Mechanical Energies of the Solar System, 423 



distance at the same time ; and 2 h the constant area described 

 by its radius vector per second. Then we have — 



M 



a)^a=-^ (centrifugal force), 



a)a^=h (equable description of areas) ; 

 from which we deduce, 



and M2 



Now, if M(j denote the mass of the sun at the epoch from which 

 time is reckoned, since the annual augmentation is about 

 72^ oVolTo ^^ *^^ ™^^® itself, we have 



and 



^"~^n^ "^32,400,000/' 

 ^"""^^^^V^"^ 32,400,000/' 



Hence, if Hq and Ht denote the angular velocities at the epoch 

 and at the present time, T ; the angular velocity, which is uni- 

 formly accelerated during the interval, will have a mean value, 

 ii, expressed as follows : — 



n=i(n„+a.) ="T{i-i^«}=n.(i-3^^^) ; 



and if S denote the angle described in the time T, we have 

 / T^ \ 



^^^H''' ""32,400,000/- 



To test this conclusion for the case of the earth, let T' denote 

 the number of revolutions round the sun in the time T. Then, 

 if the unit in which T is measured be the time of a revolution 

 with the angular velocity Ot^ we have 



rvl rp ^ 



~ 32,400,000* 

 Thus, if T be 4000 years, we have 



or only 3999^ actual years in a period of 4000 times the present 



