27 ' • 



should have to ascribe a certain negative value of the energy to a 

 field without gravitation, in such a way (comp. § 57) that the 

 energy in the shell between the spheres described round the origin 

 with radii r and r -\- dr becomes 



Ajtc 

 dr. 



X 



The density of the energy in the ordinary sense of the word 

 would be inversely proportional to r^, so that it would become 

 infinite at the centre. 



It is hardly necessary to remark that, using rectangular coor- 

 dinates we find a value zero for the same case of a field without 

 gravitation. The normal values of (^nb are then constants and their 

 derivatives vanish. 



§ 60. Using rectangular coordinates we shall now indicate the 

 form of t'\ for the field of a spherical body, with the approximation 

 specified in § 57. Thus we put 



g^^=-{l + /) + _^(;._|^), etc. 



I 



• • • (110) 



1 



By (109) and (110) we find ^) 



1) Of the laborious calculation it may be remarked here only that it is convenient 

 to write the values (110) in the form 



5'n = — 1 + « + ^-^, e«('. 



where x and (3 are infinitesimal functions of r. We then find 

 c { Z' da\^ dv da 



+ i2{aik) 



{a,{,k= 1,2,3) 



which reduces to (111) if the relations between i-, (3 and /., f/, viz. 



« + -/?'=- A , _i^' + |3" = ;-fi 

 r r 



and the equality x' = v' involved in (109) are taken into consideration. 



