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

### From inside the book

Page 23

... ratio of the pencil OP , OQ , OR , OS , will be equal to that of the range P , Q , R ,

S. * This notation is due , I believe , to Dr. Salmon . See his

273 ( third edition ) . Fig . 9 . For sin POQ sin POQ sin ANHARMONIC RATIO . 23.

... ratio of the pencil OP , OQ , OR , OS , will be equal to that of the range P , Q , R ,

S. * This notation is due , I believe , to Dr. Salmon . See his

**Conic**Sections , p .273 ( third edition ) . Fig . 9 . For sin POQ sin POQ sin ANHARMONIC RATIO . 23.

Page 33

We shall first prove that every curve , represented by such an equation , is what is

commonly called a

consideration of the general equaall investigate the nature of the curve ...

We shall first prove that every curve , represented by such an equation , is what is

commonly called a

**conic**section ; and then , before proceeding further with theconsideration of the general equaall investigate the nature of the curve ...

Page 34

We shall now inquire what are the relations of the

reference , when certain relations exist among the coefficients of the equation .

First , suppose u , v , w , all = 0 . The equation then assumes the form u By + v'rya

+ ...

We shall now inquire what are the relations of the

**conic**section to the triangle ofreference , when certain relations exist among the coefficients of the equation .

First , suppose u , v , w , all = 0 . The equation then assumes the form u By + v'rya

+ ...

Page 35

To determine the position of the centre of the

, B , C of the triangle of reference draw the tangents EAF , FBD , DCE . Bisect Fig .

14 . E H B D AC , AB respectively in H , I , join EH , FI , and produce them to ...

To determine the position of the centre of the

**conic**. Through the angular points A, B , C of the triangle of reference draw the tangents EAF , FBD , DCE . Bisect Fig .

14 . E H B D AC , AB respectively in H , I , join EH , FI , and produce them to ...

Page 36

We may hence deduce the relation which must hold between , ui , v , in order that

the

distance , its co - ordinates will satisfy the equation as + b + cy = 0 . We hence

obtain ...

We may hence deduce the relation which must hold between , ui , v , in order that

the

**conic**may be a bola . For , since the centre of a parabola is at an infinitedistance , its co - ordinates will satisfy the equation as + b + cy = 0 . We hence

obtain ...

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### Common terms and phrases

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