An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 12
... give the values of a , B , y , at the point of intersection . 10. To find the equation of the straight line , passing through two given points . Let f , g , h ; f ' , g ' , h ' , be the co - ordinates of the two given points , and ...
... give the values of a , B , y , at the point of intersection . 10. To find the equation of the straight line , passing through two given points . Let f , g , h ; f ' , g ' , h ' , be the co - ordinates of the two given points , and ...
Page 14
... gives , eliminating a , B , y by cross - multipli- cation , .. - Im ̧n − 1 ̧m ̧n + l ̧ ̧ ̧− 1 ̧m ̧n + m ̧n — 1m ̧n ̧ = 0 , - the required condition . - The identity of form between the conditions that three straight lines should ...
... gives , eliminating a , B , y by cross - multipli- cation , .. - Im ̧n − 1 ̧m ̧n + l ̧ ̧ ̧− 1 ̧m ̧n + m ̧n — 1m ̧n ̧ = 0 , - the required condition . - The identity of form between the conditions that three straight lines should ...
Page 33
... give us a quadratic for the determination of , to each of the γ two values of this ratio , real , equal , or imaginary , one value of a will correspond ; whence it appears that the straight line and the curve cut one another in two real ...
... give us a quadratic for the determination of , to each of the γ two values of this ratio , real , equal , or imaginary , one value of a will correspond ; whence it appears that the straight line and the curve cut one another in two real ...
Page 43
... ratios of L , M , N in order that the conic may become the inscribed circle . At the centre of this circle we have , as , we know , a = B = y . This gives , by the result of Art . 8 CONDITION OF TOUCHING A GIVEN STRAIGHT LINE . 43 .
... ratios of L , M , N in order that the conic may become the inscribed circle . At the centre of this circle we have , as , we know , a = B = y . This gives , by the result of Art . 8 CONDITION OF TOUCHING A GIVEN STRAIGHT LINE . 43 .
Page 44
... gives , by the result of Art . 8 , Nb + McLc + Na = Ma + Lb. To solve these equations , put each member equal to r , we then get M N + 210218 = r N L с L a + + M b = = bc ' r са r ab • Adding together the last two of these equations ...
... gives , by the result of Art . 8 , Nb + McLc + Na = Ma + Lb. To solve these equations , put each member equal to r , we then get M N + 210218 = r N L с L a + + M b = = bc ' r са r ab • Adding together the last two of these equations ...
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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