182 On a Theorem relating to a Triangle, Line, and Conic. 

 If this equation break up into factors, the form must be 



that is, we must have 



ka+B0 +Cy =2{abc-fgft), 



Bu = b% Ca=c<&, 



Cp = ctf, A/3=«&, 



Ay=a&, By = b$; 

 and the last six equations give without difficulty 



A- J, ,= }«*, 



where k is arbitrary ; the first equation then gives 



«m + m + cw 2{abc _ fghy> 



or, reducing by the equations <&^ = &$+ak &c, this is 



%a = Kb + ec-2abc + 2fgh+ (f + | + |)k = 0; 

 which, substituting for % 33, C their values, becomes 



Hence if K = 0, that is, if the conic break up into a pair of lines, 



in either case the equation of the curve of the third class becomes 



that is, the curve breaks up into a point, and a conic inscribed 

 in the triangle. 



In the case where the conic breaks up into a pair of lines, then 

 we have 



(a, b, c,f,g, hjx, y, s;) 2 =2(px-{-qg + rz){p , x-{-q , g-\-r , z) f 

 and thence 



