8 Trans. Acad. Sci. of St. Louis. 



which an equal mass of water would be elevated by the heat 

 annually lost by the sun, we shall obtain about eighteen mill- 

 ion years as the past duration of the sun's heat, computed on 

 the hypothesis of homogeneous density and uniform radia- 

 tion. 



It will of course be understood that the heterogeneity of 

 the actual sun renders this result merely an approximation to 

 the phenomenon of nature. The potential upon itself of a 

 sphere whose density increases towards the center is greater 

 than if the mass be homogeneous by an amount correspond- 

 ing to the potential energy given up by the particles of a 

 homogeneous sphere in falling towards the center to produce 

 the heterogeneous one. Thus the past duration of the sun is 

 really much greater than is indicated by the hypothesis of 

 homogeneity, as will be shown in the next section. 



Let us now consider the energy of the motions of the 

 planets. The vis viva of motion of revolution about the sun 



of any body of mass m, is ^ ^' v'^^, where v is the velocity; 

 and hence if Ej^ denotes the kinetic energy of a planet we 



shall have .£'*= ^ ^»'^'^- 



If Ep be the potential energy, and the system be supposed 

 to be a conservative one, as if composed of rigid bodies re- 

 volving in empty space, we shall have a constant C = Ej,-\- E^. 

 In the planetary system the orbits are of course somewhat 

 eccentric. It is evident that for any planet E^ is a maximum 

 at perihelion and a minimum at aphelion, while the potential 

 energy is just the reverse at the two points. The general 

 formula for the velocity of a planet * is 



F'' = A;Ml+»»') {,7.-^1 (14) 



where k is the Gaussian constant, r' the radius vector, and a' 

 the semi-axis major of the orbit. From this formula we see 

 that if r' = 2a', the velocity is zero, and all of the energy of 



• cf. WaUonH Theoretical Astronomy, p. 49; or any work on Celestial 

 Mechanics. 



