136 Proceedings of the Royal Irish A cademy. 



meets these planes is also K, and this quantity may be called the A.E. of the 



curve. 



Section 4. — Let AA', Bh' be two pairs of corresponding points of the 

 first kind, theh- coordinates being 



(^, y, z, w), {x, -y,-z, v:), (.-?', /, s\ v/), {x, - >/', - z, w"). 

 We have seen (§ 3) that the chords AB, A'JS are sjenerator-s of the same 

 system of the same quadric, and, since we may interchange B, B' at will, the 

 chords AB' and A'B are also generators of the same system of the same 

 quadric. Let the parameters of the qnadries be A, and A', and let us call the 

 generator's A^, A/. Then if (j)qrstu), (^pqr's't'u') be the line coordinates of 

 AB and AB, 



p = ijyz'), s = ixw'), J)' = - {yz'), / = {xw'), .: ps + p's = 0. 

 But (§ 3) ^ 



P _ ;AaS J}' fy^ ,T r-7— ,—. T^ „ 



„ t K^ : ~' ~AVr~' •■ '^ AaSA ySy + V . A 87A ao = U, 



■^ V A;3-/ S \ A B-y 



whence /XasA'sy = As'/A'aS: 



or 03 -f Y - a - Sj AA' - (/37 - «S) (A -f A') + /Sy (a + S) - aS (/3 + 7) = 0. (1) 



or, if Gti = 'u,xV\Vj\ be the sextie covariant of the discriminant, and u\ the 

 quadratic factor which corresponds to the grouping ySy ; uh, max' = 0. 



Hence A, A' are eorrespoudeuts in an involution of which llie roots of 

 u\ = are the foci : if A be given, X' is uniquely determined, and the quadrics 

 (A), (A') may be called conjugate quadrics in the first correspondence. If the 

 chord AB be a generator A, of (A), we can draw from A two generators of (A'). 

 One of these, which I have called A/, passes through the point B' ; the other, 

 X,', joins A to a point P^ which, since A'B is A/, is coplanar with A A'B. 

 We will call P the residual of the points A A'B. 



11 we t-ake another pair of points CB, connected by a generator A] of (X), 

 and join C to B' and to Q, the residual of CC'B ; since CChQ are coplanar, 

 and CB is A,. C"§is A,. But AB is A„ .-. ABC'Q are coplanar. Hence (§ 3) 

 AB'C'Q are coplanar. But AB' is A/, .-. CQ is A^' and CI)' is A/- Hence the 

 system of generators A^ and A^ of (A) are related each to a definite system A/ 

 and A/ of (A'), so that we may say that A, and A^' belong to the same system 

 of generators, the property being that, if from any point be drawn a pair 

 of generators of the same system of a pair of conjugate quadrics, they will 

 pass through a pair of corresponding points. (2) 



Prom this result we can immediately deduce two fundamental properties 

 of pairs of corresponding points of the same kind. 



(a) If a pair of corresponding points be joined to any point B on the curve 



