SECT. 1.] TO POSITION IN THE ORBIT. 17 



15. 



Since, as we have seen, the mean anomaly M is completely determined by 

 means of v and y, in the same manner as v by 3/ and y, it is evident, that if all 

 these quantities are regarded as variable together, an equation of condition ought 

 to exist between their differential variations, the investigation of which will not 

 be superfluous. By differentiating first, equation VII., article 8, we obtain 



dE dv d9&amp;gt; 



sm-E sinf cos&amp;lt;p 



by differentiating likewise equation XII.-, it becomes 



dM=(l ecosE)AE sin E cos y d y. 

 If we eliminate d^from these differential equations we have 



r sin E (1 e cos E) , 



smt&amp;gt; 



or by substituting for sin E, 1 e cos E, their values from equations VIII., III., 



j iir rr j r (r -4- p) sm v -, 



dM= - dv -- v ~^% dcp, 



a a cos &amp;lt;p a a cos cp 



or lastly, if we express both coefficients by means of v and &amp;lt;p only, 

 * M = n 1 SS(P v dv- (2 + e v} Sin ^ s2qi dy . 



(1 -(- e cos vy (1 -j- e cos )* 



Inversely, if we consider v as a function of the quantities M, (p, the equation has 

 this form : 



cos &amp;lt;p 



or by introducing E instead of v 



^ (2 e cos E e e) sin Ed y. 



&quot;s 



