An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory Projections |
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Page 5
... ( Y1 − Y1⁄2 ) cos A , whence r2 – ( ẞı — ẞ2 ) 2 + ( Y1 − Y1⁄2 ) 2 + 2 ( ẞ1 — B2 ) ( Y1 − Y2 ) cos A = - sin2 A a rational integral function of the second degree . where l , m , n are certain functions of DISTANCE BETWEEN TWO POINTS . 5.
... ( Y1 − Y1⁄2 ) cos A , whence r2 – ( ẞı — ẞ2 ) 2 + ( Y1 − Y1⁄2 ) 2 + 2 ( ẞ1 — B2 ) ( Y1 − Y2 ) cos A = - sin2 A a rational integral function of the second degree . where l , m , n are certain functions of DISTANCE BETWEEN TWO POINTS . 5.
Page 11
... whence Ágain , If + mg + nh = 0 , . l ( a − f ) + 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 bn - cm = B - g cl- -an = 7 - h am - bl ' ...
... whence Ágain , If + mg + nh = 0 , . l ( a − f ) + 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 bn - cm = B - g cl- -an = 7 - h am - bl ' ...
Page 12
... whence L Lf ' + Mg ' + Nh ' = 0 ; = M N gh ' - g'h hf " - h'ƒ ̄ ̄ƒg ' —ƒ'g ' giving , as the equation of the required line , ( gh ' — g'h ) a + ( hf " — h ' ƒ ) ẞ + ( ƒg ' — ƒ'g ) y = 0 . - 11. To find the general equation of a straight ...
... whence L Lf ' + Mg ' + Nh ' = 0 ; = M N gh ' - g'h hf " - h'ƒ ̄ ̄ƒg ' —ƒ'g ' giving , as the equation of the required line , ( gh ' — g'h ) a + ( hf " — h ' ƒ ) ẞ + ( ƒg ' — ƒ'g ) y = 0 . - 11. To find the general equation of a straight ...
Page 13
... , = 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 POINTS IN A STRAIGHT LINE . 13.
... , = 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 POINTS IN A STRAIGHT LINE . 13.
Page 16
... whence = a ar ī a aq c am b an or = 1 - 1 " bl q cl r = n ; n ī It hence appears , that by making the ratios : m : n suf- ficiently nearly equal to the ratios a : bc , the values of q and r may be made as great as we please , in other ...
... whence = a ar ī a aq c am b an or = 1 - 1 " bl q cl r = n ; n ī It hence appears , that by making the ratios : m : n suf- ficiently nearly equal to the ratios a : bc , the values of q and r may be made as great as we please , in other ...
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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