Examples of the Processes of the Differential and Integral Calculus |
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Page 102
... origin . Let OP , and let the co - ordinates of A and B be a , b , a1 , b . Then u = AP + BP = { b2 + ( x − a ) 2 } 1 + { b , 2 + ( a , − x ) 2 } 1 = minimum . Whence x - a = - -x { b2 + ( x − a ) 2 } } ̄ ̄ ̄ { b , 2 + ( a ̧ − x ) 2 } ...
... origin . Let OP , and let the co - ordinates of A and B be a , b , a1 , b . Then u = AP + BP = { b2 + ( x − a ) 2 } 1 + { b , 2 + ( a , − x ) 2 } 1 = minimum . Whence x - a = - -x { b2 + ( x − a ) 2 } } ̄ ̄ ̄ { b , 2 + ( a ̧ − x ) 2 } ...
Page 123
... origin , CA , CB as the axes of x and y . AC = a , BC = b , ACB = 0 . • The general equation to an ellipse is Ax2 + Bxy + Cy2 + Dx + Ey + 1 = = 0 , which involves five arbitrary constants ; three of these may be determined by the ...
... origin , CA , CB as the axes of x and y . AC = a , BC = b , ACB = 0 . • The general equation to an ellipse is Ax2 + Bxy + Cy2 + Dx + Ey + 1 = = 0 , which involves five arbitrary constants ; three of these may be determined by the ...
Page 124
... origin gives Aa + 2Baß + C'ẞ2 + 1 = 0 . ( 1 ) The condition that the curve shall pass through the point aa , y = 0 , gives A ( a − a ) 2 - 2 B ( a − a ) ß + CB2 + 1 = 0 . - - Subtracting ( 1 ) from ( 2 ) we have ( 2 ) A ( 2a - a ) + ...
... origin gives Aa + 2Baß + C'ẞ2 + 1 = 0 . ( 1 ) The condition that the curve shall pass through the point aa , y = 0 , gives A ( a − a ) 2 - 2 B ( a − a ) ß + CB2 + 1 = 0 . - - Subtracting ( 1 ) from ( 2 ) we have ( 2 ) A ( 2a - a ) + ...
Page 132
... origin be taken at the middle point between them , the equation to the curve is { y2 + ( a + x ) 2 } { y2 + ( a − x ) 2 } = c1 . - When ca , the equation is reduced to ( x2 + y2 ) 2 = 2 a2 ( x2 − y2 ) . This was the curve used by ...
... origin be taken at the middle point between them , the equation to the curve is { y2 + ( a + x ) 2 } { y2 + ( a − x ) 2 } = c1 . - When ca , the equation is reduced to ( x2 + y2 ) 2 = 2 a2 ( x2 − y2 ) . This was the curve used by ...
Page 135
... origin . The farther these points are removed from the origin the more nearly is the curve perpendicular to the axis of x , the value of π at the intersection being ( 2n − 1 ) dy dx - - " 2na being the 2 abscissa of the point where the ...
... origin . The farther these points are removed from the origin the more nearly is the curve perpendicular to the axis of x , the value of π at the intersection being ( 2n − 1 ) dy dx - - " 2na being the 2 abscissa of the point where 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