Roche — The Quadratic Vector Function. 



101 



III. 



21. The general function 



will be binomial if /SaiaoO;, = 0. This property is invariantal. 



It will also be binomial if ki4>i + hi'-. + hi^s = 0, identically. This 

 condition implies 3 scalar equations 



Swij/fi' + S (ffip./fiV.-. + a,,h%) + d,.,hl^,h = 0, 



2/,-iWi", = 0. [Ai, Bi, Ci] 



On eliminating the Jcs between these, we get 







m. 



nu 



111:, 



«12 



«21 



^23 



«32 



ttsi 



an 



0^123 



n\ 











h. 



Mi's 











m'i 



hi 



&23 







Wl'„ 







m\ 



h,. 



&23 



m'-, 











631 











Hi's 











m'. 



h. 



^31 



»?'i 



612 



«"i 











ni"z 



















«*"3 











in'\ 











in"i 



»*";) 



























in":, 















m"« 



w/'i 























m"i 



m"^ 



















m'l 























m"^ 



m" i 











m'\ 































m"i 



W>"l 



m"i 



Consistent with these relations, the invariants above take special forms. 



22. The function ^ {pp) may be monomial, of the form VOipdip. In^this 

 case, we have the following connexions between all the functions : — 



Putting /3i = 



Va,<, 



^2 = 





183 = 



Va,a, 



Ve,pd.p = ti,S(3Ap0.p + a,Sl3,9,pe,p + a,Sf3,drp9,p'] ■ 



e\(V0,p(5,) - 6\(r0,p(i,) = 2i>,p 

 o\{ve,p(3,) - e',{rd,pi3-^ = 2^,p 

 e\(r9,p(i,) - d', ( ro,p/3,) = 2^3^. 



It can easily be verified that the functions on the left are self-conjugate. 

 The relations are complicated, but it is worth while following them up a bit. 

 We write them 



m,-' Ve\-'9^p6\-'(5, - 7K,-i Ve',-'9,pe\-'^, = 2,p,p 



