460 PROCEEDINGS OF THE AMERICAN ACADEMY. 



w at the vertex, and equal in magnitude to the reciprocal of the 

 interval R along the perpendicular drawn from the point Q to that 

 tangent produced (Figure 21). On account of analogies which will 

 soon become apparent we shall call this vector function the extended 

 vector potential of the given (5)-curve.^^ We shall write 



P = ^- (66) 



We shall next consider the 2-vector field 



P = Oxp = (0;^)xw + ;^(Oxw). (67) 



We shall consider the evaluation of O^P in two steps. First we shall 

 assume that the original (5)-curve is a straight line. In this case w 

 is constant and O^W = 0. If we arbitrarily take k4 along w, we 

 may write 



/\ 1 _ 1 , a 1 1 



for it is clear that a displacement parallel to w does not change R. 

 It is evident that R becomes a radius vector in the 3-space perpendicu- 

 lar to w. If n represents a unit vector from the point Q normal to w, 

 that is, in the direction in which R was measured, then by the well 

 known formula, VR~^ = n/R^. Hence 



^R i?2 



And hence _ y. nxw ,„„, 



P = OxP = -^- (68) 



The determination of <^'p follows in precisely the same way; 

 in each of the above formulas the symbol of inner multiplication will 

 replace that of outer multiphcation, and we find that 



0'V=^= 0, (69) 



for n is perpendicular to w. 



Of all the geometrical vector fields which might have been con- 

 structed from a given (5)-curve, we shall show later that those which 

 we have just derived are the most fundamental (footnote § 44). The 



39 The vector fields produced at a point by two or more («)-curves may be 

 regarded as additive. The locus of all possible singular lines 1 drawn (as 

 in Fig. 21) from (a)-curves to a given point is the backward hypercone of which 

 that point is the apex. 



