HiTCHCOCK — On the Formulation oj two Linear Vector Functions 9 



a theorem, which, as I have elsewhere shown," is fundamental in the theory 



of quadratic vector functions. 



Theorem: The torus of the IRllEDUCIBLE\ rector V$p()p cannot h <> 

 fixed plane. 



To prove the theorem, we note that by the preceding investigation all 

 possible function-pairs may be written in one or the other of the six cases, 

 (1), (6), (10), (12), (13), or (14). 



Forming the product V<p/>0p from (1), we have 



Vfpdp = (c - b)VpvS(3pSyp + {h-OjVXpSapSfip + (a - c)Vv\SypSap, (15) 



where A, ^u, and v were by hypothesis diplanar, since <j> has an inverse. 

 Hence Vpv, Vv\, and VX/x are also diplanar. Furthermore, a, b, and c are 

 unequal, since, in this case, $ _1 6 has distinct roots. And a, j3, 7 are diplanar 

 for the same reason. Hence the locus of (15) cannot be a fixed plane, and 

 the theorem is proved for this case. 



Forming the product V$pQp from (6), we have 



Vrppttp = (//, - g) VpvSftpSyp + ((] - g x ) VvXSypSap 



+ < : [VXnS 2 ap - VflvSypSap], (16) 



where similar reasoning holds, viz., X, p, and v are diplanar because (j> has an 

 inverse : hence the vectors Vpv, VvX, and V\p. are diplanar ; g - g x is not 

 zero, since </>"'# has two unequal roots; c, was by hypothesis not zero; and, 

 finally, the scalars Sup, Sj3p, and Syp cannot be multiples one of another, since 

 a, j3, and 7 are diplanar. Hence the locus of (16) cannot be a fixed plane. 

 Forming the product VfpOp from (10), we have 



V<j>p6p = C,[ VpvS'yap - VvXSfiypSyap] + c[VX/nS : j3yp - Vp.i'S(3ypSyap]. 



(17. 

 Here neither c nor c, can be zero under the hypothesis for this case. The 

 rest of the reasoning is as before. Hence the locus of (17) is not a fixed plane. 

 Passing to (12), which is equivalent to removing the restrictive hypotheses 

 on the c's and g's of the former cases, it is evident that we have VippOp 

 divisible by Sip, hence reducible. 



From (13) we have V<f>pHp parallel to k, and so divisible by a quadratic 

 scalar, that is reducible. 



* Proc. Nat. Acad. Sci., loc. cit. For a more detailed study of quadratic vectors see 

 "A Classification of Quadratic Vectors." Proc. American Acad, of Arts and Sciences, 

 52-7 (January, 1917), p. 369. 



t An irreducible vector is one which cannot be factored into a variable scalar and a 

 vector of lower degree,— a term of the form pt being added tu the vector if necessary. As 

 an equivalent definition wo may say that a vector Fp is reducible when, and only when, 

 the vector product V?Fp i.s divisible by a scalar variable. 



K.I. A. PROO., VOL. XXXIV., SECT. A. [2] 



