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of motion about a fixed centre of force, where the force is directly as 

 the distance ; or, in other words, the system of equations not ex- 

 ceeding three in number, of the form 



whose solutions are represented under the form 



a?=X a a cos (nt+p) +fji a b sin (nt+p), 

 y=\b a cos (nt + p) + p b b sin (nt + p), 

 z=\ c a cos (nt+p) +p c b sin (nt + p) ; 



where 



A a = cos cos ^ sin sin ^ cos /, 

 Aj= cos sin ^ -f sin <f> cos \fr cos I, 

 X c = sin sin / ; 



/i a = sin cos ^ cos </> sin // cos , 



/L*6= sin sin ^/ + cos <f> cos \L cos /, 



/u c = cos sin / ; 



to which are afterwards added, 



v u = sin ^/ sin /, 



vj= cos \]s sin /, 



v c cos Z. 



These are the equations of an ellipse whose centre is at the force, 

 and situated in a plane inclined at the angle I to the plane of x y, and 

 the longitude of whose node is \$> ; and (f> is the angular distance of 

 the major axis of the ellipse from the node ; a and b are the semi- 

 axes of the ellipse ; and p is the angular distance, from the major 

 axis, of the zero-point of the motion, measured on the circle described 

 on the major axis. A uniform motion around the circle represents 

 the place of the body by the corresponding point on the ellipse, 

 where it is cut by a perpendicular dropped on the major axis. 



If the force be not situated at the origin, but at the point (X, Y, Z), 

 we have merely to substitute x X for x, &c. in the above equations 

 of motion and solutions. 



Applying the method of tangential variation to the system 



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 



