JoLY— Vector Expressions for CurDes. i 381 



throuah the pole of the points at infinity. Again^ the locus of poles 

 for the system of planes -,,.-.., 



Sp {t\ + sfji) = t + s, 

 which pass through a fixed line, i&the line 



V{t\ + S/Ji) K + {f + S) L 



^ [tX -» Sfj.) t 



8. When n is even, the pole of the points in a plane is the same as the 

 pole of the plane ivith respect to afixedquadric. 



In the second place, for curves of even order, 

 (//)» = 2 (ffoff.„ - na^ar,.^ + &c.) = 2/, suppose ; 



■ . C^/)'. = (/</>)» = «o«^« + "«^o - ^.* {p-\(^n-\ + a«-i«'i) + &c. = 2i ; . ; - - . - 

 and '^ J 



(_c}iSX(f>)n = ag«SXa„ + a„SX.aQ — w(ai<SAa„_i + a„_iSX.ai) + &c = 2dX, 



where ^ ia a seli-con jugate linear vector function defined by the equa- 

 tion just given. And now the pole of the points in SXp - 1 is, by 

 Art. 6, 



ex-i 

 ""-s^^r -^ 



and the pole of the points at infinity is 



Just as in the last article, the locus of poles of the system of parallel 

 planes SXp = t is 



ex-ti 

 ^~ sxi-tr 



and, as X varies, all these locus lines pass through the point w^, the 

 pole of the points at infinity. 



For curves of even order, it is possible, by taking the origin at the. 

 point OTg to render the vector l zero — at least, when / is not zero. 

 This may be verified directly by changing the origin, and then form- 

 ing the invariant t ; but it is otherwise obvious that this is the case, 

 since /(a-y) is unaltered by a change of oiigin, and therefore /remains- 



