J 084 



All quantities g''' and X,-/ for which /ti =|= r are equal to zero. 

 Consequently we have for points on the A\-axis 



du 11^ ' öt' öü r' ' bit' ?c' 



the other derivatives oi' y''' with respect to m, v, iü are zero. According 

 to the formulae (33) equation (2) gives 



öitlï d^^l 1 ^ d^l ÖW 



r' 



further we have because of (35) 



All these equations hold for points on the Jf,-axis and consequent!}^ 

 the two first equations (32) are valid for these points. The general 

 validity of the equations follows from the above. The proof for the 

 third formula (32) is given in the same way ; this latter proof directly 

 holds for points not on the A^-axis, as every where </'' = ^" = g^* = 0. 



Because of the equations (32) the righthand side of f28) can be 

 written in the form 



r, rj 



xcf j ^^? r' dr = ^^ 1 ( 5:/— + 2 l^f — + 5:/ — j r' dr. . (36) 



r, '-I 



Introducing the expressions (31 j and (36) for both sides of equation 

 (28) and dividing by 2 the variation principle for a field with 

 spherical symmetry finally becomes 



di \-uicldr=xn i,'— f 2 i^/ — ^- i/— JrV/r. (37) 



'■i ' n 



As the variations óii,(fv,(fiü are independent of each other, and 

 as u, V, ID, v', w' are not varied at the limits i\ and i\, we find 

 (comp. Droste, Het zwaartekrachtsveld, equations (24) which hold 

 for the field outside the matter) 



w ü'' 4" 2u v'w' r" ^ 

 \- 10-=^ — y. i;'", 



tov" -^vw -^ vw" ^ , ,u' r"" ^f, 



-\-{viv +wv')—=-x%p ) . . (38) 



U U' V 



2vv" 



- ^u-\- 2vv'— = -xZ,* 

 u u xo 



These equations are the fundamental formulae for a gravitation 



