## An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory Projections |

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Page ix

... Centre Equation of the

Conditions for a Circle All Circles pass through the same two points at infinity All

Conics , similar and similarly situated to each other , intersect in the same two

points ...

... Centre Equation of the

**Asymptotes**Condition for a rectangular HyperbolaConditions for a Circle All Circles pass through the same two points at infinity All

Conics , similar and similarly situated to each other , intersect in the same two

points ...

Page 80

... V'a + U'b + Wc २ ॥ II II 11 c , nu , 2 с u , w ' , o ris с с w ' , v , or Uvw + 2u'v'w'-

uu ? 12 VU 12 ww These equations determine the centre . 12 . To find the

equation of the

Centre.

... V'a + U'b + Wc २ ॥ II II 11 c , nu , 2 с u , w ' , o ris с с w ' , v , or Uvw + 2u'v'w'-

uu ? 12 VU 12 ww These equations determine the centre . 12 . To find the

equation of the

**asymptotes**. 80 MODERN GEOMETRY . Co-ordinates of theCentre.

Page 81

To find the equation of the

investigation of Art . 10 , and paying regard to equations ( A ) of Art . 11 , the

) – { ( aa + ...

To find the equation of the

**asymptotes**. Writing a , b , y , for f , g , h , in theinvestigation of Art . 10 , and paying regard to equations ( A ) of Art . 11 , the

**asymptotes**will be found to be represented by the equation lā , B , 7 ) $ ( Q , B , v) – { ( aa + ...

Page 82

... that the conic may be a rectangular hyperbola . If the equations of the

perpendicularity is Il + mm ' + nn ' - ( mn ' + m'n ) cos A - ( nl ' + n'l ) cos B - ( lm ' + I'

m ) cos C = 0 .

... that the conic may be a rectangular hyperbola . If the equations of the

**asymptotes**be la + mß + ny = 0 , l'a + m'p + n'y = 0 , the condition of theirperpendicularity is Il + mm ' + nn ' - ( mn ' + m'n ) cos A - ( nl ' + n'l ) cos B - ( lm ' + I'

m ) cos C = 0 .

Page 97

The equation of the

) ' = 0 . 2 ' , v , w , 1 w ' , v , u o ' , u ' , w , 1 v ' , u ' , w 1 , 1 , 1 , 0 Other formulæ may

be adapted in a similar manner . CHAPTER VI . RECIPROCAL POLARS . 1.

The equation of the

**asymptotes**is u , w ' , u ' , 1 $ ( x , y , z ) + u , w ' , u ' | ( x + y + z) ' = 0 . 2 ' , v , w , 1 w ' , v , u o ' , u ' , w , 1 v ' , u ' , w 1 , 1 , 1 , 0 Other formulæ may

be adapted in a similar manner . CHAPTER VI . RECIPROCAL POLARS . 1.

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angle angular points appears Author become BOOKS called Cambridge centre Chap chapter chord circle cloth co-ordinates coefficients coincide College common condition conic conic section considered constant corresponding Crown 8vo curve denoted described determine distance draw drawn eliminating equal equation Examples expressed find the equation fixed point focus follows given point given straight line gives harmonic Hence imaginary line at infinity locus meet observed obtained opposite sides pair parabola parallel passing pencil perpendicular point of intersection points of contact polar pole positive produced projection prove ratio reciprocal relation represented respect result right angles satisfy Schools second degree sides similar Similarly student taken tangents term theorem third three points tion touch Treatise triangle of reference values whence writing written