12 Frederick C. Ferry. 



consequently no curve which is a complete intersection of 

 S and 2 can have an odd number of points on the double 

 director. 



Since v in œ . cp is replaced only in the combinations 

 ÀV and [JLV, the degree of any term of œ . cp in \ and \x 

 together must be at least as great as the degree of that 

 term in v; thus if n^ be the degree in v of that term 

 of o) in which v occurs to the highest power, then must 

 P — ci + ïî' — ^t > n^ -}- q; and, in particular, if ^p <; q. the 

 degree of any term of a> in X and jjt together must exceed 

 the degree of that term in v by at least 2q — p, and especially 

 is this true of that term in which v occurs to the highest 

 power. 



The terms of œ . cp that do not contain v, if there be 

 any such, must contain only even powers of X and even 

 powers of \).; in such a case v is not a factor of to; and 

 hence either the aggregate of terms of m that do not contain v 

 must contain only even powers of X and even powers of [x 

 and likewise the terms of U^ contain only even powers of 

 each variable, or the aggregate of terms of oi that do not 

 contain v must be of the form 00^ . U' where (o^ is a poly- 

 nomial in X"' and jx"' and U'^ is what U^ becomes when \i 

 is replaced by — [x; or if U ^ V^^ . V.,^ then need U' be 

 only what V ^t becomes when [x is replaced by — \i, since 

 Vo^ is already a polynomial in X" and «x (p . 9); then will 

 (D(,. U^ . Up or («Q. Yp_2f Vgf Vp_2t become a f (X", [x^) and the 

 substitution be possible at once. In this case to every point 

 in the one sheet of 2 on the double director will correspond 

 the point in the other sheet at the same point as a part of 

 the entire locus of intersection, though one or both points 

 may be a part of the residual intersection in some cases. 

 Thus oJq gives pairs of points on the double director which 



