364 Mr Bromwich, The Classification of Conies and Quadrics. 



(2£ — B ) will be a linear function of the y's and zs and a constant 

 term, consequently B has now the form it would have if y r , z s 

 formed an orthogonal set of Cartesians in space of (w— 1) dimen- 

 sions ; and this must therefore be the case. 



Thus to complete the reduction of A we have only to find £i; 

 but the highest power of X in £ is A, 0-1 , so if a % 1, ^ = 0. Hence 

 we find 



a = l, A = Zy 1 ?/0 r -8 1 



r (r = 1 2 h) 



r 



for if a < 1, S x = 0. 



The case remaining is that of a ■= 2 ; here we shall see that ^ 

 can be expressed by means of a z. For, using the fundamental 

 transformation of Darboux's paper, we find 



F = \- C rs , ZbUt -£p 



A n 



Xb st x t , 



The highest power of X in A is X h+ ' 2 (as a = 2) and conse- 

 quently (using the same method as used before to reduce F) the 

 part independent of X in the first term on the right is of rank 

 (Rang) = (n—h — 2), i.e. it can be expressed as the sum of 

 (n — h — 2) squares. The term independent of X in — (A^A,,) £ 2 

 is clearly — £?/&!, so we have 



F = 2U r 2 = (n — h — 2) squares — £yyS 2 . 



Thus £7 = — So^ 2 , selecting z x to be the particular one of the 

 z's to go with Zi. 



Hence if a = 2, J. = % r 2 /0 r + 2 (- 8 2 )* ^ + S x , 



r 



and clearly the term ^ may be combined with z x without altering 

 the form of B. 



Hensel uses a number J to classify his results, and defines 



J = (rank of A) — (rank of quadratic terms in A), 



thus we find the correspondence 



a = 2, J = 2 ; a = 1, / = 1 ; a < 1, Jo = 0. 



It should be remarked that corresponding to J — we may 

 have a = 0, — 1 or — 2. Without giving the details of the proof, 

 I may indicate its lines ; — B is a form of rank n and signature 



