Examples of the Processes of the Differential and Integral Calculus |
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Page 46
... equation , we have d'y da2 - dy 2 m + ( n2 + m2 ) y = 0 . dx ( 15 ) Eliminate the arbitrary function from the equation ≈ = xyp ( y ) . Differentiating with respect to a only , dz dx = y ( y ) ; and therefore c ( 16 ) Eliminate the ...
... equation , we have d'y da2 - dy 2 m + ( n2 + m2 ) y = 0 . dx ( 15 ) Eliminate the arbitrary function from the equation ≈ = xyp ( y ) . Differentiating with respect to a only , dz dx = y ( y ) ; and therefore c ( 16 ) Eliminate the ...
Page 47
... equation to cylindrical surfaces . ( 17 ) If y Ꮗ a = - by the elimination of the function we find da dz ( x − a ) - + ( y - dx b ) = ≈ - C . dy This is the differential equation to conical surfaces . ( 18 ) Eliminate from the equation ...
... equation to cylindrical surfaces . ( 17 ) If y Ꮗ a = - by the elimination of the function we find da dz ( x − a ) - + ( y - dx b ) = ≈ - C . dy This is the differential equation to conical surfaces . ( 18 ) Eliminate from the equation ...
Page 50
... equation ( 2 ) x f ' ( a ) + y p ' ( a ) + ≈ f ' ( a ) = 0 ; f ' , ' , being the differential coefficients of f , p ... equation there results dx ( ) ( ) d2 z 2 - = 0 . dx dy This is the differential equation to developable surfaces ...
... equation ( 2 ) x f ' ( a ) + y p ' ( a ) + ≈ f ' ( a ) = 0 ; f ' , ' , being the differential coefficients of f , p ... equation there results dx ( ) ( ) d2 z 2 - = 0 . dx dy This is the differential equation to developable surfaces ...
Page 115
... equation in k whence a maximum and a mini- P mum value may be found . This is the equation for determining the radii of maximum and minimum curvature in a curved surface . ( 6 ) Let u = ay ( c - x ) = bx ( a − x ) = cx ( b − y ) ...
... equation in k whence a maximum and a mini- P mum value may be found . This is the equation for determining the radii of maximum and minimum curvature in a curved surface . ( 6 ) Let u = ay ( c - x ) = bx ( a − x ) = cx ( b − y ) ...
Page 140
... equations to the hypotrochoid , if we make h = b and b = a we obtain by the elimination of 0 the equation ( a2 - x - y3 ) 3 = 27 a2x2 y2 ; which may be put under the form æ3 + y3 = a3 . This hypocycloid occurs in the solution of many ...
... equations to the hypotrochoid , if we make h = b and b = a we obtain by the elimination of 0 the equation ( a2 - x - y3 ) 3 = 27 a2x2 y2 ; which may be put under the form æ3 + y3 = a3 . This hypocycloid occurs in the solution of many ...
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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