11 



obtained by differentiating the factor \^ — g only and the other part 

 by keeping this factor constant. 



For the calculation of the first of these parts we can use the 

 relation 



d%(l/=^) 



._ i 



gac (70) 



and for the second part we find 



è V^:E{pq)gP9^ap^cq- 

 If (32) 1915 is used (67) and (68) finally become 



2 {q) y^cq^^cq + ^ («) ^ac^a'c' = 2L, 

 2 (q) X^Jcq ^^bq -j- -^(a)lp«6 V'a'r' = . 



These equations are really fulfilled. This is evident from : rpan = 0, 



\^jan = 0, xl'ba = — ^af> and %pf,a = — ^ah', besides, the meaning of rpU 

 (§ iJ, 1915) and equation (35) 1915 must be taken into consideration. 



^ 43. In nearly the same way we can treat the gravitation field 

 of a system of incoherent material points ; here the quantities lOa 

 and iia (§§ 4 and 5, 1915) play a similar part as \^af> and tf'a^ in 

 what precedes. To consider a more general case we can suppose 

 "molecular forces" to act between the material points (which we 

 assume to be equal to each other) ; in such a way that in ordinary 

 mechanics we should ascribe to the system a potential energy 

 depending on the density only.. Conforming to this we shall add 

 to the Lagrangian function L (§ 4, 1915) a term which is some 

 function of the density of the matter at the point P of the field- 

 figure, such as that density is when b}' a transformation the matter 

 at that point has been brought to rest. This can also be expressed 

 as follows. Let dn be an infinitely small three-dimensional extension 

 expressed in natural units, which at the point P is perpendicular to 

 the world-line passing through that point, and o da the number of 

 points where da intersects world-lines. The contribution of an element 

 of the field-figure to the principal function will then be found by 

 multiplying the magnitude of that element expressed in natural units 

 by a function of q. Further calculation teaches us that the term 

 to be added to L must have the form 



