QUADRATIC VECTORS. 427 



We may therefore write 



r = «^i + gir, (184) 



and Fp becomes 



7r(a:ia;2 + gxz^ + S'iar2a:3) + 11^1X2X3 + fjiX2-, (185) 



the most general quadratic vector having j8i a quadruple axis, a deter- 

 minate tangent plane to the cones (3) at /Si, and /3i not an inflectional 

 element of these cones. It is evident that if tt lies in the tangent plane 

 it is an axis and, conversely, if an axis other than j3i exists in the 

 tangent plane, this axis coincides with tt. Let us suppose, as a very 

 special case, that tt is a double or triple axis, which we may take 

 as /Ss, (since /Ss is any vector in the tangent plane). The tangent 

 plane to (3) at /Ss must be distinct from the tangent plane at /3i since 

 cubics have no double tangent, and Fp is assumed not reducible. 

 Applying these conditions according to the methods already exem- 

 plified, we have, as the polar vector of VpFp at ^3, 



V03{u^ix'2 - gxi'^i - gX2'^2] 



which obviously cannot determine, by its vanishing, a unique tangent 

 plane at 183. Thus the supposed case is impossible; and tt cannot 

 be a double axis. 



This possibility disposed of, the vector (185) must always have at 

 least two axes not in the plane (31p) = 0. Let 182 be one of these. 

 The vector becomes 



T(a:ia;2 + gx3^ + giXiXs) + u^iX2X3 + a^'zX2^, (186) 



where a is a scalar constant. This form, therefore, is equally general 

 with (185), j8i being of order not higher than four. 



If we consider the vector tt, and the scalars u and a, as determined 

 by assigned axes 184 and 185, 1 shall now show that the determination is 

 uniquely possible, aside, obviously, from a scalar factor. If we write 



a:ia;2 + gxi^ + gr1.r2.r3 = Cp 



the conditions that F0i and Fj3;, shall be parallel, respectively, to 0i, 

 and /Ss may be written 



F^4[C47r + w/3i(314) (124) + 0^.(314)2] = . . 



V^,[C,T + vMSl5) (125) + 0/32(315)2] = 0, ^''''^ 



