July 8, 1921] 



SCIENCE 



31 



TABLH I 



Secular Motions of the Elements of the Four Inner Planets 



Computed 



Perihelia: 



Mercury . . 



Venus .... 



Earth 



Mara 



Inclinations 



Mercury . . 



Venus .... 



Mars 



Nodes : 



Mercury. . 



Venus .... 



Mars 



+ 579.2" 

 + 42.4 

 + 1161.5 

 + 1605.9 



+ 7.14" 

 + 3.87 



- 2.26 



- 753.0" 

 -1780.7 

 -2248.9 



+ 537.6" 

 + 49.7 

 + 1155.6 

 + 1597.8 



+ 6.76" 

 + 3.49 



- 2.25 



- 758.1" 

 -1790.9 

 -2249.8 



+41.6" ± 1.4" 



- 7.3 d=22.3 

 + 5.9 ± 5.6 

 + 8.1 it 2.6 



+ 0.38" ± 0.54" 



+ 0.38 ± 0.22 



- 0.01 ± 0.14 



+ 5.1" 

 +10.2 

 + 0.9 



2.8" 



2.0 



4.6 



+ 7.2% 

 -17.2 



+ 0.5 



+ 0.5 



+ 5.3% 



+ 9.8 



- 0.4 



+ 6.8% 



+ 0.6 



+ 0.0 



over by those who accept the relativity theory 

 as proved. 



In order to understand fully this question 

 of the motion of Mercury and the difficulties 

 of finding a satisfactory explanation, refer- 

 ence should be had to the secular motions of 

 the elements of the planets, as determined by 

 Newcomb. These motions are given in 

 Table I. 



The first column in the above table gives 

 the actual motions in one century as deter- 

 mined from observations of the actual planets; 

 the second column gives the corresponding 

 motions as calculated by the formulas of ce- 

 lestial mechanics, deduced from the Newtonian 

 law of gravitation. It is, however, well known 

 to every mathematical astronomer that these 

 calculations are not complete ; that they do not 

 take fully and completely into account all of 

 the bodies of the solar system. In the theories 

 and formulas upon which these calculations 

 depend, the sun has been considered as a per- 

 fect sphere and all space between the sun 

 and the various planets as free from all gravi- 

 tational matter. These are necessary mathe- 

 matical simplifications; without them the 

 equations of motion would be impossible of 

 solution. These simplifications approximate 

 very closely to the truth and the results ob- 

 tained by their use very closely represent the 

 piotions of the planets, but they are approxi- 

 mations and it, therefore, necessarily follows 

 that the results do not accurately represent 

 the actual motions. 



The column of differences contains the un- 

 explained portions of the potions of the 

 planets, together with the " probable error " 

 as determined by Newcomb. That is, in one 

 century the perihelion of Mercury moves 

 41.6" of arc more than the approximate cal- 

 culations indicate it should; whilst that of 

 Venus does not move quite as swiftly as these 

 computations would lead one to expect. These 

 unexplained portions of the motions are the 

 so-called " discordances " or " discrepancies." 

 That of the perihelion of Mercury is espe- 

 cially well known and has figured prominently 

 in all attempts to prove false the law of New- 

 ton. The perihelia of Venus and Mars show 

 large discrepancies, as do also the nodes of 

 both Mercury and Venus, 



The probable errors give some idea as to 

 the relative accuracy of the various determina- 

 tions, but it must be remembered that the as- 

 signment of these probable errors is very 

 largely a matter of judgment, and that these 

 values may have been over- or underestimated. 

 In every step of the long and complicated 

 computations an estimate, rather than an exact 

 calculation, has to be made as to the value of 

 the probable error, and the final value, as 

 given in the table, thus depends upon many 

 separate estimations or judgments. 



It is known to every astronomer that the 

 assumptions, upon which are based the simpli- 

 fications used in the calculations, are not true. 

 Neither the sun nor any one of the planets 

 is a perfect sphere. The sun-spots, which 



