Egan — Lineal' Complex, and a certain class of Twisted Curves. 35 



II. — Geneealities. 



1. If the poiut {x) and the plane (a) are specified by the coordinates 

 Xi, oi, (i = 1, 2, 3, 4), the Plucker coordinates of the Une {xy) or (afi) are the 

 two sets of twelve numbers 



Pv = 



ViVJ 



Aft- 



A linear complex is defined by 



Sa^;-^;,- = 0, or 2ayOTH=0, (1) 



the ciij being constants such that «y + aji = 0. 



2. The properties of curves whose tangents belong to a linear complex 

 have been studied by Appell (Annales de I'Ecole Normale, 1876, pp. 245 sq.) ; 

 Picard {ibid., 1877) ; Koenigs (Annales de la Faculte des Sciences de Toulouse, 

 1887) ; Lie (v. " Geometrie der Beriihrungstransformationen," ch. 6, § 4, where 

 other references are given). 



A short account of these curves is given by Jessop (" Treatise on the Line 

 Complex," pp. 47-50). Other references are given in the course of this paper. 



3. M. Appell has shown that the osculating plane at a point on such a 

 curve is the polar plane of the point with respect to the complex. From this 

 it follows that the plane-coordinates of an element are proportional to linear 

 functions of the point-coordinates. In fact, taking («) as the point on the 

 curve and {ij) a variable point, the polar plane of {x) with respect to the 

 complex (1) is 



S.aijiyiccj - ijfXi) = 0. 



Hence the coordinates of the osculating plane (a) at {x) satisfy the 

 equations 



(H\3h + 031X3 + anXj , _ ciiiXi + 032X3 + ttiiXi _ 



where Oij = - aji. 



It follows that the class of the curve is equal to its degree ; and further, 

 that if the curve has a singularity at a point, expressed in a given form ui 

 terms of the a-,-, it wdl have at the same point the singularity represented by 

 similar expressions in terms of the at, 



III. — Extension of a Theorem of Picard. 



4. There are in general on a curve in space a certain number of points at 

 which the osculating plane has stationary (four-point) contact with the curve. 

 The tangent line at such a point may meet the curve in two or in three conse- 

 cutive points. M. Picard has shown (loc. cit.) that if the curve belongs (by its 



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