  2 r3  ৻⥸ߦ
r1r rnr  ( n  1) rn

  ߔߥࠊߜ(n  1) r1 rn  r r1  r rn0    ߹ߚ㧘 ࿑ ࠃࠅTT1TA  T1 A

  ߎࠇࠃࠅrr1a r1
    2 2ߔߥࠊߜT1 A

2 r r1  r1  a࿑╙৻໧ߩ⸃ᴺ࿑


−37−

0    涌田

和芳・外川 㧘  ࠃࠅ r ߩᑼࠍᓧࠆ㧚(n  1) r1 rn  ( r1  rn ) r ( a  r1 )  2 r1 r

0

AO10

   㧘  ࠃࠅ r ࠍᶖ෰ߔࠆ㧚2(n  1)r1 rn  (a  r1 )( r1  rn )0 

  N 2(n  1) ߣ߅ߊ㧚ߎࠇ߇╬౞ࠍㅙ߁Ꮐฝߩ౞ߩ୘ᢙߢ޽ࠆ㧔╬౞ߪ㒰ߊ㧕㧚  ࠍ rn ߢഀࠆ㧚  

u1 㧘 AO2s1r r1 㧘 s2
2 2r r2 㧘 s32 2r r3
2 2 

 t1

r1 r2
 㧘 t 22 2r
r1
 r2  322   t3r2 r3 㧘 t42 2r2r
 r3  422 

s12  u12s32  (t 2  u1 ) 2  ߎࠇࠃࠅ  

t 2 2  s12  s3 2  2t 2u1

N

'OO1O3 ߦߟ޿ߡ

᡿ߦ

·§ r
(a  r1 )¨ 1  1¸¸¨ r¹ © nr1u2ߣ߅ߊ㧚ߎߩߣ߈  

§ r
·Nr1  (a  r1 )¨ 1  1¸ 0¨ r¸© n¹一仁ห᭽ߦ㧘 'OO1O2 ߦߟ޿ߡ  t12  s12  s2 2  

2t1u1

ߎࠇ߇ⴚߦㅀߴߡ޽ࠆ㧚  㨪  㧘  㧘  ߪ⵬ߞߚ㧚
r1 4 㧘 rn 1 㧘 a 28 ߥߩߢ㧘  ࠃࠅ㧘 N 6ࠍᓧࠆ㧚㧠㧚㧞 ╙ੑ໧ߦߟ޿ߡ 㧘  ࠃࠅt 2 2  s12  s3 2 t12  s12  s2 2

2t 22t10   ߦ  ࠍઍ౉ߒߡᢛℂߔࠆ㧚

Or2 2 r3  r1r2 r3  r2 3  r1r2 2  rr1r3  rr2 20

  'OO2O4 㧘 'OO2O3 ߦߟ޿ߡ߽ห᭽ߩ⚿ᨐ߇ᓧࠄࠇࠆ㧚ߔߥࠊߜ㧘  ߩ r1 㧘 r2 㧘 r3 ࠍ㧘ߘࠇߙࠇ㧘r2 㧘 r3 㧘 r4 ߦ឵߃ߡ

O4O2 AO3O1r32 r4  r2 r3 r4  r33  r2 r32  rr2 r4  rr32
2 2  ( r1r3  r2 ) 㧙   ( r2 r4  r3 ) ࠍ⸘▚ߒߡ㧘 rࠍᶖ෰ߔࠆ㧚࿑╙ੑ໧ߩ⸃ᴺ࿑࿑ ߩࠃ߁ߦ㧘╬౞ࠍ O 㧘↲౞ࠍ O1 㧘ਸ౞ࠍO2 㧘਑౞ࠍ O3 㧘ৼ౞ࠍ O4 ߣߒ㧘 O ߆ࠄ  ߟߩ╬౞ߩ੤ὐࠍ⚿߱⋥✢ߦု✢ OA ࠍᒁߊ㧚r : ౞㧔ᒐ㧕 O ߩ⋥ᓘ㧘 rn : ౞ On ߩ⋥ᓘ



0

OO1

s1 㧘 OO2

s2 㧘 OO3

s3 㧘 OO4

s4 

O1O2

t1 㧘 O1O3

t2 㧘 O2O3t 3 㧘 O 2 O4t4 

r1r2 2 r32  2r1r2 r33  r1r34  r1r2 r32 r4  r1r33r422343

 2

 r2 r3 r4  2r2 r3r4  r2 r4  r1r2 r4  r1r2 r3 r40
 
ߎࠇࠍᢛℂߒߡ

−38−

r1r32 ( r2  r3 ) 2  r1r32 r4 ( r2  r3 )
222

 

 r2 r4 ( r2  r3 )  r1r2 r4 ( r2  r3 )

0



 

小千谷二荒神社の紛失算額

 ࠍ r2  r3 ߢഀߞߡ㧘 r4 ࠍ᳞߼ࠆᑼࠍᓧࠆ㧚

r1r32 ( r2  r3 )  {r1r3 2  ( r1  r2  r3 )r2 2 }r40

৻ᣇr12 b  r1r  r1r1r32 ( r2  r3 )( r1  r2  r3 )r2 2  r1r32      r2  r3( r1  r2  r3 )r2 2r1r32

 

1b  r1    㧘  ࠃࠅߎࠇࠃࠅ㧘 r4 ߇ᓧࠄࠇࠆ㧚

r4s

 



0    ࠃࠅ r ߩᑼࠍᓧࠆ㧚 br1  (b  r1 )r0  ߹ߚ㧘 'OAD ߦߟ޿ߡߎࠇ߇ⴚߦㅀߴߡ޽ࠆ㧚  ߪ⵬ߞߚ㧚
r1 18.2 㧘 r2 27.3 㧘 r3 9.1 ߥߩߢ㧘  ࠃࠅ㧘
r4 1.4 ࠍᓧࠆ㧚㧠㧚㧟 ╙ਃ໧ߦߟ޿ߡ
ar( ) 2  (  b) 222

 ࠃࠅ r ߩᑼࠍᓧࠆ㧚a 2  4b 2  4brCEA0

   㧘  ࠃࠅ r ࠍᶖ෰ߔࠆ㧚

O2O1r( )2 2  

O3D(a 2  4b 2 )b  {(a 2  4b 2 )  4b 2 }r1B0



 

ߎࠇࠍ a 2  4b2 ߢഀߞߡ㧘 r1 ߩᑼࠍᓧࠆ㧚§ 4b 2·b  ¨¨ 2
 1¸¸r12© a  4b¹O0

 ࿑╙ਃ໧ߩ⸃ᴺ࿑࿑ ߩࠃ߁ߦ㧘౞㧔ᒐ㧕ࠍ O 㧘╬౞ࠍ O1 㧘 O2 㧘O3 㧘ᒏࠍ AB 㧘⍫ࠍ CD ߣߒ㧘 O1 ߆ࠄ⍫ߦု✢O1E ࠍᒁߊ㧚r : ౞㧔ᒐ㧕 O ߩ⋥ᓘ㧘 r1 : ╬౞ߩ⋥ᓘaAB 㧘 b CD ߣߒ㧘 O2 E(r  r1 2 r  r1) ( s) 2
22 ߎࠇࠃࠅ

sr12r  r1 r1b2


4b
1a 2  4b 2

 ߎࠇ߇ⴚߦㅀߴߡ޽ࠆ㧚  㧘  ߪ⵬ߞߚ㧚
a 4.8 㧘 b 1.2 ߥߩߢ㧘 r1 1 ࠍᓧࠆ㧚㧡㧚߅ࠊࠅߦs ߣ߅ߊ㧚'O1O2 E ߣ 'O1OE ߟ޿ߡr12  s 2

ߎࠇ߆ࠄ r1 ߇ᓧࠄࠇࠆ㧚   ᄤ଻ 㧔㧕ᐕ㧘૒⮮⸃⸥ࠄߦࠃߞߡ㧘ዊජ⼱ੑ⨹␹␠ߦᄺ⚊ߐࠇߚ▚㗵ࠍ㧘ឬ↹࠰ࡈ࠻ࠍ↪޿ߡᓳరߒߚ㧚ᒰᤨ૶ࠊࠇߡ޿ߚṽሼߪᣥሼ૕ߢ޽ࠆ߇㧘ᣥሼ૕ߦߪ㧘޿ࠊࠁࠆᱜሼߣ๭߫ࠇࠆ߽ߩએᄖߦ߽᭽‫⇣ߥޘ‬૕ሼ߇޽ࠆ㧚߹ߚ㧘౮ᧄ߇ᱜ⏕ߦ౮ߒߚ߽ߩ߆ਇ᣿ߢ޽ࠆ㧚ߒߚ߇ߞߡ㧘ߎߩ▚㗵ߩᓳరߦᒰߚߞߡߪ㧘ේೣߣߒߡᣥሼ૕ߩᱜሼࠍ↪޿ߚ߇㧘ᣥሼ૕ߩᱜሼߣᣂሼ૕ߣ߇௖߆ߦ⇣ߥࠅ㧘ᣂሼ૕ߣห
−39−涌田和芳・外川ߓ߽ߩ߇ᒰᤨ߽૶ࠊࠇߡ޿ࠆ႐วߦߪ㧘ᣂሼ૕ࠍ૶߁ߎߣߦߒߚ㧚࿑ߩ⦡ᓀߦߟ޿ߡߪ㧘⃻ሽߔࠆ▚㗵ࠍෳ⠨ߦ㧘ฎߊ߆ࠄ૶ࠊࠇߡ޿ࠆ‫ޟޠ⊕ޟ‬㤛࿯‫ޠ‬‫ޟ‬ㄖ⍾‫ޟޠ‬ᵻ⫐‫✛ޟޠ‬㕍‫⟲ޟޠ‬㕍‫  ߩޠ‬⒳㘃ߩ⦡ࠍ↪޿ߚ㧚ṽሼߣ⦡ᓀߦߟ޿ߡߪ㧘੹ᓟ߽ᬌ⸛ࠍ㊀ߨߚ޿㧚 ▚㗵ߩ໧㗴ߩ⸃ᴺߪ㧘ᣣᧄቇ჻㒮⬿ߩ‫▚ޡ‬ᴺ⸃㓸‫ߦޢ‬ၮߠ޿ߡ⸃⺑ߒߚ㧚ᒰᤨߩ๺▚ᦠߪ◲ẖߦᦠ߆ࠇߡ޿ࠆߩߢ㧘ᐞࠄ߆⵬޿㧘ಽ߆ࠅᤃߊߒߚ㧚ᦠ㧘 ᐕ㧚 ዊኹ⵨㧦ߛ߆ࠄᭉߒ޿ᳯᚭߩ▚㗵㧘⎇ᚑ␠㧘ᐕ㧚 ᪉੗ㅴ㧦ᳯᚭߩᢙቇᢎ⑼ᦠ㧘㓸⧷␠㧘 ᐕ㧚 ዊኹ⵨㧦ᳯᚭߩᢙቇ๺▚㧘ᛛⴚ⹏⺰␠㧘 ᐕ㧚 ጊᧄ⾐೨▚ᴺഥⴚ㧘ᄤ଻ 㧔㧕ᐕ㧘᧲ർᄢቇ๺▚⾗ᢱ࠺࡯࠲ࡌ࡯ࠬ⬿㧚ᷓᎹ⧷ବᩞᵈ㧦▚ᴺഥⴚ㧔ᓳೞ㧕㧘ᦺᣣᣂ⡞␠੐ᬺㇱฬฎደડ↹੐ᬺ࠴࡯ࡓ㧘 ᐕ㧚ฃઃ ᵈ‫▚ޡ‬ᴺഥⴚ‫౏ߩ  ╙ߩ ޢ‬ᑼߦ޽ࠆ㧚ߎࠇߪࠃߊ⍮ࠄࠇߚ౏ᑼߢ޽ࠆ㧚‫▚ޡ‬ᴺഥⴚ‫౏ߩ  ╙ߩޢ‬ᑼߦ޽ࠆ㧚ߎࠇߪ⃻ઍᢙቇߩ૛ᒏቯℂߣห╬ߢ޽ࠆ㧚ߔߥࠊߜ㧘s32

s12  t2 2  2 s1t2  2 s1t2 cos O1 ෳ⠨ᢥ₂ ૒⮮⸃⸥㧦▚ᴺ⸃㓸㧘ᚑ┙ᐕਇ⹦㧘ᣣᧄቇ჻㒮⬿㧚 ⷏⣁ᷣਃ㇢㧦૒⮮㔐ጊ⇛વ෸▚ᴺ౞ℂਃบ⪺⠪⠨㧘⑳ኅ 㧘࿖┙࿖ળ࿑ᦠ㙚ㄭઍ࠺ࠫ࠲࡞࡜ࠗࡉ࡜࡝࡯⬿㧚 ૒⮮⸃⸥㧦▚ᴺ౞ℂਃบ㧘ᒄൻ 㧔㧕ᐕ㧘᧲ർᄢቇ๺▚⾗ᢱ࠺࡯࠲ࡌ࡯ࠬ⬿㧚 ૒⮮ஜ৻࡮ᄢ┻⨃㓶࡮ዊኹ⵨࡮’㊁ᱜඳ㧦๺▚ผᐕ⴫㧘᧲ᵗᦠᐫ㧘 ᐕ㧚 ੖ච፲⑲ᄥ㇢㧦⹏વ૒⮮㔐ጊ㧘ᕡᢥ␠㧘 ᐕ㧚 ਃ਄⟵ᄦᢥൻผ਄ࠃࠅ⷗ߚࠆᣣᧄߩᢙቇ㧘ጤᵄᢥᐶ㧘 ᐕ㧔 ᐕೋ಴㧕㧚 ᷓᎹ⧷ବ㧘࠳ࡦ࡮ࡍ࠼࡯㧦ᣣᧄߩᐞ૗㧙૗㗴⸃ߌ߹ߔ߆㧫㧘᫪ർ಴ 㧘 ᐕ㧚 ᷓᎹ⧷ବ㧘࠳ࡦ࡮࠰ࠦࡠࡈࠬࠠ࡯ᣣᧄߩᢙቇ㧙૗㗴⸃ߌ߹ߔ߆㧫㧔਄㧕 ਅ 㧘᫪ർ಴ 㧘 ᐕ㧚 ᷓᎹ⧷ବ㧘6QP[4QVJOCP5CETGF/CVJGOCVKEU㧙,CRCPGUG 6GORNG )GQOGVT[ 㧘 2TKPEGVQP 7PKX 㧚2TGUU㧘 ᐕ㧚㧔㇌⸶㧕⡛ߥࠆᢙቇ㧦▚㗵㧘᫪ർ಴ 㧘 ᐕ㧚 ዊୖ㊄ਯഥᣣᧄߩᢙቇ㧘ጤᵄᣂᦠ㧘 ᐕ㧚 ᐔጊ⺼㧦๺▚ߩᱧผ㧙ߘߩᧄ⾰ߣ⊒ዷ㧘ߜߊ߹ᢥᐶ㧘 ᐕ㧔 ᐕೋ಴㧕㧚 ᐔጊ⺼㧦๺▚ผ਄ߩੱ‫ޘ‬㧘ߜߊ߹ᢥᐶ㧘 ᐕ㧔 ᐕೋ಴㧕㧚 ᄢ⍫⌀৻๺▚౉㐷㧘ᣣᧄ⹏⺰␠㧘 ᐕ㧚 ᐔጊ⺼㧦๺▚ߩ⺀↢㧘ᕡᤊ␠ᕡᤊ㑑㧘 ᐕ㧚 ૒⮮ஜ৻ᣣᧄੱߣᢙᳯᚭᐼ᳃ߩᢙቇ㧘᧲ᵗᦠᐫ㧘 ᐕ㧚 ᷓᎹ⧷ବᣣᧄߩᢙቇߣ▚㗵㧘᫪ർ಴ 㧘 ᐕ㧚 ૒⮮ஜ৻ᣂ๺▚౉㐷㧘⎇ᚑ␠㧘 ᐕ㧚 ዊᎹ᧤㧘ᐔ㊁⪲৻⻠ᐳᢙቇߩ⠨߃ᣇ  ᢙቇߩᱧผ㧘ᦺୖᦠᐫ㧘 ᐕ㧚 દ⮮ᵗ⟤ᚻߠߊࠅㆬᛯᢙቇ߅߽ߒࠈ๺▚㧘᣿ᴦ࿑一仁
−40−