° 
199 
The equations of the field being covariant for all transformations 
of the coordinates whatever, we are at liberty to choose instead of 
r a new variable which will be such a function of 7, that in ds? 
the coefficient of the square of its differential becomes unity. That 
new variable we name 7 again and we put 
ds? = w° dt? — dr’ — v* (dd + sin? B dp°) . . . . (4) 
w and v only depending on 7. We now find 
408 i HDE 4v'w' 4 4p” 
Gy — —, Gs = Gr= en pn ey Gip= - = rr il 
w v vw v v 
In these equations accents represent differentiations with respect 
to 7. So 
A Ap? Syl! Bo. Aw! 
RS Ln 
vy v vw v Ww 
Now, as } ge wsin dS, the function to be integrated in the 
principle of variation becomes 
4 (w—wv" — 2vv'w'—2vwv" —v?w") sin D-. 
We now apply the principle to the region &, SESt, r, Sr Sr, 
By effecting the integrations with respect to 7, ® and g we find 
the condition 
ra 
afc -- wo? — Zvw — 2vwv'—v?w") dr = 0, 
ry 
This gives us 
ENDS Te ete oh eee A 1:2 
and 
Et Ve 
These are the equations of the field required. 
2. To solve (6), we introduce instead of r the quantity w =v 
as an independent variable by which, on taking account of (5), (6) 
changes into 
dw dw - 
2a + Qi 0. 
ow 
da? da 
This equation is satisfied by wa. The other particular solution 
is now also easily found, viz. 
(1 —z*) 
lg 
w= 1 —ta log ——. 
+2 
But we want w to be a finite constant if v =1 (for * = oo). 
Then w must be equal to x, if we take the constant to be 1 (the 
speed of light then approaches to 1 at large distances from the 
centre). 
