28 



,■.=-.■•+; (A-,) 



r 



(111) 



Thus we see (comp. § 58) that at a distance from the attracting 



sphere t'/ decreases proportionally' to — . Further it is to be noticed 



that on account of the indefiniteness pointed out in § 58, there 

 remains some uncertainty as to the distribution of the energy over 

 the space, but that nevertheless the total energy of the gravitation field 



E = é7t ft'/r'dr 



= '"!'' 



has a definite value. 



Indeed, by the integration the last terra of (111) vanishes. After 

 multiplication by r'' this term becomes namely 



(A — i^y 4- 2r {X - ft) (X — (i') = ^ [r (A - iiYl 



dr 



The integral of this expression is because (comp. §§ 57 and 58) 

 r{X—liy is continuous at the surface of the sphere and vanishes 

 both for r = and for r = oo. 



We have thus 



00 



jrc r 

 E^z—iv'-'r^dr, (112) 







where the value (107) can be substituted for v'. If e.g. the density 



Q is everywhere the same all over the sphere, we have at an internal 

 point 



V =: \ TiQr 



and at an external point 



- a* 



From this we Dnd 



E z= ^ jt C7i{) a*. 



§ 61. The general equation (99) found for t'\ can be transformed 

 in a simple way. We have namely 



^^, ^^ d / ÖQ \ d / ÖQ \ 



^ (a6/e) _ g^^ J. — ^ (ab/e) ~- gaf,,/ — 



ÖXe \Ogab,feJ OXg \Ogab,fe J 



— 2 (ab/e) gabje 



Ogab,fe 



and we may write — Q^ (§ 54) for the last term. Hence 



