37. Having given the length of a string, the density of which at every point is a given function of the distance from one extremity, determine the form which when suspended at two given points it must assume, in order that its centre of gravity may be the lowest possible. 38. The three edges of a triangular pyramid which meet, and the angles which they make with each other, being given, find its content. 39. The ratio of the apparent axes of Saturn's ring being known from observation, investigate an equation for determining its inclination to the ecliptic. Since Saturn's centre does not coincide with that of the ring, what probable supposition will account for the stability of the equilibrium ? 40. The greatest probability of correctness in solving a small spherical triangle from three observed angles and a measured side, is when the angle opposite to the known side is less than a right angle, and the other two sides are nearly equal. If two angles only be observed, for what value of the third angle will the errors probably be smallest ? 41. Out of three white balls and two black ones, two are placed at random in one bag and three in another, after which a white ball is accidentally lost ; find the probability that a person going to either bag will draw a white ball. 42. To an eye placed at the centre, every section of an ellipsoid whose projection on a tangent plane at the extremity of one axis is a circle, passes through a fixed point in the less of the remaining axis. 43. If u = f (x, y) = F (r, z), and p = $ (ax + cz) 4 (ax by), 1 du 1 du 1 du shew that + + C dz a dx ī dy 44. Integrate the following equations : dạy dy =avy, x2 dix + y = a log, 4, dx2 dx and shew that if A =VI - (sin 6)?, and 0 = 0, 45. If a string without weight be stretched between two points on any surface, the pressure at any point varies inversely as the radius of absolute curvature. 46. A cask full of wine contains (a) gallons, (b) gallons are drawn from it, and the void filled with a mixture of wine and water in a given ratio ; (6) gallons are again taken from this and replaced as before. How much wine will remain after this process has been repeated (n) times. Solve the problem also when the strength of the mixture, which replaces the void in each process, increases as the numbers 1, 2, 3 ...n. 47. If the base of a right pyramid be a regular polygon of n sides, having given the length of one of these sides and the length of one of the edges of the pyramid, find the inclination of any two of its adjacent faces, and the solid angle at the vertex. 48. In instrumental observations, explain the advantage of taking the mean of several observations. If the only errors of observation that can arise are a, 0, and - a, and if it is equally probable that any one of those shall arise, what is the probability that one-third of three observations will be the true value of the quantity sought? 49. Having given the edges of a parallelopiped and the angles they make with each other, find the diagonal. 50. If H and h be the small altitudes of two objects near the horizon, and a the angle which these two objects subtend at the station of the observer, and a + x the value of a reduced to the horizon, prove that H tan cot 1" a are 51. Each of a series of parabolas is described to pass through two given points, and have its axis parallel to a given fixed line. Prove that the locus of their foci is an hyperbola, and determine the ratio of its axes. 52. From each point of a parabola a straight line is drawn perpendicular to the chord joining it with the vertex. Determine the locus of the ultimate intersections of all such lines. 53. Which is greater, n'+1, or n + n2; n, or log. (1+n); chord 1089, or chord 36° + chord 60°? In the equations 2 (ab + xy) + (a + b) (x + y) = 0 shew 2 (cd + xy) + (6 + d) (x + y) = 0; that the values of r and y (a + b) - (c + d) also shew that if a, b, c, d, are the roots of any biquadratic equation, the values of x and y are real. 54. Shew that a square greater in area than the face of the cube in the proportion of 9 to 8, can be just placed within the cube. 55. If two circles touch one another internally, and any circle be described touching both, prove that the sum of the distances of its centre from the centres of the two given circles will be invariable. Also, if a series of such circles be described touching each of the given circles and one another, and a, b, , be respectively the radii of the given circles, and of the first circle in the series, prove that the radius of the (n + 1) circle will be ab (a - b), abr + {n (a - b) vr + Vab (a – 6 —r)}2 56. If the three sides of a triangle be tangents to a parabola, its area will be half of that of the triangle whose angular points are the points of contact. 57. From the middle point of the hypothenuse of a rightangled triangle draw a line so as to be equally inclined to it and the base, and terminated in the base produced, and from the middle point of this line draw a second equally inclined to it and the base produced, and terminated in the base produced, and so on; prove that the ultimate value of the length of the last line is a • a, a being the altitude of the triangle, and a the opposite angle. 58. Having given that in the year 1600, the 1st of March fell on the fourth day of the week (Wednesday), shew that in the year s. 102 + m it falls on the ath day, x being the remainder after dividing by 7 the quantity 4 + m + 2m + 5s + is, where the integral parts only of įm, is are to be taken. Enumerate the remaining steps for finding when Easter falls in the same year. 59. The arcs of great circles joining the angular points of a spherical triangle and the poles of the opposite sides, meet one another in the same point, and by their intersection with the sides determine the angular points of a spherical triangle whose perimeter is less than that of any other which can be inscribed in the first. 60. Find the locus of the extremity of a line drawn from the centre of a conic section such that the rectangle contained by it, and the diameter perpendicular to it, is equal to the rectangle under the axes. 61. Find the curve which touches all circles whose centres are in a given curve, and peripheries pass through a given point. Apply the method to shew that when the curve and point are a conic section and its centre, the touching curve is that constructed in the preceding problem. 62. Find the permanent temperature at any point of a fine metallic wire of indefinite length, exposed to a uniform current of air of given temperature, and having one of its extremities subjected to a constant source of heat. 63. If there be n quantities forming a geometric progression whose common ratio is r, and Sm denote the sum of the m first terms of such a series, prove that the sum of their products taken two and two together = Sn. Sn_1. rt 1 64. Straight lines are drawn from a fixed point to the several points of a straight line given in position, and on each as base is described an equilateral triangle. Determine the locus of the vertices. r 65. Find the volume of any tetrahedron of which the lengths, the inclination, and the least distance of two opposite edges are given. 66. If in any one trial an event may happen m ways and fail in n ways, prove that in r(m + n) trials it is most likely to happen rm times and fail rn times. Find the probability that in the same number of trials it will happen not fewer than r(m - 1) times, nor more than r(m + 1) times; and shew that this probability continually approximates to certainty, as the number of trials increases. 67. Expand (sin-lx)2 in a series of ascending powers of x by Maclaurin's theorem, and prove that x2 / dłu dnu + + 1.2.3...nl dir" xn+1 dn 1.2...n dzi where u, and u, denote the same functions of the independent variables x and z. 68. Give a construction for drawing normals to a parabola from a given point. Shew that in certain situations of the point it will be possible to draw three, and in others one ; and that these situations are separated by a curve from any point of which two normals can be drawn. 69. By what experiments is it shewn that terrestrial bodies are attracted by the earth and by each other, proportionally to their quantities of matter. 70. Having given the number of years which an individual wants of 86, find the present worth of an annuity to be paid during his life; it being supposed that out of 86 persons born, one dies every year till they are all extinct. 71. When mn is of the form a + BV – 1 find the relation between a, b, a, b, in order that the expression (a + bv — 1)m may be exhibited in a possible form. 72. The sides of a plane triangle are arcs of equal circles, radius = 1, prove that the algebraical sum of its sides and angles is always equal to 180°, those sides being considered negative which are concave to the interior of the triangle. If the three |