Page:The Mathematical Principles of Natural Philosophy - 1729 - Volume 1.djvu/279

 prop. 38) they will in equal times either decribe ellipes in that plane about the centre C, or move to and fro in right lines paing through the centre C in that plane; compleating the ame periods of time in all caes.  Q. E. D.

The acent and decent of bodies in curve uperficies has a near relation to thee motions we have been peaking of. Imagine curve lines to be decribed on any plane, and to revolve about any given axes paing through the centre of force, and by that revolution to decribe curve uperficies; and that the bodies move in uch ort that their centres may be always found in thoe uperficies. If thoe bodies reciprocate to and fro with an oblique acent and decent; their motions will be performed in planes paing through the axis, and therefore in the curve lines by whoe revolution thoe curve uperficies were generated. In thoe caes therefore it will be ufficient to conider the motion in thoe curve lenes.

If a wheel ands upon the out-ide of a globe at right angles thereto, and revolving about its own axis goes forward in a great circle; the length of the curvilinear path which any point, given in the perimeter of the wheel,