96 Proceedings of the Royal Irish Academy. 



Finally 



[7?iis, m\i, m"ii refer to ;;^,2, not to (j>\<^'i'^, as in C3.] 



6. It is not necessary to write down all the relations between the 

 invariants got from these identities. But we have 



(As) m"i2 = 5ij, m',2 = ai2w"2 + a^ivfi - Cu, 



mi2 = %n^m^ + la^^a^^ - rfi2(,2). 

 Suppose we find in^ directly. Consider the expression 

 SV{fxv' + ix'v) V{vX + v'\) VX,/ + AV). 

 Eeduciug, it becomes 



= SV(,uv' + iii'v)[^'SvX'X - fiSv'X'X -X(SX'n'v + SX/ii'v') - X' {SX'iJ-v - SX^v') 

 = S/i'fxvSvXX' — \_SX'fj.v + SX/x'v + SXfJiv][S\f/v' + SX'/iv' + SX'fx'v^ 



- Sv/x'hSv'X'A + SX' /LivSX/x V + SX'fiv'SXi/v - SX'/j-'vSXfiv . 

 Since 



S Vfi'v V{ Vnv' VX'X) - S VX'X Vfx'v Vfiv' = 0, 



and S/x'fj,vSvX X = Sf/vX'SfMvX + Sf/v'iiiSX'inX + Sfx'vXSX I'/u, 



we get the required expression 



(Be) = SXfivSX'ju v' — {SX'f.iv + S/ivX + Sv'Xfi) (SX/x'v' + S/xv'X' + SvX'/Li'). 



Put A = (j)\a, fx - <p'if3, V = (j>'\y, X' = (^'ja, fi = ^'z/S, 7 = <P ij \ 



then yifJ^v' + pi'v) = xn V^l > 



so finally rivn = a^aii - wiima. 



Cor. — £^12(12) = 3m,m2 + a^ia^. 



(Ce) Sought cubic is 



X12' - iiiX^i + ic<-2im"i, + a,2a"2 - c,,) X12 + («i2«2i - miVh) = 0. 



7. Suppose now <p„ (jji, <pi are self-conjugate; let p and a be two vectors 



(pidSXp + (piaSjxp + (pz(ySvp 



is a particular case of $, ki = SXp, etc. Its discriminant is 



"Zm^SXp^ + 2 [anS'Kp''Sf.ip + ttiiSXpSfip^] + dmSXpS/ipSvp. 



If ^1, ^2, ^s are transformed by any linear substitution, A, p, v are 

 transformed by the contragredient transformation. Suppose, in fact, the 

 ^'s transformed 



01 = aiii/-! + an(j>i + ai3./)s 



02 = ^2101 + «2202 + «2303 



03 = asi'/-! -I- «3202 + «3303, 



