Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
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Сторінка 9
... John Bernoulli ; and , in the course of his investigations , he discovered , by induction , the Theorem which bears his name . He also conceived the existence of differentials with fractional or irrational indices , but he made no steps ...
... John Bernoulli ; and , in the course of his investigations , he discovered , by induction , the Theorem which bears his name . He also conceived the existence of differentials with fractional or irrational indices , but he made no steps ...
Сторінка 79
... John Bernoulli , Acta Eruditorum , 1704 , p . 375 . The expression P ( r ) Q ) > that is , any one of the series of fractions which present themselves in the operation above described , may be replaced by any equivalent fraction re ...
... John Bernoulli , Acta Eruditorum , 1704 , p . 375 . The expression P ( r ) Q ) > that is , any one of the series of fractions which present themselves in the operation above described , may be replaced by any equivalent fraction re ...
Сторінка 137
... John Bernoulli : 1st , that it is the curve along which a body will , under the action of gravity , fall in the shortest time from one given point to another not in the same vertical 2nd , that if any arc of a curve as AB ( fig . 21 ) ...
... John Bernoulli : 1st , that it is the curve along which a body will , under the action of gravity , fall in the shortest time from one given point to another not in the same vertical 2nd , that if any arc of a curve as AB ( fig . 21 ) ...
Сторінка 138
... John Bernoulli , Opera , Vol . iv . p . 98. Euler , Commen . Petrop . 1766 . Legendre , Exercices du Calcul Integral , Tom . 11. p . 491 . ( 11 ) Epitrochoids and Hypotrochoids . When the generating 138 GENERATION OF CURVES .
... John Bernoulli , Opera , Vol . iv . p . 98. Euler , Commen . Petrop . 1766 . Legendre , Exercices du Calcul Integral , Tom . 11. p . 491 . ( 11 ) Epitrochoids and Hypotrochoids . When the generating 138 GENERATION OF CURVES .
Сторінка 147
... John Bernoulli . ( See his Works , Vol . 111. p . 447. ) For the perpendicular from the origin on the tangent we find p = ( axy ) 3 . ( 4 ) In the cissoid of Diocles , y2 = 2 a X3 - whence the subtangent and the subnormal ( 5 ) In the ...
... John Bernoulli . ( See his Works , Vol . 111. p . 447. ) For the perpendicular from the origin on the tangent we find p = ( axy ) 3 . ( 4 ) In the cissoid of Diocles , y2 = 2 a X3 - whence the subtangent and the subnormal ( 5 ) In the ...
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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