An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 48
... give ẞ = 0 , y = 0 . Hence PQ , RS intersect in A. - Similarly , PR , QS intersect in B , and PS , QR intersect in C. Hence , the angular points of the triangle of reference coincide with the intersections of the line joining each pair ...
... give ẞ = 0 , y = 0 . Hence PQ , RS intersect in A. - Similarly , PR , QS intersect in B , and PS , QR intersect in C. Hence , the angular points of the triangle of reference coincide with the intersections of the line joining each pair ...
Page 50
... gives one straight line on which the centre lies . It may be similarly proved to lie on the straight line L'a M2ß b a = Therefore the co - ordinates of the centre are given by the equations L'a M2ß N2y a b C Combining therewith aa + bB ...
... gives one straight line on which the centre lies . It may be similarly proved to lie on the straight line L'a M2ß b a = Therefore the co - ordinates of the centre are given by the equations L'a M2ß N2y a b C Combining therewith aa + bB ...
Page 51
... gives ( see Art . 5 , p . 8 ) M2 N2 b cos B = c cos C Similarly , since the lines joining the centre with B , C are respectively perpendicular to B = 0 , y = 0 , we shall have or N ' C cos C = L ' L ' Ms a cos Д ' a cos Ab cos B Hence ...
... gives ( see Art . 5 , p . 8 ) M2 N2 b cos B = c cos C Similarly , since the lines joining the centre with B , C are respectively perpendicular to B = 0 , y = 0 , we shall have or N ' C cos C = L ' L ' Ms a cos Д ' a cos Ab cos B Hence ...
Page 73
... gives ( up + w'q + v'r ) ƒ + ( w'p + vq + u'r ) g + ( v'p + u'q + wr ) h = 0 , or , ( uf + w'g + v'h ) p + ( w'f + vg + u'h ) q + ( v'f + u'g + wh ) r = 0 . Hence , since , for every point on the line required , = a - ...
... gives ( up + w'q + v'r ) ƒ + ( w'p + vq + u'r ) g + ( v'p + u'q + wr ) h = 0 , or , ( uf + w'g + v'h ) p + ( w'f + vg + u'h ) q + ( v'f + u'g + wh ) r = 0 . Hence , since , for every point on the line required , = a - ...
Page 76
... gives , as the necessary and sufficient relation among the coefficients , u , w ' , v ' , a = 0 w ' , v , u ' , b , v ' , u ' , w , c a , b , C , 0 or Ua2 + Vb2 + Wc2 + 2 U'bc + 2 V'ca + 2 W'ab = 0 . 7. To find the condition that the ...
... gives , as the necessary and sufficient relation among the coefficients , u , w ' , v ' , a = 0 w ' , v , u ' , b , v ' , u ' , w , c a , b , C , 0 or Ua2 + Vb2 + Wc2 + 2 U'bc + 2 V'ca + 2 W'ab = 0 . 7. To find the condition that the ...
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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