Elements of Conic Sections And Analytical Geometry
Elements of Conic Sections And Analytical Geometry
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31.
• L. , 3, 15, Cor. ; W. , 3, 14, Cor.
72 CONIC SECTIONS.
Therefore R^= ™-^^ (AC. DC)i F7 7V In the same manner it may be shown that r'as . P' ^ ^ Therefore R^ : r^ :: FlOiV^ : FZTZV^.
(132«) Cor. The radius of curvature varies as the cube of the liameter conjugate to that w^hich passes through the point of con- tact. For, in the equation R= . ^ j^^, tje denominator of the fraction is constant ; and therefore R varies as CH^.
(133) Prop. VL Theorem.
In the parabola the squares of the radii o...f curvature at different points of the curve, are to each other as the cubes of the distances from the focus.
T'lat is, putting R and r for the radii of curvature at M and Z, R2 : r2 :: FM« ; FZ«.
(Fig. 61. ) OF CURVATURE. 78 Draw MP the diameter of the osculating circle, join RP, and draw FL perpendicular to MT. Then will the triangles MRP and FML be similar. * Hence MP : MR=(129) 4FM :: FM : FL.
And squaring, MP^ : 4FM :: FM^ : FL2=;(68c) AF. FM.
leFSP Therefore MF : 4FM' :: FM : AF, and'MP2= AF In the same manner it may be shown that the square of the IfiFT* diameter of curvature at Z= .
What to read after Elements of Conic Sections And Analytical Geometry?
You can find similar books in the "Read Also" column, or choose other free books by James H James Henry Coffin to read onlineMoreLess
To quickly assess the difficulty of the text, read a short excerpt:
31.
• L. , 3, 15, Cor. ; W. , 3, 14, Cor.
72 CONIC SECTIONS.
Therefore R^= ™-^^ (AC. DC)i F7 7V In the same manner it may be shown that r'as . P' ^ ^ Therefore R^ : r^ :: FlOiV^ : FZTZV^.
(132«) Cor. The radius of curvature varies as the cube of the liameter conjugate to that w^hich passes through the point of con- tact. For, in the equation R= . ^ j^^, tje denominator of the fraction is constant ; and therefore R varies as CH^.
(133) Prop. VL Theorem.
In the parabola the squares of the radii o...f curvature at different points of the curve, are to each other as the cubes of the distances from the focus.
T'lat is, putting R and r for the radii of curvature at M and Z, R2 : r2 :: FM« ; FZ«.
(Fig. 61. ) OF CURVATURE. 78 Draw MP the diameter of the osculating circle, join RP, and draw FL perpendicular to MT. Then will the triangles MRP and FML be similar. * Hence MP : MR=(129) 4FM :: FM : FL.
And squaring, MP^ : 4FM :: FM^ : FL2=;(68c) AF. FM.
leFSP Therefore MF : 4FM' :: FM : AF, and'MP2= AF In the same manner it may be shown that the square of the IfiFT* diameter of curvature at Z= .
What to read after Elements of Conic Sections And Analytical Geometry?
You can find similar books in the "Read Also" column, or choose other free books by James H James Henry Coffin to read onlineMoreLess
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