120 



DETERMINATION OF ELLIPTIC ORBITS. 



first and second positions, and r t for that between the second and 

 third, and set t = for the second position, 



for t = TO , 



for = 0, 

 for ^ = T X , 



dt 2 ' TV*' 



dt 2 ~ r 3 ' 



n 



We may therefore write with a high degree of approximation 





From these six equations the five constants 51, 33, 

 eliminated, leaving a single equation of the form 



where 



[, $), (S may be 

 = 0, (1) 



This we shall call our fundamental equation. In order to discuss 

 its geometrical signification, let us set 



7? \ / 7? \ / J5 \ 



jLf-t \ / -i "^"^9 A A I "1 "^"^^ \ /rt\ 



^y> 3 / ' 2 \ /y 3 / O\ /ya O / 



so that the equation will read 



This expresses that the vector 7i 2 SR 2 is the diagonal of a parallelogram 

 of which 7i 1 9? 1 and %$R 3 are sides. If we multiply by $R 3 and by 9^, 

 in skew multiplication, we get 



whence % ^ = 1 ? = - 



/io /vj /yi 



/ti /to *' 



