446 HITCHCOCK. 



we shall have (233) identically equal to Vcjipdp aside from the scalar 

 factor -S;8ii3o|33. 



Returning to (225), if ^21 = 0, the vector can have no axis except 

 /3i. If we take 



y = , 2 = , (235) 



c c 



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

 term p(yxo + zxz), the resulting quadratic vector may be written 



^i{BnX2^-\-BnX2X3+BuX3^—yxiX2 — zx 1X3) -{-^oiB 22X2X3 — zx2Xz—yX'?) 



-^z{yx2Xz+zxi). (236) 

 If we now take 



# = {X2B22-^X3B2Z) l%^3-{XiB22+X2BnC) l%Pi ] 



+ McB22-B2z)+X2{c'Bn-cB,2)-cBnX3]V^i^2, > (237) 



dp = - X3V^3^1+(X2 + CX3)1%^2 } 



we shall have (236) identically equal to V(})pdp aside from the scalar 

 factor cS|8i/32|83. 



This completes the factorization into V(t>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 jSi is an 

 ordinary, or an inflectional, element. 



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

 ^iu are diplanar. For if we take 



<^p = X2Vpir — UXiV-K^i, 1 



6p = XiVtt^i - X2V(3ip + .r3 r~ Vpw + cjiVtM I (238) 



we shall have (185) identically equal to V4)pdp aside from the factor 

 Sirp^i. 



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

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

 constant a cannot be zero. If we now subtract the term op.r2, the 

 resulting quadratic vector may be written 



a;i.T2(7r — a/Si) + .T2.r3(<7i7r - ajSs) + Xs^gir, (239) 



