CONIC SECTIONS, ANALYTICAL. 169 



The anharmonic ratio of four points, or four straight lines, can 

 never be equal to 1 ; as that leads immediately to the condition 

 AD.BC = 0, or sin APD . sin BPC = ; making two of the points, 

 or two of the lines, coincident. 



If A y B, C, D be four fixed points on a conic, and P any 

 other point on the same conic, P [A BCD} is constant for all posi- 

 tions of P, and is harmonic when BC, AD are conjugates to the 

 conic. Also, if the tangents at A, B, C, D meet the tangent 

 at P in a, b, c, d t the range [abed] is constant and equal to the 

 former pencil. 



The range formed by four points in a straight line is equal to 

 the pencil formed by their polars with respect to any conic. 



If the equations of four straight lines can be put in the form 

 Mrr^v, u = n t v, u = fJL 9 v, u^p 4 v, the anharmonic ratio of the 

 pencil formed by them, or of the range in which any straight line 

 meets them, is 



, 

 Os -/*) Os -/**)' 



806. Two fixed straight lines meet in A ; B, C, D are three 



fixed points on another straight line through A ; any straight lino 



through D meets the two former in #, (7; BK, CO' meet in P, 



' // in Q : prove that the loci of P, Q are straight lines 



through A which form with the two former a pencil equal to 



807. ABC is a triangle, two fixed straight lines intersect in 

 a ]>oint on BC, any point is taken on AO, and the straight lines 



.ng it to Bj C meet the two fixed lines in 2?,, /?,, (7,, C t , 

 respectively; prove that Bf 9 and B t C\ |ass each through a 

 point on BG. 



808. Chorda are drawn through a fixed point on a conir, 

 making equal angles with a given direction ; prove that tin 

 straight line joining t icmities passes through a fixed 

 point 



