P. HooTce — Law of Densities of Planetary Bodies. 399 



An interesting application of the law of density, under the 

 adopted hypothesis, is to determine, by computation, the ulti- 

 mate diameters and mean densities of the sun and the outer 

 planets, that is, the diameters and mean densities they will 

 have when they have become solid like the earth and moon. 

 That they will ultimately reach this condition, if they undergo 

 no other change except that due to the radiation of heat, can- 

 not be questioned. The application referred to will, first, be 

 made to the sun. 



Let d be the diameter of an hypothetical planet of the class 

 to which the inner planets have been assumed to belong, and 

 whose mean density is 2P, P being its surface density, and 

 let the diameter of this hypothetical planet, which is found by 

 the law of density to be 6632 miles, be taken as the unit of 

 diameter, and its mass as the unit of mass. Now if we make 

 M the mass of another planetary body whose diameter is 1+x, 

 we have, by the law of density 



M= (1 +£C) 3^1-? 



or 2M=2 + 7x-\- 9z 2 +5cc 3 +a: 4 . 



Making If the ratio of the mass of the sun to that of the 

 hypothetical planet (610400) which is taken as the unit of mass, 

 and finding the root of the equation, we have 



l+x= 1+31-9928 = 32-9928, 



from which we get the following value for the ultimate diame- 

 ter of the sun, in miles : 



32-9928x6632=218808; 



and for the sun's mean density we have from equation (3) 



218808 k+P=8l-69. 



The same application of the law of density has been made 

 to the outer planets, and the following are the results obtained 

 for their ultimate diameters and mean densities, the list being- 

 headed with the values for the sun for the purpose of com- 

 parison : 





Ultimate diameters. 



Ultimate mean deosities, 



Sun 



218,808 miles. 



8769 



Jupiter 



37,183 



17-04 



Saturn 



27,128 



13-13 



Ui-anus 



16,552 



9-00 



Neptune 



17,220 



9-28 



The foregoing results are based on the following values of 

 the masses of the sun and of the planets mentioned : — 



