ANALYTIC GEOMETRY P 



The two forms of the (piadric, not already di>cussed,* 

 have therefon- for their equations, when simplified (dropping 

 the accents), 



and 1 //.- -'A :<>.... [88] 



A center of a surface is a point such that it 1- iv 



chord of the surface which passes through it. It is clear 

 that the locus of equation [:':>] is a central qnadrie. while 

 the locus of equation [33] is non-central (cf. Art. 178). 



227. Standard forms of the equation of a quadric. 1 <>r 

 convenience of discussion, the intercepts of the locus of 

 equation [32] on the coordinate axes may be represented ly 

 a, b, c, respectively, so that 



Then, since A, 2?, (7, and K cannot be all of the same 

 there will he three types of equation [32], according to tin- 

 signs of A, BI (7, and K\ viz.: 



++-'- - 



Similarly, equation [33] may he written for convenience in 

 the typical forms 





An exceptional case occurs where the general equation can be factored 

 into linear factors, and therefore represents two planes. 



