446 HITCHCOCK. 



we shall have (233) identically equal to V(t)pdp aside from the scalar 

 factor S/3i|82i83. 



Returning to (225), U B^i = 0, the vector can have no axis except 

 /3i. If we take 



y=-^,, = -?!!, (235) 



c c 



where c is any non-vanishing scalar, and subtract from (225) the 

 term p{yx2 + 2x3), the resulting quadratic vector may be written 



l3i{BnX2^-\- BnXoXsi- EnXz"^ —yxiX2 — ZX1X3) -\-^2{B22X2X3 — zx2Xz—yxi) 



-Uyx2X,+zxi). (236) 

 If we now take 



# = (.T2522+.T3523) F/32/33 - (x i522+X2Buc) F,33^i 



+ [xx{cB22-B2z)+X2{c'Bn-cBn)-cBnXz]V^ifi2, } (237) 



Op = - X3V^z^i-\-{x2-\-CXs)V^i02 



we shall have (236) identically equal to V(f)p9p aside from the scalar 

 factor cS/3i/32/33. 



This completes the factorization into V4>pdp for all possible cases 

 where all cones (3) have /3i a double element. 



If the cones (3) have /3i an axis of the fourth or higher order, with a 

 uniquely determined tangent plane, it has been shown that we may 

 throw into one of the two forms (185) or (195), according as /3i is an 

 ordinary, or an inflectional, element. 



The sub-case (185) may be at once factored if the vectors tt, ju, and 

 ^iii are diplanar. For if we take 



<f>p = XoVpw — nxsVir^i, 



dp = ariFTT/Si - X2l%fJi + Xz (^ F/xx + ^iFx/3i j { (238) 



we shall have (185) identically equal to Vcppdp aside from the factor 

 Sirp^i. 



If w = 0, this method fails. If there exists an axis not in the plane 

 a-o = we may suppose, as already shown, that jjl = a^2- Then the 

 constant a cannot be zero. If we now subtract the term apx2, the 

 resulting quadratic vector may be written 



a;ia;2(7r — OjSi) + X2X3{gnr — a^s) + -I's^gTr, (239) 



