QUADRATIC VECTORS. 399 



(a57) {Pa,P,Pp) . (37^ 



(a56) (PasPsPr) 



The factor (PaiiPiPp) differs from Ci235(bcij p) only in the presence of a 

 factor (123)2- (Cf. (32)). 



Considering the remaining terms of (31), the coefficient of a after 

 the substitution of ma + n^^ for ^i is 



(567) {P,P,Pp) 

 (a56) (PtPePi) 



By the aid of (85), the coefficient of /Ss may be written 



n(567) (P,P,Pp) 

 m(a56) (PsPePy) 



[7»(6a7) - n(567)] [m%PaP,Pp) + nm{Pa,P,Pp) + n^(P,PePp)] 

 m(a56) [vi^PaPePi) + mniPa^P^Pi) + n^PtP^Pi)] 



which, if we let m approach zero, approaches the limit 



(567) [jPJ'ePp) (PaBPePi) - jPoP^Pi) {PasPePp)] + (6a7) {P,PeP7) {PfP^Pp) 



(a56) (PePePv)* 



a result rendered more compact by the identity 



{P,P,Pp) {Pa,P,P-,) - {P,P,Pt) {Pa,P,Pp) = (PsPePae) {P.PiPp) 



Collecting results, we find that as m approaches zero, (31) approaches 

 the limiting form 



' fc7a(567) (P,P,Pp) 

 (a56) (P5P6P7) 

 kM{5Q7) {P,P,Pa,) {P,P,Pp) + (6a7) (P5P6P7) {Pf,P,Pp)] 



(a56) {P,P,P,f 

 kMa57) {Pa,P,Pp) . 



(a56) (PasPsPr) ' ^ 



a normal form for a quadratic vector having the vectors /3i, ^2, ^3, 

 jSe, and jSy as ordinary axes, but jSs a double axis, the cones (3) being 

 tangent to the plane (a5p) = along the vector jSs. We may, if we 

 wish, verify directly by polarization that VpFp = gives at most one 



