16 



which is a measure for the "distance" to the centre. As to ^-j and 

 x^, we shall put x^ = cos d^, .i\ = (p, after tirst having introduced 

 polar coordinates i9^, ff (in such a way that the rectangular coor- 

 dinates are r cos vf, r sin {) cos ff, r sin 0- sin(f). It can be proved that, 

 because of the sj^mmetrj about the centre, gab = for a ^zjzzzib, 

 while we may put for the quantities gaa 

 u 



9ll = — -. i' 922 = — U (1— ^-f), .^38 = — ^'. ^44 = ^^. (80) 



1 .^j 



where u, v, id are certain functions of r. Ditferentiations of these 

 functions will be represented by accents. We now find that of the complex 

 t only the components ti\ r,' and t/ are different from zero. The 

 expressions found for them may be further simplified by properly 

 choosing r. If the distance to the centre is measured by the 

 time the light requires to be propagated from to the point in 

 question, we have w = v. One then finds 



1 / ic'^ ,. uv'^ uv' 



1 / u' u'v'\ 



1 / u" „ uv"\ 



^* = — {—2v h 2w" H . 



2>eV 2m ^ vj 



§ 49. We must assume that in the gravitation fields really existing 

 the quantities gnO have values differing very little from those which 

 belong to a field without gravitation. In this latter we should have 



U = 7'^, V = W ZZZ 1, 



and thus we put now 



u = r- (1 -\- ft), Ü z= w' = 1 + 1', 



where the quantities (i and v which depend on r are infinitely small, 

 say of the first order, and their derivatives too. Neglecting quantities 

 of the second order we find from (81) 



«,' = — (2 + 2fi + 6V + 2rV" + r'v'% 

 2k 



«,2 = — (ft — D + rn' + rv'), 



t' = — {2(j, — 2r -f Qrii' + 2rV' + r^v"). 

 2x 



For our degree of approximation we may suppose that of the 

 quantities Tab only 7\^ differs from 0. If we put 



