Mr Bromwich, The Classification of Conies and Quadrics. 371 



Qaadi^ics (n = 4<). 



(i) Xa rs p r p s =j= 0. 



(1) A = avf + bv£ + cv./ + (%a rs p r p s ) v 2 , central quadric, 



{ellipsoid, hyperboloid, sphere) 



(2) A = av? + bv£ + (Za rs p r p s ) v 2 , central conic. 



(3) A = avj 2 + (2,a rs p r p s ) v -, pair of points. 

 (ii) 2a l . s p r p s = 0. 



(4) A = av 2 - + bv 3 2 + 2v v-i (— ajk)^, paraboloid. 



(5) A = av 2 2 + 2v V! (— a/Jc)$, parabola. 



a = Lt (A„/X 2 ) 



A=oo 



Here cases (1), (4) appear as (4), (1) in the former list, but the 

 other cases are not common to the two lists. 



As | B | = it is possible that we may meet with the singular 

 case |^4 — \B\ = for all values of X. But on consulting Kronecker's 

 reduced forms of singular quadratic forms, we readily see that 

 (owing to the nature of B) they only arise when A is a (generalized) 

 conic in the hyper-plane at infinity. 



In reducing these no special point occurs, for if we are in 

 a space of (n — 2) dimensions | B j 4= and B is still a definite form, 

 so that Weierstrass's reduction may be applied directly. 



