380 Proceedings of the Royal Irish Academy. 



Thus the pole of tlie n coplanar points is given by 



'SA(/<^)„ -(//)„' 



In particular, as the poiats at infinity are determined by f{xy) - 0, 

 the pole of the points at infinity is 



^ (//)V 



7. Distinction between curves of odd and even order. WTien n is odd, the 

 pole of n copJaiiar points lies in their plane. Thehctis of poles of 

 paralld planes is a right line parallel to a fixed direction. 



It is now convenient to consider separately curves of odd and curves 

 of even order. Taking in the first place curves of odd order, (//)„ = 0, 



(<;5)/;„ = a,<7„ -o^a^-n (aiff„_i - a^-xa^ -i- &c. = - (/^)„, 

 and 



(^<fiSX(f)]n = a(,<S'Xa„ - a„SXan — «(ai5Aa^i - a„_iiSAai) + &c. 



= V. V{a^a„ - naiaL„.i + ^n {n - 1) ootv-s - &c.) X. 



Thus the pole of the points in the plane S\p = 1 is given by the 

 equation 



TXk-^l 



in which 



t = ttofif, - a„ffo - n (aiff^i - a„-iffi) ^- &C. 

 and 



*« = ^("ott,. - WaiOT.-! + &C.)- 



In particular, the pole of the points at infinity is situated at the point 

 at infinity on the line parallel to t, since {ff]n = 0- 



Again, the pole of n coplanar points lies in their plane ; for 



oAt 



Further, the locus of the poles of a system of parallel planes SXp = t, 

 is found by replacing A by ^^A, and is the right line 



P 



Ski 



These locus lines are all parallel to the vector i ; that is, they all pass 



