Mr Bromwich, The Classification of Conies and Quadrics. 365 



(n — 2) (Signatw, introduced by Frobenius, Berl. Berichte, 1894 

 = Crelle, 114, p. 187). For clearness we give the definition, 



signature = (number of positive squares) — (number of negative 



squares). 



We can now apply a theorem due to F. Klein (Inaug. 

 Diss. Bonn, 1868 = Math. Annalen, 23, p. 539) 1 and deduce that at 

 most one invariant-factor of A is not linear : and this one may be 

 squared or cubed. Further the invariant-factors of Aj are all 

 linear. These facts will connect a and J Q . 



Hensel's second classifying number is J x = h (the rank of the 

 quadratic terms in A). 



We add a classified list of conies (n = 3) and quadrics (n = 4) ; 

 and we should remark that in drawing these out we have 



< h < (n — 1) or (n — «)• 



Conies (n = 3). 



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



(i) a = 2, Jo = 2. 



(1) h = 1, A = y 1 2 /0 1 + 2kz 1 , parabola. 



(2) h = 0, A= 2fe„ (pair of lines, one 



(at infinity. 

 k=Lt (- Ao/VAO*. 



(ii) a = l, Jo = l. 



(3) h = 2, A = 2/i 2 /#i + 2///^ 2 + k', central conic. 



(4) h = l, A = y 1 2 /d 1 +k', parallel lines. 



(5) h = 0, A= k\ 



k' = Lt (- Ao/XAj). 



two lines, both at 

 infinity. 



(iii) a < 1, J = 0. 



(6) h =2, A = yi"/0i + y?ld 2 , V a ^ r °f lines- 



(7) h = l, A = y^/Oi, two coincident lines. 



(8) h = 0, A = 0. 



1 It may he pointed out that Klein's results have been considerably extended by 

 A. Loewy [Math. Annalen, 52, p. 588 and Crelle, 122 (1900), p. 153). These papers 

 give iuequalities connecting the numbers of various classes of invariant-factors of 

 \A-B\ with the characteristic and signature of B. 



