. The first three sections of Newton's Principia; with an appendix, and the ninth and eleventh sections. Edited by John H. Evans . SECTION IX. ON THE rOSITION OF THE APSIDES IN ORBITS VERYNEARLY CIRCULAR. Prop. XLIII. The orbit i7i which a body moves re-volves round the center of force with an angular velocity^which always bears a fixed ratio to that of the body ; to shewthat the body may be made to move in the revolving orbitin the same man7ier as in the orbit at rest by the action of aforce tending to the same center. Let C be the center of force, andwhen the body in the fixed orbit VCPhas d
. The first three sections of Newton's Principia; with an appendix, and the ninth and eleventh sections. Edited by John H. Evans . SECTION IX. ON THE rOSITION OF THE APSIDES IN ORBITS VERYNEARLY CIRCULAR. Prop. XLIII. The orbit i7i which a body moves re-volves round the center of force with an angular velocity^which always bears a fixed ratio to that of the body ; to shewthat the body may be made to move in the revolving orbitin the same man7ier as in the orbit at rest by the action of aforce tending to the same center. Let C be the center of force, andwhen the body in the fixed orbit VCPhas described the arc VP, let vCp bethe position of the revolving orbit, andp that of the body moving in it; thenZ vCp ^ ^ VCP. Also let the angu-lar velocity of the orbit be to that ofPasG- F : F. The angles VCv, TCP begin to- ^ gether at F, and their contemporary increments are as the angular velocities of Cu and CP, that is, as G — F : F,therefore the anoles themselves arc in that ratio, or. VCv : VCP or vCp ^ G - F : F;.-. componendo VCp : VCP = G : F; hence, if the angle VCp be always taken = t7^ angle VCP, r G4 and C]t = CP, Vj) tlie locus of jt will \)l- the curve tracedout in fixed space by a body p moving in the revolvingorbit in the same manner as P in the fixed orbit. Also the body may describe the orbit Vp by the actionof a force placed in C For let PCK, p Ck be the areas described by CP, Cp inthe same small increment of time; draw AT, kt perpendi-cular to CP, Cp; then the contemporary increments of theareas, described by p and P, are ultimately as Cp . kt : CP . KT = Cp. smpCk : CP. sin PCX = /. pCk : z PCX = L vel. of r/> : z vel. of CP = G : F, and the whole areas begin together at F, therefore they arethemselves in the same ratio; hence area VCp oc area VCPoc the time (Prop. ]); and therefore (Prop. 2) a body maybe made to move in the orbit Vp by a proper centripetal forceplaced in C. Def. An apse or apside is a point in an orbit at whichthe direction of t
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Keywords: ., bookauthornewtonisaacsir16421727, bookcentury1800, bookdecade184
