JoLY— Vector Exjjressions for Curves: 385 



As an example, take the case of the general twisted cubic 



Here 



t = ttQ^s - o.->fl,Q — 3 [o-xa-i — a^(l\), 

 and 



K = Faottg — 3 Vo-xa-i. 



The origin may be supposed to be taken on the curve, so that ag = 0. 

 Then if i = 0, a,,, ai, and a^ are coplanar, and the curve must be plane ; 

 if 8iK = 0, 8a^p.xa.<i = 0, and again the curve is plane ; if k = 0, ai is 

 parallel to a2, and here again the curve is plane. 



11. But when n is even, it is scalar, and its vanishing determines the 

 director sphere of the quadric of Art. 8. 



"When n is even, this invariant is a scalar, and its half is 



^ySoi8„ - nSPxIB^^^x + &c., 



in which the last coefficient must be halved. 

 The binary quantic 



p {a^ax . . . a„){xyY - {a^ax . . . a„){xy)" 



affords the invariant 



S{a^p - aQ){a„p - a„) - nS{axp - ax){a„.xp - a„_i) + &c. 



= p^I — 2SpL + {Sa^an - nSaxa^-i + &c.), 



using the notation of the 8th Article. 



If this invariant vanishes, the vector p must terminate on a sphere 

 whose centre is the point /~^t - the pole of the points at infinity. For 

 this point as origin, the equation of the sphere is 



p^/+ Sa^a^ - nSaxa„_x + &C. = 0. 



Consider an ellipse referred to its centre as origin with a and /3 for its 

 axes major and minor ; the equation of the ellipse is 



p = acosu + B Binu= —^- ^- — — , it t = tan ht. 



For this curve, /= 1, and the equation of the sphere is 



p~ = a- + (3-; 



