QUADRATIC VECTORS. 415 



from zero. It may well happen that the triple axis, the double axis, 

 and one of the single axes are in the same plane. 



This special case, while it cannot be written in the form (113), is 

 easily studied by the methods of Art. (19). Assume 0i for the 

 triple axis, /32 for the double axis, 183 and ^4 for the single axes. 

 We cannot have both /Ss and 184 coplanar with /3i and B2. Suppose Si 

 coplanar with the two multiple axes. Then (123) cannot vanish if 

 Fp is irreducible. Let the vector be thrown into the form (61). The 

 conditions that Si be coplanar with I3i and ^2 but distinct are given 

 by (93). We have thei-efore case (c) of Art. 19. Moreover the condi- 

 tion that 182 be a double axis becomes Asi = 0. This, together with 

 (93) and (133) yield, as necessary and sufficient for the present case 



A31 = A32 = A33 = A2ZA12 — A13A22 — 0; An and ^21 not 

 both zero; with A22, A23, An, and Au all different from zero. 



(146) 



We thus have ^i = Ai3J3i + A2302, and may write 



X2X3(An0i + ^21/32) + (/3i + 6/32) (Anxsxi + A13X1X2), (147) 



(where 6 is a constant different from zero), as a normal form for a 

 quadratic vector having a triple, a double, and a single axis in the 

 same plane, and one other axis. As perhaps the simplest example of 

 this case we may put ^i = i, j32 = j, 03 = k, An = 0, and the remain- 

 ing constants equal to unity. The equation (147) then becomes 



i{zx + xy) + j{yz -\- zx -{- xy) (148) 



The cones (3) become 



(a) zx{y + 2) = 0, ) 



(b) z{yz +zx-\-xy) = 0, > (149) 



(c) x{zx -^ xy — y^) = 0. ) 



The axes are (1, 0, 0), (0, 1, 0), (0, 0, 1) and (1, 1, 0). The quadrics 

 iyz + zx + xy) = and {zx -{- xy — y^) = pass through the vector 

 (1, 0, 0) and have the same tangent plane there, viz. y -\- z = 0. 

 Therefore (c) meets both (a) and (b) three times in the element i. 

 This element is a double line for both (a) and (b). Hence it is a 

 triple axis. At j = 02, or (0, 1, 0), we have also a double line for (a) 

 and for (b) ; giving a double axis. The axes i, j, and i + j are co- 



