Examples of the Processes of the Differential and Integral Calculus |
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Page 137
... evolute of the cycloid is an equal cycloid in an inverted position , and that the radius of curvature is double of the chord of the generating circle which is perpendicular to the tangent . He also discovered the important dynamical ...
... evolute of the cycloid is an equal cycloid in an inverted position , and that the radius of curvature is double of the chord of the generating circle which is perpendicular to the tangent . He also discovered the important dynamical ...
Page 141
... evolute of the epicycloid is a similar figure , the radii of the fixed and generating circles being a2 and a + 2b ab a + 2b respectively . An analogous theorem holds for the hypo- cycloid . ( 12 ) The Spiral of Archimedes . While the ...
... evolute of the epicycloid is a similar figure , the radii of the fixed and generating circles being a2 and a + 2b ab a + 2b respectively . An analogous theorem holds for the hypo- cycloid . ( 12 ) The Spiral of Archimedes . While the ...
Page 143
... evolute and involute of this curve are both spirals equal to the original one , and differing from it in position only ; its caustics both by reflexion and refraction ( the pole being the origin of light ) are also spirals equal to the ...
... evolute and involute of this curve are both spirals equal to the original one , and differing from it in position only ; its caustics both by reflexion and refraction ( the pole being the origin of light ) are also spirals equal to the ...
Page 160
... evolute of the spiral , its equation being Ꮎ 12 = - = - a ( 6 ) The equation to the Cardioid is - r = a ( 1 − cos 6 ) . If r ' be a radius in the direction of r produced backwards , r ' = a { 1 - cos ( 0 + π ) } = a ( 1 + cos 0 ) ...
... evolute of the spiral , its equation being Ꮎ 12 = - = - a ( 6 ) The equation to the Cardioid is - r = a ( 1 − cos 6 ) . If r ' be a radius in the direction of r produced backwards , r ' = a { 1 - cos ( 0 + π ) } = a ( 1 + cos 0 ) ...
Page 192
... Evolutes of Curves . When a curve is referred to rectanglar co - ordinates , the co - ordinates ( a , ß ) of its centre of curvature are given by the equations ' dy 2 dy 2 1 + 1 + dy d α = x " B = y + ; d2y dx d2y or , if u = 0 be the ...
... Evolutes of Curves . When a curve is referred to rectanglar co - ordinates , the co - ordinates ( a , ß ) of its centre of curvature are given by the equations ' dy 2 dy 2 1 + 1 + dy d α = x " B = y + ; d2y dx d2y or , if u = 0 be the ...
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a² b2 a²x² angle arbitrary constant assume asymptote becomes branches C₁ Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz dz dz eliminate ellipse equal Euler factor formula fraction function Geometry gives Hence hypocycloid infinite intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral Multiply negative origin parabola perpendicular radius SECT singular points singular solution spiral Substituting subtangent surface tangent plane theorem triangle University of Cambridge vanish whence x²)³