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 



/37(452) = /35(247) + 13,(^57) 



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

 stant), 



/37(23p) (45p) + (aii36 + bi^^) (52p)2, 



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

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

 may also write, identically, 



p(237) = ^2(37p) + i33(72p) + ^7(23p), 



which gives us our quadratic vector as 



- /32(37p) (45p) - ^\72p) (45p) + (ai^s + b^2) (52p)2 (209) 



The axes /So, 185, and 187 are coplanar, so that we may write 



iSs = m^2 + n^7. 



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



If we now put xi = (37p), 2:2 = (45p), and xs = (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 Vcppdp + 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 j8i 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 /3i, 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<l>(£p. 



If, however, we subtract from (165) the term pAuXs, the resulting 

 quadratic vector may be written 



