RocHic — The Quadratic Vector Function. 



97 



«22 Chi 





a,, 



«,, 



a. 



^23 ^d3 



D = 



ai2 



«13 



«23 



a. 



D 



a 



and suppose Xi = ^4iiA + ^^/u + A^^v, etc., we have 3 sets of equations like 

 ] = ctnAu + chiAii + «3i-43, 

 = a, 2^1, + «.22^ji + cinA-ii 



= «|3^11 + «23^2l + «33-4s,. 



Giving ^1 



8. Vectors o, j3, y exist such that c^oy = <^3^, ^3a = <i>iy, i^i/3 = t/jja for 

 ^2^r'^3a = ^3</>fV2"' oi' " is the spin-vector of ^2f/),"'i^3 (or (^30r^</)2) and 

 j3 = (/>r'^3a, y = (pi'^fiO. We see 



(jCjic^i + a;22^2 + 0:23(^3) (a-'sia + «32i3 + a;3jy) 



= (X3i(j)i + *'32<^2 + *'3303) (3^210 + «22^ + ^23j), 



or a, /3, y, are co-gredient with ^i, ^3, ^3. If now we put 



X = rj3y, fx = Tya, z/ = Va^, 

 we have, e.g., 



F)3y = Fa/3 (a;2iiC32 - a^^s'^'si) + V^y {XiiXsi, - aJajS^o) + Fyn (a;239;3i - «2ia.'33), 

 or. A, fjb, V are conti'agredient to 0,, 02, </)3 [v. § 7.] G-ive the X, /a, z/ those 

 particular values in <1>. The nature of 4> will depend on the pencil 



a^ii^i + «i203 + a''i3</>3- 

 Calling the function -^ (ap) we have 



^P (<T|o) - iP (/orr) = 201 ( TA Tpa) = S<^i (yS'i^p^ - iS'S'r/O'T) = 0. 

 (A«) .-. ^ (ap] = ;^ (p<t). 



The third invariant of the former is 



m = — = bcpipcpipcpzp . OA/J-V, 



so that the cubics S<j)ip<j}2p(^3p = 0, and 



S^iSAp^ + 5 [a^iSXp'Spp + a^iiSXpSpp'') + di^^SXpSppSvp = 

 are identical. 



Also since (j)i, ^2, ^^ are self-conjugate, if p, a, r, are any vectors 



(Be) Srxf/ [pa) = Spip (ar) = Saip [rp). 



9. If S(T(pi(T = 0, S(7(j)2(T = 0, 8(T^3<r = arc three quadrics, the equations 

 of polars of p with respect to each are 



Sp(pia = 0, Sp<h2(T = 0, Sp<j>3(T = 0. 



