An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page ix
... Equation of the pair of Tangents , drawn to the Conic from a Co - ordinates of the Centre 13 . 14 . 23+ 12 . Equation of the Asymptotes Condition for a Rectangular Hyperbola Conditions for a Circle 33 ARTS . PAGE 15 . 16 . All Circles.
... Equation of the pair of Tangents , drawn to the Conic from a Co - ordinates of the Centre 13 . 14 . 23+ 12 . Equation of the Asymptotes Condition for a Rectangular Hyperbola Conditions for a Circle 33 ARTS . PAGE 15 . 16 . All Circles.
Page 80
... asymptotes , for the coefficients of a will be found , by actual expansion , to be u ( uf2 + vg2 + wh3 + 2u'gh + 2v'hƒ + 2w'fg ) = = - 2 — ( u3f2 + w'2g2 + v'3h2 + 2v'w'gh + 2uv'hƒ + 2uw'fg ) · ( uv — w'2 ) g3 + ( wu — v'2 ) h2 + 2 ( uu ...
... asymptotes , for the coefficients of a will be found , by actual expansion , to be u ( uf2 + vg2 + wh3 + 2u'gh + 2v'hƒ + 2w'fg ) = = - 2 — ( u3f2 + w'2g2 + v'3h2 + 2v'w'gh + 2uv'hƒ + 2uw'fg ) · ( uv — w'2 ) g3 + ( wu — v'2 ) h2 + 2 ( uu ...
Page 81
... b + Wc k 12 . uvw + 2u'v'w ' — uu " — vv2 — ww'2 ' These equations determine the centre . - 12. To find the equation of the asymptotes . Writing F. 6 Condition of Tangency Condition for a Parabola Co-ordinates of the Centre.
... b + Wc k 12 . uvw + 2u'v'w ' — uu " — vv2 — ww'2 ' These equations determine the centre . - 12. To find the equation of the asymptotes . Writing F. 6 Condition of Tangency Condition for a Parabola Co-ordinates of the Centre.
Page 82
... 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 , 7 ) & ( a , 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 , 7 ) & ( a , ß , y ) ...
Page 83
... asymptotes be la + mB + ny = 0 , l'a + m'B + 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'B + 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
aa+bB+cy angular points anharmonic ratio asymptotes b₁ b₂ centre Chap chord coefficients common tangents condition of tangency conic section Crown 8vo curve determine directrix distance escribed circles Fcap find the equation fixed point foci focus four points given conic given point given straight line harmonic pencil Hence imaginary internal bisectors investigated Let the equation line at infinity line joining locus meets the conic nine-point circle obtain opposite sides pair parabola parallel Pascal's Theorem perpendicular point f point of intersection points at infinity points of contact pole prove radical axis reciprocated with respect rectangular hyperbola represented right angles second degree sin POS tangents tangents drawn theorem three points three straight lines touches the line triangle of reference ua² V'ca values vß² W'ab whence wy² λα
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