An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 43
... chap . I. 10 . To find the equations of the four circles which touch the three sides of the triangle of reference . These may be obtained most readily by the employment of the equations for the determination of the centre , obtained in ...
... chap . I. 10 . To find the equations of the four circles which touch the three sides of the triangle of reference . These may be obtained most readily by the employment of the equations for the determination of the centre , obtained in ...
Page 70
... Chap . II . Again , by supposing D to coincide with B , and E with C , we readily obtain the follow- ing theorem : " If the opposite sides of a quadrilateral , in- scribed in a conic , be produced to meet , and likewise the pairs of ...
... Chap . II . Again , by supposing D to coincide with B , and E with C , we readily obtain the follow- ing theorem : " If the opposite sides of a quadrilateral , in- scribed in a conic , be produced to meet , and likewise the pairs of ...
Page 72
... Chap . III ) , represents a conic section . 2. To find the point in which a straight line , drawn in a given direction through a given point of the conic , meets the conic again . Let f , g , h , be the co - ordinates of the given point ...
... Chap . III ) , represents a conic section . 2. To find the point in which a straight line , drawn in a given direction through a given point of the conic , meets the conic again . Let f , g , h , be the co - ordinates of the given point ...
Page 76
... Chap . III . to be u , w ' , v ' = = 0 w ' , v , u ' v ' , u ' , w 12 or uvw + 2u'v'w ' — uu'2 — vv'2 — ww'2 = 0 . - - 8. To find the equation of the polar of a given point . If through a given point any straight line be drawn cutting a ...
... Chap . III . to be u , w ' , v ' = = 0 w ' , v , u ' v ' , u ' , w 12 or uvw + 2u'v'w ' — uu'2 — vv'2 — ww'2 = 0 . - - 8. To find the equation of the polar of a given point . If through a given point any straight line be drawn cutting a ...
Page 77
... Chap . 1. ) Again , the equations of the tangents at f g h1 ) , f 92 ha ) respectively , are f1 ( ua + w'ß + v'y ) + 91 ( w ́a + vß + u'y ) + h2 ( v'a + u'B + wy ) = 0 , f2 ( ua + w'ß + v'y ) + 92 ( w'a + vß + u ́y ) + h2 ( v'a + u'ß + wy ) ...
... Chap . 1. ) Again , the equations of the tangents at f g h1 ) , f 92 ha ) respectively , are f1 ( ua + w'ß + v'y ) + 91 ( w ́a + vß + u'y ) + h2 ( v'a + u'B + wy ) = 0 , f2 ( ua + w'ß + v'y ) + 92 ( w'a + vß + u ́y ) + h2 ( v'a + u'ß + wy ) ...
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Common terms and phrases
a₁ aa+bB+cy Algebra angular points anharmonic ratio Arithmetic asymptotes auxiliary conic b₁ c₁ CAMBRIDGE CLASS BOOKS centre Chap chord cloth co-ordinates coefficients College common tangents condition conic passing conic section conics intersect Crown 8vo curve determine directrix ellipse find the equation fixed point focus four points given conic given point given straight line Hence hyperbola imaginary investigated line at infinity line joining locus meets the conic obtain pair parabola Pascal's Theorem perpendicular point f point of intersection points at infinity points of contact PQRS projection prove reciprocal polars represented respect right angles second degree Second Edition Similarly sin POS system of conics tangents tangents drawn theorem three points three straight lines touches the line Treatise triangle of reference U'bc ua² V'ca Vb² vß² W'ab wy² λα