Examples of the Processes of the Differential and Integral Calculus |
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Page 141
... Spiral of Archimedes . While the straight line OM ( fig . 25 ) revolves uniformly round O , let the point P move uniformly along OM : the locus of the point P is the spiral of Archimedes . To find its equa- tion let AOP = 0 , OP = r ...
... Spiral of Archimedes . While the straight line OM ( fig . 25 ) revolves uniformly round O , let the point P move uniformly along OM : the locus of the point P is the spiral of Archimedes . To find its equa- tion let AOP = 0 , OP = r ...
Page 142
... Spiral . The definition of this spiral is , that the radius increases in a geometric while the angle increases in an arithmetic ratio . 0 Hence its equation will be of the form r = cea , rcea , = a . or , as it is usually written , r ...
... Spiral . The definition of this spiral is , that the radius increases in a geometric while the angle increases in an arithmetic ratio . 0 Hence its equation will be of the form r = cea , rcea , = a . or , as it is usually written , r ...
Page 143
... spirals equal to the primary one ; and if another equal spiral be made to roll on the first , the pole of the rolling spiral will trace out another spiral equal to the original . This property of the logarithmic spiral of constantly ...
... spirals equal to the primary one ; and if another equal spiral be made to roll on the first , the pole of the rolling spiral will trace out another spiral equal to the original . This property of the logarithmic spiral of constantly ...
Page 158
... Spirals may have asymptotic circles : these are found by the condition that an infinite value of gives a finite value for r . Ex . 1. The equation to the spiral of Archimedes is 1 ' = ав . The angle between the radius and tangent is tan ...
... Spirals may have asymptotic circles : these are found by the condition that an infinite value of gives a finite value for r . Ex . 1. The equation to the spiral of Archimedes is 1 ' = ав . The angle between the radius and tangent is tan ...
Page 159
... spirals , the equations to which are a 03 " a 01 = " " " = 1.2 1.2.3 ( n ) = a Ꮎ 1.2 ... ( n - 1 ) ' the angle in each case ... spiral is λ = a Ꮎ A - > or 2 = a therefore the subtangent de = = a . du The locus of the extremity of the ...
... spirals , the equations to which are a 03 " a 01 = " " " = 1.2 1.2.3 ( n ) = a Ꮎ 1.2 ... ( n - 1 ) ' the angle in each case ... spiral is λ = a Ꮎ A - > or 2 = a therefore the subtangent de = = a . du The locus of the extremity of 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