29 



1 i ö / do \ 



where we must give the values 1, 2, 3 to d and ƒ. 



The gravitation energy lying within a closed surface consists therefore 

 of two parts, tlie first of which is 



E^ = -—iQdx^dx^dx, ..... (114) 



while tiie second can be represented by surface integrals. If namely 

 ^1. ^2' ^z ^i'6 the direction constants of the normal drawn outward 



1 r do 



E^=--:S{abf,-) \~ gahjqedö . . . . (115) 



In the case of the infinitely feeble gravitation field represented 

 by X, n, V (§ 57) both expressions E^ and E, contain cpian titles of 

 the first order, but it can easily be verified that these cancel each 

 other in the sum, so that, as we knew already, the total energy 

 is of the second order. 



From Q = l"^ — g G and the equations of § 32 we find namely 

 do 



5 =-> \/—9i'^9''''ff^'—9''-^ff'''-9''-^9^'% . . • (116) 



09ab, fe 



so that we can write 



^, = — ƒ l/-^-^ (abfe) C2g'^^> gf'^-gV goe -gaf gbe) g^^j.^^ do. 



The factor gabj is of the first order. Thus, if we confine ourselves 

 to that order, we may take for all the other quantities these normal 

 values. Many of these are zero and we find 



2{ae) I ^«« {gaa.e — 9ae,a )qeda. . . . (117) 



Here we must take a = l,2, 3, 4; é^ = l,2, 3, while we remark 

 that for a=z e the expression between brackets vanishes. For a = 4 



rdv 



the integral becomes 1^ — qe do, which after summation with respect 



to e gives 



dv 



c 



ƒ 



.do, (118) 



On 



n representing the normal to the surface. If a and e differ from 

 each other, while neither of them is equal to 4, we can deduce 

 from (110) and (109) 



9aa,e 9ae,a ^ • 



