124 SEE— THE PAST HISTORY OF THE EARTH. [April 23, 



fine cosmical dust. In the later periods of the earth's history, the 

 medium has been so rare that but little matter has been added to 

 our globe; so that not only is the whole history very long, but the 

 latter part longer than the earlier part, as measured by the accretion 

 then going on. In other words, the accretion now taking place is 

 so slow as to give us by calculation, based on the observed rate, an 

 exorbitant age of the earth; while that once going on was so large 

 as to give too short a duration for the genesis of our planet. All 

 estimates on the age of the earth must therefore be subject to a 

 wide margin of uncertainty. But we may feel entirely confident 

 that we have at length recognized the true process by which the 

 earth was formed. 



There is, however, a modifying cause which should be taken 

 into account, in our final judgment of the process involved. It 

 cannot be assumed that the sun was of its present mass at the start; 

 on the contrary, we must suppose this mass to have steadily in- 

 creased. The result of the augmentation of the sun's mass would 

 be a decrease in the length of the year. Thus while the resisting 

 medium reduced the major axis and eccentricity of the planetary 

 orbits, the growth of the sun's mass also shortened the periodic 

 times, without, however, decreasing the mean distance of these 

 masses to any appreciable extent.^ 



In the actual history of our system, these two causes have there- 

 fore conspired together and the results now observed must be 

 ascribed to both causes combined. If we wish to inquire at what 

 rate a change of a given percentage in the sun's mass would affect 

 the length of the year, we may proceed as follows. By a well known 

 law for circular motion we have 



M^m = ^. (3) 



If we differentiate this expression, considering M and t alone to be 

 variable, we shall get 



dM{t'') + {M^m')2tdt = o, 

 or 



dM _ 2dt 



^ Cf. Laplace, " Mecanique Celeste," Liv. X., Chap. VH., § 21. 



