Vol. 6, 1920 MATHEMATICS: L. P. EISENHART 
679 
is stationary along the curve, the curve is called a "world-line," or a 
geodesic, in the four-space. 
Einstein^ considered the case when Xi, X2, x^, are rectangular coordinates 
and Xi^ represents the time, and assumed that the field was produced by 
a mass at the origin which did not vary with the time. In order to obtain 
the equations of the world-lines in the form which enabled him to es- 
tablish his well-known expression for the precession of the perihelion of 
Mercury, Einstein made also the following assumptions: 
A. The quantities gik are independent of i. 
B. The equations gi^ = 0 for z = 1, 2, 3. 
C. The solution is spacially symmetric with respect to the origin of 
coordinates in the sense that the solution is unaltered by an orthogonal 
transformation of X], x^, Xz. 
D. The quantities g^-^ = 0 for z =t= ^ at infinity and also g44 = ~g\\ = 
— g22 = — g33 = 1. Schwarzschild,^ using the first three of these assump- 
tions and certain others integrated the equations Gik = 0, and obtained 
(1) in the form 
ds^ = ^1 - -^^ dt^ - - R^(dd^ + sin2 dd<p^), (3) 
^ ~^ R 
where o; is a constant depending on the mass at the origin. lyevi-Civita"^ 
has given three solutions of the equations Gik = 0, one of which includes 
the above, and Weyl^ has given still another solution. Later, Kottler^ 
obtained the form (3) not by the solution of the equations Gik = 0 
but as a consequence of certain postulates. It is the purpose of this 
paper to accomplish the same result by the following set of postulates: 
I. Assumptions A and B of Einstein, in accordance with which we 
write (1) in the form 
= Vdt' - dso', (4) 
where 
1,2,3 
dso^ = ajkdXjdXk, (5) 
ik 
the functions V and being independent of t. 
II. The function V is a solution of 
A2V = 0, (6) 
where is the Beltrami differential parameter formed with respect to 
the form (5), and is defined by 
A,e ^ 1^ . \ 
i,k k 
where a is the determinant of the functions a^^ and a^^^^ is the co-factor 
