40 ON THE GENEBAL EQUATION OF CURVES 



^^_ / — F"im^+ 2 m COS y-^ 1) .jg. 



^^-y/ Am^+2Bm + C ^ '' 



(a.) The factor m^-{-2m cos y-\-l in the numerator of the 

 fraction under the radical sign in this equation is always posi- 

 tive, since it is equal to 



{m + cos yY + (sin y)^, 

 and consequently the numerator has the same sign as — F." 

 The denominator admits of the form 



B\2 , AC— B^) 



A{(. + f)-H 



A2 ] 



and therefore when B^ < AC the denominator has the same 

 sign as A and can never vanish. Hence, since A is supposed 

 to be positive, No. I., when — F' is positive the value of CP 

 given by eq. (12) is real and finite for all values of 7n, and the 

 curve (1) is an oval limited in every direction ; but when F' 

 is positive CP is imaginary and the equation has no locus. In 

 the former case the curve is called an ellipse, and thus we see 

 that the conditions in order that the equation (1) may repre- 

 sent an ellipse are that B* be less than AC and F' negative. 



(^.) When B^ > AC the roots of the equation Am^ + 2 Bw 

 + C=o are real. If m' and m" denote these roots it is evi- 

 dent from equation (12) that the lines CS and CT (Fig. 4) 

 whose equations are 



y—y^^m' {x—x^ and y~y^—rd' [x—x^ 

 meet the curve (1) in four points at infinity. The infinite 

 diameters SCS' and TCT* of a curve of the second degree are 

 called the asymptotes of the curve. Hence we see that for 

 any curve (1) of the second degree, the direction indices of 

 the asymptotes are the roots of the quadratic equation 



Aw^ + g Bw + C=o (13). 



By a well known property of quadratics, 



AmH2 B»i + C=A (»»— w') (w — m"). 

 Now when m is intermediate between ni and to", one of the 

 factors TO — to' and to — to" is positive and the other negative, 



