154 BOOK OF MATHEMATICAL PROBLEMS. 



(4) to which the triangle is self-conjugate 



The general equation of a conic passing through four given 

 points (la = m{$ = ny) is 



with the condition -rr H -- H ? = ; 



a m 2 n* 



and the general equation of a conic touching four given straight 

 lines (la m/3 ny=Q) is 



Za'-f 



with the condition -= + -r> + -r=.= 0. 



L * M JV 



The form (4) of the equation of a conic admits of our denoting 

 any point on the conic by a single variable, analogous to the 

 eccentric angle in the case of a conic referred to its axes, which 

 is indeed a particular case of this form. 



Thus, the equation of the conic being la? + mf? + ny* = 0, any 

 point on it may be represented by the equations 



which we may call the point 0. 



The equation of the tangent at 6 is 



fjla cos + Jmfi sin = N /(- n) y ; 

 the equation of the chord through 0, <j> is 



. + d> 

 + Jmftmn -Lm. n)ycos 



and the point of intersection of the tangents at 0, <j> is 



cos _- sin ~- cos 



