長岡工業高等専門学校研究紀要

論第49巻（2013）文୚ᯈඵᖭᐑࡢ⣮ኻ⟬㢠㸦㸱㸧
1ᾛ⏣ ࿴ⰾ ࣭እᕝ ୍ோ122୍⯡ᩍ⫱⛉㸫ᩘᏛ㸦Liberal Arts-Mathematics, Nagaoka National College of Technology㸧㟁ᏊไᚚᕤᏛ⛉㸦Department of Electronic Control Engineering, Nagaoka National College of Technology㸧

The Sangaku Lost from the Yoita Hachiman Shrine (3)Kazuyoshi WAKUTA1 and Kazuhito TOGAWA2Abstract
A mathematical tablet called sangaku was dedicated to the Yoita Hachiman Gǌʊa Shinto shrine in theEdo period of Japan. Although the sangaku was lost, fortunately it was recorded in a book “SaishiShinzan”ʊthe collection of sangaku problems. The sangaku is a historical work that shows thewasanʊthe traditional Japanese mathematics in the Edo period. Then, we try to restore the sangaku
dedicated in 1808 through the drawing by computer. Also, from the literature,we infer the solutions in theEdo period to the sangaku problems.
Key Words : wasan, sangaku, restoration by computer, solution㸰㸬⟬㢠᚟ඖᅗ㸯㸬ࡣࡌࡵ࡟Ụᡞ᫬௦ྛᆅࡢ⚄♫࣭௖㛶࡟ዊᥖࡉࢀࡓ⟬㢠ࢆ㞟㘓ࡋࡓ୰ᮧ᫬୓⦅ࡢࠗ㉏⚆⚄⟬࠘ ࡀ࠶ࡿ㸬ࡇࡢ୰࡟㸪୚ᯈඵᖭᐑࡢ⟬㢠ࡀ  㠃㞟㘓ࡉࢀ࡚࠸ࡿࡀ㸪 㠃࡜ࡶኻࢃࢀ࡚ࡋࡲࡗࡓ㸬ᐶᨻ 㸦㸧ᖺ࠾ࡼࡧᩥ໬ඖ㸦㸧ᖺࡢ⟬㢠ࡣ㸪᪤࡟᚟ඖࢆ⾜ࡗࡓ 㸬ᮏ✏࡛ࡣ㸪ᘬࡁ⥆ࡁ㸪ᩥ໬ 㸦㸧ᖺ࡟ᯇᾆᏒ㔜㸪➉ෆᗘ㈆ࡢᥖ㢠ࡋࡓࡶࡢࢆࠗ㉏⚆⚄⟬࠘㸦ᕳஅᅄ㸧ࡼࡾᥥ⏬ࢯࣇࢺࢆ⏝࠸࡚᚟ඖࡋ㸪ᙜ᫬ࡢゎἲࢆ࿴⟬᭩➼࠿ࡽ᥎ ࡍࡿ㸬࿴⟬ࡣ㸪᫂἞௨ᚋ㸪⛉Ꮫྐࡢ❧ሙ࠿ࡽ◊✲ࡀ⾜ࢃࢀ㸪ᾏእ࡟ࡶ⤂௓ࡉࢀ཯㡪ࢆ࿧ࢇࡔ  㹼 㸬㏆ᖺࡣ㸪Ꮫ⾡ⓗ࡞◊✲ࡔࡅ࡛࡞ࡃ㸪୍⯡ࡢேࠎ࡬ࡢ࿴⟬ࡢ⤂௓ࡸᩘᏛᩍ⫱ࡢ❧ሙ࠿ࡽࡢ◊✲ࡶ┒ࢇ࡛࠶ࡿ  㹼 㸬ࠗ㉏⚆⚄⟬࠘ࡣ✏ᮏ࡛࠶ࡾ㸪ࡑࡢ෗ᮏ࡟⟬㢠ࡢㄝ᫂ᩥ࡜ᅗࡀ࠶ࡿ㸬⟬㢠ࡢ኱ࡁࡉ㸪࠾ࡼࡧㄝ᫂ᩥ࡜ᅗࡢ㓄⨨➼ࡣ୙࡛᫂࠶ࡿ㸬₎Ꮠࡣ෗ᮏ࡟ࡼࡾ␗ྠࡀ࠶
ࡾ㸪ཎ๎࡜ࡋ࡚ᪧᏐࡢṇᏐࢆ⏝࠸ࡓ㸬ᅗࡢᙬⰍ࡟ࡘ࠸࡚ࡣ㸪⌧Ꮡࡍࡿ௚ࡢ⟬㢠ࢆཧ⪃࡟ࡋࡓ㸬ࢯࣇࢺࡣ㸪ᥥ⏬ࢶ࣮ࣝ㸦$GREH,OOXVWUDWRUFV㸧ࢆ⏝࠸ࡓ㸬సᡂࡋࡓ᚟ඖᅗࢆ♧ࡍ㸦ᅗ㸫㸧㸬㸱㸬㢠ᩥࡢゎㄝ㸱㸬㸯㢠ᩥࡢ᭩ୗࡋᩥ 㢠ᩥࡢ᭩ୗࡋᩥࢆ୍᱌࡜ࡋ࡚᭩ࡁ㸪ࡑࡢ⌧௦ㄒヂࢆ♧ࡍ㸬࿴⟬ࡢ⏝ㄒ࡟ࡘ࠸࡚ࡣㄝ᫂ࢆࡋ࡞࠸ࡀ㸪⌧௦ㄒヂ࡜ẚ㍑ࡍࢀࡤ㸪ࡑࡢព࿡ࡀศ࠿ࡿ㸬

―1―

和芳・外川➨୍ၥ௒㸪ᅗࡢዴࡃ㸪㕍⫤ࡢෆ࡟኱ᘼ㸪ᑠᘼ㸪෇ࢆᐜࡿࡿ᭷ࡾ㸬኱ᘼࡣ㸪⫤ᘻࡢ㝮࡟࡚⫤࡟ษࡋ࡚ࡼࡾ㕍ᘻࡢ㝮࡟⮳ࡿ㸬ᑠᘼࡣ㸪㕍ᘻࡢ㝮࡟࡚኱ᘼ࡟ษࡋ࡚ࡼࡾ㕍⫤ࡢ㝮࡟⮳ࡿ㸬෇ࡣ㸪୧ᘼཬࡧ⫤࡟ษࡋ஺ࡣࡿ㸬ྈபࡩ㸬㕍஧༑ඵᑍඵศ୕ཊ㸪⫤୕༑ඵᑍᅄศᅄཊ㸬෇ᚄ㸪ᗄఱ࡜ၥࡩ㸬⟅࡬࡚᭣ࡃ㸪෇ᚄ୍༑஧ᑍ㸬⾡࡟᭣ࡃ㸪ᘻࢆồࡵ㸪௨࡚⫤ࢆ㝖ࡋᴟ࡜ྡ࡙ࡃ㸬⫤ࢆ௨࡚㕍ࢆ㝖ࡋᴟࢆຍ࡬㸪ࡇࢀࢆ⮬ࡽࡍ㸬௨࡚㕍ࢆ㝖ࡋ㸪෇ᚄࢆᚓ࡚ၥࡦ࡟ྜࡍ㸬➨஧ၥ௒㸪ᅗࡢዴࡃ㸪┤ࡢෆ࡟⏥㸪எ㸪୤㸪୎㸪ᠾࡢ஬෇ࢆᐜࡿࡿ᭷ࡾ㸬ྈபࡩ㸪ᖹ஬ⓒ஑༑஬ᑍ㸬⏥෇ᚄ㸪ᗄఱ࡜ၥࡩ㸬⟅࡬࡚᭣ࡃ㸪⏥෇ᚄᅄⓒ஧༑භᑍ᭷ወ㸬⾡࡟᭣ࡃ㸪ᅄಶ஬ศࢆ⨨ࡁ㸪ᩳ⋡஧ẁࢆῶࡌᴟ࡜ྡ࡙ࡃ㸬஧ศ஬ཊࢆῶࡌᖹ᪉࡟㛤ࡁᴟࢆຍ࡬㸪ᅄ࡟࡚ࡇࢀࢆ㝖ࡍ㸬ᖹࢆ஌ࡌ㸪⏥෇ᚄࢆᚓ࡚ၥࡦ࡟ྜࡍ㸬㸱㸬㸰⌧௦ㄒヂ➨୍ၥᅗࡢࡼ࠺࡟㸪┤ゅ୕ゅᙧࡢ୰࡟㸪኱ᘼ㸪ᑠᘼ㸪෇ࡀ࠶ࡿ㸬኱ᘼࡣ㸪ᗏ㎶࡜ᩳ㎶ࡢ㝮࡛ᗏ㎶࡟᥋ࡋ㸪ᑐ
㎶㸦⦪ࡢ㎶㸧࡜ᩳ㎶ࡢ㝮ࡲ࡛࠶ࡿ㸬ᑠᘼࡣ㸪ᑐ㎶࡜
ᩳ㎶ࡢ㝮࡛኱ᘼ࡟᥋ࡋ㸪ᑐ㎶࡜ᗏ㎶ࡢ㝮ࡲ࡛࠶ࡿ㸬෇ࡣ㸪 ࡘࡢᘼ࠾ࡼࡧᗏ㎶࡟᥋ࡍࡿ㸬ᑐ㎶  ᑍ㸪ᗏ㎶  ᑍ࡛࠶ࡿ࡜ࡁ㸪෇ࡢ┤ᚄࡣ࠸ࡃࡽ࠿㸬 ⟅㸬෇ࡢ┤ᚄࡣ  ᑍ㸬⾡㸬ᩳ㎶ࢆồࡵ㸪ࡇࢀ࡛ᗏ㎶ࢆ๭ࡾᴟ࡜ྡ࡙ࡅࡿ㸬
ᗏ㎶࡛ᑐ㎶ࢆ๭ࡾᴟࢆຍ࠼࡚  ஌ࡍࡿ㸬ࡇࢀ࡛ᑐ㎶ࢆ๭ࡾ෇ࡢ┤ᚄࢆᚓࡿ㸬➨஧ၥᅗࡢࡼ࠺࡟㸪㛗᪉ᙧࡢ୰࡟㸪⏥㸪எ㸪୤㸪୎㸪ᠾࡢ  ෇ࡀ࠶ࡿ㸬⦪  ᑍ࡛࠶ࡿ࡜ࡁ㸪⏥෇ࡢ┤ᚄࡣ࠸ࡃࡽ࠿㸬⟅㸬⏥෇ࡢ┤ᚄࡣ  ᑍ࡜వࡾࡀ࡛ࡿ㸬⾡㸬 ࠿ࡽ 2 2 ࢆᘬࡁᴟ࡜ྡ࡙ࡅࡿ㸬ࡇࢀ࠿ࡽ ࢆᘬࡁᖹ᪉᰿ࢆྲྀࡾᴟࢆຍ࠼㸪 ࡛๭ࡿ㸬ࡇࢀ࡟⦪ࢆ᥃ࡅ࡚㸪⏥෇ࡢ┤ᚄࢆᚓࡿ㸬㸱㸬㸱ዟ௜࡟ࡘ࠸࡚⟬㢠ዊᥖ⪅ࡣ㸪๓⾡ࡢᯇᾆᏒ㔜㸪ᚋ⾡ࡢ➉ෆᗘ㈆ࡢ  ྡ࡛࠶ࡿ㸬ࡇࡢ  ྡ࡟ࡘ࠸࡚ࡣヲ⣽ࡣ୙࡛᫂࠶ࡿ㸬ࡑࡢᖌࡢኴ⏣ṇ൤ࡣ㸪㛗ᒸ⸬ࡢṊኈ࡛࠶ࡾ㸪⵬ᰘ⚄♫ࡢ⟬㢠࡞࡝࡟⟬㢠ዊᥖ⪅ࡢᖌ࡜ࡋ࡚ྡ๓ࡀグࡉࢀ࡚࠸ࡿ  㸬
―2―一仁ᅗ᚟ඖᅗ涌田与板八幡宮の紛失算額（３）㸲㸬⾡ࡢゎㄝ

O1( a 2  b 2 )b  
a 2  b2r2 ➨୍ၥ㸪➨஧ၥ࡜ࡶ࡟ᙜ᫬ࡢゎἲࡣ୙࡛᫂࠶ࡿࡢ࡛᥎ ࡍࡿ㸬㸲㸬㸯 ➨୍ၥ࡟ࡘ࠸࡚


kO2 F ࡜࠾ࡃ㸬෌ࡧ 'AO1E ң 'O2 AF ࡼࡾ
r1:a2

r2:k

2  ࡋࡓࡀࡗ࡚

AEF

'O2OH ࡟ࡘ࠸࡚ O2OHHOBDCGBC 㸪 b(a ( r1 k2r12  4 2a ( r1  r2 )b r2r )r 2 2
r1a 2 ( r1  r2 ) 2r12

2b 2 r2
440

 r2 2 b 2
4 4a 2 r2 2r12 ࡇࢀࢆ  ࡢ᭱ᚋࡢ㡯࡟௦ධࡋᩚ⌮ࡍࡿ㸬

r1 ( 2r1  b  r2 )r  4a r1 ( r1  r2 ) r
0   ࡇࡇ࡟  㸪  ࢆ௦ධࡋ࡚ᩚ⌮ࡍࡿ㸬

( a 4  b 4  a 2 b 2 ) r  2a 3 b a 2  b 2 r'AO1E ң 'O2 AF ࡞ࡢ࡛  a 2 b(a 2  b 2 )b r2:2 2  ࡋࡓࡀࡗ࡚

r2r r(  2 )22 2

 2a 2 ( r1  2r2 )a 2  b2  brr( 1  b) : 122ar2
 rr1   
r1'O2 AF ࡟ࡘ࠸࡚

ࡋࡓࡀࡗ࡚

r1ar2b r
 rr1 )2  (  )2r12 2

 r1 㸸෇ O1 ࡢ┤ᚄ㸪 r2 㸸෇ O2 ࡢ┤ᚄra 2  ( 1  b) 22aࡇࢀࡼࡾ r ࡢᘧࢆᑟࡃ㸬AC 㸪 r 㸸෇ O ࡢ┤ᚄ࡜࠾ࡃ㸬 'AO1E ࡟ࡘ࠸࡚㸪୕ᖹ᪉ࡢᐃ⌮ࡼࡾBC  CG  BD࡞ࡢ࡛㸪୕ᖹ᪉ࡢᐃ⌮ࡼࡾᅗ ᅗ ࡟࠾࠸࡚㸪┤ゅ୕ゅᙧࢆ 'ABC 㸪኱ᘼࡢ෇ࢆ O1 㸪ᑠᘼࡢ෇ࢆ O2 ࡜ࡋ㸪෇ࢆ O ࡜ࡍࡿ㸬෇ O ࠿ࡽ BC ࡟ᘬ࠸ࡓᆶ⥺ࢆ OD 㸪 A ࠿ࡽ O1 B ࡟ᘬ࠸ࡓᆶ⥺ࢆ AE 㸪 O2 ࠿ࡽ AC ࡟ᘬ࠸ࡓᆶ⥺ࢆ O2 F 㸪 O2 ࠿ࡽ BC ࡢᘏ㛗ୖ࡟ᘬ࠸ࡓᆶ⥺ࢆ O2G 㸪 O ࠿ࡽ O2G࡟ᘬ࠸ࡓᆶ⥺ࢆ OH ࡜ࡍࡿ㸬aar2  r1kxr ࡜࠾ࡁ㸪 x ࡢ  ḟ᪉⛬ᘧࢆゎࡃ 㸬x

br1  r1  2b  ࡟  ࢆ௦ධࡋ࡚―3―0 ra 3 b a 2  b2  ab(a 2  b2 ) b
a 4  b 4  a 2b 2a b a 2  b2 (a 2  b a 2  b2 )
a 4  b2 (a 2  b2 )

 

涌田

和芳・外川

r2  r3a b a 2  b2a 2  b a 2  b2bab22aa b2a    㸪  ࡼࡾ r3 ࡢᘧࢆᑟࡃ㸬(a  2 ar1  r1 )r3  2 2a r1 r3  ar1

  ^( 2a  a  r ) r  ar `u ^( 2a  a  r ) r  ar ` 0  1b
§
ab·¨
 ¸
¨ 2
2a ¸¹
© a b0   ࡇࢀࢆᅉᩘศゎࡍࡿ㸬

ᨾ࡟
r

一仁

  
23

1

131

ࡋࡓࡀࡗ࡚
r3ࡇࢀࡀ⾡࡛㏙࡭ࡽࢀ࡚࠸ࡿ㸬 ௒㸪 a 38.44, b 28.83 ࡞ࡢ࡛㸪⟅࠼ࡣ㸪 r 12࡛࠶ࡿ㸬
㸲㸬㸰 ➨஧ၥ࡟ࡘ࠸࡚
A

ar1
2a  a  r1,

ar1
2a  a  r1  ࡇࡇ࡛
ar1
2 a  a  r1
ar1D2 a  a  r1

!

ar1

r12 2aar1

r12 2aO3
ᅗࡼࡾO1

r3O2

ar12a  a  r1 ࢆ  ࡟௦ධࡍࡿ㸬
O5O4B

r2C2 ( 2  1)a  ( 2  1) ar1( 2  1) a  r1ᅗᅗ ࡟࠾࠸࡚㸪⏥෇㸪எ෇㸪୤෇㸪୎෇㸪ᠾ෇ࢆ㸪ࡑࢀࡒࢀ㸪 O1, O2 , O3 , O4 , O5 ࡜ࡋaa ( 2a  r1 )a  (1  2 ) r 1

2 ar1r4 r5࡜࠾ࡃ㸬 ෇ O1, O2 , O3 ࡟ࡘ࠸࡚a ( r1  r3 )  r1r3   ࡀᡂࡾ❧ࡘ 㸬 ABr2r
 r2 r3  322  a ( r4  r5 )  r4 r5   ࡀᡂࡾ❧ࡘ㸬ࡇࢀࡼࡾ r5 ࡢᘧࢆᑟࡃ㸬( a  r4 ) r5  2 ar1r4 r5  ar4a ࡼࡾ㸪  ࡜ྠᵝ࡟a   

୍᪉㸪 ෇ O1, O4 , O5 ࡟ࡘ࠸࡚㸪  ࡜ྠᵝ࡟AB 㸪 rn 㸸෇ On ࡢ┤ᚄ

2 ar1r2 r3  0   ࡲࡓ㸪 ෇ O1, O2 , O5 ࡟ࡘ࠸࡚㸪  ࡜ྠᵝ࡟2 ar1r2 r5ࡀᡂࡾ❧ࡘ 㸬ࡍ࡞ࢃࡕ

a ( r2  r5 )  r2 r5   ࡀᡂࡾ❧ࡘ㸬  ࡜  ࡼࡾ r1 ࢆᾘཤࡋ㸪 r5 ࡟ࡘ࠸―4―与板八幡宮の紛失算額（３）࡚ゎࡃ㸬r2 r4 ar5a  r2 r4x  ᅗࡼࡾ㸪 ࢆ  ࡟௦ධࡍࡿ㸬( r2  r4 )a ar5991
2 2 r2 2 
24 , 9  17   6  4 2, 248

  2 r1 (a  r2 r4 ) 㸪  ࡼࡾ991
2 2 
2 2 224
  4x
ᨾ࡟

991
2 2 
2 2 
224 a  
44r1 r2 r4 (a  r2 r4 ) ( r2  r4 )2 a 2   r1

୧㎶ࢆ a2r2 r4 ๭ࡿ㸬r§
r r ·4  1 ¨1  2  4 ¸a ¨©
a a ¸¹r2a r4ar4a 2  r2aࡇࡢᘧࡀ⾡࡛㏙࡭ࡽࢀ࡚࠸ࡿ㸬ὀ ࿴⟬࡛ࡣ㕍⫤ᘻࡢ⾡㸦⌮㸧࡜ࡋ࡚▱ࡽࢀ࡚࠸ࡓ㸬 ┦ఝグྕࢆ⏝࠸࡚࠸ࡿࡀ㸪࿴⟬࡛ࡣ┦ఝ࡜࠸࠺ᴫᛕࡣ⾲࡟ࡣฟࡉࡎ㸪┤᥋㸪ẚ౛㛵ಀࢆ㏙࡭ࡿ㸬 ෇ࡢඹ㏻እ᥋⥺ࢆồࡵࡿබᘧࡣ㸪ࡼࡃ▱ࡽࢀ࡚࠸ࡓ㸦ᒣᮏ㈡๓⦅ࠗ⟬ἲຓ⾡࠘  ࡢබᘧ 㸧㸬 ḟ᪉⛬ᘧࡢゎࡢබᘧࡀ▱ࡽࢀ࡚࠸ࡓ㸬」ྕࡢ
ෆ㸩ࡢ࡜ࡁࡣ㸪  ࡟࠾࠸࡚r1࡜࠾ࡃ࡜㸪  ࡼࡾax

2 x

r2a
CD

1  (1  2 ) x

  a ࡼࡾr1r
 r1r4  422a2a ba   b 1    a( r1  r2 )h  r1r2  2 r1r2 r3h2a   0   ࠗ⟬ἲຓ⾡࠘ࡣබᘧࡢࡳ࡛࠶ࡿࡀ㸪ḟࡢࡼ࠺࡟ド᫂ࡉࢀࡿ㸬2  x    㸪  ࢆ  ࡟௦ධࡋ࡚ᩚ⌮ࡍࡿ㸬4 x 3  4(1  2 ) x 2 x  (23  2 2 ) x 2
 (16  14 2 ) x x  26 x  (8  8 2 ) x  8X

ᒣᮏ㈡๓⦅ࠗ⟬ἲຓ⾡࠘ࡢබᘧ  ࡟㸪ୗࡢᅗ
࡟࠾࠸࡚㸪ḟࡢᘧࡀᡂࡾ❧ࡘ࡜࠶ࡿ㸬ࡋࡓࡀࡗ࡚

r4a2

 ࡞ࡢ࡛ r ! b ࡜࡞ࡾ୙㐺ᙜ࡛࠶ࡿ㸬ࡀᡂࡾ❧ࡘ㸬ࡍ࡞ࢃࡕ

r1  r4

1
 x  1 ࡞ࡢ࡛20

  x ࡢ  ḟ᪉⛬ᘧࢆᅉᩘศゎࡋ࡚ඖ࡟ᡠࡍ㸬( x  2)( x  (2  2 ))u (2 x  x  (2  2 ))(2 x  x  2)

0

 
ࡇࡢゎࡣ

AO3hO1tsBᅗ―5―O2涌田h和芳・外川AB 㸪 rn 㸸෇ On ࡢ┤ᚄs 㸸 O1 ࠿ࡽ AB ࡟ᘬ࠸ࡓᆶ⥺ࡢ㛗ࡉt 㸸 O2 ࠿ࡽ AB ࡟ᘬ࠸ࡓᆶ⥺ࡢ㛗ࡉ ෇ O1 㸪 O2 ࡟ࡘ࠸࡚str1r2   ୕ᖹ᪉ࡢᐃ⌮ࡼࡾs2r rr r( 1  3 ) 2  (h  1  3 ) 22 22 2   2h( r1  r3 )  hྠᵝ࡟t2h ( r2  r3 )  h 2    㸪  ࡼࡾ
s2  t 2h ( r1  r2 )    㸪  ࡼࡾsth( r1  r2 )r1r2   㸪  ࡼࡾ2sr1r2 h( r1  r2 )r1r2  h( r1  r2 )r1r2

r1r23UHVV㸪 ᖺ㸬 㑥ヂ ⪷࡞ࡿᩘᏛ㸸⟬㢠㸪᳃໭ฟ∧㸪 ᖺ㸬 ᑠ಴㔠அຓ᪥ᮏࡢᩘᏛ㸪ᒾἼ᪂᭩㸪 ᖺ㸬 ᖹᒣㅉ㸸࿴⟬ࡢṔྐ㸫ࡑࡢᮏ㉁࡜Ⓨᒎ㸪ࡕࡃࡲᩥᗜ㸪 ᖺ㸦 ᖺึฟ㸧㸬 ᖹᒣㅉ㸸࿴⟬ྐୖࡢேࠎ㸪ࡕࡃࡲᩥᗜ㸪 ᖺ㸦 ᖺึฟ㸧㸬 ኱▮┿୍࿴⟬ධ㛛㸪᪥ᮏホㄽ♫㸪 ᖺ㸬 ᖹᒣㅉ㸸࿴⟬ࡢㄌ⏕㸪ᜏᫍ♫ᜏᫍ㛶㸪 ᖺ㸬 బ⸨೺୍᪥ᮏே࡜ᩘỤᡞᗢẸࡢᩘᏛ㸪ᮾὒ᭩ᗑ㸪 ᖺ㸬 ῝ᕝⱥಇ᪥ᮏࡢᩘᏛ࡜⟬㢠㸪᳃໭ฟ∧㸪 ᖺ㸬 బ⸨೺୍᪂࿴⟬ධ㛛㸪◊ᡂ♫㸪 ᖺ㸬 ᑠᕝ᮰㸪ᖹ㔝ⴥ୍ㅮᗙᩘᏛࡢ⪃࠼᪉  ᩘᏛࡢṔྐ㸪ᮅ಴᭩ᗑ㸪 ᖺ㸬 ఀ⸨ὒ⨾ᡭ࡙ࡃࡾ㑅ᢥᩘᏛ࠾ࡶࡋࢁ࿴⟬㸪᫂἞ᅗ᭩㸪 ᖺ㸬 ᑠᑎ⿱㸸ࡔ࠿ࡽᴦࡋ࠸Ụᡞࡢ⟬㢠㸪◊ᡂ♫㸪ᖺ㸬 ᱜ஭㐍㸸ỤᡞࡢᩘᏛᩍ⛉᭩㸪㞟ⱥ♫㸪 ᖺ㸬 ᑠᑎ⿱㸸ỤᡞࡢᩘᏛ࿴⟬㸪ᢏ⾡ホㄽ♫㸪 ᖺ㸬 ᾛ⏣࿴ⰾ㸪እᕝ୍ோ㸸⵬ᰘ⚄♫ࡢ⟬㢠㸪㛗ᒸᕤᴗ㧗➼ᑓ㛛Ꮫᰯ⣖せ➨  ᕳ➨  ྕ㸪SS㸪 ᖺ㸬 ᒣᮏ㈡๓㸸⟬ἲຓ⾡㸪ኳಖ 㸦㸧ᖺ㸪ᮾ໭኱Ꮫ࿴⟬㈨ᩱࢹ࣮ࢱ࣮࣋ࢫⶶ㸬῝ᕝⱥಇᰯὀ㸸⟬ἲຓ⾡㸦᚟้㸧㸪ᮅ᪥᪂⪺♫஦ᴗ㒊ྡྂᒇ௻⏬஦ᴗࢳ࣮࣒㸪 ᖺ㸬ཷ௜    㸪  ࡼࡾ r3 ࡟ࡘ࠸࡚ゎࡃ㸬
r3一仁( h( r1  r2 )  r1r2 ) 2  
4r1r2hࡇࢀࡼࡾබᘧࡀᚓࡽࢀࡿ㸬ཧ⪃ᩥ⊩୰ᮧ᫬୓㸸㉏⚆⚄⟬㸪ኳಖ 㸦㸧ᖺ㸪ᮾ໭኱Ꮫ࿴⟬㈨ᩱࢹ࣮ࢱ࣮࣋ࢫⶶ㸬ᾛ⏣࿴ⰾ㸪እᕝ୍ோ㸸୚ᯈඵᖭᐑࡢ⣮ኻ⟬㢠㸪㛗ᒸᕤᴗ㧗➼ᑓ㛛Ꮫᰯ⣖せ➨  ᕳSS㸪 ᖺ㸬 ᾛ⏣࿴ⰾ㸪እᕝ୍ோ㸸୚ᯈඵᖭᐑࡢ⣮ኻ⟬㢠㸦㸰㸧㸪㛗ᒸᕤᴗ㧗➼ᑓ㛛Ꮫᰯ⣖せ➨  ᕳSS㸪 ᖺ㸬 ୕ୖ⩏ኵᩥ໬ྐୖࡼࡾぢࡓࡿ᪥ᮏࡢᩘᏛ㸪ᒾἼᩥᗜ㸪 ᖺ㸦 ᖺึฟ㸧㸬 ῝ᕝⱥಇ㸪ࢲ࣭ࣥ࣌ࢻ࣮᪥ᮏࡢᗄఱ㸫ఱ㢟ゎࡅࡲࡍ࠿㸽㸪᳃໭ฟ∧㸪 ᖺ㸬 ῝ᕝⱥಇ㸪ࢲ࣭ࣥࢯࢥࣟࣇࢫ࣮࢟᪥ᮏࡢᩘᏛ㸫ఱ㢟ゎࡅࡲࡍ࠿㸽㸦ୖ㸧 ୗ 㸪᳃໭ฟ∧㸪 ᖺ㸬 ῝ᕝⱥಇ㸪7RQ\5RWKPDQ6DFUHG0DWKHPDWLFV㸫-DSDQHVH 7HPSOH *HRPHWU\ 㸪 3ULQFHWRQ 8QLY 㸬―6―