. The first three sections of Newton's Principia; with an appendix, and the ninth and eleventh sections. Edited by John H. Evans . the cubes of the maior axes. ^ Prop. XVI. To find the velocity at any point of aconic section y described about a center of force in the focus. Let V be the velocity at the point P, ^ SP 2 Now in the ellipse and hyperbola, PV CD- I SP = ^-i>.^.^ AC .AC V . AC j w a 11(1 in tliL parabola, Hence in the ellipse V^ >/^(^2 -^j , ]> r ^^^ ^ in hyperbola V = V^fs + ^) , in parabola V -^ \/ ^ - . Cor. To compare the velocity at P witii that of a bodymoving i
. The first three sections of Newton's Principia; with an appendix, and the ninth and eleventh sections. Edited by John H. Evans . the cubes of the maior axes. ^ Prop. XVI. To find the velocity at any point of aconic section y described about a center of force in the focus. Let V be the velocity at the point P, ^ SP 2 Now in the ellipse and hyperbola, PV CD- I SP = ^-i>.^.^ AC .AC V . AC j w a 11(1 in tliL parabola, Hence in the ellipse V^ >/^(^2 -^j , ]> r ^^^ ^ in hyperbola V = V^fs + ^) , in parabola V -^ \/ ^ - . Cor. To compare the velocity at P witii that of a bodymoving in a circle, radius = SP, and described round thesame center of force. Let U = velocity in the circle,then (Prop. vi. Cor. 5), V / SP — .-. in ellipse — = \/2 - —^ which is less than \/^2, ~n hyperbola— = V 2 + -^ greater y _ in parabola — = \/2. APPENDIX. Note to Lemma II. 1. To find the area of a plane curve. Let the area ABC be boundedby the curve AC and the straightlines AB, BC. Let AB be dividedinto n equal parts, and let MN bethe r^ part from A; draw MP, NQparallel to BC, and complete the pa-rallelogram Let AB = h, then MN = - ,n z ABC = i, M N- area of parallelogram RN = - w, sin i. n Therefore giving to r the values 1, 2, , the sum of theparallelograms described on all the parts ^ •/ V Therefore area of curvilinear figure = h sin i. limit 2 — when 11 is 50 Ex. 1. T(i fiiul the urea of a portion of a j)aral)olacut off by a tliaimtcr and one of its ordinatcs. Let yiBC be the parabolic area ^cut off l)y the (lianictcr JB and asemi-ordinate BC Complete the ^^parallelogram ABCD\ then AD isa tangent at A. Let AD = h, AB = k, and let aAN be the abscissa, and XQ, paral-lel to AB, the ordinate to tlie point Q; then by a propertyof the parabola,
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