Examples of the Processes of the Differential and Integral Calculus |
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Page ix
... Curvature of Curved Lines ......... 188 XIII . Application of the Differential Calculus to Geometry of Three Dimensions 200 XIV . Envelops to Lines and Surfaces XV . General Theorems in the Differential Calculus ......... 224 237 PART ...
... Curvature of Curved Lines ......... 188 XIII . Application of the Differential Calculus to Geometry of Three Dimensions 200 XIV . Envelops to Lines and Surfaces XV . General Theorems in the Differential Calculus ......... 224 237 PART ...
Page 29
... expression for the radius of curvature when a is the independent variable is dy + da - d'y dx2 When y is made the independent variable , it becomes { 1+ (税) dy dx dy ( 3 ) Transform d2 y dx2 ' dy 3 CHANGE OF THE INDEPENDENT VARIABLE . 29.
... expression for the radius of curvature when a is the independent variable is dy + da - d'y dx2 When y is made the independent variable , it becomes { 1+ (税) dy dx dy ( 3 ) Transform d2 y dx2 ' dy 3 CHANGE OF THE INDEPENDENT VARIABLE . 29.
Page 45
D. F. Gregory. This is the expression for the square of the radius of curvature of any curve . ( 9 ) Eliminate m from the equation ( a + mẞ ) ( x2 - my3 ) = my2 ; the result is axy - dx ' + ( Ba2 — ay3 — y3 ) dy – Bay = 0 . - dx 2 ' dy ...
D. F. Gregory. This is the expression for the square of the radius of curvature of any curve . ( 9 ) Eliminate m from the equation ( a + mẞ ) ( x2 - my3 ) = my2 ; the result is axy - dx ' + ( Ba2 — ay3 — y3 ) dy – Bay = 0 . - dx 2 ' dy ...
Page 115
... curvature in a curved surface . ( 6 ) Let u = ay ( c - x ) = bx ( a − x ) = cx ( b − y ) . Then x = a , y = 1b , ≈ = c , give z = u = abc , a maximum . ( 7 ) Let u = a cos2x + b cos2y ; - - x and y being subject to the condition y ...
... curvature in a curved surface . ( 6 ) Let u = ay ( c - x ) = bx ( a − x ) = cx ( b − y ) . Then x = a , y = 1b , ≈ = c , give z = u = abc , a maximum . ( 7 ) Let u = a cos2x + b cos2y ; - - x and y being subject to the condition y ...
Page 137
... curvature is double of the chord of the generating circle which is perpendicular to the tangent . He also discovered the important dynamical property of the tautochronism of a cycloidal pendulum ; that is to say , that a body under the ...
... curvature is double of the chord of the generating circle which is perpendicular to the tangent . He also discovered the important dynamical property of the tautochronism of a cycloidal pendulum ; that is to say , that a body under 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