376 HITCHCOCK. 



II. Thefnumber of distinct axes is less than seven, some being 

 multiple. 



III. The number of axes is infinite. It will be convenient to speak 

 of this as the reducible type. 



3. If the number of axes is infinite, there exists either a plane or a 

 quadric cone (according as the common factor of (3) is linear or quad- 

 ratic), such that any direction of p in the plane or the cone gives an 

 axis of the vector. 



Conversely, that a quadratic vector have an infinite number of axes, 

 it is sufficient that it have six distinct axes on a quadric cone, proper 

 or degenerate. For suppose the number of axes finite. Let the 

 coordinate system be so taken that no axis lies in the plane 2=0. 

 If the first two equations (3) have six solutions on a quadric, the 

 remaining three are linearly related.^ That is, the seventh axis of the 

 vector lies in the same plane with the two lines of intersection of 

 z = and Z = 0, contrary to hypothesis. Hence the number of 

 axes cannot be finite. 



4. The axes are not altered by adding to the vector a term of the 

 form tp, where t is a scalar. This is geometrically obvious, for if p 

 and jP(p) are parallel, extending F(p) in the direction p will not disturb 

 the parallelism. Analytically, t must, in the present case, be a linear 

 form in .r, y, z. If we write 



t = px -^ qy -i- rz (4) 



p, q, and r being constants, the addition of tp to F(p) is the same as 

 putting X + x{px -\- qy -^ rz) for X, with similar expressions put for 

 Y and Z. If we make these substitutions in (3), the equations are 

 invariant.^ 



It is equally obvious that the axes of a vector are not altered if we 

 multiply it by any non-vanishing scalar constant. 



2 Salmon, Higher plane curves, Art. 24, 1st Ed. 



3 This is directly seen if we introduce vector multipHcation, (see part II, 

 below), for equations (3) are equivalent to the vector equation VpFp = 0. 

 We may change Fp into Fp + tp at will, since p^ is a scalar. 



The results of Arts. 2-4 are evidently applicable with shght modification 

 to homogeneous vectors of any degree, the components being polynomials. 



