I88i.] -*■"* [Kirkwood. 



primitive relations above developed may we not hope to trace some of the 

 unexplained facts of the solar system ? As has been remarked by an 

 eminent writer,* "the plan of the coming universe must have resided in 

 the initial chaos, as certainly as the eagle is in tlie egg, or the leviathan in 

 its primitive germ." 



I. To find the relative mean densities of the earth and moon at the 

 epoch of their separation. 



With the notation used in equation (1) the ratio sought will evidently be 



p M m 



3? ■ (a — a;)^' 



where p =r the ratio of the equatorial to the polar radius of the terrestrial 

 spheroid. The value of this ratio is not known. An approximate value 

 may be found, however, by a tentative process. 



We have a = 240,300 miles, x = 200,450, a — x = 39,850, M= 81, and 

 m= 1. Hence the ratio is 0.636o : 1. 



But during the cooling period the ratio of the densities would probably 

 be nearly constant ; or, if the moon contracted more rapidly its solidifica- 

 tion would occur earlier and the increase of its density practically cease. 

 The present ratio of the mean densities is 5.67 : 3.57, and assuming this to 

 have been constant we obtain 



0.636^ : 1 : : 5.67 : 3.57, 

 or, f) = 2.498 ; 



tliat is, the ratio of the earth's equatorial to its polar radius at the epoch 

 of the moon's separation was nearly 5 : 2, and this may be regarded with 

 some probabilitj' as nearly the ellipticitj- in other cases at the respective 

 epochs of separation. 



II. To find the relative mean densities of Jupiter and his first satellite 

 at the epoch of the latter' s origin. 



Here a = 260,000 miles, x = 254,750, a — x = .5250 ; and therefore the 

 ratio is 



59240jo 1 



(254750)3 (5250)3 ^■^'^' • ^' 



and assuming the constancy of the ratio, 



0.52 p -.1 :: 121 : 100 ; or, p = 2.33. 



This value of p is nearly equal to that found for the earth ; the difference 

 being no greater than might result from the probable error in the elements 

 used. 



The present density of Phobos is unknown ; but with p = 2.5, the value 

 found for the earth, the ratio of the original densities of Mars and Phobos 

 was 1.27 : 1. These results seem to indicate that the ratio of the equatorial 

 to the polar radius of the central mass, at the epoch of a planet's or satel- 



* Prof. Pierce. 



