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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* + sin 4/ P y by the symbol P, 

 and 



cos i (cos ^P x sin i//P y ) + P^, sin t by the symbol P,, ; 

 and let 



P sin + P,, cos 0= P^ e P% sin m + P,, cos zzr= P| >w ; 

 cos + P,, sin 0=P r))e ; P^ cossr + P,, sin tBf=P, |W ; 



rcosd 



then 



*'' 



r sin 6 



= 2 _ 2 * sin(8-Br) P ff -(! + <? cos (0- 



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

 by means of the ordinary expressions for r, 6, and w 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 

 of t, and their initial values, are to be substituted in the ordinary 

 expressions for the coordinates, so as to obtain their values at the 

 time t. 



The author exhibits the application of the preceding formulae to 

 certain simple examples, and then proceeds to apply them to the 

 planetary theory. For two planets (distinguished by the suffixes 2 

 and 3) supposed to move in the same plane, the following are the 

 rigorous expressions for the variations. Let a 2 and a 3 be the ratio 

 of the mass of each planet to that of the central body. Let P denote 

 the cube of r a -s-r 23 , and let (P 1) sin (0 3 2 ) be called Q, and 



