Mathematical Questions with Their Solutions, from the "Educational Times" Volume 66 (Paperback)

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. 1868 edition. Excerpt: ...it in A, B, C, D. Prove that the three opposite pairs of sides of the tetragon ABCD will respectively intersect in I, Q, R. Solution by F. D. ThoMSoN, M.A. I is the point of inflexion, P, Q, R the points of contact of the three tangents from it, A, B, C, D the points of contact of the four tangents from P. Then since III is a straight line, the satellites PQR are in a straight line. Again, since PAA, and also PBB, are in a straight line, therefore the satellites of PP, AB, AB are in a straight line. But I is the satellite of PP or P; therefore if x be the satellite of AB, I is the satelhte of x. But the only points of which I is the satellite are I, P, Q, R; therefore x must coincide with I, Q, or R. Similarly, each of the six chords joining A, B, C, D passes through either I, Q, or R. If, for example, AB passes through I, BC through R, and therefore BD through Q, then the satellites of AC, BB, IR are in a straight line; but PQR is a straight line, hence AC passes through Q. Hence BD.CA meet in Q; AD, BC in R; and BA, CD in I. 2694. (Proposed by G. O. Hanlon.)--A rope without weight passes over a smooth pulley, having a monkey at one end and a weight equal to the monkey at the other. The monkey suddenly runs up the rope to a height h with a uniform velocity; what is the resulting motion of the weight? Solution ly Septimus Tebat, B.A. Let P be the mass of the monkey, Q the mass of the weight at the other end of the rope, x the space through which Q has ascended at the time t, and T the tension of the rope. Then if the monkey begins to ascend with a velocity a, its position at the time t will be at--x. Now tho accelerating forces ou P, Q are?, i9. 2614. (Proposed by Capt. Clarke, R.E., P.R.S.)--Three points are taken at random, one on each...