L 374 ] 



IX. 



VECTOR EXPRESSIOIs^S FOR CURVES. Br CHARLES JASPER 

 JOLT, M.A., F.T.C.D. 



Paet I. — Unicursal Cuetes. 



[Eead December 14, 1896.] 



1 . Vector equation for a unicursal curve of the n*^ degree. 



Let ao, tti, 02 ... a„ be any given and constant vectors, and «o, a^, a-i 

 . . . a„ be given scalars ; tben the general vector expression for a 

 unicursal curve of order n may be written in the form 



a^^" + naxV^^ + . . , + a„ 



ft = , 



«o^" + "ffi'^"'^ + . . . + a^ 



in which Hs a variable scalar parameter, and p is the vector to a point 

 on the curve. For, consider the number of points on this curve locus 

 which lie in an arbitrary plane 8\p - 1. This number is equal to the 

 order of the scalar binary equation in t, which is obtained by substi- 

 tuting in the equation of the plane the vector to a point on the curve 

 expressed in terms of the parameter t. Arranged in powers of t, the 

 result is 



(SXao - flo) ^" + « (-SXai - ffi) f "-1 + \n {n-\) {SXoo - a^) t"-- + &c. = 0, 



which gives n values of the parameter, or determines n points in the 

 plane, where t" is the highest power of t in the numerator or in the 

 denominator of the given vector expression. 



2. Vector equation for a tangent line and osculating plane. 



It is sometimes convenient to suppose the numerator and the deno- 

 minator of the vector expression to be rendered homogeneous in x and y, 

 where ty = x. In this case 



^ (apai • • • «, . )(^y)" ^ <^(^y) 



and here ff){xg) is a binary quantic with vector coefficients, and/(:ry) 

 a binary quantic with scalar coefficients, and in general both quantics 



