qutsjjratic vectors. 445 



-XiX2B2l^l-\-X2Xs(Bv28y+Bo,0o-B2,03) + Xi'(Bu0l+B23P2), (227) 



the axis 182 being made a zero, and, consequently, the term in a-2^ dis- 

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

 if the vector coefficients are diplanar. We may not have B21 = 0, 

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

 is that B23 shall vanish. If so, the quadratic vector (225) becomes 



^liBnXoXz + BnXi") + 182(^21.1:22 + B22X2XZ); (228) 



and if /3i2 is not zero we may subtract the vector term 



P I 52,0:2+^x3 I (229) 



writing the resulting quadratic vector as 



Pl{Bi2X2X3-hBi3X3^ — B2lXiX2 — bXiX3)+^2(,B22 — b)x2Xz — ^3iB2lX2X3 



+bx3') (230) 

 where 6 stands for the coefficient of xs in (229). Taking ) (9'i^\ 



<t>P = X2V0MB2:Bi3-B22Bn)-V^3^l(x3B2lBi3+X2Bi2B2l) j 

 dp = B2lX3V^2^3-\-{Bl2X3-B2lX{)V^^2, 



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



52i5i2-S/3ij32i33* 



If Bn = (225) becomes 



^iBi3X3^ + ^2{B2xX2^ + ^220:23:3) ; (232) 



if we add the term —cp{x2B2\ + 0:3522), where c is a scalar constant 

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



^\{Bi3X3^ — cB2\XlX2 — cB22XlX3)-\-^20- " c) (521X2^+522.1:23:3) — Ci33(52lX2.T3 



+ ^22X3^) (233) 



If we take 



*P = -r^^ F/33/3i+c((x2521+X3522)F^2^3+(X35l3-CXi522)F|Su82, 



1 — c 



Bp = ^^^X2F'/3ii82 + X3F/33/3i, 



(234) 



