292 ON THE KEUUCTI0>" OF TRE GENEBA.L EQUATION. 



which is always real aud finite. The curve in this case resolves into 



two imaginary straight lines which have a real point of intersection. 



If A=B, which requires w=0, and therefore a=b, and c=0, the 



curve becomes a circle, the co-ordinates of the centre reducing to 



( . — — ), and the square of its radius being— (d^ + e 2 — af). 



As before, thi3 reduces to a point if a^ + t 2 — af vanish, and i3 wholly 

 imaginary if d*+e 2 —af be negative. 



II. Iu the hyperbolic class, where ab—c l is negative. 



Here either sign of m is admissible; (a + b — m) and (a + b+m) 

 are both finite but of different sign3, and of the two quantities A and 

 B, one i3 real and the other imaginary : the curve is therefore always 

 real, and we must take that sign for m which renders A real and B 

 imaginary ; the other sign having reference to the ' conjugate' hyper- 

 bola : that is, m must be taken of the same sign as the quantity at? + 

 Id" —2cde—(ab—c*)f As in the previous class, A and B may van- 

 ish together, but neither can vanish separately, nor can they become 

 infinite except by passing into the parabola. When they both van- 

 ish, which will be when 



ae* -\-ld-2cde-(ab-c*) f=0. 

 the curve is reduced to two real straight lines, whose intersection is 

 given by (10), and which are equally inclined to the transverse axis 

 (whose direction remains determinate): in this case, both foci and 

 centre coincide with this point, and both directrices coincide with 

 the direction of the conjugate axis : hence from the mode of genera- 

 tion, the angle of inclination of each of these lines to the transverse 



axis is sec c or tan } ~^r b \ * - tnat ^g 11 of m Dein g taken, 

 which makes this quantity real. 



If -4=1?,/— 1, which requires a=—b, the hyperbola is known a« 

 the ' equilateral.' 



III. In the parabolic class, where ab — c 2 =0. 



This may be treated as the limiting case of the foregoing classes. 



Here m=a + b, A becomes infinite, ;\nd B takes the form — but is 







really also infinite (since / , „ = — . „ — r ) unless at the 



same time ae 2 +bd 2 — 2cde=0. 



Since ab=c 2 . this requires ae % =bd 2 and therefore bd—ce and ae=od, 



and then 



ae % +bd 9 — 2cde=ae i —cde 

 . c 2 e* 

 b 



