131 Proceedings of the Royal Irish A cadcinjj. 



The following ta 



ble gives 



the 



CO 





X 



y 



z 



IV 



A 



+ 



+ 



+ 



+ 



A' 



+ 



- 



- 



+ 



A" 



- 



+ 



- 



+ 



A'" 



- 



- 



+ 



+ 



groups : — 



A 

 A' 

 A" 

 A" 



X y z w 



+ + ^- - 



- + + + 



+ - + + 



+ H- - + 



(1) 



There are three kinds of correspondence, which are related each to one of 

 the three groupings of the roots of the discriminant, /By, aS ; 7 a, /sS ; a/3, 78. 

 These will be called the first, second, and third kinds of correspondence. If 

 we call the four points whose tangents are generators Aj of (A) A A' A" A'", 

 and their coUinears 'AA'A"A"'. the coordinates being as given in the 

 table, the pairs AA', A" A'" are corresponding points of the first kind, 

 AA", A' A'" of the second, A A'", AA" of the third.* 



If the vertices of the tetrahedron of reference be X^, Y,,, Z„, Wg, tlie pairs 



AA, A' A', A" A", A'" A'", wliich differ only in the sign of the coordinate w, 

 are collinear with W^, (2J 



' A A', A' A, A" A.'", A'" A" with X, ; AA'\ A" A, A' A'", A'" A' with F„ ; 

 AA'", A"\A, A' A", A" A' with Z,. 

 Thus we liave a configuration of three groups of four points, including the 

 vertices of the tetrahedron, which are joined by sixteen lines, three points 

 lying on each line, and four lines passing through each point. 



Any plane through TF;,meets L' Fin four points ^i?^5. The lines AB, AH 

 intersect, and are, therefore, generators of opposite systems of the same 

 quadric (A), which is touched by the plane at their intersection. Hence 



AB, AB intersect on the plane w = 0, which is the polar plane of Wo, with 

 respect to all quadrics of the system \U - V. 



Similarly AB, AB intersect ou vj = 0. 



The plane joining ^-6 to Xo passes through the points A'B', which are 

 correspondents of the first kind to A and B (v. (2) above), and the chords AB 

 A'B' intersect on the plane x = 0, and so with the other correspondences. 

 Thus, if AB be any two points on UV, the chords AB, A'B', A"B", A"'B'", 

 are all generators of the same quadric (A) of the opposite systems to AB, and 

 meet that chord in the four points in which it meets the coordinate planes. 



Four such chords joining any two sets of cotangential points are called 

 by Harnack a " quadruple." We see from the above that two collinear 



* When not otherwise stated, a correspondence is understood to be of the first kind. 

 AH statements apply, mittatis miitandis, to the other two correspondences. 



