178 THE MATHEMATICAL PRINCIPLES [BOOK I ellipsis, in descending from the upper to the lower apsis, describes an angle, if I may so speak, of ISO deg., the other body in a movable ellipsis, and there fore in the immovable orbit we are treating of, will in its descent from 180 the upper to the lower apsis, describe an angle VCjt? of — ^ deg. And this \/o comes to pass by reason of the likeness of this orbit which a body acted upon by an uniform centripetal force describes, and of that orbit which a body performing its circuits in a revolving ellipsis will describe in a quies cent plane. By this collation of the terms, these orbits are made similar ; not universally, indeed, but then only when they approach very near to a circular figure. A body, therefore revolving with an uniform centripetal 180 force in an orbit nearly circular, will always describe an angle of — » deg/, or v/o 103 deg., 55 m., 23 sec., at the centre; moving from the upper apsis to the lower apsis when it has once described that angle, and thence returning to the upper apsis when it has described that angle again ; and so on in infinitwn. EXAM. 2. Suppose the centripetal force to be as any power of the altiAn tude A, as, for example, An — 3, or-r^ ; where n — 3 and n signify any inA. dices of powers whatever, whether integers or fractions, rational or surd, affirmative or negative. That numerator An or T — X