306 T ran ■■^actions. 



It may also be noticed that according as P is within or 

 without the triangle ABC so is its inverse point P' within or 

 without that triangle. 



2. The line whose equation is 



la + m{3-\-ny = 

 will invert into the conic having for equation 



l(Sy-{-mya+na(3 = 



Also, any conic circumscribed to the triangle ABC will invert 

 into a line : in particular the circumcircle of the triangle ABC 

 will invert into the line at infinity. 



If a point P (ai^iyi) be determined by the intersection of 

 the circle ABC with the conic l^y-\-7nya + 7iaft = o, it may be 

 at once shown that the lines 



/3(3i-yyi = 0, yyi-aai = O, aaj — ^/3i = 



which determine the position of the inverse of P, are all parallel 

 to the line la-\-mf3 + ny = o. 



A line passing through a vertex of the triangle ABC inverts 

 into a line passing through the same vertex. 



3. The conic iPy + mya-\-na/3 = o will be a hyperbola, para- 

 bola, or ellipse according as 



\7a+ v'mb+ Vnc > = or < o 

 but this is the condition that the line la+m(3 + ny = o shall 

 intersect, touch, or not intersect the circle ABC : hence the 

 theorem that a line inverts into a hyperbola, parabola, or 

 ellipse according as it cuts, touches, or does not cut the cir- 

 cumcircle of the triangle of reference. 



4. The asymptotes of the conic Z/3y + wya -f na/3 = o are 

 given by 



hnn (aa + 6/3 + cy)'^+ A(//3y + wya + n«/3) = 

 where 



A = dH^+hhn^-^d^n^ — lhcmn — ^canl — lahlm 



It is easily shown that the angle (</>) between the asymptotes 

 is given by _ 



i/a 



*'* "~ 2E {I cos A-fm cos B+w cos C) 

 K being the radius of the circle x\BC. Hence 



I cos A-fm cos B-l-71 cos C 

 cos <^ = ^ 



where O'-^ = i^ + m'^-f n'^ — 2mn cos A - %il cos B — 2/??i cos C. 



