Examples of the Processes of the Differential and Integral Calculus |
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Page 102
... tangent is also a maximum . Let P be the point , AP = x , AC = a , AB = b ; = tan BPC tan ( APC – APB ) = ( a - b ) x - ; ab + x2 x therefore is to be a maximum . ab + x2 2 Whence ab - = 0 and x = ( ab ) . ( 24 ) Through a point M ( fig ...
... tangent is also a maximum . Let P be the point , AP = x , AC = a , AB = b ; = tan BPC tan ( APC – APB ) = ( a - b ) x - ; ab + x2 x therefore is to be a maximum . ab + x2 2 Whence ab - = 0 and x = ( ab ) . ( 24 ) Through a point M ( fig ...
Page 132
... tangents to a rect- angular hyperbola with perpendiculars drawn to them from the centre , and its form is that of the figure . Of the properties of the arcs of this curve , which have been in- vestigated by Fagnani and Euler , we shall ...
... tangents to a rect- angular hyperbola with perpendiculars drawn to them from the centre , and its form is that of the figure . Of the properties of the arcs of this curve , which have been in- vestigated by Fagnani and Euler , we shall ...
Page 133
... tangent at its extremity and the axis of x ; and the solid formed by the revolution of the curve round . its asymptote is equal to a cylinder , the radius of whose base is the bounding ordinate , and whose height is the tangent at its ...
... tangent at its extremity and the axis of x ; and the solid formed by the revolution of the curve round . its asymptote is equal to a cylinder , the radius of whose base is the bounding ordinate , and whose height is the tangent at its ...
Page 137
... tangent at any point P ( fig . 19 ) is perpendicular to the corresponding chord BQ of the generating circle , and con- sequently that it is parallel to CQ : from this also it readily follows that if QR be a tangent to the generating ...
... tangent at any point P ( fig . 19 ) is perpendicular to the corresponding chord BQ of the generating circle , and con- sequently that it is parallel to CQ : from this also it readily follows that if QR be a tangent to the generating ...
Page 142
... tangent be drawn at the extremity of the are formed by one revolution of the radius , the subtangent will be equal to the circumference of the circle whose radius is a . If at the extremity of the arc formed by two revolutions , it will ...
... tangent be drawn at the extremity of the are formed by one revolution of the radius , the subtangent will be equal to the circumference of the circle whose radius is a . If at the extremity of the arc formed by two revolutions , it will ...
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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