Examples of the Processes of the Differential and Integral CalculusJ and J. J. Deighton, 1846 - 529 стор. |
З цієї книги
Результати 1-5 із 100
Сторінка 47
... differential equation to cylindrical surfaces . ( 17 ) If y b - -c = - by the elimination of the function we find d % dz ( x − a ) - + ( y - b ) = - C. dx dy This is the differential equation to conical surfaces . ( 18 ) Eliminate and ...
... differential equation to cylindrical surfaces . ( 17 ) If y b - -c = - by the elimination of the function we find d % dz ( x − a ) - + ( y - b ) = - C. dx dy This is the differential equation to conical surfaces . ( 18 ) Eliminate and ...
Сторінка 51
... equation there results dx2 - d2 % 2 = 0 . ' d2s dy2 dx dy This is the differential equation to developable surfaces . ( 24 ) If u = ƒ ( x , y ) = F ( r , ≈ ) , and dr dx dr 1 - r = $ ( ax + cx ) = ↓ ( ax – by ) , 1 du then du dx du + ...
... equation there results dx2 - d2 % 2 = 0 . ' d2s dy2 dx dy This is the differential equation to developable surfaces . ( 24 ) If u = ƒ ( x , y ) = F ( r , ≈ ) , and dr dx dr 1 - r = $ ( ax + cx ) = ↓ ( ax – by ) , 1 du then du dx du + ...
Сторінка 195
... equation to the evolute - - 81aß2 = 16 { 2a ± ' ( a2 − 6aa ) 3 } 2 { ± ( a2 – 6a a ) 3 — a } . ( 5 ) In the hypocycloid , the equation to which is x3 + y } = a3 , a = x + 3x3y3 , · B = y + 3 x3yś . Adding these a + B = x + y + 3 ( xy ) ...
... equation to the evolute - - 81aß2 = 16 { 2a ± ' ( a2 − 6aa ) 3 } 2 { ± ( a2 – 6a a ) 3 — a } . ( 5 ) In the hypocycloid , the equation to which is x3 + y } = a3 , a = x + 3x3y3 , · B = y + 3 x3yś . Adding these a + B = x + y + 3 ( xy ) ...
Сторінка 196
... equation x dß dx = - and we have a da dy " differential equation in a or ß , which is the differential equa- tion to the evolute . ( 6 ) In the cycloid dy ( 2ay - y ^ ) } dx 2 2 a = dx y y y 1 + whence yẞ = 2y , or y = - B ...
... equation x dß dx = - and we have a da dy " differential equation in a or ß , which is the differential equa- tion to the evolute . ( 6 ) In the cycloid dy ( 2ay - y ^ ) } dx 2 2 a = dx y y y 1 + whence yẞ = 2y , or y = - B ...
Сторінка 197
... differential equation to the evolute . ( 9 ) The equation to the logarithmic curve is y = ac " ; a2 0 . B ± ( B2 - 8 a2 ) } whence y - B = a2 + y2 y " or y2 - B y + = 2 From this y = 4 da and 4 a dß + ß ± ( ß3 − 8 a2 ) § = 0 is the ...
... differential equation to the evolute . ( 9 ) The equation to the logarithmic curve is y = ac " ; a2 0 . B ± ( B2 - 8 a2 ) } whence y - B = a2 + y2 y " or y2 - B y + = 2 From this y = 4 da and 4 a dß + ß ± ( ß3 − 8 a2 ) § = 0 is the ...
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a² b2 a²x² angle arbitrary constant asymptote axis becomes C₁ c²x² Cambridge circle co-ordinates condition curvature curve cycloid cylinder determine differential coefficients differential equation dx dx dx dy dx dx² dy dx dy dy dy dy dz eliminate ellipse equal Euler find the value formula function Geometry gives Hence hypocycloid infinite Integrating with respect intersection John Bernoulli Let the equation lines of curvature locus logarithmic logarithmic spiral maximum minimum Multiply negative origin parabola perpendicular radius radius of curvature singular solution spiral Substituting subtangent surface tangent plane theorem tractory triangle vanish whence x²)³