72 Proceedings of the Royal Irish Academy. 



obtain an infinity of surfaces and curves. All the surfaces have the same 

 cuspidal line p, and touch the same hyperboloid .ff along /;. All the directrices 

 are generators of H of one system (a), since they meet the generators p, I, J 

 of the other system (/3). Hence all the linear congruences to which the gene- 

 rators of the surfaces belong contain the regulus (/3). The surfaces S fall into 

 two systems, / being a generator of those of one system, and Joi those of the 

 other. 



System of Quadrilatercds and Tdrahedra in Space connected with a Chirve h. 



76. Take the equations of the curve in the form 



x.lit' - it) = xjt' = .rj/f = xj(3t' - 1), (36) 



which is easily derived from (34) or (35). 



The complex is C = j^is + 2^ii = 0- 



The congruence (t, t'^) is constituted by C, and C" = pij + pu = 0. 



The congruence (t, -t'^) is constituted by C, and C" = p,. - j^m = 0- 



The lines (+ t) belong to the congruence C" = C" = 0. 



The three complexes C, C\ 0" are in involution, and therefore determine 

 a system of tetrahedra in space such that any vertex is the pole of each of the 

 adjacent faces with respect to one of the three complexes. Each edge will of 

 course belong to the congruence formed by two of the complexes. 



If (.'/j), iX), (v), (Z) are the points f, t'^, - f, - 1^ respectively on the curve, 

 we have easily, from (36), 



rf-i : j:2 : .^s : .r4 = - ?4 : ?3 : ?3 : - |i = - >)i : - V2 : m ■ ri^^ - Zi ■ Zs ■ -Z2 : Zi- 

 It follows that, if five points ti{i = 1, 2, ... 5) lie in a plane Hi, the sets of 

 five points tr\ - 1-,, - tr^ will lie in planes IIj, lis, IIj. 



If {y) is the pole P, of n, with respect to C, the equation of Ed is 



Z, 7/3 - X3 2/1 + X. 2/.1 - Xi y. = 0. 

 Hence ITa is - X4 yj - Xz y^ + X3 yi + X, 7/2 = 0. 



Hence n2 contains P,, and is its polar plane in ("''. 



Similarly Da contains P,, and is its polar plane in C" . 



If Pi is the pole of 11; in C, we see in this way that the tetrahedra (P) 

 and (n) coincide, and form a tetrahedron of the kind described above. 



Starting, therefore, from any point Pi in space, we get a quadrilateral 

 P1P2P3P4 such that the pairs of opposite sides belong to the congruences 

 (t.,t"^') and {t,-t~^), and the diagonals to the congruence {t,-t). Further, if 

 the plane n,(PjPiP2) meets the curve in five points A^, the plane lis will 

 meet the curve in the five vertices P; of the quadrilaterals ^,P, GiDi on the 

 curve (par. 67) (c) ; and so with IIj and Hi. 



