. The London, Edinburgh and Dublin philosophical magazine and journal of science. sothat L and 0 are inverse points with respect to the sphere,centre g and radius ga ; thence LT : TO is a constant ratio,equal to La : aO, and similarly LQ : QO is the constant ratioLA : AO round the circle AQD, and OQ = OA . A(/>. The inverse of the circle aTd with respect to 0 is anothercircle aTd parallel to AQD ; for LT : OL=LT: OT, aconstant ratio, so that LT is constant, L being the pointinverse to L. Conversely the inverse of a system of parallels of latitudeon a sphere with respect to a point 0 on the
. The London, Edinburgh and Dublin philosophical magazine and journal of science. sothat L and 0 are inverse points with respect to the sphere,centre g and radius ga ; thence LT : TO is a constant ratio,equal to La : aO, and similarly LQ : QO is the constant ratioLA : AO round the circle AQD, and OQ = OA . A(/>. The inverse of the circle aTd with respect to 0 is anothercircle aTd parallel to AQD ; for LT : OL=LT: OT, aconstant ratio, so that LT is constant, L being the pointinverse to L. Conversely the inverse of a system of parallels of latitudeon a sphere with respect to a point 0 on the sphere is asystem of dipolar circles in a plane, as the circles of latitudeon the stereographic representation of a hemisphere. 6. The line 0/T/ from 0, the centre of the sphere on thediameter OD, makes a constant angle, c1, with OL, and theangle DOT is double the angle DOT; so that if the arcDT in fig. 5 in the representation on a sphere, centre 0, isproduced to double length to V, OV will make a constantangle c with OA, which is parallel to OL, and the arcAV = c. Fi<r. Then in fig. 5, by Spherical Trigonometry,(1) cos = J(cos AD + cos AV)=-^(cosc+ cose),a constant; so that T describes another sphero-conic, interior 734 Sir G. Greenhill on Pendul um to that described by Q, and with the same cyclic arcs, sothat its tangent QQL cuts off from the cyclic arcs a triangleUJU of constant area, aud UU is bisected at T (Salmon,Solid Geometry, §§247, 248).Bat since, in figs. 2, 3, m QT_ _ QL_ QO[-} TQ2 ~ LQX ~ OQ; OT bisects the angle QOQi, and T in fig. 5 is the midpoint ofQQi, so that (Salmon, § 252) QQX cuts off a constant areafrom the outer sphero-conic. With constant ux—u=w, the area QMM1Ql is constant,so that the spherical quadrilateral QMMjQj^ is constant, andthis implies that the sum of the angles DQQ1? DQXQ or DQVis constant, and this is found in § 8 to be am(K—w)-j-^7r. As before in fig. 4, XCJ and YU are quadrants,QY = QlK=p suppose, QlY~QX = g suppose
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