267 



deduce the formulae for the variation of the elements of this system, 

 in order to arrive at the solution of the system 



The elements which have been selected, for reasons fully explained 

 in the paper, are I and \//, whose meanings are already known ; A 

 and Nr denoting respectively the mean distance, and the longitude 

 of the epoch measured in the plane of the tangential ellipse as it 

 exists at the time t, and measured from the node at that time ; and 

 e and & denoting respectively the eccentricity of the tangential 

 ellipse, and the longitude of its perihelion measured as above ; and 

 it is observed that these are strictly normal elements, according to 

 Professor Donkin's definition of normal elements. 



The variations of these elements are then rigorously found, and 

 are expressed as follows : Denote 



cos $ P.Z, + sin ;// Py by the symbol P, 

 and 



cos I (cos J//P.*. sin v/>P y ) + % sin I by the symbol P^ ; 

 and let 



Pf sin -j- P,, cos 0= P^e ; P$ sin or -f P,, cos cr = P )W ; 

 P| cos 6 + PI, sin 6= P^ ? e ; P^ cos or + P,, sin w = P^ )W ; 

 then 



r cos0 



r sin 61 



-^r)) P,,. 



which are capable of being expanded in terms of the elements, and t 

 by means of the ordinary expressions for r y Q, and 9 iff in terms of 

 the same quantities. The values of the elements at the time t being 

 supposed to be found, by the integration of these formulae, in terms 



