Mat 26, 1922] 



SCIENCE 



571 



If we write 

 (2) / = 



VO-t) 



,■2(92 + sinCeqjS) 



where dots indicate derivatives with respect to 

 s, the world-lines of particles in the gravita- 

 tional field, are the curves in the 4-spaee for 

 which the integral 

 (3) fids 



is stationary. The conditions that (3) be sta- 

 tionary are four differential equations of the 

 second order. From one of them it follows 

 that the path is plane; the coordinates may be 

 chosen so that the equation of the plane is 

 6 = 0. Two of the other equations admit as 

 first integrals 



(5) 



ds 



where A and h are constants. It is readily 

 shown that I = k, a constant, is a first integral 

 of the four equations. When k is not zero, s 

 can be chosen so that k = 1. Then we have 

 6 = 0, (4), (5), and 



(A-^ 



'ds 



2m /42 



1) + + 2m— 



for the equations of a world-line of a particle 

 in the gravitational field. 



When 7 = 0, the integral (3) is stationary 

 and the corresponding world-lines are those of 

 light in accordance with the Einstein theory. 

 Their equations are (4), (5) and 



(7) \-) +r-'KT) =^= + 2".- 

 ^ ds ^ds )-3 



Some writers have obtained these equations by 

 solving 1=0 for dt and expressing the con- 

 dition that fdt be stationary, in accordance 

 with the Fermat principle. The above method 

 was given by Professor Veblen in his lectures, 

 and appears also in Laue, Die Relativitats- 

 theorie, Vol. 2, p. 225. Putting / = in (2), 

 we see that the units are such that the velocity 

 of light is unity for r = co, and that it dimin- 

 ishes as the light approaches the sun. If the 



unit of length is taken as a kilometer, then the 

 unit of time is 1/300,000 of a second. 



3. In classical mechanics for a central force 

 of attraction f(r) the equations are 

 ,^d(f 

 dt 



n 



(Sr+^X, 



f{r)dr = E, 



(8) 

 and 



^ dt f N dt 



where h and E are constants. For planetary 

 motion about the sun, whose mass in gravita- 

 tional units is denoted by m, equation (9) as- 

 sumes the form _ 



(dr\~ (dm\- m 2m 



where a is the semi-major axis. For the solar 

 system m/a and m/r are of the order of IQ— ', 

 for the units previously defined. Thus if we 

 identify Ar — 1 in (6) with — m/a in (10), 

 A — 1 = 4 10~* approximately. Then from 



dt 

 (4), — = 1 -f 3/2 10~^ approximately, which 



ds 

 shows the order of discrepancy so far as the 



solar system is concerned in interpreting ds and 

 At as the same in (5), (6), (8) and (9) (c/. 

 Eddington, Report, p. 50). 



It is well-known that it is the term 2mh-/r^ 

 in (6) which accounts for the motion of the 

 perihelion of Mercury. Comparing (6) and 

 (9), we see that from the point of view of 

 action at a distance this is accounted for if we 

 take 



/ 1 37j= \ 



(11) f{r) = m(--^ I . 



From the preceding remarks it follows that if 

 we put 



dm 



(12) "^ = T^ 



ds 



then a) may Ije interpreted as the angular ve- 

 locity of the planet about the sun. Tlien from 

 (5), (11) and (12) we have that 



(13) The attraction =z 





= —(1 + 3-«=), 



where v is the component of the velocity per- 

 pendicular to the radius vector. 



We have remarked in the preceding that the 

 velocity of light at co is equal to 1 in the units 



