JoLY — Vector Expressions for Curves. 383 



poles. More generally, the pole of the plane SXp = t with respect to 

 this quadric is the point 



TO = - t-'i-'dx, 



and this is precisely the point which is the pole of the points in 

 which the twisted curve meets the plane with respect to that twisted 

 curve of even order. 



9. Standard vector expression for curves of even ordsr. 



Hemembering the definition of {(pSXcf))^ = 26 (A.) in Art. 6, it follows 

 that, if A and /a satisfy the relation SKOfx = 0, the (12)" invariants 

 derived from the two scalar quantics SXcji [xy) and aS^^ {xy) vanishes. 

 Hence, if X, fj., and v satisfy 



SixBv = Svex = sxefx, 



and if 



cfi {xy) SXfxv = VfjLv A {x, y) + VvX B {x, y) + VXjx C{x, y), 



where 



A {x, y) = SX<j> {x, y), &e., 



the (12)" invariants (-5C)„, {CA)„, and {AB\ of the scalar binaries all 

 vanish. If, further, SXOX = -I, the (12)" invariants 



{AA\ = {££)„ = {CC%^-2I. 

 If, again, 



/a = - ex, 1/3 = - Ofx, and ly = - Ov, 



it follows at once, since 



SXa = 1, and Sfxa = Sva - 0, that aSX/xv - Vfxv, &.Q., 



and that a, /5, and y are conjugate radii of the quadric surface 



ISpe-'p = -l. 



Hence the vector equation of the curve of even order may be written 

 in the form 



_ aA {xy) + IBB {xy) + yC{xy) 



where 



{AA),, = {BB),, = ( CC\ = - (//)„ ^ - 2/. 

 and 



(i?C), = (C^),. = (^5), = 0; 



