366 Mr Bromwich, The Classification of Conies and Quadrics. 



It may be remarked that here (\ — 0^, (A, — 2 ) are factors of 

 A 2 ; and that further subdivisions of each class may be made, by 

 examining the signs and equalities of 6 1 , 2 . This is done in the 

 ordinary text-books on conies and may be omitted here. Hensel 

 gives the subdivision according to the possible arrangements of 

 signs (by using Sylvester's " law of inertia" of quadratic forms), 

 but his method will not give the absolute values of l3 2 and so 

 does not give the subdivision 1 = 2 . 



Quadrics (n = 4). 

 Note, a is the highest power of X in the expansion of (A /A I7 in powers of 1/X. 



(i) « = 2, J =2. 



(1) h = 2, A = yi 2 /0 1 + y 2 2 /0 2 + 2kz 1} paraboloid. 



(2) h = 1, A = y 1 2 /0 1 + 2kz 1} parabolic cylinder. 



/n\ 7 /% a «7 {two planes, one 



(3) h = 0, A = 2kz 1} \ . r 



{at infinity. 



k = Lt (- Ao/VAi)*. 



A=oo 



(ii) a = 1, J = 1. 



(4) h = n, A = y 1 >l0 1 + y 2 */0 2 +.y>/0 s + k', 



[central 

 \quadric. 

 (5) h = 2, .4 = 3/j 2 /^ + y 2 2 /0 2 + A;', cylinder. 



two parallel 

 planes. 



(6) A=l, A=y 1 >/0 1 +k', I 



(fayo planes, 



(7) /i = 0, ^4 = &', j6o^a« 



[infinity. 

 k'= Lt (-Ao/XAO- 



(iii) a < 1, i/o = 0. 



(8) /i = 3, A^y^ + y^ + y*^, cone. 



(9) /i = 2, ^4 = 2/r/^i + ^V^, £wo planes. 



(10) A = l, A = y?\9^, two coincident planes. 



(11) ^ = 0, ^1 = 0. 



