QUADRATIC VECTORS. 399 



(a57) (PasPaPp) 



(a56) (Pa,P,P^) 



(87) 



The factor (Pa^P^Pp) differs from 01235(50, 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 7na + 11^5 for ^i is 



(567) (P,P,Pp) 

 (a56) (P5P6P7) 



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



rt(567) {P,PePp) 

 m(a56) (P5P6P7) 



^ [m(6a7) - 71(567)] WiPaP^Pp) + mn{Pa,P,Pp) + n2(P5P6Pp)] 

 7/i(a56) [m\PaP,P^) + mn{Pa,P,Pj) + n^iP,P,Pj)] 



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



(567) [{PJ'.Pp) (Ptt^PePT) - (P5P6P7) (PaJPePp)] + (6a7) jP^PePi) (PJ'J'p) 



(a56) (PtPeP^y 



a result rendered more compact by the identity 



(P,PePp) (Pa^PeP:) - (P.PePi) {Pa,P,Pp) = {PlP.Pa,) {P,P-,Pp) 



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

 the limiting form 



A;7a(567) {P,P,Pp) 

 {a5Q) {P,P,Pr) 



, ^•7^6[(567) {P,P,Pa,) (PePyPp) + (6a7) (P.PePy) (P.PePp)] 



(a56) (P5P6P7)^ ~ 



kMa57) (Pa,P,Pp) 



ia5Q) {Pa,P,P,) ' ^^^^ 



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

 /Je, and ^7 as ordinary axes, but /Ss a double axis, the cones (3) being 

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

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



