Mathematical Questions and Solutions, from the "Educational Times": With Many Papers and Solutions in Addition to Those Published in the "Educational Times", Volume 15

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W. J. C. Miller
Hodgson, 1871
 

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Página xi - Show that the lines joining the middle points of the opposite sides of the quadrilateral whose vertices are (6, 8), (— 4, 0), (— 2, — 6), (4,— 4) bisect each other.
Página xiii - ... 0; and find the values of в, ф ................ 80 3357. If four points lie in a plane, prove that the locus of the centres of all conies which pass through them has for its centre the centre of gravity of the four points, and that the asymptotes of the locus are diameters of the parabolas of the system 86 3358. Show that the formulae, by which the rectangular tangential coordinates of a plane in space are transformed, establish...
Página 33 - A simple solution is not easily found by the ordinary methods for NEWTON'S theorem, that the line which joins the middle points of the diagonals of a quadrilateral passes through the centre of an inscribed conic. Let OQAB be the quadrilateral, let OA = «, OС = e, OB = b, ÖD = d.
Página 91 - If a triangle is formed by joining three points taken at random in the circumference of a circle, prove that the odds are 3 to 1 against its being acute-angled.
Página 72 - The radius of curvature at any point of a parabola is double of the portion of the normal intercepted between the curve and the directrix.
Página 51 - ... viz., this is either (as in the foregoing example) the system of all the substitutions (or say the entire group), or else it is a system the number of whose terms is a submultiple of the whole number of substitutions. The interesting question is, to determine those two or more substitutions which, by their combination as above, do not give the entire group ; for in this way we should arrive at all the forms of a submultiple group. 3263
Página 19 - Solution ay the PROPOSER. Let the numbers be 1, 2, 3, 4, 5, 6, 7, 8, 9. Any number, say 1, enters into three triads, no two of which have any number in common. We may take these triads to be 123, 145, 167. There remain the two numbers 8, 9 ; and these are, or are not, a duad of the system. First Case — 8 and 9 a duad. In the triad which contains 89, the remaining number cannot be 1 ; it must therefore be one of the numbers 2, 3 ; 4,6; 6, 7 ; and it is quite immaterial which ; the triad may therefore...
Página x - It is required, with nine numbers each taken three times, to form nine triads containing twenty -seven distinct duads (or, what is the same thing, no duad twice) , and to find in how many essentially distinct ways this can be done 17 3279. If a quadratic equation with real coefficients be written down at hazard, find the probability of its roots being imaginary ... 22 3280. A plank, upon which at the upper end a dog is standing, is placed directly along a smooth inclined plane.
Página 20 - ... (series of numbers not divisible by 3), such that for any triad (such as 145) where the sum of the numbers, one taken negatively, = 0, the three points are in linea ; and so also that, if two of the points become identical, in the figure 13 = 14, then there is not any new point, but the preceding points are indefinitely repeated ; thus, 2, 14, 16 being in linen, and 14 being =13, 16 must be =11, and so on.
Página 21 - ... triads combined with AE ; viz., it is 123, 145, 167 ; 824, 837, 856 ; 926, 935, 947, which, it is to be observed, consists of three triads of triads, each triad of triads containing all the nine numbers ; viz., the system is 123, 479, 668 ; 145, 269, 378 ; 167, 248, 359.

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