198 
motion of a body, so small that it does not produce any observable 
change in the original field. 
1. The equations for the calculation of the field can be got from 
a principle of variation. Where matter is absent (7;;== 0) the varia- . 
| l J 
tion of the integral 
{ff GY —o da, de, dx, dx, 
must be zero, if the variations of all g’s and their first derivatives 
be zero at the threedimensional limits of the fourdimensional region 
over which the integral is extended. Here G represents the quantity 
er ze (afta aa i{ tl fe 
ijk , y ; ‘| 
i Oa; (i) das i 
ij | a, ij Ogu gs: 09:5 
= > gk! ; == _ - EE 
k | l J i l | | l | ‘ (= z Ow; 02) 
For a centre at rest and symmetrical in all directions it is easily 
seen that 
id 
ds? = w? di? — u° dr? — v? (A9 + sin? dp’), . . . (2) 
w, u, v only depending on 7, and (9, p) representing polar coordi- 
nates. Now, if g;; and therefore also gij are all zero, if 1=/-j, G 
breaks up into six pieces, each of them relating to two indices. We 
collect the terms belonging to @ and 8 and name their sum Giz ug. 
Now, if a, 6, c represent three different indices, 
ab ues, aal , Waa apa |e Odea [aa sce Ògaa 
re: as Ce MK B Ee dn Be aye Cn aL 
AE Ògaa 
zi = — tg — 
¢. 
C a 
Let the first sum in (1) contribute to Gx, a3 the terms, in which. 
i=a, j=8, or i= —, j=a. By taking for @ and @ successively 
the six couples of indices and adding the expressions, we get exactly 
the first sum of (1). 
Let the second sum in (1) contribute to Grez those terms in 
‚which one of the differentiated gs contains the index a, the other 8. 
So that sum too will have been broken up into six pieces, one of 
which relates to «a and 8. 
In that way we obtain 
0 Ogas 0 Ovus es) Odun 
Grap = 9 gef JR Lg ge? dae + gÊf ge Lis + 
OMAN “LOE “ Oma" Ome Òzz Òzz 
0 5 0 Ld. 0 35 
(: ax ee), ge gef = gt ihe zak Go! to ERR (3) 
aise (Oa Oa; 
