An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 5
... ( ẞ1 − ẞ2 ) ( Y1 — Y2 ) cos A , ( B1 - B2 ) + ( 1-2 ) +2 ( B1 - B2 ) ( Y1-79 ) cos A sin2 4 whence 2- a rational integral function of the second degree . where l , m , n are certain functions of DISTANCE BETWEEN TWO POINTS . 5.
... ( ẞ1 − ẞ2 ) ( Y1 — Y2 ) cos A , ( B1 - B2 ) + ( 1-2 ) +2 ( B1 - B2 ) ( Y1-79 ) cos A sin2 4 whence 2- a rational integral function of the second degree . where l , m , n are certain functions of DISTANCE BETWEEN TWO POINTS . 5.
Page 11
... whence Again , If + mg + nh = 0 , - 1 ( a −ƒ ) + m ( B − g ) + n ( y − h ) = 0 , aa + bB + cy = 2A , af + bg + ch = 2A ; - - :: a ( a − ƒ ) + b ( B − g ) + c ( y − h ) = 0 ; a - f B - g bn - cm = cl -an y - h = am - bl Hence , ...
... whence Again , If + mg + nh = 0 , - 1 ( a −ƒ ) + m ( B − g ) + n ( y − h ) = 0 , aa + bB + cy = 2A , af + bg + ch = 2A ; - - :: a ( a − ƒ ) + b ( B − g ) + c ( y − h ) = 0 ; a - f B - g bn - cm = cl -an y - h = am - bl Hence , ...
Page 12
... whence Lf ' + Mg ' + Nh ' = 0 ; - M L N - fg ' - f'g gh ' — g'h ̄ ̄ hf ' — h'ƒ ̄ ̄ƒg ' —ƒ'g ' giving , as the equation of the required line , ( gh ' —g'h ) a + ( hf ' — h'f ' ) B + ( fg ' —ƒ'g ) y = 0 . 11. To find the general equation ...
... whence Lf ' + Mg ' + Nh ' = 0 ; - M L N - fg ' - f'g gh ' — g'h ̄ ̄ hf ' — h'ƒ ̄ ̄ƒg ' —ƒ'g ' giving , as the equation of the required line , ( gh ' —g'h ) a + ( hf ' — h'f ' ) B + ( fg ' —ƒ'g ) y = 0 . 11. To find the general equation ...
Page 13
... γ = 0 , λα + β + γ = 0 , λας + μβ + γ = 0 , whence , eliminating λ , μ , v , by cross multiplication , ― the required condition . - - - 13 . To find the condition that three straight lines THREE POINTS IN A STRAIGHT LINE . 13.
... γ = 0 , λα + β + γ = 0 , λας + μβ + γ = 0 , whence , eliminating λ , μ , v , by cross multiplication , ― the required condition . - - - 13 . To find the condition that three straight lines THREE POINTS IN A STRAIGHT LINE . 13.
Page 16
... whence or bc 2133 2018 a C 012 ar = m ī bc | = n ī C " α aq am b an 1 1 bl q cl 2 It hence appears , that by making the ratios 7 : m : n suf- ficiently nearly equal to the ratios ab : c , the values of q and may be made as great as we ...
... whence or bc 2133 2018 a C 012 ar = m ī bc | = n ī C " α aq am b an 1 1 bl q cl 2 It hence appears , that by making the ratios 7 : m : n suf- ficiently nearly equal to the ratios ab : c , the values of q and may be made as great as we ...
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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