A Treatise on Conic Sections: Containing an Account of Some of the Most Important Modern Algebraic and Geometric MethodsLongmans, Green, Reader, and Dyer, 1869 - 377 страници |
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Страница 9
... of polar co- ordinates . θ B It is very easy , being given the x and y co - ordinates of a point , to find its polar ones , or vice versâ . C First , let the fixed line Y coincide with the POLAR CO - ORDINATES . 9 Polar Co-ordinates PAGE.
... of polar co- ordinates . θ B It is very easy , being given the x and y co - ordinates of a point , to find its polar ones , or vice versâ . C First , let the fixed line Y coincide with the POLAR CO - ORDINATES . 9 Polar Co-ordinates PAGE.
Страница 10
... coincide with the axis of x , then we have OP : PM :: sinPMO : sinPOM ; denoting OP by p , POM by 0 , and YOX by w ; then P PM or y ρ = sin 0 ; sin w M X and similarly , OM = = x = p sin ( w - 0 ) sin w For the more ordinary case of ...
... coincide with the axis of x , then we have OP : PM :: sinPMO : sinPOM ; denoting OP by p , POM by 0 , and YOX by w ; then P PM or y ρ = sin 0 ; sin w M X and similarly , OM = = x = p sin ( w - 0 ) sin w For the more ordinary case of ...
Страница 55
... coincides with the third . It is cos ( aẞ ' ) cos ( By ' ) cos ( ya ' ) = cos ( a'ß ) cos ( B'y ) cos ( y'a ) . But the symmetry of this equation shows that this is also the condition that the perpendiculars from the vertices of the ...
... coincides with the third . It is cos ( aẞ ' ) cos ( By ' ) cos ( ya ' ) = cos ( a'ß ) cos ( B'y ) cos ( y'a ) . But the symmetry of this equation shows that this is also the condition that the perpendiculars from the vertices of the ...
Страница 69
... coincide ; and when a and b are imaginary , that the lines are imaginary . It may seem to the student a matter of in- difference which mode of speaking we adopt ; we shall find , how- ever , as we proceed , that we should lose sight of ...
... coincide ; and when a and b are imaginary , that the lines are imaginary . It may seem to the student a matter of in- difference which mode of speaking we adopt ; we shall find , how- ever , as we proceed , that we should lose sight of ...
Страница 78
... coincide . We shall therefore , not say that the tangent meets the circle in only one point , but rather that it meets it in two coincident points ; just as we do not say that the corre- sponding quadratic has only one root , but rather ...
... coincide . We shall therefore , not say that the tangent meets the circle in only one point , but rather that it meets it in two coincident points ; just as we do not say that the corre- sponding quadratic has only one root , but rather ...
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anharmonic ratio asymptotes ax² axes bisected bisectors chord of contact circumscribing coefficients common tangents condition conic section conjugate diameters corresponding cos² denote determine directrix double contact drawn ellipse equal find the co-ordinates find the equation find the locus fixed lines fixed point foci focus four points given circles given line given point Hence hyperbola imaginary points infinite distance inscribed intercept joining the points last Article length line at infinity line joining line meets meet the curve middle points origin parabola parallel Pascal's theorem perpendicular point of contact point x'y points at infinity points of intersection polar polar equation pole proved quadratic quadrilateral radical axis radius vector rectangle right angles right line second degree sides sin² square substituting tangential equation theorem values vanish vertex vertices