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 and becomes
The density of the energy in the ordinary sense of the word would be inversely proportional to , so that it would become infinite at the centre.
It is hardly necessary to remark that, using rectangular coordinates we find a value zero for the same case of a field without gravitation. The normal values of are then constants and their derivatives vanish.
§ 60. Using rectangular coordinates we shall now indicate the form of for the field of a spherical body, with the approximation specified in § 57. Thus we put
By (109) and (110) we find
- Of the laborious calculation it may be remarked here only that it is convenient to write the values (110) in the form
where and are infinitesimal functions of . We then find
which reduces to (111) if the relations between and , viz.
and the equality involved in (109) are taken into consideration.