Examples of the Processes of the Differential and Integral CalculusJ and J. J. Deighton, 1846 - 529 стор. |
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Сторінка 139
... Hypocycloid respectively . When a and b are commen- surable the curve will re - enter after a number of revolutions of the generating circle equal to the least common multiple of a and b in such cases the curve is expressible by an ...
... Hypocycloid respectively . When a and b are commen- surable the curve will re - enter after a number of revolutions of the generating circle equal to the least common multiple of a and b in such cases the curve is expressible by an ...
Сторінка 140
... hypocycloid occurs in the solution of many problems . If in the equations to the hypotrochoid we put b = a 2 , then @ = + h h ) COS Ꮎ , - y = ( − h ) sin 0 . a 2 hx2 2 h = h2 2 Whence ( -4 ) + ( + ) y ' – ( ~ * − 4 ' ) ' , 4 which is ...
... hypocycloid occurs in the solution of many problems . If in the equations to the hypotrochoid we put b = a 2 , then @ = + h h ) COS Ꮎ , - y = ( − h ) sin 0 . a 2 hx2 2 h = h2 2 Whence ( -4 ) + ( + ) y ' – ( ~ * − 4 ' ) ' , 4 which is ...
Сторінка 141
... hypocycloid it is ( 3a - 2b ) . a a2 of the fixed and generating circles being and a + 2b The evolute of the epicycloid is a similar figure , the radii ab a + 2b respectively . An analogous theorem holds for the hypo- cycloid . ( 12 ) ...
... hypocycloid it is ( 3a - 2b ) . a a2 of the fixed and generating circles being and a + 2b The evolute of the epicycloid is a similar figure , the radii ab a + 2b respectively . An analogous theorem holds for the hypo- cycloid . ( 12 ) ...
Сторінка 146
... of the tangent intercepted between the axes = 2 ( x2 + y2 ) 3 = a ; or the hypocycloid is constantly touched by a straight line of given length which slides between two rectangular axes . The con- 146 TANGENTS TO CURVES .
... of the tangent intercepted between the axes = 2 ( x2 + y2 ) 3 = a ; or the hypocycloid is constantly touched by a straight line of given length which slides between two rectangular axes . The con- 146 TANGENTS TO CURVES .
Сторінка 147
... hypocycloid , was first shewn by John Bernoulli . ( See his Works , Vol . 111. p . 447. ) For the perpendicular from the origin on the tangent we find p = ( axy ) 3 . ( 4 ) In the cissoid of Diocles , y2 = 2 a x3 - x whence the ...
... hypocycloid , was first shewn by John Bernoulli . ( See his Works , Vol . 111. p . 447. ) For the perpendicular from the origin on the tangent we find p = ( axy ) 3 . ( 4 ) In the cissoid of Diocles , y2 = 2 a x3 - x whence the ...
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a² b2 a²x² angle arbitrary constant asymptote axis becomes C₁ c²x² Cambridge circle co-ordinates condition curvature curve cycloid cylinder determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz eliminate ellipse equal Euler find the value formula function Geometry gives Hence hypocycloid infinite Integrating with respect intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral maximum minimum Multiply negative origin parabola perpendicular radius radius of curvature singular solution spiral Substituting subtangent surface tangent plane theorem tractory triangle vanish whence x²)³