QUADRATIC VECTORS. 445 



-XiXoB2lfi,-\-X2X,(B,28i-^B22P2-B2^3)+Xi'{Bn0l-\-B23^2), (227) 



the axis 02 being made a zero, and, consequently, the terra in Xo"^ dis- 

 appearing. (227) is in the form (203) and can be factored into V4>pdp 

 if the vector coefficients are diplanar. We may not have Bn = 0, 

 for if so the vector is reducible. Thus the condition for coplanarity 

 is that B2Z shall vanish. If so, the quadratic vector (225) becomes 



HBiiX2Xz + Bnxz') 4- HBiix.^ + B22X2XZ) ; (228) 



and if 0n is not zero we may subtract the vector term 



p|5.,x,+ ^a:3| (229) 



writing the resulting quadratic vector as 



0\{Bi2X2Xi-\-BnX^ — B2iXiX2 — hXiXz)+02{B21 — h)x2Xz — 0z{B2\X2Xz 



-\-hxz^) (230) 



where 6 stands for the coefficient of ^3 in (229). Taking ) /oqn 

 <i>p = X2V0MB2iBiz-B22Bn)-V030,{xzB2iBn+X2BnB2i) j 



dp = B2lXzV020z+{BnXz-B2iX,)V0i02, 



we have (230) identically equal to V<l)p6p aside from the scalar factor 



-B21B12 -5/31)32/33 • 



If B12 = (225) becomes 



^1513x32 + 182(521x22 + ^22x2x3) ; (232) 



if we add the term —cp{x2B2\ + X3B22), where c is a scalar constant 

 neither zero nor unity, the resulting quadratic vector may be written 



/3i(5i3X32 — c52iX 1X2 — c522XiX3)+i32(l —C) (521X2^+522X2X3) —0/33(521X2X3 



+522X3^) (233) 

 If we take 



^p = th^ F|83^i+c((x2521+X3522)F^2i83+(X35l3-CXi522)F/3li32, 



1 — c 



ep = -^ X2F'j8u82 + X3F/33i3i, 



(234) 



