An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 5
... ) ( Y1-72 ) cos A , ( B1 − B2 ) 2 + ( Y1 − Y1⁄2 ) 2 + 2 ( ẞ1 − B2 ) ( Y1 — Y2 ) cos A sin2 A whence 2 = a rational integral function of the second degree . where l , m , n are certain functions of DISTANCE BETWEEN TWO POINTS . 20 5.
... ) ( Y1-72 ) cos A , ( B1 − B2 ) 2 + ( Y1 − Y1⁄2 ) 2 + 2 ( ẞ1 − B2 ) ( Y1 — Y2 ) cos A sin2 A whence 2 = a rational integral function of the second degree . where l , m , n are certain functions of DISTANCE BETWEEN TWO POINTS . 20 5.
Page 11
... whence Again , lf + mg + nh = 0 , - - 1 ( a −ƒ ) + m ( B − g ) + n ( y − h ) = 0 . ax + bB + cy = 2A , af + bg + ch = 2A ; - :: a ( a − ƒ ) + b ( B − g ) + c ( y − h ) = 0 ; a - f B - g $ y - h = bn = ст cl . an am — bl ' - ...
... whence Again , lf + mg + nh = 0 , - - 1 ( a −ƒ ) + m ( B − g ) + n ( y − h ) = 0 . ax + bB + cy = 2A , af + bg + ch = 2A ; - :: a ( a − ƒ ) + b ( B − g ) + c ( y − h ) = 0 ; a - f B - g $ y - h = bn = ст cl . an 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 ) B + ( ƒg ' —ƒ'g ) y = 0 . 11. To find the general equation ...
... 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 ) B + ( ƒg ' —ƒ'g ) y = 0 . 11. To find the general equation ...
Page 13
... γ = 0 , λα2 + β + γ = 0 , λαg + μß3 + vY3 = 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.
... γ = 0 , λα2 + β + γ = 0 , λαg + μß3 + vY3 = 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 218 b = bc m C bc ar τ a or - r = n ; n -- = a aq с 018 am b an 1 1 : 1 bl ' 2 cl 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 r may be made as ...
... whence 218 b = bc m C bc ar τ a or - r = n ; n -- = a aq с 018 am b an 1 1 : 1 bl ' 2 cl 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 r may be made as ...
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Common terms and phrases
angular points anharmonic ratio asymptotes auxiliary conic b₁ b₂ C₁ centre Chap chord co-ordinates coefficients common tangents conic section determinant directrix find the equation fixed point fixed straight line follows four points given conic given point given straight line given triangle Hence investigated Let the equation line at infinity line joining locus meets the conic nine-point circle obtain opposite sides parabola Pascal's Theorem perpendicular point f points at infinity points of contact points of intersection PQRS prove radical axis reciprocal curve reciprocated with respect rectangular hyperbola represented right angles second degree shewn similar and similarly sin POS tangents drawn term a,b,c theorem three points three straight lines triangle of reference ua² uf+w'g+v'h ux² V'ca va² Vb² vß² W'ab whence wy² λα