Vol. 6, 1920 MATHEMATICS: S. D. ZELDIN 
543 
independent functions in r + g variables. It should be observed, however, 
that one of those functions will also be 'invariant to the adjoint of G' 
and will consist of the first r variables alone. The following theorem may 
therefore be stated: 
// the adjoint of G' has one invariant, the adjoint of G has q -\- i inde- 
pendent invariants, one of which is the invariant of the adjoint of G' . 
We assume first that q of those invariants, not involving the one com- 
mon to both, G and G' , are all flats of order r + g— 2 in the r + g— 1 
space and that their common intersection is an f— 1 flat. It is then proven 
that if that flat does not pass through any of the points of the space of 
the adjoint of G, corresponding to the exceptional transformations of G, 
we can form new operators, such linear functions of the old ones, of which 
f, not involving the exceptional operators, form an invariant subgroup 
of G of order r. This means that the structural constants, whose last 
subscripts are greater than r, will all become zero. If, however, those q 
invariants are not flats but algebraic spreads, not necessarily of the same 
order, none of which passes through the points of the space of the adjoint 
of G described above, then by considering the polars of the q invariant 
spreads we prove that the structural constants which have for their last 
subscripts numbers greater than r can also be made all zero. If, finally, 
each of the q invariant spreads passes through only one of those points, 
not necessarily the same for all spreads, then the structural constants 
described above can also be made all zero. 
Detailed proofs and references are given in a paper accepted for publica- 
tion by the Annals of Mathematics. 
