382 Proceedings of the Royallrish Academy. 



unchanged. Or directly, changing the origin, the expression for the 

 curre of order n (odd or even) becomes, when the origin is at po, 



_ ("o - guPo> «i - Cipw •••«»- c-npo) {^yf _ 

 («o«i . . . <?„) {xyf 

 The invariant 



(ctfl - ttoPo) «» - « («! - «iPo) S'i + • • • + (-)" (a„ - fi?„Po) ^i^o 



may, when n is even, by choice of p^ be made to vanish ; but when n 

 is odd, it is independent of po, and cannot be made to vanish by chang* 

 ing the origin. 



Thus, for curves of even order, the pole of the points in SXp - 1 is, 

 if the origin is taken at the pole of the points at infinity, 



zy = - 1-^6 (X), 



and the locus of the poles of the system of parallel planes SXp = ^ is 

 the line 



p = - ir' 1-^6 {\). 



Let the quadric SpOp = const, be constructed, then the locus of the 

 poles of points in a system of planes at right angles to a given radius 

 vector to the quadric is the central perpendicular to the corresponding 

 tangent plane. 



In this case, also, the locus of poles of the system of planes 



S {tk + Sfji) p = t + s, 



which pass through a given line, is the line 



__6{t\ + sp.) 

 ^~~ I{t + s) ' 



The pole of the plane S\p = 1 being given by t? = - IWX, will not 

 lie in the plane (as in the case of curves of odd order) unless 



skTs = - i-'sxex = - iSzye~'zT = i . 



Thus the locus of poles which lie in the corresponding planes is the 

 quadric surface ISp6~^p = - 1. The tangent plane at ot to this surface 

 is 



ISpO~''z} = - 1 or SpX =1, 



and the quadric is also the envelope of the planes which contain their 



