In the particular case in which a relativistic gravitational vacuum field is axially symmetric, the solution of the Einstein field equations can be reduced to the solution of a pair of simultaneous nonlinear differential equations. Nakamura, (1983), has shown that there exist infinite families of solutions of these equations which he denotes by Pn', n=1,2,3, ., and Pn, 2,3,4, ., whose members are interrelated by two Backlund transformations denoted by beta and gamma . He finds each of the solutions P'1, P2, P'2, P'3, P'3 explicitly in the form of the ratio of two determinants. The object of this paper is to state the general solutions Pn, Pn' in terms of the first and second cofactors of the corner elements of a certain determinant of order n and to prove that they are correct by showing that the application of transformation gamma to Pn gives P'n and that the application of transformation beta to P'n gives Pn+1.