Empty Space-Times Admitting a Three Parameter Group of Motions

where RV is the Ricci tensor, R is the scalar curvative and K is Einstein's gravitational constant. Equations (1.1) are second order partial differential equations for the g,, and their solutions involve arbitrary functions which are to be determined from physical data expressed as initial values or boundary conditions. On the basis of Mach's principle one would like to conclude that if T, = 0 and there are no singularities anywhere in the g9, then equations (1.1) which take the form