An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory Projections |
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Page xiii
... SATISFY FOUR CONDITIONS . 11 . Four points given 12 . Three points and a tangent 13 . Two points and two tangents 14 . One point and three tangents • 15 . Four tangents . • 165 167 · 168 169 171 SUPPLEMENTARY PROBLEMS . ARTS . PAGE 16 ...
... SATISFY FOUR CONDITIONS . 11 . Four points given 12 . Three points and a tangent 13 . Two points and two tangents 14 . One point and three tangents • 15 . Four tangents . • 165 167 · 168 169 171 SUPPLEMENTARY PROBLEMS . ARTS . PAGE 16 ...
Page 13
... satisfy the following equations : λα + μβ + vy , = 0 , λα + β + γ = 0 , λα + β + γ = 0 , whence , eliminating λ , μ , v , by cross multiplication , the required condition . = 13. To find the condition that three straight lines may THREE ...
... satisfy the following equations : λα + μβ + vy , = 0 , λα + β + γ = 0 , λα + β + γ = 0 , whence , eliminating λ , μ , v , by cross multiplication , the required condition . = 13. To find the condition that three straight lines may THREE ...
Page 16
... satisfy the equation aa + bß + cy = 0 , yet we can always satisfy the equation la + mß + ny = 0 , where the ratios l m n ap- proach as nearly as we please to the ratios a : b : c . : By referring to the investigation of Art . ( 7 ) it ...
... satisfy the equation aa + bß + cy = 0 , yet we can always satisfy the equation la + mß + ny = 0 , where the ratios l m n ap- proach as nearly as we please to the ratios a : b : c . : By referring to the investigation of Art . ( 7 ) it ...
Page 36
... its co - ordinates will satisfy the equation aa + bB + cy = 0 . We hence obtain the following equation : λ2a2 + μ2b2 + v2c2 — 2μvbc — 2vλca — 2λμab = 0 , - which is equivalent to ± ( λa ) * ± 36 TRILINEAR CO - ORDINATES .
... its co - ordinates will satisfy the equation aa + bB + cy = 0 . We hence obtain the following equation : λ2a2 + μ2b2 + v2c2 — 2μvbc — 2vλca — 2λμab = 0 , - which is equivalent to ± ( λa ) * ± 36 TRILINEAR CO - ORDINATES .
Page 37
... satisfies the analytical condition of touching the straight line aa + bB + cy = 0. This is generally ex- pressed by saying that every parabola touches the line at infinity . 6. To investigate the equation of the circle , circumscribing ...
... satisfies the analytical condition of touching the straight line aa + bB + cy = 0. This is generally ex- pressed by saying that every parabola touches the line at infinity . 6. To investigate the equation of the circle , circumscribing ...
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Common terms and phrases
aa+bB+cy angular points asymptotes auxiliary conic b₁ Brianchon's Theorem C₁ centre Chap chord co-ordinates coefficients common tangents conic passing conic section determinant directrix find the equation fixed point fixed straight line focus follows four points given conic given point given straight line Hence imaginary internal bisectors investigated Let the equation line at infinity locus meets the conic nine-point circle obtain opposite sides pair parabola Pascal's Theorem perpendicular point f point of intersection points at infinity points of contact polar reciprocal projection prove radical axis reciprocal curve reciprocated with respect rectangular hyperbola represented right angles second degree similar and similarly system of conics tangents drawn theorem three points three straight lines triangle of reference ua² uf+w'g+v'h Vb² vß² W'ab Wc² whence wy²