268 



become variable. In such a case, the elements are those of an 

 ellipse osculating with the actual curve of motion, always of course 

 having its centre at the moveable point (XYZ). The following 

 formulae are obtained for the variation of these elements : 

 Let 



then 



l(nab) = - n((X') b cos T + ( Y') a sin T), 



b Bi 



( 2 -& 2 ) (fy + cos i ty)= -((X') b sin T + (Y f ) a cos T) + 2a b sin T cos T-, 



in which -^ and -77 are the differential coefficients of the expressions 



for and r), taken explicitly with regard to t. 



This method is denominated the method of Osculating Variation. 

 Applying the method of tangential variation to the system 



*" + *=<), &c., 



we perceive that this system admits of complete solution in finite 

 terms, leading in fact to the usual theory of elliptical motion. Taking 

 this system, therefore, as a normal system, the author proceeds to 

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

 in order to arrive at the solution of the system 



*" + *=? & c . 



The elements which have been selected, for reasons fully explained 

 in the paper, are t and \fs, 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 w denoting respectively the eccentricity of the tangential 



