An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 15
... obtain the condition under the form ( mn ' — m'n ) a + ( nl ' — n'l ) b + ( lm ' — l'm ) c = c = 0 ...... ( 4 ) . - This is the necessary condition of parallelism , and is generally the most convenient form which can be employed . It is ...
... obtain the condition under the form ( mn ' — m'n ) a + ( nl ' — n'l ) b + ( lm ' — l'm ) c = c = 0 ...... ( 4 ) . - This is the necessary condition of parallelism , and is generally the most convenient form which can be employed . It is ...
Page 30
... obtain a demonstration of the statements made in Art . 6 ; that the points in which the external bisectors of each angle of a triangle respectively intersect the sides opposite to them , lie in the same straight line ; and that the ...
... obtain a demonstration of the statements made in Art . 6 ; that the points in which the external bisectors of each angle of a triangle respectively intersect the sides opposite to them , lie in the same straight line ; and that the ...
Page 36
... distance , its co - ordinates will satisfy the equation aa + bB + cy = 0 . We hence obtain the following equation : - X3u2 + μ3b2 + v3c2 - 2μv bc - 2vλca — 2λμ ab = 0 , which is equivalent to ± ( λa ) * ± 36 MODERN GEOMETRY .
... distance , its co - ordinates will satisfy the equation aa + bB + cy = 0 . We hence obtain the following equation : - X3u2 + μ3b2 + v3c2 - 2μv bc - 2vλca — 2λμ ab = 0 , which is equivalent to ± ( λa ) * ± 36 MODERN GEOMETRY .
Page 37
... obtained by eliminating a between the equations λβγ + μγα + ναβ = 0 , la + mß + ny = 0 , must be coincident . The equation which determines these is − xlBy + ( μy + vß ) ( mß + ny ) = 0 , - and the condition that the two values of By ...
... obtained by eliminating a between the equations λβγ + μγα + ναβ = 0 , la + mß + ny = 0 , must be coincident . The equation which determines these is − xlBy + ( μy + vß ) ( mß + ny ) = 0 , - and the condition that the two values of By ...
Page 39
... obtained by writing a = 0 in the above may be a perfect square . This requires that u2 = vw , or u ' = ± ( vw ) * . Similarly , v ′ = ± ( vw ) 3 , w ' = ± ( uv ) * , are necessary conditions that the conic should touch the lines B = 0 ...
... obtained by writing a = 0 in the above may be a perfect square . This requires that u2 = vw , or u ' = ± ( vw ) * . Similarly , v ′ = ± ( vw ) 3 , w ' = ± ( uv ) * , are necessary conditions that the conic should touch the lines B = 0 ...
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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