184 THE ROYAL SOCIETY OF CANADA 



Now p and q are independent, and therefore each member of (54) 

 must be equal to a constant, say qi, q^. If the functions k{p), l{p), 

 m(p) are such that 



Hp) + k(p)m{p), 

 fhere are two values of g, namely Çi and 52 for which A vanishes for all 

 values of p. In this case (S) has two distinct straight line directrices. 



If 



Hp) = Hp)m{p), 

 (/i and 52 are respectively equal to qo. In this case the two directrices 

 of (S) become coincident. 



In conclusion we may note certain properties of the complex (28) 

 corresponding to the cases we have just discussed. 



When (aia^as) is unequal to zero each regulus of the quadric is 

 contained in a net of complexes. 



When (aia2a3) is equal to zero, let Çi and q-2 be two values of q for 

 which the minors (aia2), (a2a3), (asai) are unequal to zero. Then the 

 vanishing of (aia2a3) implies that 



(55) Aai{q)-{-Ba2{q)-\-Caz{q) = 0. 



From (45) we see that the axis of the complex (28) is determined 

 in direction by ai, ao, as. It therefore follows from (55) that axis of (28) 

 describes a surface having a plane directrix. If ai, 02, as be such chat 

 A, B, C all vanish, then the axis of (28) must be parallel to a fixed 

 direction. Equation (45) now leads to the relation 



(56) aei-\-bd2+cds = Hiq), 



where a, b, c are constants. Since 0i, Qi, Qz are solutions of (A), the 

 function Hiq) must also be a solution; and therefore X(^, q) vanishes. 

 The corresponding surfaces have been discussed previously. 



McGill University, 

 Montreal. 



