An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page v
... writing on this subject has mainly been to place it on a basis altogether independent of the ordinary Cartesian system , instead of regarding it as only a special form of Abridged Notation . A desire not unduly to increase the size of ...
... writing on this subject has mainly been to place it on a basis altogether independent of the ordinary Cartesian system , instead of regarding it as only a special form of Abridged Notation . A desire not unduly to increase the size of ...
Page 39
... writing a = 0 in the above may be a perfect square . This requires that u2 = vw , or u ' = ± ( vw ) . Similarly , v ′ = ± ( wu ) * , w ' = ± ( uv ) 1 , are necessary conditions that the conic should touch the lines B = 0 , y = = 0 . We ...
... writing a = 0 in the above may be a perfect square . This requires that u2 = vw , or u ' = ± ( vw ) . Similarly , v ′ = ± ( wu ) * , w ' = ± ( uv ) 1 , are necessary conditions that the conic should touch the lines B = 0 , y = = 0 . We ...
Page 40
... writing for convenience , L , M , N , instead of u , v , w , the equation of the conic which touches the three sides of the triangle of reference becomes L'a2 + M2ß2 + N2y2 — 2MNßy — 2NLyx – 2LMαß = 0 , which is equivalent to - - ± ( La ) ...
... writing for convenience , L , M , N , instead of u , v , w , the equation of the conic which touches the three sides of the triangle of reference becomes L'a2 + M2ß2 + N2y2 — 2MNßy — 2NLyx – 2LMαß = 0 , which is equivalent to - - ± ( La ) ...
Page 45
... writing , for convenience of future investigations , L , M , N2 for u , v , w respectively , our equation assumes the form 12 . - - L2x2- M2ß2 - N2y2 = 0 . We have now to enquire how this conic is related to the triangle of reference ...
... writing , for convenience of future investigations , L , M , N2 for u , v , w respectively , our equation assumes the form 12 . - - L2x2- M2ß2 - N2y2 = 0 . We have now to enquire how this conic is related to the triangle of reference ...
Page 52
... writing La2 + 2xBy = 0 , L -k for 2λ ' ka2 = By . This equation , it will be observed , involves only one arbi- trary constant , as ought to be the case , since when a tangent and its point of contact are given , the conic is thus ...
... writing La2 + 2xBy = 0 , L -k for 2λ ' ka2 = By . This equation , it will be observed , involves only one arbi- trary constant , as ought to be the case , since when a tangent and its point of contact are given , the conic is thus ...
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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