Hogg. — On Isogonal Transformations. 307 



If^ be the length of the perpendicular from the centre of the 

 circle ABC on la-{-m^-\-ny = o, then 



E U cos A+m cos B-f m cos C) 

 p= ^ 



therefore cos (/> = ^ : but ^ is the cosine of the angle between 



the choi-d la-\-m(^^ny — o of the circle ABC and the tangent 

 to the circle at the extremity of the chord, hence the angle 

 between the asymptotes of the conic lf3y-{-7nya-\-na^ = o is 

 equal to the angle at which the Ime la-\-ml3-\-ny = o cuts the 

 circle ABC. 



Moreover, since the excentricity (e) of the conic is con- 

 nected with (f> by the relation e = sec ^ and ^^ = E cos (/>, it 



follows at once that tangents to circles concentric with the 

 circle ABC invert into sniiilar conies. 



5. Suppose that a curve S inverts into a curve 8' : then to 

 any two points P and Q on S will be two corresponding 

 inverse points P' and Q' on S'. If now the point Q move up 

 to P and become infinitely close to it, the point Q' will become 

 infinitely close to P'. Hence if the tangent to S at the point 

 P be inverted, it will become a circumconic touching S' at the 

 point P'. 



If the line la -\- vift + wy = o be inverted, then any tan- 

 gent to the conic If^y + mya + na^ = o will invert into a 

 conic touching la -\- m/3 -f ^ly = o, and a pair of tangents to 

 the conic l/3y -f niya + 7ia/3 = o will invert into a pair of cir- 

 cumconics intersecting in the point which is the inverse of 

 that from which the tangents were drawn and having the 

 hue /a + vifi -f- ?iy = as a common tangent. 



6. Let two lines Lj = l^a + in^ft -f Wjy = o and L^ = 4" + 

 71I2I3 + it.fy = be taken : these will invert into the conies 



Si = lifSy + Vl^ya + n^a/S = 



S2 = k/3y + '»2ya- + ''-a/^ = 

 Let L = Xa -f /x/? -f i/y = be a common tangent of Sj and 

 S.2 : then L will invert into the conic 



S = X.(3y + /xya + va/3 = 



which will have double contact with the line pair Lj h.^ : its 

 equation therefore will be of the form 



L.L^- {pa +g/3+ ry^o 



