788 



NATURE 



[February 17, 192 1 



addition of a small superposed force, outward in 

 the outer half of the orbit, and inward in the inner 

 half. These additional forces clearly cause the 

 outward or inward motion to persist a little longer 

 than it would otherwise do, thus delaying the 

 attainment of the apses. It is proved in Tait and 

 Steele's "Dynamics of a Particle," p. 127, that if 

 the force vary as r", the apsidal angle in a nearly 

 circular orbit is ir(3 + n)-l. Put n= — 2(1 +d), 

 where d. is very small, then the apsidal angle is 

 jr(i — 2d)-i = ir(i + d). Or, in a complete revolu- 

 tion, the apse advances 27rd, which is a constant 

 for all orbits. 



This was the hypothesis advanced by Prof. 

 Asaph Hall to explain the motion of Mercury's 

 apses. The shift in the case of Mercury is o-i" 

 per revolution. Applying this to the moon, it 

 would give an apsidal advance of 135" per century 

 above the amount indicated by Newtonian theory. 

 At the time Prof. Newcomb adopted the Hall 

 formula, there appeared to be such an excess of 

 motion of the lunar perigee ; but further research, 

 both on the side of theory (Prof. Brown) and on 

 that of observation (Dr. Cowell and others), has 

 emphatically proved that the actual excess is far 

 smaller, and quite consistent with the 2" per cen- 

 tury resulting from Einstein's theory. Thus Brown 

 found 14643527" and 14643511" for the theo- 

 retical centennial motion on two different assump- 

 tions of the earth's oblateness (Mon. Notices, 

 vol. Ixiv. , p. 532). His discussion gave as the 

 observed value 14643523", while Cowell found 

 14643538". In any case, the difference between 

 theory and observation is very much smaller than 

 that required by Hall's law. The latter is thus 

 definitely put out of court, and it becomes a matter 

 of regret that Newcomb adopted it in his 

 tables of the four inner planets. It meant a 

 more drastic alteration of the Newtonian law than 

 that effected by Einstein; the former alters the 

 law in all circumstances, while-the latter leaves it 

 unaltered for bodies at rest, but introduces a term 

 that increases the force where there is relative 

 motion. 



The second suggestion in explanation of the 

 motion of Mercury's apse is oblateness of the 

 sun. It is easy to show that the attraction of 

 an oblate body falls off more rapidly than the 

 inverse square, thus producing advance of the 

 apse of a satellite. Most of the satellites of the 

 solar system afford examples ; the most striking 

 case is Jupiter V., the apse of which makes two 

 entire revolutions in a terrestrial year. Where 

 the satellite does not revolve in the equatorial 

 plane of its primary, there is a second effect of 

 oblateness ; it causes the satellite's orbit-plane to 

 shift, its pole describing a circle round the planet's 

 pole. We may refer, for example, to the system 

 of Mars. H. Struve determined the position of 

 the pole of Mars and the amount of oblateness 

 from observations of its satellites ; similarly the 

 position of Neptune's pole can be approximately 

 inferred from the change in the orbit-plane of its 

 satellite. 



The amount of oblateness of the sun necessary 

 NO. 2677, VOL. 106] 



to account for the motion of Mercury's perihelion 

 is not great. Newcomb deduced that the necessary 

 excess of the equatorial diameter over the polar 

 would be slightly more than \" ; the existence of 

 even this small excess is rendered highly improb- 

 able by the very numerous measures of the solar 

 disc, notably the heliometer measures in con- 

 nection with the Venus-transits of 1874 and 1882; 

 these were discussed by Dr. Auwers, and seemed, 

 if anything, to indicate that the polar diameter is 

 the greater. A further objection is that the solar 

 equator is inclined 3° 21' to Mercury's orbit, and 

 its oblateness would produce a diminution of 2-6" 

 per century in the inclination of Mercury's orbit. 

 Observation, if anything, indicates a shift of the 

 inclination in the opposite direction, and the 

 amount 26" is so far beyond the probable error 

 as to render the theory of solar oblateness un- 

 tenable. The above points were established by 

 Newcomb, "Elements of the Four Inner Planets," 

 in 1895, so that it is strange to find this untenable 

 hypothesis still freely suggested in the United 

 States. 



It is fairly obvious that the portion of the 

 zodiacal light that is inside Mercury's orbit would 

 produce effects of the same general kind as those 

 arising from solar oblateness. Now observations 

 of the zodiacal light indicate a smaller inclination 

 to the ecliptic than 7°, the inclination of Mercury's 

 orbit. Thus J. F. J. Schmidt found values rang- 

 ing from 4O to 0°, and Prof. Douglas's photo- 

 graph, taken at Flagstaff, Arizona, on March 19, 

 1901, shows the light almost symmetrical on each 

 side of the ecliptic. Now, unless the mean plane 

 of the light agreed with that of Mercury's orbit, its 

 gravitational effect on the apse of that planet 

 would be accompanied by a shift of its orbital 

 plane, not verified by observation. It has been 

 found impossible to assign any position to hypo- 

 thetical perturbing matter that would explain the 

 apse motion of Mercury without causing other 

 anomalies in the elements of that planet and of 

 Venus, which are negatived by observation. 

 Moreover, there is the difficulty that the zodiacal 

 light, if of sufficient mass to produce such an 

 effect, should give us more light than it does. 

 Dr. H. Jeffreys examines the question in Mon. 

 Not., vol. Ixxx., p. 138. He shows that if the 

 light arises from reflection by the molecules of a 

 gas, the effect on Mercury would be only 1/3000 

 of that observed. If the light arises from re- 

 flection bv solid particles, he takes 10 km. as their 

 maximum admissible diameter (probably far in 

 excess of what is really tenable) ; he still finds 

 that their gravitational effect would be only 1/20 

 of that required. He makes similar calculations 

 for the corona, reaching like conclusions. 



Hence we seem to be driven bv exhaustion to 

 the Einstein law as the only satisfactory explana- 

 tion. It clearly can have no effect on orbital 

 planes, so it produces accordance in apsidal 

 motion without introducing other anomalies. 

 Further, it was not an ad hoc hypothesis ; it was 

 reached on independent grounds, and it was an 

 undesigned coincidence that it fitted so well for 



