A treatise on the integral calculus and its applications1857 |
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Page 71
... ordinates . 68. Let P be any point on the curve APQ , and let x , y be its co - ordinates ; let s denote the length of the arc AP measured from a fixed point A up to P ; T M R then , ( Dif . Cal . Art . 307 ) Hence , ds dx = dy / { 1 ...
... ordinates . 68. Let P be any point on the curve APQ , and let x , y be its co - ordinates ; let s denote the length of the arc AP measured from a fixed point A up to P ; T M R then , ( Dif . Cal . Art . 307 ) Hence , ds dx = dy / { 1 ...
Page 72
... co- ordinates of the variable extremity of the arc , the curve is said to be rectifiable . 70. Application to the Parabola . The equation to the parabola is y = √ ( 4ax ) ; hence ds X dy dx = √ + a = x ' dx ( x + a thus S = dx X ...
... co- ordinates of the variable extremity of the arc , the curve is said to be rectifiable . 70. Application to the Parabola . The equation to the parabola is y = √ ( 4ax ) ; hence ds X dy dx = √ + a = x ' dx ( x + a thus S = dx X ...
Page 75
... Co - ordinate Geo- metry , Art . 168 ) . Therefore , by the preceding article , ds аф = √ ( a2 cos2 + b2 sin2 4 ) , and s = √√ ( a2 4 + b2 = [ √ ( a2 cos2 6 + b2 sin2 p ) dô = a ... Co - ordinates . 79. Let 2 LENGTHS OF CURVES . 75.
... Co - ordinate Geo- metry , Art . 168 ) . Therefore , by the preceding article , ds аф = √ ( a2 cos2 + b2 sin2 4 ) , and s = √√ ( a2 4 + b2 = [ √ ( a2 cos2 6 + b2 sin2 p ) dô = a ... Co - ordinates . 79. Let 2 LENGTHS OF CURVES . 75.
Page 76
Isaac Todhunter. Plane Curves . Polar Co - ordinates . 79. Let r , be the polar co - ordinates of any point of a curve , and s the length of the arc measured from any fixed point up to this point ; then ( Dif . Cal . Art . 311 ) hence ds ...
Isaac Todhunter. Plane Curves . Polar Co - ordinates . 79. Let r , be the polar co - ordinates of any point of a curve , and s the length of the arc measured from any fixed point up to this point ; then ( Dif . Cal . Art . 311 ) hence ds ...
Page 79
... co - ordinates of P , and x and y ' the variable co - ordinates . Hence it will be found that the per- pendicular p from the origin on the tangent at P is given by αθ p = ( a + 2b ) sin ; 26 also p2 = a2 + 4b ( a + b ) sin2 αθ ; 26 thus ...
... co - ordinates of P , and x and y ' the variable co - ordinates . Hence it will be found that the per- pendicular p from the origin on the tangent at P is given by αθ p = ( a + 2b ) sin ; 26 also p2 = a2 + 4b ( a + b ) sin2 αθ ; 26 thus ...
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Application c² sin² Cambridge circle cloth co-ordinates constant cos² cos³ Crown 8vo curve cycloid definite integral denote differential coefficient double integral dx dx dx dy dz dy dx dz dx element ellipse equal Eulerian integral example expression Fcap Fellow of St Find the area find the volume formula function Hence indefinitely Integral Calculus integrate with respect intrinsic equation John's College latus rectum length limits M.A. Fellow MACMILLAN & CO.'S numerous obtain ordinate parabola partial fractions perpendicular plane positive preceding article radius vector result revolution revolve round round the axis shew sin² solid suppose surface tangent tion tractory transformed integral Trinity College unity vanishes variables vertex x₁ Y₁ πα аф
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