. The first three sections of Newton's Principia; with an appendix, and the ninth and eleventh sections. Edited by John H. Evans . rately in the same periodic time. For if P[^V be the circle of curvature at P, the expression . QR .for F, viz. 2 limit _- , is the same in the curve and circle, and therefore what has been proved in the one case is truein the other. Hence p^encrally in any orbit described in thesame time round two centers of force. F to R : F to .S = SG RP. SP. If the periodic tinies are nut the same. ^ „ ,, . Vrr RP . SP F to R : /• to .V = „, . : P- round R P- round S Prop, VIII
. The first three sections of Newton's Principia; with an appendix, and the ninth and eleventh sections. Edited by John H. Evans . rately in the same periodic time. For if P[^V be the circle of curvature at P, the expression . QR .for F, viz. 2 limit _- , is the same in the curve and circle, and therefore what has been proved in the one case is truein the other. Hence p^encrally in any orbit described in thesame time round two centers of force. F to R : F to .S = SG RP. SP. If the periodic tinies are nut the same. ^ „ ,, . Vrr RP . SP F to R : /• to .V = „, . : P- round R P- round S Prop, VIII. To find the law of force by which a Imdymay describe a semicircle, the center of force beijig so distant,that all lines drawn from it to the body may be consideredparallel. 35 Let PQ be an arc of the semi-circle, C the center; draw PS, QSparallel to each other towards the cen- ter of force, and CM perpendicular toPS; then CM produced both wayswill determine the semicircle QT perpendicular, and QR pa-rallel to SP, and produce PR, TQto meet in Z ; join CP. The trianglesPZT, CPM are evidently QR . {RN + QN) _ RP^ _ ZP^ _ CPQT ~ QT ~ ZT ~ pW ,. QR CP ••• l^t Q^ = 2^1j^3 -> «i^e limit {RN+ QN) =2PM CP SP . PM M 1—? , and . •. oc . PM Coll. To find the velocity at any point. Tr-2 r. ^^ ^ • CP h^ . CP^SP\ PliP SP . PM PM 3—2 Mi ScHOMlM TO PkoI. \ III, //? AQP he any conic sec//(f)i, if uuiy he desrrihed hythe action of a force tending to a point at an infinite dis-tance^ and varying itwersely as the cube of the ordinate. Let PO, the diameter of curvature atP, cut the axis of the conic section in K;draw OV perpendicular to PS, then PVhthe chord of curvature at P in direction ofthe force ; and complete the constructionas in the proposition. By similar triangles ZPT, PMK, [ N M \k \ ^ --TTi^ = ZT : ZP = PM : PIP,QB QR and this being true always will be true when Q moves upto P, .-. L. R. I! : ^^ . PM : PA-,QR QR and PV PO = PM : PA, .. R
Size: 1612px × 1550px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookauthornewtonisaacsir16421727, bookcentury1800, bookdecade184
