1240 



We shall however shorten the calcnlatioii by application of the 

 variation principle in the form (28) I. In these variations we must 

 keep <p and q constant and vary il, in and v = rp. On the right- 

 hand side of (28)1 we have of course to introduce ^Dï = '^ïW = >XK"'), 

 so that this form is split up into two parts which we shall consider 

 separately. First we consider the part containing '':SR'"). We then 

 obtain, attending to (4rz) and (5), 



r\ '1 'i 



'2 'a 



ƒ__ Cuwe^ f ÖU 6v 6ro\ 



m<^) r' dr = 2 — 4- — dr. ... (6) 

 J 4 jrr" \^ M V 10 J 



As to the part containing 3)?'^'"^ we still have our former formula 

 (36) I, if only we keep in mind that, W being replaced by ^DZ('«\ 5: 

 gets the significance of a stress-energy-tensor for the matter, if the 

 electric field is considered as not belonging to the matter. In this 

 ^ % will keep this altered signilicance. 



Considering the equation (31)1 and (6) together with (36)1, in 

 which ^33K"') has been introduced instead of ^^, we obtain for the 

 variation formula: 



(7) 



r I uj u" -f 2u V iv' i r{ fuw e- ^r \ ffu 



'l '•l 



/ wo e^ ^i, \ <fv /moe* ^4^^'") 



\ 4:t V / ^ yiJi v j w ) 



Executing the variations we find the following set of equations, 

 which take the place of the set (38) 1 



W* -j- 2 VVIO Y. ( HW e" ,- \ 



U U \V>71 V^ J 



lov" -\- v'w' -\- mo" u' X / mo e? i,\ \ 



+ (W + i..')- ^ --— , + r' ^\, \. (8) 



u u v\ ^nv / [ 



2üv" 4- u' m' X f wo e* ^4 

 + « + 2.t.'- = - ^ + r-t4 



U U 10 \OJl V 



These equations determine the gravitation field, when the tensor 

 Ï for the matter and the distribution of the electricity are given ^). 



1) Comparing this set of equations with (38)1 we see from the right-hand sides 

 of (8), how the components of the stress-energy-tensor for the electro-magnetic 



