\\(L\\) \u3092\u6b63\u5b9a\u6570\u3068\u3059\u308b. \u5ea7\u6a19\u5e73\u9762\u306e \\(x\\) \u8ef8\u4e0a\u306e\u6b63\u306e\u90e8\u5206\u306b\u3042\u308b\u70b9 P \\(( t , 0 )\\) \u306b\u5bfe\u3057,\r\n\u539f\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3057\u70b9 P \u3092\u901a\u308b\u5186\u5468\u4e0a\u3092, P \u304b\u3089\u51fa\u767a\u3057\u3066\u53cd\u6642\u8a08\u56de\u308a\u306b\u9053\u306e\u308a \\(L\\) \u3060\u3051\u9032\u3093\u3060\u70b9\u3092 Q \\(\\left( u(t) , v(t) \\right)\\) \u3068\u8868\u3059.<\/p>\r\n
(1)<\/strong>\u3000\\(u(t)\\) , \\(v(t)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(0 \\lt a \\lt 1\\) \u306e\u7bc4\u56f2\u306e\u5b9f\u6570 \\(a\\) \u306b\u5bfe\u3057, \u7a4d\u5206\r\n\\[\r\nf(a) = \\displaystyle\\int _ a^1 \\sqrt{\\{ u'(t) \\}^2 +\\{ v'(t) \\}^2} \\, dt\r\n\\]\r\n\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (3)<\/strong>\u3000\u6975\u9650 \\(\\displaystyle\\lim _ {a \\rightarrow +0} \\dfrac{f(a)}{\\log a}\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(\\angle\\text{POQ} = \\dfrac{L}{t}\\) \u306a\u306e\u3067\r\n\\[\r\nu(t) = \\underline{t \\cos \\dfrac{L}{t}} , \\quad v(t) = \\underline{t \\sin \\dfrac{L}{t}}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \\[\\begin{align}\r\nu'(t) & = \\cos \\dfrac{L}{t} +t \\left( -\\dfrac{L}{t^2} \\right) \\left( -\\sin \\dfrac{L}{t} \\right) \\\\\r\n& = \\cos \\dfrac{L}{t} +\\dfrac{L}{t} \\sin \\dfrac{L}{t} \\\\\r\nv'(t) & = \\sin \\dfrac{L}{t} +t \\left( -\\dfrac{L}{t^2} \\right) \\cos \\dfrac{L}{t} \\\\\r\n& = \\sin \\dfrac{L}{t} -\\dfrac{L}{t} \\cos \\dfrac{L}{t}\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\{ u'(t) \\}^2 & = \\cos^2 \\dfrac{L}{t} +\\dfrac{2L}{t} \\cos \\dfrac{L}{t} \\sin \\dfrac{L}{t} +\\dfrac{L^2}{t^2} \\sin^2 \\dfrac{L}{t} \\\\\r\n\\{ v'(t) \\}^2 & = \\sin^2 \\dfrac{L}{t} -\\dfrac{2L}{t} \\cos \\dfrac{L}{t}\\sin \\dfrac{L}{t} +\\dfrac{L^2}{t^2} \\cos^2 \\dfrac{L}{t}\r\n\\end{align}\\]\r\n\\(\\sin^2 \\dfrac{L}{t} +\\cos^2 \\dfrac{L}{t} = 1\\) \u306b\u7740\u76ee\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\sqrt{\\{ u'(t) \\}^2 +\\{ v'(t) \\}^2} & = \\sqrt{1 +\\dfrac{L^2}{t^2}} \\\\\r\n& = \\dfrac{\\sqrt{L^2 +t^2}}{t} \\quad ( \\ \\text{\u2235} \\ t \\gt 0\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nf(a) = \\displaystyle\\int _ a^1 ! \\dfrac{\\sqrt{L^2 +t^2}}{t} \\, dt\r\n\\]\r\n\u3053\u3053\u3067 \\(t = L \\tan \\theta\\) \u3068\u7f6e\u63db\u3059\u308b\u3068\r\n\\[\r\ndt = \\dfrac{L}{\\cos^2 \\theta} d \\theta \\\\\r\nt \\ : \\ a \\rightarrow 1 \\text{\u306e\u3068\u304d} \\quad \\theta \\ : \\ \\theta _ 1 \\rightarrow \\theta _ 2 \\\\\r\n\\left( \\ \\text{\u305f\u3060\u3057, } \\tan \\theta _ 1 = \\dfrac{a}{L} , \\ \\tan \\theta _ 2 = \\dfrac{1}{L} \\quad ...[1] \\ \\right)\r\n\\]\r\n\u3053\u308c\u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\nf(a) & = \\displaystyle\\int _ {\\theta _ 1}^{\\theta _ 2} \\dfrac{L \\sqrt{1 +\\tan^2 \\theta}}{L \\tan \\theta} \\cdot \\dfrac{L}{\\cos^2 \\theta} \\, d\\theta \\\\\r\n& = L \\displaystyle\\int _ {\\theta _ 1}^{\\theta _ 2} \\dfrac{1}{\\sin \\theta \\cos^2 \\theta} \\, d\\theta \\\\\r\n& = L \\displaystyle\\int _ {\\theta _ 1}^{\\theta _ 2} \\dfrac{\\sin \\theta}{\\cos^2 \\theta \\left( 1 -\\cos^2 \\theta \\right)} \\, d\\theta \\\\\r\n& = -L \\displaystyle\\int _ {\\theta _ 1}^{\\theta _ 2} \\left( \\dfrac{1}{\\cos^2 \\theta} +\\dfrac{1}{2( 1 -\\cos \\theta )} +\\dfrac{1}{2( 1 +\\cos \\theta )} \\right) \\, ( \\cos \\theta)' d\\theta \\\\\r\n& = -L \\left[ -\\dfrac{1}{\\cos \\theta} -\\dfrac{1}{2} \\log( 1 -\\cos \\theta ) +\\dfrac{1}{2} \\log( 1 +\\cos \\theta ) \\right] _ {\\theta _ 1}^{\\theta _ 2} \\\\\r\n& = L \\left( \\dfrac{1}{\\cos \\theta _ 2} -\\dfrac{1}{\\cos \\theta _ 1} +\\dfrac{1}{2} \\log \\dfrac{1 -\\cos \\theta _ 2}{1 +\\cos \\theta _ 2} -\\dfrac{1}{2} \\log \\dfrac{1 -\\cos \\theta _ 1}{1 +\\cos \\theta _ 1} \\right)\r\n\\end{align}\\]\r\n\u3053\u3053\u3067 [1] \u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{1}{\\cos \\theta _ 1} & = \\sqrt{1 +\\tan^2 \\theta _ 1} = \\dfrac{\\sqrt{L^2 +a^2}}{L} , \\\\\r\n\\dfrac{1}{\\cos \\theta _ 2} & = \\sqrt{1 +\\tan^2 \\theta _ 2} = \\dfrac{\\sqrt{L^2 +1}}{L}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308b\u3068\r\n\\[\\begin{align}\r\nf(a) & = L \\left( \\dfrac{\\sqrt{L^2 +1}}{L} -\\dfrac{\\sqrt{L^2 +a^2}}{L} \\right. \\\\\r\n& \\qquad \\left. +\\dfrac{1}{2} \\log \\dfrac{\\sqrt{L^2 +1} -L}{\\sqrt{L^2 +1} +L} -\\dfrac{1}{2} \\log \\dfrac{\\sqrt{L^2 +a^2} -L}{\\sqrt{L^2 +a^2} +L} \\right) \\\\\r\n& = \\underline{\\sqrt{L^2 +1} -\\sqrt{L^2 +a^2}} \\\\\r\n& \\qquad \\underline{+L \\log \\left( \\sqrt{L^2 +1} -L \\right) -L \\log \\dfrac{\\sqrt{L^2 +a^2} -L}{a}}\r\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n \\(A = \\sqrt{L^2 +1} -\\sqrt{L^2 +a^2} +L \\log \\left( \\sqrt{L^2 +1} -L \\right)\\) , \\(B = L \\log \\dfrac{\\sqrt{L^2 +a^2} -L}{a}\\) \u3068\u304a\u304f.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(\\displaystyle\\lim _ {a \\rightarrow +0} \\log a = -\\infty\\) \u3067\u3042\u308b\u3053\u3068\u304b\u3089, \\(a \\rightarrow +0\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\n\\dfrac{A}{\\log a} & \\rightarrow 0 \\\\\r\n\\dfrac{B}{\\log a} & = L \\cdot \\dfrac{\\log \\frac{a^2}{a \\left( \\sqrt{L^2 +a^2} +L \\right)}}{\\log a} \\\\\r\n& = L \\left\\{ 1 -\\dfrac{\\log \\left( \\sqrt{L^2 +a^2} +L \\right)}{\\log a} \\right\\} \\\\\r\n& \\rightarrow L\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\r\n\\displaystyle\\lim _ {a \\rightarrow +0} \\dfrac{f(a)}{\\log a} = \\displaystyle\\lim _ {a \\rightarrow +0} \\dfrac{A-B}{\\log a} = \\underline{-L}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(L\\) \u3092\u6b63\u5b9a\u6570\u3068\u3059\u308b. \u5ea7\u6a19\u5e73\u9762\u306e \\(x\\) \u8ef8\u4e0a\u306e\u6b63\u306e\u90e8\u5206\u306b\u3042\u308b\u70b9 P \\(( t , 0 )\\) \u306b\u5bfe\u3057, \u539f\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3057\u70b9 P \u3092\u901a\u308b\u5186\u5468\u4e0a\u3092, P \u304b\u3089\u51fa\u767a\u3057\u3066\u53cd\u6642\u8a08\u56de\u308a\u306b\u9053\u306e\u308a \\(L\\) \u3060\u3051\u9032 … \u7d9a\u304d\u3092\u8aad\u3080