jSee — 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- 

 tsmce 2a'. 



Substituting this value of v\ we have 



I!,= lm'k^(l + m-)0-l^) (15) 



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

 ki^m' the mass of the planet. Using M for Jc^ in this formula 

 we may write 



/ 2 2 \ / 2 2 \ 



^' = ^-' (? - 2^') + ^^-^7 -2^-) (1«) 



Now suppose the planet at perihelion to touch the surface 

 of the sun; then r' = B, and JS^. will become a maximum. 

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

 wi'*; and therefore may* be disregarded. In the remaining 



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

 pending on ^, and thus we have approximately 



^. = ^' (17) 



Comparing this expression with (4) we see that 



i:,: ^=5m' : SM. (18) 



But Sj^ is the vis viva of a single planet only, and hence we 

 shall have 



^ r - SM^"^' ~ 3 746 * 

 and thus for the solar system 



1 = 8 



2^" = 44Y76^ (19) 



»=1 



