ANALYTIC GJTO.v /://,!' [CM. v. 



Solving equation ( 1) for x in terms of y. 1>\ 

 the square of tin- r-tiTins. it 



Hut since # is, by hypothesis, expressible rationally in 

 terms of y, therefore the expression under the radical sign 

 is a perfect square, and therefore 



(EG - AFyt-dP- ABXO 3 - -Itf )= 0, 

 i.e., ABC + 2FGH-AF*-BG*-CH* = 0. . . [17] 



If this condition among the coefficients is fulfilled, thm 

 equation (1) has for its locus two straight lines. 



The expression ABC+ 2FGH- AF*- BG 2 - CH~ 

 called the discriminant of the quadratic, and is usually 

 represented by the symbol A. 



NOTE. The analytic work just given fails if A = 0. In that case 

 equation (1) may be solved for y instead of solving it for r, and the same 

 condition, viz. A = 0, results. If, however, both A and B are zero, then the 

 above method fails altogether. In that case equation (1) reduces to 



J Uxy -I- 2 Gx + 2 Fy + C = ....... 



If the first member of equation (3) can be factored, then evidently the 

 ; -in must take the form 



....... (4) 



which shows that equation (3) is satisfied for all finite values of y provi<l-,l 



x = , a constant. Let be represented by &, then equation ( 1) 

 a o 



becomes 2 Hky + 2 Gk + 2 Fy + C = 0, 



