Let \(m>3\) be an integer, \(q>1\) a power of a prime \(p>2\), and consider the polynomials \(F^-= Y^n+T^{q^2} Y^{u'} +X^qY^u- XY^w- TY^{w'}-1\) and \(F^*= Y^{n*}+ XY+1\) in indeterminates \(X,Y,T\) over an algebraic closed field \(k\) of characteristic \(p\), where \(n=1+q+ \cdots +q^{2m-1}\), \(u'=1+q+ \cdots +q^{m+1}\), \(u=1+q+ \cdots +q^m\), \(w=1+q+ \cdots+ q^{m-2}\), \(w'=1+q+ \cdots +q^{m-3}\), \(n^*=1+q+ \cdots +q^{m-1}\), and consider their respective Galois groups \(\text{Gal} (F^-, k(X,T))\) and \(\text{Gal} (F^*,k(X))\). In a previous paper [Isr. J. Math. 88, 1-23 (1994; Zbl 0828.14014)] the author has proved that \(\text{Gal}(F^*,k(X))\) is isomorphic to the projective special linear group \(\text{PSL} (m,q)\). In the paper under review the author shows that \(\text{Gal} (F^-,k(X,T))\) is isomorphic to the projective negative orthogonal group \(P\Omega^- (2m,q)\). As a consequence of this, he proves that the Galois group of a more general polynomial \(f^-\) is also \(P\Omega^- (2m,q)\), and then by slightly changing \(f^-\) and \(F^-\) he obtains two other polynomials \(\varphi^-\) and \(\varphi^-_2\) whose Galois group is the negative orthogonal group \(\Omega^- (2m,q)\).