31 



and thus to derive the result (1J2) from (113), we should have to 

 determine the quantity T (comp. 120)), accurately to the order x. 

 The surface integrals in (115) too would have to be considered 

 more closely. We shall not however dwell upon this. 



§ 62. From the expression for t'/ given in (113) and the value 



derived from ir, it can be inferred that, though t' is no tensor, we yet may 

 change a good deal in the system of coordinates in which the pheno- 

 mena are described, without altering the value of the total energy. 

 Let us suppose e.g. that .v^ is left unchanged but that, instead of the 

 rectangular coordinates x^, .i\, x^ hitherto used, other quantities 

 üj\, .r'j, x's are introduced, which are some continuous function of 

 ^',,^!j,^'8, with the restriction that x\ = x^, x'^=x^, .t' 3 =: a'j outside 

 a certain closed surface surrounding the attracting matter at a 

 sufficient distance. If we use these new coordinates, we shall have 

 to introduce other quantities g'nb instead of gab- As however outside 

 the closed surface the quantities t/ab and their derivatives do not 

 change, the value of E^ will ai)proach the same limit as when we 

 used the coordinates .c^, .i',, .^3, if the surface <j for which it is calculated 

 expands indefinitely. The value which we find for E^ after the 

 transformation of coordinates will also be the same as before. Indeed, 

 if dt is an element of volume expressed in a-j, *',, *j-units and c/t' the 

 same element expressed in a;\, x\, .I'Vunits, while Q represents the 

 new value of Q, we have 



Qdr =z Q'dr'. 



It is clear that the total energy will also remain unchanged if 

 x\, x\, .v\ differ from a;^, x,, x^ at all points, provided only that these 

 differences decrease so rapidly with increasing distance from the 

 attracting body, that they have no influence on the limit of the 

 expression (115). 



The result which we have now found admits of another inter- 

 pretation. In the mode of description which we first followed (using 

 a?i, A'j, a;,), q ^) and c/ab are certain functions of .Cj, ti',, .^, ; in the new 

 one q', g'ab are certain other functions oïx\,x\,x\. If now, without 

 leaving the system of coordinates x^,x^,x^, we ascribe to the density 

 and to the gravitation potentials values which depend on x^, ^,, .r, 

 in the same way as q\ g'ab depended on x\,x\,x\']uai now, we 

 shall obtain a new system (consisting of the attracting body and 

 the gravitation field) which is different from the original system 



^) By I) we mean here what was denoted by « in § 56. 



