Some special solutions of the equations of axially symmetric gravitational fields

The problem of axially symmetric fields was first treated by Weyl, who succeeded in obtaining solutions for a static field in terms of the Newtonian potential of a distribution of matter in an associated canonical space. He also solved the more general problem involving the electric field. Levi Civita, by slightly different methods, obtained solutions differing from those of Weyl in one respect, and discussed fully the case in which the field is produced by an infinite cylinder. R. Bach has discussed the special case of two spheres and has calculated their mutual attraction. Bach also considered the field of a slowly rotating sphere, and obtained approximate solutions, taking the Schwarz child solution as his zero-th approximation. The same field was discussed earlier by Leuse and Thirring, who considered, the linear terms, only, in the gravitational equation. Kornel Lanczos has also considered a special case of stationary fields and applied the results to cosmological problems. The more general case of gravitational fields produced by matter in stationary rotation has been treated by W. R. Andress and E. Akeley. Both these authors obtain approximate solutions of the general problem, and the latter treats at length the field of a rotating fluid. The object of this paper is to present some special, but exact, solutions which the author obtained some years ago and, also, two methods of successive approximation for obtaining solutions of a more general type, which behave in an assigned manner at infinity and on a surface of revolution enclosing the rotating matter to which the field is due. Our solutions include as special cases the solutions of Weyl, Levi Civita and others which pertain to static fields. Also, the approximate solutions for stationary fields obtained by Leuse and Thirring, Bach and Andress are contained in our solutions when appropriate choice of boundary conditions is made and higher order terms are neglected.