See — Temperature of the Sun and Ages of Stars and Nebulae. 9 



the planet becomes potential energy. Thus the velocity at 

 any instant is equivalent to that which would be produced 

 by letting the planet fall to its position from rest at a dis- 

 tance 2a. 



Substituting this value of v', we have 



E k = - 2 ^(l + w')(7- 2^) (15) 



In astronomical units k 2 expresses the mass of the sun and 

 k 2 m' the mass of the planet. Using M for k 2 in this formula 

 we may write 



/2 2 \ / 2 2 \ 



*. = ■«■•(? -») + **» (7-5?) (IB) 



Now suppose the planet at perihelion to touch the surface 

 of the sun ; then r = li, and E k will become a maximum. 

 The second term of ( 16) is very small on account of the factor 

 m' 2 ; and therefore may be disregarded. In the remaining 



term the part depending on — is small compared to that de- 

 pending on jrr, and thus we have approximately 



„ Mm' 



E * = 1T (17) 



Comparing this expression with (4) we see that 



E k : T-hm' : 3M. (18) 



But E k is the vis viva of a single planet only, and hence we 

 shall have 



i =i _ i = n 



\Ek ° ^1 5 1 





i=\ i = 



and thus for the solar system 



i = 8 



wi = - 



3 746 



2 £ « = 447T6 r < 19 » 



»=1 



