QUADRATIC VECTORS. 441 



possible to throw this vector into the form (201), because we have not 

 three diplanar axes. The form (203) is still possible. By the iden- 

 tity 



^7(452) = 185(247) + ^2(457) 



the vector (208) may be written, (neglecting a multiplicative con- 

 stant), 



^7(23p) (45p) + (au85 + 61^2) (52p)2, 



the constants ai and 61 being arbitrary but not zero. By subtraction 

 of the vector term p (237) (45p) the axis 187 is made a zero. We 

 may also write, identically, 



p(237) = /32(37p) + ^3(72p) + ^7(23p), 



which gives us our quadratic vector as 



- |82(37p) (45p) - /33(72p) (45p) + (a^, + ^uSs) (52p)2 (209) 

 The axes ^2, ^5, and ^j are coplanar, so that we may write 



/35 = m^2 + n^7. 



We therefore have (52p) = n(72p). 



If we now put xi = (37p), 22 = (45p), and X3 = (72p), we shall have 

 thrown (209) into the form (203). It is evident that the vector coeffi- 

 cients are not coplanar and the factorization proceeds as in Art. 34. 



All quadratic vectors, therefore, having a triple axis, but no axis 

 of higher order, can be thrown into the form V^f^p -\- pS8p. 



37. Taking, finally, vectors having axes of order higher than the 

 third, it has already been shown that these differ in their properties 

 according as the cones (3) have, all of them, a double element at the 

 multiple axis; or have a uniquely determined tangent plane there. 



If /3i is an axis of the fourth or higher order, and if we are dealing 

 with the case of vanishing polar vector, so that the tangent plane is 

 not unique or determinate; and if, also, there exist two other axes 

 diplanar with j8i, the form (201) is not possible. For we can throw at 

 once into the form (165), obviously a vector of constant plane, in- 

 capable, therefore, of being factored as V(f>pdp. 



If, however, we subtract from (165) the term p A 12X3, the resulting 

 quadratic vector may be written 



