QUADRATIC VECTORS. 421 



uniquely determined tangent plane at the triple axis. If, on the 

 other hand, the axis is quadruple, we may distinguish two cases, — 



1°. The polar vector of VpFp, viz. Vp'Fp + Vp^(p, p), vanishes 

 identically when the axis, (as /3i), is written for p; this is equivalent 

 to the condition that all cones (3) have the axis for a double element, — 

 which therefore counts for four intersections. 



2°. The polar vector of VpFp does not vanish identically when 

 the axis is written for p. 



27. Case 1" is easily disposed of, for the four elements of the 

 determinant (63) vanish. Choose for ^<> and 183 any two of the three 

 single axes such that (123) does not vanish. The vector Fp may now 

 be written, (after reducing j8i, /So, and jSs to zeros by a term pSbp), 



^\{AuX'iXi + AnXzXi + ^i3a;ia;2) + 032^21 + ^zA-s^x-iZz (165) 



The remaining axis is 



^^AvlA2lA,l - [AnAzi + .'li3^2i - A^iAzi) (jSoA^i + ^3^31), (166) 



as may be directly verified. Conditions for coplanarity of this axis 

 with a pair of the other axes are respectively, An = 0; yl2i = 0; 

 and ^431 = 0. In these cases the fourth axis becomes, respectively, 

 1^2 A2\ + 183^31; 03] and 182. Thus we cannot have two of the three 

 single axes, (distinct), coplanar with the multiple axis, but the three 

 may themselves be coplanar. In fact the conditions A21 = or 

 ylsi = agree with those already foimd that /S2 or 183, respectively, 

 may be double axes, for we may not have An = nor yet Au = if 

 the vector (165) is to be irreducible. 



Without assuming the existence of three diplanar axes, we may 

 throw the given vector into the form (92). If 182 is a double axis we 

 have as above Azi = 0, for, by inspection of the polar vector (96), 

 this condition is independent of the terms in Xz^. The vector now 

 takes the form 



^\{AnX2Xz + AuXzXl + A1ZX1X2 + BiXz^) + j82(^2ia;2.T3 + B2X3^) 



+ ^zBzXz^ (167) 



Let us now suppose 183, till this time arbitrary, to lie in the tangent 

 plane to the cones (3) at 182. This is the same as requiring the tangent 



