16. Three candles are placed in a room, and the two shorter being lighted throw shadows of the third upon the ceiling; if the directions of these shadows be produced, where will they meet? The shadow of any straight line, caused by a luminous point, is in the plane passing through the luminous point and the line. Therefore the two shadows on the ceiling are the intersections of the ceiling by the two planes passing through the longer candle and the two flames respectively; the shadows if produced will meet in the line in which these two planes meet, that is, in the point when the direction of the longer candle meets the ceiling, that is, the point directly over the longer candle. 17. Within a reflecting circle on the same side of the centre are two parallel rays, one dividing the circumference into arcs which are as 3 to 1, the other dividing it into arcs which are as 8 to 1; find the least value of n such that, after each ray has suffered n reflections, they may be again parallel. Let AB (fig. 9) be the original direction of the first ray, BC its direction after one reflection; the deviation of the ray = π LABC = LAOB. Now the arc ADB = three times the arc AEB, therefore AEB is one fourth of the whole circumference; therefore the angle And since the deviation at each successive reflection is always the same, the deviation after n reflections = n.. Similarly, for the other ray, the deviation at each reflection Now after n reflections the rays are parallel to each other; therefore the deviation of one must exceed the deviation of the other by some multiple of two right angles; therefore and, since p is an integer, the least value of n is 18. 18. One asymptote of an hyperbola lies in the surface of a fluid: find the depth of the centre of pressure of the area included between the immersed asymptote, the curve, and two given horizontal lines in the plane of the hyperbola. Let BB'C'C (fig. 10) be the included area. Draw PM, horizontally, equidistantly from BB', CC'. Take any two strips PM, P'M', of equal breadths, and equidistant from PM. Then, 7 denoting the breadth, Pressure on PM 7. PM.OM.sina = Hence PM, P'M', balance about PM. Similarly for all like pairs of strips. Hence the centre of pressure of BB'C' C lies in the line PM. 19. A cone is totally immersed in a fluid, the depth of the centre of its base being given. Prove that, P, P', P", being the resultant pressures on its convex surface, when the sines of the inclination of its axis to the horizon are s, s', respectively, P2 (s′ − s′′) + P12 (s' − s) + P112 (s — 8') = 0. = Let R the resultant pressure on the whole surface of the cone, the base included; P= the resultant pressure on the convex surface, when the axis is inclined at an angle a to the horizon; B = the pressure on the base; h the altitude of the cone; the depth of the centre of its base; r = the radius k of its base; σ = the density of the fluid. = Then Now hence Similarly, P12 = ƒœ2π2r2 (h2 — 6hks' +9k2), P''2 = }ơ3ñ3r1 (h2 — 6hks" + 9k2). Multiplying these three equations in order by s' s-s', and adding, we have P2 (s' — s'') + P12 (s" − s) + P'' (s — s′) = 0. 20. Light emanating from a luminous circular disk, placed horizontally on the ceiling of a room, passes through a rectangular aperture in the floor: ascertain the form and area of the luminous patch on the floor of the room below. Shew that neither the shape nor the area of the patch will be affected by any movement of the disk along the ceiling. Let 0 (fig. 11) be the centre of the disk, M any point in its circumference. Through P, any point in a side of the aperture ABCD, draw OPO' to meet the floor of the lower room in O'. Draw MP and produce it to M', a point in the floor. With O' as centre and radius O'M' describe a circle on the floor. This circle will be the area illuminated by the rays which pass through the point P. Again, lines drawn from 0 through A, B, C, D, to meet the floor will form a rectangle A'B'C'D' on the floor. The form of the patch is therefore such as represented in (fig. 12). Let AB = a, BC = b, r = radius of disk, A'B' a', B'C' = b', r' = = O'M', and let h, h', denote the heights of the higher and lower rooms. This result shews that the form and area of the patch are independent of the position of the disk on the ceiling of the upper room. 21. If c1, C2, C3, be the lengths of the meridian shadows of three equal vertical gnomons, on the same day, at three different places on the same meridian, prove that the latitudes λ, λ λ of the places are connected together by the equation 27 = 0. = the sun's declination Let x = when on the meridian. =tan (sun's zenith distance when on the meridian), х €3 Similarly, = tan (λ, - 8). 3 and therefore (1). Multiplying (1), (2), (3), in order by c1 (c-c2), c, (cc), adding, and observing that TUESDAY, Jan. 17, 1854. 9 to 12. m 1. IF C denote generally the number of combinations of m m things s together, and C be taken to denote unity for all values of m; prove that, if then n 0 n-1 n-2 n-r+1 n-r S = 1'. C + 21. C + 3′′2. C + 4′′ ̃3.C +........+ r2. C + C, n-3 r-3 S+S+S+S+...+ S= 1′′+1+2′′ +3′′¬1 +...+ (n−1)3+n2+ (n+1)'. |