1311 



I use this occasion to point out that, as is well known, Prof. 

 Klein was the tirst to call attention to the elliptical space and its 

 relation to and difference from the spherical space, and generalij 

 to investigate and explain the different possibilities of non-Enclidean 

 geometry ^). In fact all geometrical concepts used in the different 

 stages of the development of modern physical theory are contained 

 in Klein's general scheme as given in the second of the papers 

 quoted in the footnote. 



3. If we start from the assumption, that the gravitational field is 

 of such a nature that it is possible, by introducing a suitable system 

 of coordinates, to bring the line-element into the form 



ds' z=~a dr' — b ((/»|)' + sin' if> dd') -\- f dt\ . . . (2) 

 then we can call r the "radius- vector" and t the "time". If now 

 we add the condition that a, b, f must be functions of /• only, and 

 not of t, \p, x>, then these conditions may be briefly expressed by 

 saying that the field is static and isotropic. Then the line-element of 

 three-dimensional space is 



do' = a dr' i- b [d^^' i- siti^ i]r diV] (3) 



and consequently we have 



ds' =z — do' -{- f de (2') 



If now we add the hypothesis that d(f shall be the line-element 

 of a space of constant curvature, thus 



do' = R' \ dy' -\- sin' X [d\\>* + sin' if) di)-'] }, i ' * 

 then the field-equations (J) reduce to one equation for /, of which 

 the solutions A and B are 



/=''' • . . . (4^) 



f=c'cos'/ ....... (4^) 



If we drop the condition of isotropy, then ƒ may be a function 

 of r, \]\ iy. For this case Levi-Civita ') has given the general solu- 

 tion of the differential equation for ƒ. He starts from Einstein's 

 original equation, i.e. the equation (1) with ^==0. It is however 

 not difficult to extend the proof to the general case. Then the 

 equation (11') of Levi Civita [I.e. page 530J re}ilaces (11) [p. 526J, 



(3') 



^) Ueher die sogenannte Nicht- Euclidische Geometrie, Math. Annalen, Band 4 

 and 6 (1871 and 1872). 



Programm ziim Eintritt in die j) hilosophische Faculidt, Er]a.ngen 1872, reprint- 

 ed Math. Annalen, Band 43, p. 63. 



^) Reolta flsica di alcuni spazi normali del Biunchi, Rendiconti della R. 

 Accad. dei Lincei, Vol. XXVI, p. 519 (May 1917). 



90* 



