. The London, Edinburgh and Dublin philosophical magazine and journal of science. vertical plane AQD inverts into a sphere on the diameterOD, and the circle AQD into another circle AQD in aplane perpendicular to OA, so that these circles are circularsections of the cone, vertex 0 and base AQD. Now AQ is perpendicular to the plane ODQ, so that theplanes OAQ, ODQ are at right angles, and the angle AQDon the sphere in fig. 4 is a right angle. If DX is the perpendicular on the tangent at Q, QDX =QDA = in figs. 1 and 4 ; so also in fig. 4, if AY is the per-pendicular from A on the tangent at Q, QAY
. The London, Edinburgh and Dublin philosophical magazine and journal of science. vertical plane AQD inverts into a sphere on the diameterOD, and the circle AQD into another circle AQD in aplane perpendicular to OA, so that these circles are circularsections of the cone, vertex 0 and base AQD. Now AQ is perpendicular to the plane ODQ, so that theplanes OAQ, ODQ are at right angles, and the angle AQDon the sphere in fig. 4 is a right angle. If DX is the perpendicular on the tangent at Q, QDX =QDA = in figs. 1 and 4 ; so also in fig. 4, if AY is the per-pendicular from A on the tangent at Q, QAY=QAD = (/>,but is the angle QAD in fig. 2, or ADQ in fig. 1. 3. In fig. 4, by Spherical Trigonometry, (1) sin AQ = sin AD sin ADQ = k sin , cos AQ = A<£, (2) sin DQ = k sin , cos DQ = A, (3) cos AQ cos DQ = cos AD = cos c, A0 Acf>/ = k, (4) <j) = am (K—u), with = am u. Motion and Spherical Trigonometry. 7ol Fig. With EJ the polar circle o£ A, and EJ of D, and from(3) § 1, when Q makes a small advance to q, and M, M tom, m\ d Afidxf) cos DQ (5) dU = -t^t = —i-f-z = d v y A> v A /c cos c T _ spherical area QMmq cos c (7) area AEMQ _ j irea DEMQ cos c cos c (8) K = area AEJDQA cos c The point Q describes a sphero-conic, with EJ, EJ thecyclic arcs, since (9) cos AQ cos DQ = sin QM sin QM = cos c, (Salmon, Solid Geometry, Chap. X.), and the tangentXJQTJ intercepted by the cyclic arcs is bisected at Q, and 732 Sir G-. Greenliill on Pendulum cuts off a constant area UJU = 7r — 2c ; so that the anglesJUU, JUU are equal to the angle UJU7, and then JQ=QU=QU, JM=MU, JM=MU. Since AU is a quadrant and AYU a right angle, YU isa quadrant, and so also is XU ; and XU= YU, QX = QY. If DX cuts AM in W, the spherical triangles XQW,YQA are equal, and (10) DWQ=QAY=DAQ=£, DW=DA, (11) DW=DX+XW=DX+AY=DA=
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Keywords: ., bookcentury1800, bookdecade1840, booksubjectscience, bookyear1840
