Egan — Linear Complex, and a certain class of Twisted Curves. 47 



Value of IV on a Ctcrve of a Linear Complex. 



26. In this case, let f = be the polar plane of the origin with respect to 

 the complex. ? (t) is a function of t of order n, having the poles of aj, y, z 

 as poles, and having as zeros the poles of Ajuv : in effect, the points whose 

 osculating planes pass through the origin lie in the plane ^ = 0. Hence 



f = hiv = ki Zja^, 

 where k and k^ are constants. 



It will be noticed that, for a general P-curve, we have only been able to 



see that w is a function having the same poles as », y, z, and having given 



zeros : for a complex curve it is a linear function of x, y, z. The equation 



? = hZjryi 



is a particular case of a theorem due to Professor McWeeney, which holds for 

 any curve of a linear complex, algebraic or not [vii, equation (19)]. 



VI. — Sufficient Conditions that an Algebraic Curve should belong 



TO A Linear Complex. 



27. Every curve belonging to a linear complex is a P-curve, as we have 

 seen (section iii), and is characterized by the identity (22) = 0. In this 

 section we investigate the further condition required, in order that an 

 algebraic P-curve should belong to a linear complex. (A geometrical 

 interpretation of this condition will be given in section vii.) 



The following lemma is required : — 



Let A, B, C, D be four points in space forming a tetrahedron of non-zero 

 volume, and let P = be the equation of a quadric. If R passes through 

 A, B, and 0, and if the tetrahedron ABCB is self -conjugate with respect to it, 

 P must reduce to the square of the plane ABC. 



This is easily proved, either analytically or geometrically. 



28. Consider an algebraic P-curve represented in homogeneous coordinates 

 by two sets of functions Xi and o;. We have, in the notation of section rv, 



(??m) + {nm) = 0, if m + n < 6. 



We define sixteen functions Wy of t by the sixteen equations 



{d = d/dt, -i = 1, 2, 3, 4, m = 0, 1, 2, 3). I 



If the functions [x) and (a) are rational either in the i^-plane or on a 

 liiemaun surface, so likewise will be the ay. 



