LINEAR GROUPS WITH CERTAIN INVARIANTS OF DEGREE q > 2. 219 



r 



r 

 AJ.tJ ^ [JuijA,,-j')nij c -\-Lijl/ij[Jii} c -{- "kijlij JJuLik) == V) 



218) 

 219) 2 



220) 



+ 



+ ^ JVJ*) 



i= 1 



LI j LI % LH 



hij &ik ^it 

 l/ij l/ik In 



lit {*.'* V, 



* 



ik pit 



ik m it 



1=1 



-0, 



(if j-*-o 



(unless j h = j), 



where, throughout, i, j, k = 1, . . ., r, while Jc =%=j in 216) and 219), 

 and t=^=j 7 k in the first of the relations 219); together with relations 

 derived from 216), 217), 218) and 219) upon interchanging L, A, I 

 with M y [i, m or with N, v, n. But relation 216) must also hold for 

 Jc =j, being then derived from the first one of set 215) upon multiply- 

 ing the latter by 3. Similarly 219) must hold for k=j, being then 

 derived from 216), 217), 218). Lastly, the first of relations 219) 

 must hold for t = k == j, being then derived from the first of the 

 set 215) upon multiplying by 6. Hence the above conditions must 

 hold for i, j, k = 1, . . ., r independently. 



Let j be any fixed integer ^ r and consider the 3r equations 

 216), 217), 218) for k = 1, . . ., r. Taking as unknowns the 3r products 



221) L^, Lijlij, Wit (* = !,..., r), 



the determinant of their coefficients is seen to equal the determinant 

 of S and is therefore not zero by hypothesis. Hence the products 221) 

 are all zero. From the analogous conditions, 



Qfj^\ -n/r -n/r /~v f/tn 1 V~\ 



Expanding the symbols in 219) according to the last columns 

 and applying a similar reasoning to the resulting equations, we find 



224) 



y = 0. 



We obtain similar identities 225) and 226) between the M, p, m and 

 the N,v,n. From 220) for j 4= ft and the foUowing of type 219), 



