so that of the stress-energy-components of the matter only one is different from zero, namely

Further (66) involves that, also of the quantities , only one, namely , is not equal to zero. As we may put we have namely

Finally we are led to the three differential equations

(104) |

(105) |

(106) |

It may be remarked that , represents the "mass" present in the element of volume . Because of the meaning of (§ 48) the mass in the shell between spheres with radii and is found when is integrated with respect to between the limits —1 and +1 and with respect to between 0 and . As depends on only, this latter mass becomes , so that is connected with the "density" in the ordinary sense of the word, which will be called , by the equation

The differential equations also hold outside the sphere if is put equal to zero. We can first imagine to change gradually to near the surface and then treat the abrupt change as a limiting case.

In all the preceding considerations we have tacitly supposed the second derivatives of the quantities to have everywhere finite values. Therefore and will be continuous at the surface, even in the case of an abrupt change.

§ 58. Equation (106) gives

(107) |

where the integration constant is determined by the consideration that for all the quantities and their derivatives must be finite, so that for the product must be zero. As it is natural to suppose that at an infinite distance vanishes, we find further