442 HITCHCOCK. 



Al3^lXiX2-Anl33X3^+(Au^l-^A2lPo-An^2+A3lP3)x2X3, (210) 



which is in the form (203) and can therefore be factored into VcjipOp, as 

 in Art. 34, provided the vector coefficients are not in the same plane. 

 It was shown in Art. 27 that neither ^13 nor An can be zero. The 

 condition that the vector coefficients be coplanar is therefore 



A21 - A12 = 0. (211) 



If this condition is satisfied, (210) cannot be factored into V<l>pdp. 

 If, however, we subtract from (165) the term pAnX2, the resulting 

 quadratic vector may be written 



^i2|8ia-3a:i-^i3i82X2'+(^ii|8i+^2ii32+^3i|83-^i3i33)x2a;3, (212) 



which may be factored like (203) if the vector coefficients are diplanar, 

 that is unless 



A31 - ^13 = 0. (213) 



Suppose (211) and (213) both satisfied. Subtract from (165) the 

 term p^AnXs + A13X2). The resulting quadratic vector may be 

 written 



- An^2Xo^ - .412183.T3' + Aii^iX2X3. (214) 



Neither An nor A13 can be zero, the vector being irreducible by 

 hypothesis. If An is not zero we may put 



0p = AnXoV^i^o + AnX3V^2(33 



ep = ^^^^^ F^2/53 + ,r^^ F^3^. >- ^^^^^ 



AnS^i^2^3 Si3i^2^3 



and we then have (214) identically equal to Vcppdp. If, on the other 

 hand, An = 0, (214) cannot be factored into VcppOp. The resulting 

 simple vector 



^i3^2.r2- + An^sxi" (216) 



may, however, be factored by first adding a term pS8p, where 5 may 

 be chosen in an infinite number of ways. It can be shown that any 

 term 



— p{bx2 + CX3) 



will suffice, if b and c satisfy 



A12A13 — bAu — cAi3 = 0. 



