100 NEWTON S PEINCIPIA. 



is reduced to that formerly explained, for finding motions 

 and trajectories when the centre is in the same plane with 

 the motion. Hence in the case first put, of the force to 

 wards S being as D, and the force towards C being, con 

 sequently, as d } it follows from what was formerly shown 

 respecting motion in the same plane, that the curve de 

 scribed on the plane of the centre C, or P B, in this case is 

 an ellipse ; that the times in which the ellipse is described 

 will be the same in whatever plane the bodies move; and 

 that if the ellipse, by lengthening its axis indefinitely, 

 becomes a straight line, the vibrations of the body in that 

 line will be performed in equal times to and from the centre 

 on both sides of it. 



By a somewhat similar process, we find the motion and 

 trajectory of a body moving on a curve surface, by a force 

 directed towards a given centre in the axis of the solid of 

 revolution which forms that curve surface. It is first 

 shown, that if from any point of the trajectory P g H on 

 the curve surface (which being a curve of double curvature 

 we shall call the double curve), a perpendicular g o be 

 drawn to the axis C S, and from any other point of the 

 axis there be drawn a line equal and parallel to g o, as C p 9 

 C p will describe areas proportional to the times. By 

 means of this proposition and the former ones respecting 

 motion in the same plane, we are enabled to find the curve 

 P p h on the plane P B E, the points of which curve are, 

 as it were, a projection on that plane of the trajectory, or 

 double curve, P g H ; and having found P p h, the double 

 curve is found by drawing perpendiculars to the plane 

 P B E, from the curve P p h to the curve surface P G E, 

 whose form is given. Thus suppose the solid to be a cy 

 linder, in \vhich case the curve P p h will be the circle which 

 is the section of the cylinder ; then if the central force 

 acts (by S being removed to an infinite distance) in lines 



