An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page ix
... Conic from a given external Point · 11 . Co - ordinates of the Centre 12 . Equation of the Asymptotes • 13. Condition for a Rectangular Hyperbola 14. Conditions for a Circle . 888 * 8 80 82 83 84 86 ARTS . 15 . 16 . 17 . 18 . CONTENTS . ix.
... Conic from a given external Point · 11 . Co - ordinates of the Centre 12 . Equation of the Asymptotes • 13. Condition for a Rectangular Hyperbola 14. Conditions for a Circle . 888 * 8 80 82 83 84 86 ARTS . 15 . 16 . 17 . 18 . CONTENTS . ix.
Page 81
... asymptotes , for the coefficients of a2 will be found , by actual expansion , to be u ( uf2 + vg2 + wh2 + 2u'gh + 2v'hf + 2w'fg ) = − ( u2ƒ2 + w ” 3g2 + v22h2 + 2v'w'gh + Quv'hƒ + 2uw'ƒg ) = ( uv — w'2 ) g2 + ( wu — v'2 ) h2 + 2 ( uu ...
... asymptotes , for the coefficients of a2 will be found , by actual expansion , to be u ( uf2 + vg2 + wh2 + 2u'gh + 2v'hf + 2w'fg ) = − ( u2ƒ2 + w ” 3g2 + v22h2 + 2v'w'gh + Quv'hƒ + 2uw'ƒg ) = ( uv — w'2 ) g2 + ( wu — v'2 ) h2 + 2 ( uu ...
Page 82
... γ or = Ua + W'b + V'c 2 Wa + Vb + Uc V'a + U'b + Wc k - uvw + zu'v'w ' — uu " - vv2 — ww2 These equations determine the centre . √12 . To find the equation of the asymptotes . 82 TRILINEAR CO - ORDINATES . 11 Co-ordinates of the Centre.
... γ or = Ua + W'b + V'c 2 Wa + Vb + Uc V'a + U'b + Wc k - uvw + zu'v'w ' — uu " - vv2 — ww2 These equations determine the centre . √12 . To find the equation of the asymptotes . 82 TRILINEAR CO - ORDINATES . 11 Co-ordinates of the Centre.
Page 83
... asymptotes . Writing a , B , y , for f , g , h , in the investigation of Art . 10 , and paying regard to equations ( A ) of Art . 11 , the asymptotes will be found to be represented by the equation or & ( α , B , y ) & ( α , ß , y ) ...
... asymptotes . Writing a , B , y , for f , g , h , in the investigation of Art . 10 , and paying regard to equations ( A ) of Art . 11 , the asymptotes will be found to be represented by the equation or & ( α , B , y ) & ( α , ß , y ) ...
Page 84
... asymptotes be la + mB + ny = 0 , l'a + m'ß + n'y = 0 , the condition of their perpendicularity is ll ' + mm ' + nn ' — ( mn ' + m'n ) cos A- ( nl ' + n'l ) cos B - - - ( lm ' + l'm ) cos C ' = 0 . Writing , for shortness , Ua2 + Vb2 + ...
... asymptotes be la + mB + ny = 0 , l'a + m'ß + n'y = 0 , the condition of their perpendicularity is ll ' + mm ' + nn ' — ( mn ' + m'n ) cos A- ( nl ' + n'l ) cos B - - - ( lm ' + l'm ) cos C ' = 0 . Writing , for shortness , Ua2 + Vb2 + ...
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Common terms and phrases
angular points anharmonic ratio asymptotes auxiliary conic b₁ b₂ C₁ centre Chap chord co-ordinates coefficients common tangents conic section determinant directrix find the equation fixed point fixed straight line follows four points given conic given point given straight line given triangle Hence investigated Let the equation line at infinity line joining locus meets the conic nine-point circle obtain opposite sides parabola Pascal's Theorem perpendicular point f points at infinity points of contact points of intersection PQRS prove radical axis reciprocal curve reciprocated with respect rectangular hyperbola represented right angles second degree shewn similar and similarly sin POS tangents drawn term a,b,c theorem three points three straight lines triangle of reference ua² uf+w'g+v'h ux² V'ca va² Vb² vß² W'ab whence wy² λα