Hogg. — Ou Isogonal Tramformations. 311 



It is the locus of points whose axes of homology are parallel 

 to la-\-mji-\-ny = 0. 



Let this conic cut in the point (a'/S'y') the curve 



'a p y 



Then the axis of homology of {a'(3'y') will be a tangent to 

 the circle ABC and parallel to the line la-\-m/3-\-ny = o. 

 Eliminating a'fS'y' between the equations 



a' y8' ^ y' 



VIC — nh na —Ic lb — 7na 

 ; — + — pT — + ; — = 



a li y 



we have for the equation of the pair of tangents 



a ^J- {hp + Cy) - {m^ + ny) + b aJ^^ (Cy + aa) - (Wy + la) 



+ c\/-{aa + bp)-{la + ml3) = 

 ^ c 



The equation of the two parabolas may be written down 



from the above by substituting in it for a y8 y respec- 



a ^ y 



tively. 



11. Any line parallel to a = o will invert into a conic of the 

 form 



K/Sy + a^y + 6ytt + Ca/3 = 



All conies of this family touch each other and the circle 

 ABC at the vertex A of the triangle of reference. 



The two tangents to the circle ABC parallel to a = o invert 

 into the pair of parabolas 



{b±cY(iy + aa{by-^cfi) = 



The two tangents to the same circle drawn parallel to the 

 diameter of the circle through A invert into the pair of 

 parabolas 



a/Sy + bya + Ca(3 + 4 E sin B sin C a (/S cos B — y cos C) = o 

 where R is the radius of the circle ABC. 



