20 Prof. F. 1). Murnaghan on the 



Pythagorean theory that is not tenable for non-Euclidean 

 spaces in general. Since the only existing experimental 

 verifications of the relativity gravitational theory are based 

 on the expression for (ds) 2 in a permanent symmetrical 

 gravitational field, it is desirable that the assumptions in 

 the mathematical treatment should be clearly stated. In 

 the following discussion Einstein and Schwarzschild's form 

 as well as others are derived on certain definitely stated 

 assumptions as to the meaning of the term symmetry, 

 and, peculiarly enough, the differential equations of the 

 theory prove easily integrable, without making any use 

 of the form of the Euclidean element of length on the 

 unit sphere. 



According to Einstein the fundamental space which 

 has physical reality (that is, with reference to which the 

 laws of physics must have the tensor form) is of four 

 dimensions. All coordinate systems are, without doubt,, 

 equally valid for the description of physical phenomena, 

 but it is reasonable to suppose that for a given observer 

 a certain coordinate system may have a direct and simple 

 relationship to the measurements he makes ; such a system 

 is termed a natural coordinate system for that observer. 

 For the problem under discussion, one of the natural co- 

 ordinates # 4 is a time coordinate which is such that 

 the coefficients g u , g 2 ±, g^ in the expression for {ds) 2 , 

 (ds) 2 = g rs dx r dx s ... (r, s umbral or summation symbols) 

 vanish identically whilst the other coefficients g rs do not 

 involve a? 4 . Accordingly (ds) 2 = g rs dx r dx s +gu{dx 4 ) 2 . . . 

 (r, 5 = 1, 2, 3). Now in a space of three dimensions- 

 we can always find an orthogonal system of coordinates 

 (j/i? 2/2> Vz) say ? f° r we have merely three differential 



equations q rs ^~ -^-^ = . . . (r, 5=1, 2, 3 are umbral 

 ^ v ^x r ^x s v 



symbols whilst I and m are different numbers of the 



set 1, 2, 3) (the g rs being the coefficients of the reciprocal 



form for (ds) 2 ) for the three unknown functions y 1} y 2 , g s 



of x 1} # 2 , x 3 . Hence, using these coordinates instead of 



Xi, x 2 , x s (but keeping the notation x instead of y) 



we see that there is no lack of generality in writing 



(ds) 2 =g n (dx l ) 2 -tg22{dx2) 2 + g3z(dx z y+ 9 gu(dx 4 ) 2 , where the 



coefficients g rr are functions of (x u x 2 , x 3 ) at most. [It'- 



will be observed in passing that this statical field is very 



special; it is not in general possible to. find orthogonal 



coordinates in space of four dimensions, there being six 



equations /*|^ |^ 1 = . . . (r, s = 1, ... 4) for the four- 



