JoLY — Vector Expressions for Curves. 879 



6. For any unicursal curve the vanishing of a corresponding invariant 

 determines a definite point, the ^'pole" of n given points.^ 



Take the general twisted curve of the w"* degree, and suppose the 

 vector to a point on it to be given by 



^ ^ {^, y) ^ (ao«i • ' • (^n){xyy ^ 

 ^ f (^> y) (<^o^i • • • an){xyY " 

 Take also a scalar binary of the w"* degree 



whose vanishing determines n definite points on the curve, and con- 

 sider the (12)" invariant formed between this and 



ra/(a;y) - ^{xy\ 



It may be written in the form 



^(/i^)„-(c^i^)„, 



where {fF\ is the (12)" invariant formed between the scalar binaries 

 f{xy) and F{xtj), and {<i>F)„, that formed between fj>{xy) and F{xy). 

 If this invariant vanishes, a definite poiut is determined by the 

 vector 



and, for the sake of brevity, this point may be called the pole of the 

 n points on the given curve determined by F{xy) = 0. 



In particular, when these n points lie in a given plane <SAp= 1, the 

 binary F{xy) = is replaced by the binary 



SX4>{xy)-f(xy) = 0. 

 In this case 



{fF\ = SX{fc}>)^-(ff)„, and (<^F)„ = (<^-SA<^)„ - (<^/).., 

 where 



(/<A)» = «'o«« - '>^aiO-n-i + &C. + (-)"ff„ao = {-T{<kf)n 

 and 



(^(f>S\(ji)n = ao<SAa„ — naiSXa^-i + &C. + [—Ya^SXa^f 

 and 



(//)» = «'o«'n - W«l«n-1 + &C. + (-)X«0 = (-)"(//)«• 



^ The use of the -word "pole " in this extended sense is due to Professor "W. K. 

 Clifford, who has given the theorems of Arts. 7 and 8 for curves of the >;"> degree 

 in «- dimensional space. " Classification of Loci," collected works, p. 312. 



