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 ' — I'm ) c = 0 ....... ( 4 ) . - 1 This is the necessary condition of parallelism , and is generally the most convenient form which can be employed . It ...
... obtain the condition under the form ( mn ' — m'n ) a + ( nl ′ ' — n'l ) b + ( lm ' — I'm ) c = 0 ....... ( 4 ) . - 1 This is the necessary condition of parallelism , and is generally the most convenient form which can be employed . It ...
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 ax + bB + cy = 0 . We hence obtain the following equation : λ3a2 + μ2b2 + v3c2 - 2μvbc - 2vλca - 2xμab = 0 , which is equivalent to ± ( λa ) * ± 36 TRILINEAR CO - ORDINATES .
... distance , its co - ordinates will satisfy the equation ax + bB + cy = 0 . We hence obtain the following equation : λ3a2 + μ2b2 + v3c2 - 2μvbc - 2vλca - 2xμab = 0 , which is equivalent to ± ( λa ) * ± 36 TRILINEAR CO - ORDINATES .
Page 37
... obtained by eliminating a between the equations λβγ + μγα + ναβ = 0 , lx + mB + ny = 0 , must be coincident . The equation which determines these is −XBY + ( μy + vß ) ( mß + ny ) = 0 , and the condition that the two values of B : y be ...
... obtained by eliminating a between the equations λβγ + μγα + ναβ = 0 , lx + mB + ny = 0 , must be coincident . The equation which determines these is −XBY + ( μy + vß ) ( mß + ny ) = 0 , and the condition that the two values of B : y be ...
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 ' = ± ( wu ) ' , w ' = ± ( uv ) , are necessary conditions that the conic should touch the lines B = 0 , y ...
... obtained by writing a = 0 in the above may be a perfect square . This requires that u2 = vw , or u ' = ± ( vw ) . Similarly , v ' = ± ( wu ) ' , w ' = ± ( uv ) , are necessary conditions that the conic should touch the lines B = 0 , y ...
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Common terms and phrases
angular points asymptotes ax+bB+cy b₁ b₂ C₁ Cambridge centre Chap co-ordinates coefficients College common tangents conic section Crown 8vo determinant directrix Edition equation Fcap find the equation fixed point follows four points given conic given point given straight line Hence imaginary investigated Let the equation line at infinity line joining locus meets the conic nine-point circle numerous Illustrations obtain opposite sides Owens College pair parabola Pascal's Theorem perpendicular point f points of intersection pole Professor prove radical axis ratios rectangular hyperbola represented respect right angles second degree shewn similar and similarly sin POS tangents tangents drawn term a,b,c theorem three straight lines tion touches the line triangle of reference Ua² ux² V'ca v'f+u'g+wh va² values Vb² vß² W. K. CLIFFORD W'ab whence wy² λα