Mr Bromwich, The Classification of Conies and Quadrics. 367 



Here (X — 0j), (X — 2 ), (X — 3 ) are factors of Aj ; and we can 

 subdivide each class according to possible equalities and arrange- 

 ments of sign amongst 1} 2 , S . These subdivisions may be 

 found in all text-books on Solid Geometry. The remark above 

 (referring to Hensel's further subdivisions) applies here also. 



3. Conies or quadrics given by a tangential equation. 



There are some cases in which the point-equation does not 

 give sufficient information to completely determine the conic or 

 quadric. For instance, a conic in a plane has a space tangential 

 equation, but the corresponding point-equation is simply the 

 equation to its plane squared. It is therefore convenient to have 

 a method of classifying by means of the tangential equation, 

 directly. 



Let then A = *2, a rs v r v s (r, s = 1, 2 ...n) be the given tangential 



r, s 



equation and let B = X b rs v r v s be the tangential equation to the 



r, s 



absolute with the coefficients determined so that the perpendicular 

 on the generalized plane ~%v^x r = is ~%v^ r \B l ~, when the cc's are 



r r 



subject to the relation 2pr« r = 1 as before. It follows that as the 



r 



absolute is a (generalized) conic in the hyper-plane at infinity 

 %>,.&> = 0, we shall have \B\ = and the point-equation corre- 



sponding to B — will be (5#vc r ) 2 = 0, i.e. the first minors of \B\ 



r 

 will be proportional to p r p s . So let us write kp r p s as the minor 

 of b rs ; we may note that we have further 



%b rs p r = 0, (r, 5=1, 2 . . . n). 



r 



We have now to consider the simultaneous reduction of the 

 forms A, B; and we observe that B is a definite form of rank 

 (n — 1), consequently by a theorem of Weierstrass's 1 the deter- 

 minant \kA + \B\ may have one invariant-factor k 2 but no more, 

 but under ordinary circumstances there will be the one invariant- 

 factor k. Besides this one, all the other invariant-factors must 

 be linear. 



The reduction of (A — XB) for the finite roots of | A — \B\ = 

 can be carried out by the ordinary Weierstrass or Darboux process, 

 thus if (X — c) 1 is a factor of \A —XB\, (X — c) will be a factor of 



1 Quoted in a previous footnote (p. 363). 



