Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
З цієї книги
Сторінка 47
... differential equation to cylindrical surfaces . ( 17 ) If y b - = $ ( ~ - a ) , by the elimination of the function we find dz ( x − a ) d x + ( y − b ) dx dx = - C. dy This is the differential equation to conical surfaces . ( 18 ) ...
... differential equation to cylindrical surfaces . ( 17 ) If y b - = $ ( ~ - a ) , by the elimination of the function we find dz ( x − a ) d x + ( y − b ) dx dx = - C. dy This is the differential equation to conical surfaces . ( 18 ) ...
Сторінка 50
... differential coefficients of f , p , Y. Differentiating ( 1 ) with respect to x , da { x f ' ( a ) + y p ′ ( a ) + ... equation there results d2 2 d ( d ) - ( addy ) - 0 . dx = This is the differential equation to developable surfaces ...
... differential coefficients of f , p , Y. Differentiating ( 1 ) with respect to x , da { x f ' ( a ) + y p ′ ( a ) + ... equation there results d2 2 d ( d ) - ( addy ) - 0 . dx = This is the differential equation to developable surfaces ...
Сторінка 195
... equation to the evolute ± – - 6aa ) 3 - 81aß2 = 16 { 2a + ( a2 − 6a a ) 1 } 2 { ± ( a2 — 6a a ) 1 — a } . - ( 5 ) In the hypocycloid , the equation to which is x3 x3 + y2 = a3 , a = x + 3x3y3 , B = y + 3x3yś . Adding these a + B = x + ...
... equation to the evolute ± – - 6aa ) 3 - 81aß2 = 16 { 2a + ( a2 − 6a a ) 1 } 2 { ± ( a2 — 6a a ) 1 — a } . - ( 5 ) In the hypocycloid , the equation to which is x3 x3 + y2 = a3 , a = x + 3x3y3 , B = y + 3x3yś . Adding these a + B = x + ...
Сторінка 196
... equation differential equation in a or ẞ , tion to the evolute . ( 6 ) In the cycloid 2 dy dx dy 2 a 1+ ( 1/2 ) = dx y = αβ dx da dy = - and we have a which is the differential equa- ( 2ay - y ° ) y whence yẞ = 2y , or y = - B ...
... equation differential equation in a or ẞ , tion to the evolute . ( 6 ) In the cycloid 2 dy dx dy 2 a 1+ ( 1/2 ) = dx y = αβ dx da dy = - and we have a which is the differential equa- ( 2ay - y ° ) y whence yẞ = 2y , or y = - B ...
Сторінка 197
... equation to the evolute . ac2 ; ( 9 ) The equation to the logarithmic curve is y = a ca ; B ± ( B2 − 8 a2 ) $ - whence y - B : = - a2 + y2 y " or y2 - B a2 y + = 0 . 2 From this y = da and 4 a dß 4 + ß ± ( B2 − 8 a2 ) § = 0 is the ...
... equation to the evolute . ac2 ; ( 9 ) The equation to the logarithmic curve is y = a ca ; B ± ( B2 − 8 a2 ) $ - whence y - B : = - a2 + y2 y " or y2 - B a2 y + = 0 . 2 From this y = da and 4 a dß 4 + ß ± ( B2 − 8 a2 ) § = 0 is the ...
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Загальні терміни та фрази
a² b2 a²x² angle arbitrary constant asymptote becomes C₁ c²x² Cambridge circle co-ordinates condition Crelle's Journal curvature curve cycloid determine differential coefficients differential equation dx dx dx dy 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 plane of reference radius SECT singular solution spiral Substituting subtangent surface tangent plane theorem triangle vanish whence x²)³