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work done with P. C. Gilmore on knapsack problems strongly suggested the results of this paper. Finally, we note that the following steps, (i) solve P1 obtaining the opitmal B, (ii) put M(I)/M(B) into standard form and identify the a,, (iii) solve (4) obtaining y, (iv) compute XB = B-1(b Ny), will yield an optimal solution (XB, Y) if XB 2 0. It is not necessary for b to be in KB(l(D 1)) to apply the procedure. The problems for which the procedure provides a solution are those for which those inequalities binding the solution of P1 alone determine the solution to P2.

FINITELY PRESENTED GROUP WHOSE WORD PROBLEM HAS THE SAME DEGREE AS THAT OF AN ARBITRARILY GIVEN THUE SYSTEM (AN APPLICATION OF METHODS OF BRITTON)