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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 



x=\ a aco8(nt+p) + p a bsm(nt+p), 

 y=X A a cos (nt + p) +/z fi b sin (nt+p), 



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

 where 



A = cos cos $ sin fy sin \// cos t, 



\ b =. cos sin ^ + sin ^ cos $ cos i, 

 \ c = sin sin t ; 



/*= sin <f> cos \f/ cos sin v// cos t, 

 /U4= sin ^ sin i// + cos cos \f> cos , 

 /u c = cos ^ sin t i 



to which are afterwards added, 



v a = sin ^ sin t, 



yj= cos -fy sin t, 



v c = cos i. 



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

 and situated in a plane inclined at the angle t 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 fora?, &c. in the above equations 

 of motion and solutions. 



It is then shown that a system of the form 



#" + A=P*, &c., 

 where w 2 and P,, P v> and P* are any variables, may be solved by the 



