Mathematical Questions and Solutions in Continuation of the Mathematical Columns of "The Educational Times" Volume 18

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1873 edition. Excerpt: ...probable; hence the mean range is 3817. (Proposed by M. Collins, B.A.)--A body will describe a circle by a force = v2 rl directed towards its centre. Hence prove that if a circle be drawn round any other interior point C as centre of attraction, the force of attraction towards C must vary as PC-r (the cube of the distance of P from the polar of C); P being any position of the point moving in the circumference; and show that this last property is true for all conic sections. Solution by J. J. Walker, M.A. 1. Let CT be the perpendicular on the tangent at P, and O the centre of the circle. If f be the force along PC, its component along PO must be v'2 CT 1 CP equal to--; that is, fx--oc----, or fee cp op. But in the circle CT is to the perpendicular from P on the polar of C as CO to radius; therefore, &c. 2. Let K be the centre of a conic, and KL, KM the semi-diameters conjugate to the directions KC, KP. If O be the centre of curvature at CP CP P, from above, fx Qp cc CT3 gM3-But & is readily shown that the perpendicular from P on the polar of C, a fixed point, is proportional to CT. KM; thus is proved the last elegant extension of the theorem, which is due to the late Sir Wm. Rowan Hamilton. II. Solution by the Proposer. The force urging P towards C multiplied by cos CPO is equal to the force urging the moving body P towards the centre O = 1 =; B Kp2 hence the force towards C Bp2 cos CPO Now let OC. OC= E2 = OP2; then CT perpendicular to OCC and to PT is polar of C; then since OC: OP = OP: OC, the angle OPC = OCP; that is, / PCY = CPT; hence the triangle CPY is similar to PCT; therefore CY: PT = PC: PC, which is constant, since C and C are inverse points relative to the circle. Hence the force acting on P towards C, and causing P to...show more

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• Paperback | 30 pages
• 189 x 246 x 2mm | 73g
• United States
• English
• black & white illustrations
• 1236976568
• 9781236976567