Examples of the Processes of the Differential and Integral Calculus |
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Page 102
... axis of x , O the 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 ...
... axis of x , O the 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 ...
Page 105
... axis of the prism in order that for a given content the total surface may be the least possible . = Let ABC abc ( fig . 6 ) be the base of the prism , PQRS one of the faces of the terminating solid angle passing through the angles P , R ...
... axis of the prism in order that for a given content the total surface may be the least possible . = Let ABC abc ( fig . 6 ) be the base of the prism , PQRS one of the faces of the terminating solid angle passing through the angles P , R ...
Page 106
... axis of the ellipse . CD = b , CN = x , BP being the Then the condition that the area of the ellipse shall be a maximum gives x = 2b ( a2 — b2 ) ± b ( a1 − 14a2b2 + b1 ) 3 - 3 ( a2 + b2 ) * Mémoires de l'Académie des Sciences , 1712 ...
... axis of the ellipse . CD = b , CN = x , BP being the Then the condition that the area of the ellipse shall be a maximum gives x = 2b ( a2 — b2 ) ± b ( a1 − 14a2b2 + b1 ) 3 - 3 ( a2 + b2 ) * Mémoires de l'Académie des Sciences , 1712 ...
Page 110
... axis of an ellipse which revolves round an axis parallel to the major axis . In these cases we have d'u d'u u de dy- ( dady ) " dx 2 = 0 , an equation which is usually excluded from Lagrange's con- dition . It is to be observed ...
... axis of an ellipse which revolves round an axis parallel to the major axis . In these cases we have d'u d'u u de dy- ( dady ) " dx 2 = 0 , an equation which is usually excluded from Lagrange's con- dition . It is to be observed ...
Page 121
... axes , a2 12 b2 m2 c2 n2 + + 2.2 a2 p2 - b2 2.2 - = 0 . a2 The last term of this when arranged according to powers of 2 is a2 b2 c2 a2 l2 + b2 m2 + c2n2 ' and this being equal to the product of the roots , the area of the section is ...
... axes , a2 12 b2 m2 c2 n2 + + 2.2 a2 p2 - b2 2.2 - = 0 . a2 The last term of this when arranged according to powers of 2 is a2 b2 c2 a2 l2 + b2 m2 + c2n2 ' and this being equal to the product of the roots , the area of the section is ...
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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