Astronomy. — "On the. curvature of space". By Prof. W. de Sitter. 

 (Communicated in the meeting of 1917, June 30). 



1. In order to make possible an entirely relative conception of 

 inertia, Einstein ^) has replaced the original tield equations of his 

 theory by the equations 



^y-- — 2 <7yv ^ = — y- T,j., -{- ^y^ (f,j,, 7' . . . . . (I) 



In my last paper ") I have pointed out two different systems of 

 (/uv wi)ich satisfy these equations. The system A is Einstein's, in 

 which the whole of space is filled with matter of the average 

 density q^. In a stationary state, and if all matter is at rest without 

 any stresses or pressure, then we have Ty,, = with the exception 

 of T^^ =: (7^4 ()(,. In the system B this "world-matter" does not exist: 

 we have q^ = and consequently all Ty, = 0. The line-element in 

 the two systems was there found to be 



ds^ = — R^ [dyï' -\- sin^ x C^^f'' + «'^i' ^ d^"] I + <^'' ^^'^ • • (2^) 

 ds^ = — E' \dü)^ + sin^ o) [(^x' + ^"'n' X (^»P' + «iw' »P d»^)] \ ■ (2 S) 



In the system A we have : 



and in B -. 



;=rA , ^„=0. ..... . i'óB) 



In the system ^4 x, if', ^ are real angles; in 5 if' and ,"> are also 

 real, but to and x ai'e imaginary. If, however, we put 



sin a) sl7i x =^ si7i $ , 7' = R^, 



tcm CD cos yi :=. tan. it] , t = Rij, 



1) A. Einstein, Kosmologische Beh'achtimgen zur Allgemeinen -Relativitdts- 

 theorie, Sitzungsber., Berlin 1917 Febr. 8, p. 142. 



') W. DE Sitter, On the reto^«'% o/iwer^/a, these Proceedings, 1917 March 31, 

 vol. XIX, p. 1217. 



in the footnote to page 1220 of that paper it is stated that the four-dimensional 

 world of the system B can be represented as a hyperboloïd of two sheets in a 

 space of five dimensions, which is projected on a euclidean space of four dimensions 

 by a "stereographic projection". This is erroneous. The hyperboloid has only one 

 sheet. Its projection fiUs only part of the euclidean space of four dimensions; 

 the part outside the limiting hyperboloid 1 + oh- — (which is called (a) in the 

 quoted footnote) is the projection of the conjugated hyperboloid (which is of 

 two sheets). 



