368 
equations (1) do not determine the g,, completely. To determine 
them completely an arbitrary restriction must be added, which can 
be considered as a definition of the coordinates z,..., In first 
approximation we find the ordinary mechanics according to Nrwron’s 
law, in the terms of higher order there remains an undeterminateness. 
If „we ‘take rectangular coordinates 2, SW, ds, = 2, a) =e 
(c being the velocity of light in a portion of space where there is 
no matter and no gravitation), then the gq, which determine the 
field of a sphere at rest at the origin of coordinates can be expressed 
in terms of three *) quantities «a, 8,7, which are of the first order 
of smallness. Thus : 
2 ae 
gn = (1 Ss 8) + “a (3 a at) 
Vik j 
(Ge) OT Bi RE) 4 ee 
a 
Jia == 0, WTA 1 sah 
If we introduce polar coordinates 2’; =r, 2’, = 9%, dt, =w by 
the formulae of transformation : 
Jij = 
@ — r COs YP cos 
y= r cos sind 
zr sin W 
then we find 
Ju == (dl +0) 
Ja, = — (1 + B) r° cos? yp 
733 = — (1 + 8)” 
ge == ter At. 
G,j=1,2,3) 2... (9 
The radial symmetry requires that a, ?, y are functions of 7 alone. 
The differential equations contain the quantities 7. If we neglect 
pressures ete. inside the sun, arising from the mutual gravitation of 
its parts, and if the material constituting the sun is at rest, these are 
Wi ioe 0 
je = Q (1 ++ 7) 
(i,j = 1, 2.8) 
Here o is the number of material points contained in the four- 
dimensional element of volume de,de,de,de,. We can take dv, = 
=cdt= 1, and since the matter is at rest o then becomes the 
ordinary density. 
I now write down the equations (1) of Ernstrin. Differentials with « 
respect to 7 are indicated by accents. Then I find 
' 
Ap 
ar = — EY — tal — ty’) =— tala) Tt, == Arola) B) 
1) See Droste, these Proceedings XVII (Dec. 1914) page 998. 
