338 Transactions. 



The tangents to the above hyperbola at the in- and ex- 

 centres form the standard quadrilateral 



la + mfi + lly = o 



The equation of the line joining the middle points of the 

 diagonals of this quadrilateral is 



l-a m 2 B iry 



a b c 



Hence, since l-\-m-\-n = o, the envelope of this line is the circle 

 ABC, and the line touches its envelope at the centre of the 

 corresponding hyperbola. 



The tangents to this hyperbola at its intersections with 

 the conic \(3y + ixya + va/3 = o meet the quartic lfi' 2 y 2 + viy\r 

 -fraa. 2 /3' 2 = o in four points lying on the straight line \a -j-fj.fi 

 + vy = 0. 



12. Let five points A, B, C, D, E, no three of which are 

 collinear, be taken. If any three — say, A, B, C — be taken as 

 the vertices of the triangle of reference, and the isogonal con- 

 jugates D' E' of the two remaining points be constructed, then 

 the conic through the five .given points may be regarded as the 

 isogonal transformation of D' E' with respect to the triangle 

 ABC. If, therefore, D' E' touch the circle ABC, then will 

 each of the lines A' B', A' C formed in a correspond- 

 ing manner touch the circles CDE, BDE respectively. 



If the line D' E' cut the circle ABC at an angle (f>, then 



the lines A' B', A' C will cut the corresponding circle 



CDE, BDE at the same angle <£, and the Simson lines 



of the points of intersection of each line with its associated 

 circle will form two sets of parallel lines. 



13. In connection with the theory of isogonal transforma- 

 tion are certain curves which remain unaltered when the co- 

 ordinates of any point (a, f3, y) are changed into (-, -z, -J 

 Among such curves we have the conies of the forms 



a 2 + (3y = o 



a 2 + f3y ± k-« 08 + y) = o 



Other curves are homogeneous functions of U, V, W, such as 

 U V W 



-+w+v = w 



U / b c\ V / c a\ W la b_\ 

 I \m n) m\ii I n\l m/ ^ ' 



*/a U + VW + Vy«W = o (hi) 



