126 Dr. L. Silberstein on General Relativity 



If f LK is as in (42), H is a genuine scalar or general in- 

 variant. It is in fact Beltrami's second differential para- 

 meter A 2 < I ) whose definition can, obviously, be retained for 

 a manifold of any number of dimensions. 



Dropping the operand, <I>, we thus obtain the comparatively 

 simple invariant differential operator 



ft =^rl — {fhr-1; • • • < 43 ) 



terms to be summed up as before. Operating with this 

 upon a scalar, as is the above "potential/ 5 and equating the 

 result to another given scalar T, say, we shall have in 



12$ = T (44) 



a differential equation of the second order for <J>, invariant 

 with respect to any transformations of the coordinates. The 

 scalar T can, for example, be zero outside of matter and, 

 say, proportional to appropriately measured " density of 

 mass " p within matter. I do not say that <I> is the gravita- 

 tional potential ; I am only constructing an example of an 

 abstract generally covariant law of motion of a particle. 



Having thus ascertained the general invariance of the 

 operator H it is obviously interesting to see what its form is 

 like in some simple reference system, more especially in a 

 natural system. First of all, in any system of orthogonal 

 coordinates we have, with ^.= 1/^=1/^, 



o = 1 /9 2 



ffi Id", 



vim • • • ^ 



and, in the natural system r, $, etc., for instance, developing 

 the second term of (43 a), with the tensor (2), and c=l, 



o-&- 



R 2 sin 2 ^5 . sin </> 



{U****-* 1 **) 



In the second part of this operational equation the reader 

 will easily recognize the Laplacian, v 2 = div y, f° r a space 



of constant curvature ™; thus, in any natural reference 



system the whole operator assumes the simple and familiar 



