348 SEE— THE EULER-LAPLACE THEOREM. [April 24, 



In any revolution about the sun the perihehon advances by the 

 interval 



27r(l -h U)gg 



and the aphelion regresses by the interval 



27r(l - U)^g 



the mean distance therefore decreases in the interval about 27rgg/c ; 

 and after i revolutions this decrease in the mean distance will be 

 2i7rgg/c. 



Accordingly, after i planetary revolutions, the perihelion distance 

 from the sun becomes : 



g _ 2^7r(i + jDgg 



I + r c(i + ty ' 



and the following aphelion distance : 



I - r c(i - tr 



The addition of these values, after i revolutions, effects the 

 transverse axis of the orbit : 



2g 4^7rgg x(i - U)sg 



I - rr c(i - ^tr- ^i - ly • 



Here indeed, since the time is to be defined, the time from perihelion 

 to aphelion may be omitted ; and thus after i revolutions the trans- 

 verse axis of the orbit is found to be : 



2g 4^7rgg 



I - ^^ c{i - rr)^ ' 



wherefore also the initial transverse axis is assumed equal to 



2g/{i—a). 



If, therefore, the distance from the perihelion to the sun after i 

 revolutions, which is equal to 



I + r c(i + r)' ' 



