Examples of the Processes of the Differential and Integral Calculus |
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Page ix
... Properties 129 ***** ..... IX . X. On the Tangents , Normals , and Asymptotes to Curves ...... 144 Singular Points in Curves 162 XI . On the Tracing of Curves from their Equations 175 XII . On the Curvature of Curved Lines ......... 188 ...
... Properties 129 ***** ..... IX . X. On the Tangents , Normals , and Asymptotes to Curves ...... 144 Singular Points in Curves 162 XI . On the Tracing of Curves from their Equations 175 XII . On the Curvature of Curved Lines ......... 188 ...
Page 129
... PROPERTIES . As in the following chapters frequent reference will be made to certain curves which have acquired historical im- portance , and have in consequence been distinguished by particular names , I shall here describe the mode of ...
... PROPERTIES . As in the following chapters frequent reference will be made to certain curves which have acquired historical im- portance , and have in consequence been distinguished by particular names , I shall here describe the mode of ...
Page 132
... properties of the arcs of this curve , which have been in- vestigated by Fagnani and Euler , we shall treat in the chapter on the comparison of Transcendents in the Integral Calculus . If we assume x = r cos 0 , y = r sin 0 , we find r2 ...
... properties of the arcs of this curve , which have been in- vestigated by Fagnani and Euler , we shall treat in the chapter on the comparison of Transcendents in the Integral Calculus . If we assume x = r cos 0 , y = r sin 0 , we find r2 ...
Page 133
... properties : others were discovered by Huyghens . Euler † , and more recently Vincent , in the An- nales de Gergonne , Vol . xv . p . 1 , have conceived that the equation y = a * expresses , besides the continuous curve , a series of ...
... properties : others were discovered by Huyghens . Euler † , and more recently Vincent , in the An- nales de Gergonne , Vol . xv . p . 1 , have conceived that the equation y = a * expresses , besides the continuous curve , a series of ...
Page 134
... properties are analogous to properties of the circle . Thus , the part of the normal intercepted between the curve and the axis of x is equal to the radius vector , but measured in the opposite direction ; and if we represent an ...
... properties are analogous to properties of the circle . Thus , the part of the normal intercepted between the curve and the axis of x is equal to the radius vector , but measured in the opposite direction ; and if we represent an ...
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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