300 ANALYTIC OEOME I i;Y [c n .MI. 



182. Summary. It lias been slmwn in tlie preceding 

 articles that every equation of the second degree in t\\<> 

 variables represents a conic section, whether the axes are 

 oblique or rectangular; ami tiiat its species and position 

 depend upon the values of the coefficients of tin- njuatii.ii. 

 The various criteria of the nature of the conic represented 

 by such an equation, in rectangular coordinates, appear in 

 the following table : 



The General Equation of the Second Degree 



+ (7=0 



I. 77 2 - AB < 0. The ellipse. 



(1) if A = B, and JJ=0, a circle. 



(2) if A is +, imaginary. 



(3) if A is -, real. 



(4) if A is 0, a pair of imaginary straight lines, 



or, a point. 



II. 7/ a - AB = 0. The parabola. 



(1) if H is +, axis is the new y-axis. 



(2) if H is , axis is the new ir-axis. 



(3) if A is 0, pair of parallel straight lines, which 



are real and different, real and coin< i<l nt. 

 or imaginary, according as # a AC>, 

 = , or <0. 



III. /T a - AB > 0. The hyperbola. 



(1) if A = B) a rectangular hyperbola. 



(2) if A is +, principal axis is the new #-axis. 



(3) if A is , principal axis is the new z-axis. 

 (4; if A is 0, a pair of real intersecting straight 



lines. 



