23 



1 

 of the factor - in (96) and (97) ) these functions become proportional 



to X, so that in a feeble gravitation field they have low values. 



^ 56. Because of the complicated form of equations (96) and (97), 

 we shall confine ourselves to the calculation for some cases of t\\ 

 i.e. of the energy per unit of volume. This calculation is considerably 

 simplified if we consider stationary fields only. Then all differential 

 coefficients with respect to ,v^ vanish, so that we have according 

 to (96) 



1 I d / ÖQ A j 



t'/==-- --Q, + :^(a6/.) — U-^W.^,W • • (99) 



^^ { O'Ve \Ogab,feJ " \ 



We shall work out the calculation, first for a field without gravita- 

 tion and secondly for the case of an attracting spherical body in 

 which the matter is distributed symmetrically round the centre. 



If there is no gravitation field we may take for the quantities 

 (/«ö the "normal" values. For the case of orthogonal coordinates these 

 are given by (98). When we want to use the polar coordinates 

 introduced into § 48 we have the corresponding formulae 



g^i, = 0, for a =1= h. j 



If, using polar coordinates, we have to do with an attracting sphere 

 and if we take its centre as origin, we may put 



u 

 9m = — 1 ^' 92:1 = - (1 — ^i') "' 9zz — - ^^ 9iA = "'. (^01) 



where tt, v, w are functions of r. The ^«^'s which belong f o an 

 orthogonal system of coordinates may be expressed in the same 

 functions. 



These ^«i's are 



u .^•^' fu \ 



9..=-^[-,-^y ^tc. 



9x4=924=9zi = ^^ 944='^' 



The "etc." means that for g^^,gzi we have similar expressions 

 as for (/j, and for ^33,^/31 similar ones as for g^^. 



§ 57. In order to deduce the differential equations determining 

 II, V, to we may arbitrarily use rectangular or polar coordinates ; 

 the latter however are here to be preferred. If differentiations 



