CONIC SECTIONS, ANALYTICAL. 175 



conic at P meet them in Q, (X, OP makes equal angles with OQ 

 and OQ'. If, however, the point lie on the curve, the original 

 curve was a parabola ; and one of the straight lines being the 

 reciprocal of the point at infinity on the parabola will be the 

 tangent at 0. 



Since the anharmonic ratio of the pencil formed by any four 

 Rtniight lines is equal to that of the range formed by their poles 

 with respect to any conic, it follows that in any reciprocation 

 whatever, a pencil or range is replaced by a range or pencil 

 having the same anharmonic ratio. 



Th j Method of Projections enables us to make the proof of 

 any general theorem of position depend upon that of a more simple 

 particular case of the theorem. Given any figure in a plane, we 

 have five constants disposable to enable us to simplify the pro- 

 jected figure, three depending on the position of the vertex and 

 two on the direction of the Plane of Projection. It is clear thut 

 relations of tangency, of pole and polar, and anharmonic ratio, 

 are the same in the original and projected figures. 



As an example we will take the following : " To prove that if 

 two triangles be self-conjugate to the same conic, their angular 

 points lie on a conic." 



Let the two triangles be ABC, DEF ; project the conic into 



a circle with its centre at D, then E t F will be at infinity, and 



/'/ will be at right angles to one another. Draw a conic 



through ABCDK, tlu-n since ABC is self-conjugate to a circle 



whose centre is/), D is the centre of perpi-ndirulars of the triangle 



.1 /.' , the conic is then-fore a rectangular hyperbola, and E being 



t it* points at infinity, F will be the other. The theorem is 



therefore : 



Again, retaining the centre at />, take any other conic instead 

 of tli' DE t DF will still be conjugate diameters, and 



therefore if any conic pass through A, B t C t D t its asymptotes will 

 be parallel to a pair of conjugate diameters of the conic whose 

 centre is /), anl to win s -lf ronju^ut ame must 



bo the case with respect to the four conies, each having 



