420 HITCHCOCK. 



The existence of triple axes at i and at j may be verified as in the 

 preceding example. 



24. We may bring the discussion of triple axes into close relation 

 with the theory of quadratic point-transformations by writing 



R{p) = 'i.T2^+ j{xiX2 — axz"^) + kxiXs (164) 



That R(p) has jSi for a triple axis independently of the value of the 

 constant a may be verified by either of these three methods; by the 

 rule of Art. 18; by forming the cones (3) ; or by noting that the scalar 

 components of R{p), as quadric cones, meet three times in 0i, so that 

 R has /3i for a triple zero. If, therefore, we write R for P in the general 

 normal form (31) we shall have a general form for a quadratic vector 

 with a triple axis and four other distinct axes. Similarly, by writing 

 R for P in (88), we shall have one triple and one double axis. Various 

 other forms with R in combination with preceding methods are 

 evidently possible. 



The vector R(p) becomes reducible when /3i is an element of inflection 

 for the cones (3), as may be easily shown by applying the rule of Art. 

 18; a result we might anticipate, because three proper quadric cones 

 cannot have three coincident elements in a plane. If the vector 

 F (p) has /3i a triple axis, with jSi an element of inflection for (3), we 

 may use the preceding general methods. Jg 



25. As a comprehensive set of normal forms for irreducible quad- 

 ratic vectors with a triple axis, we may select the four following, — 



(a) If there is no double axis, (136) is always possible. For the only 

 restrictions, other than irreducibility, are that the determinants (123) 

 and (156) shall be different from zero. But the axes /So, /Ss, /Ss, and jSe 

 are the four single axes. Since no four distinct axes are coplanar we 

 may evidently so choose the numbers that these restrictions hold. 



(b) If there is one double axis, use (143) provided neither single 

 axis, /Ss nor /Se, is coplanar with the triple axis 13 1, and the double axis 

 182. If so, use (147). The generality of these forms has already been 

 proved. 



(c) If there are two double axes, use (144). 



(d) If there are two triple axes, use (159). 



26. When an axis is triple, but not of higher multiplicity, it has 

 already been pointed out that at least one of the cones (3) has a 



