404 HITCHCOCK. 



16. From the foregoing results on double axes it appears that 

 any quadratic vector having less than seven distinct axes but no triple 

 axis may be written in one of the three following normal forms, — 



(a) If there is but one double axis, wTite Q for P in (31). The 

 tangent plane to (3) at the double axis can pass, at most, through one 

 of the single axes. Any other single axis may be taken as a zero of 

 Qip). The remaining four single axes cannot all lie in the same plane. 

 Any three which are diplanar may be numbered 4, .5, and 6, to cor- 

 respond with (31). 



(b) If there are just two double axes, write Q for P in (88) . Choose 

 either double axis to be a zero of Q. The tangent plane to (3) at this 

 double axis can pass at most through one of the single axes. Choose 

 either of the other single axes .to be the second zero of Qifi). At the 

 double axis not already taken, the tangent plane to (3) can pass, at 

 most, through one of the two remaining single axes. The one through 

 which it passes, (if either), must be taken as /St. Otherwise, the choice 

 of numbering is arbitrary. 



(c) If there are three double axes, (91) is always possible. 



17. It remains to consider vectors with less than six distinct axes, 

 one of which is of multiplicity three or greater. For triple axes we 

 have at our disposal a variety of methods, analogous to those used 

 above for double axes. We may, for example, assign a relation to 

 connect the constants A of (91) in order that /3i may be a triple axis. 



It may well happen that all four elements of the determinant (63) 

 are zero. If so, the cones (3) all ha^•e a doul)le line at j8i, which is 

 consequently a quadruple axis. I shall assume, for the present, that 

 such is not the case. 



This possibility excluded, the cubic cone 



S\pFp = 0, (99) 



where X is a constant vector, (so that S\pFp is linearly related to the 

 left members of (3)), wall not have a double line at /3i for all values of 

 X. To say that /Si is a triple axis of Fp is therefore equivalent to 

 saying that all cones obtained by varying X, exclusive of those with 

 double lines at /3i, will osculate along |3i. Or again, all these cones 

 give the same curvature for normal sections at any point on an ele- 

 ment i3i. The most direct way to express this condition is to say that 

 dp is independent of X, where v is a vector of unit length normal to 

 the cones at a point on an element /3i, and dp is any vector in the 



