r .i (r2 - z% 



(2) 



V — y 



(3) 





(4) 



146 Proceedings of the Royal Irish Academy. 



the curve UV is the intersection of the qnadrics 



^7, = Z= + F^ + ^ i W\ V, = ?(X= - W-} 

 where 



and the discriminant is 



D(X) = (X^- a (X^ -,'-). 



The equations (2) may he called the canonical form of the curve UV. 



In the discriminant D (X) the coefficients of X' and X vanish. These are 

 the invariants 0, 0' of the quadrics U^ and V^. Hence these quadries ai"e 

 doubly apolar, and a skew quadrilateral can be inscribed in either, two paii-s 

 of opposite edges of which are generators of the other. These quadrics, with 

 the two other pairs which are similarly related to v and u , have been 

 called the Vossian quadries of the system.* 



Conversely, if the discriminant of kU— V have the canonical form (4), 

 U and V are a pair of Yossian quadrics. For the roots of u are then zero 

 and infinity, and these values of X give the quadiics V and TJ respectively. 



It is convenient to express the coefficients of V^ in terms of the roots 

 £i, ii: f 3. of the reducing cubic or D (X). 

 From (3] above 



x' - a V - /3 

 ^ ^ r, _ C - a ' V - fi _ .^ t?p' - \{a -5- j8) (» -f »') ^ a/3 

 S- ij " c'-a _ c' - 8 ~ " ' {V-I!'){a - /3; ' 



c — a c — |3 



Substituting the values of r, c from the equation « = 0, 

 £ + jj )a - -y . /3 — o |- 7/t 



f'a - 7 , 



,/3-^ 



^/a-/3 



■ 7-0 



T7- V 



= n, 

 -', or 



I- m 



If \/e. - «3 = m, .y^e, - c, = n, m- = - Im, n- = ^jV. (5) 



n 



n ^ ^ 5 - 'J J/ m - n 



and the equations of </K in its canonical form are 

 Uf^^X-^ Y'- + 2^^ W- = 0, 



Fo ^ (m H-«) (X= - TF^) + (m - nj ^^P- - Z-) = 0. (6) 



From (2) and (4), since the coefficients of the discriminant are invariants, it 

 follows that if two quadrics U, V, whose equations are given in any form, 

 are doubly apolar, the canonical form of their curve of intersection can be at 

 once written down in the form (2), where ± 5, ± ij are the roots of the 

 discriminant of X C^ - V. 



* Toss, Math. Ann., Bd. s. See also Klayver, American Joanial of IMathematics, 



ToL six. 



