raf ree a; i AD 2; EN air ai a 
— NIE = Km. § ——— -—— 5 ye) 
dt r° r! 7? y* egt 5) 
and for polar coordinates 
dr 4 dy \* a \* | ktm / 4A? 7 q° 
— — r cos* wp} — - 7 -) +——=hk’m + 4. — - 
dt? dt dt a Th r eae 
d9 2drdd dd dy igor 
- -—— — 2 tan =4k'm —, nn 
dt? rdt dt dt dt r [ 
Gy 2 drdw Ad \* ; mp | 
+ —— — + sin Wp cos Wp -) = 4kh?m—_, 
dt? r dt dt dt 2 
where we have put 
D= ae u er? + rt cos? wo? + rab. 
The right-hand members are of the second order. The left-hand 
members put equal to zero give the motion according to NewTon’s law. 
3. Planetary motion. 
From the third equation (21) it follows, that when once 7 = 0 
and ¢ = 0, this always remains true: the orbit is plane. Then we 
have, accurate to the second order 
es aia FO BG oe pe, Seo ete ee (AK) 
. 
This equation is general: if we introduce the special values of 
[ 
8 and 7 which are here used, it is reduced to the second of (21). 
We now put 
"sG. 
Then the integral of (22) is 
Ge rn veh en NA 
This equation replaces the integral of areas. *) | now put 
EE AR 
Then, with our values of 3 and y we find 
de . | Sa. 
1) If we put 7 DES G. ds being the element of the world-line, i.e. the proper- 
ds 
time of the planet, then we find 
Gak. 
If we take B—=0, as in Einstein's system, then the law of areas is exact if 
the proper-time of the planet is taken as independent variable, as has already 
been remarked by Erster (I. c. page 837). 
