Examples of the Processes of the Differential and Integral CalculusJ. and J.J. Deighton, 1846 - 529 стор. |
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Сторінка 106
... ellipse which can be cut from a given cone . Let AC ( fig . 8 ) = a , major 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 ...
... ellipse which can be cut from a given cone . Let AC ( fig . 8 ) = a , major 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 ...
Сторінка 107
... ellipse increases continually till it coincides with the base . It may happen that the maximum value of the section is less than the base of the cone ; and this will be the case unless the vertical angle of the cone be less than 11 ...
... ellipse increases continually till it coincides with the base . It may happen that the maximum value of the section is less than the base of the cone ; and this will be the case unless the vertical angle of the cone be less than 11 ...
Сторінка 110
... 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 , however , that ...
... 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 , however , that ...
Сторінка 123
... ellipse is Ax2 + Bxy + Cy2 + Dx + Ey + 1 = = 0 , which involves five arbitrary constants ; three of these may be determined by the conditions that the ellipse shall pass through the three points A , B , C. Instead however of directly ...
... ellipse is Ax2 + Bxy + Cy2 + Dx + Ey + 1 = = 0 , which involves five arbitrary constants ; three of these may be determined by the conditions that the ellipse shall pass through the three points A , B , C. Instead however of directly ...
Сторінка 124
... ellipse may be put under the form A ( xa ) 2 + 2B ( xa ) ( y - 3 ) + C ( y - 3 ) + 1 = 0 , where A , B , C are to be determined . Now the condition that the ellipse shall pass through the origin gives Aa + 2Baß + C'ẞ2 + 1 = 0 . ( 1 ) ...
... ellipse may be put under the form A ( xa ) 2 + 2B ( xa ) ( y - 3 ) + C ( y - 3 ) + 1 = 0 , where A , B , C are to be determined . Now the condition that the ellipse shall pass through the origin gives Aa + 2Baß + C'ẞ2 + 1 = 0 . ( 1 ) ...
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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