18. Shew that cones and cylinders upon equal. bases are to one another as their altitudes. 19. Three points being given in position; it is required to 1823 draw a straight line through one of them, so that the rectangle of the perpendiculars let fall upon it from the other two may be the greatest possible. 20. Find the differential of an arc the tangent of whose half is x. 21. If in the spherical triangle ABC, c and C be constant, and the other angles and sides variable, then will AC and BC be the corresponding values of in the differential and ✓ (1-e2 sin2 ) ✓ (1 - e2 sin20) 22. Determine that point of the cubical parabola where the curvature is the greatest. 23. Find the greatest triangle that can be inscribed in a given circle. 24. Two given spheres are situated at the extremities of the diameter of a given circle: determine the position of an eye in the circumference, where the surface seen is the greatest possible. 25. Trace the curve whose equation is and determine the position of its tangent at the point where x = a. 26. In Newton's second Lemma, if the ordinate vary as the mth power of the abscissa, find the limit of the sum of the areas of the circumscribing parallelograms. 1824 27. Find x so that tan3 x tan 3x may be a maximum. 28. If u be a function of x, and in the equation there be m roots equal to a and n roots equal to b, then there will be one minimum value of u for each of the roots a and b if m and n be odd, and neither maxima nor minima values when they are even. G 1825 1826 29. Find the point of contrary flexure of a spiral, where the angle varies inversely as the nth power of the radius vector. 30. Explain the method of drawing asymptotes to spirals, and apply it to the hyperbola considered as a spiral having the pole in the focus. 31. Investigate the differential expression for the radius of curvature, and apply it to find the radius of curvature of the logarithmic curve. 32. Compare the curvatures of an ellipse at the extremities of the major and minor axes. 33. Trace the curve whose equation is y2 determine the nature of its singular points. bx2 X3 - = and x + c 34. Find the shortest line which can be drawn touching a given ellipse, and intercepted by the tangents drawn at the extremities of the axes of the ellipse. is 37. Find all the angles in which the curve whose equation cuts the axis, and determine the value of its greatest ordinate. 38. Three given points are taken in the circumference of a given circle; find its vertical position on a horizontal plane, that the sum of their altitudes may be the greatest or least possible. 39. Shew that the evolute of a cycloid is an equal cycloid, and find its position. 40. The vertex of a parabola is A and the axis AN, and in the ordinate NP a point Q is taken always equidistant from A and P; find the equation to the curve which is the locus of Q; trace it, and determine the angles in which it cuts the axis. and the arcs of the parabola. 41. Trace the curve whose equation is y = and determine the positions of its asymptotes. 42. Differentiate α a + x b+ x α 43. Given one of the angles and the perimeter of a plane triangle, to find the sides, when the area is the greatest possible. 44. A parabola and hyperbola have the same vertex and the same axis; draw a tangent to the former which shall cut the latter in a given angle. 45. If u be a homogeneous function of x, y, z, dimensions, and p, q, r . . . . the values of du du of n 1827 dx' dy' 46. Find the differentials of log (sin x) and 47. If y =203 =x3-2x2 + x + 4, find the maximum and minimum values of y, distinguish them from each other, and shew that they are not the greatest and least values that y admits of. 49. Shew that in general a parabola may be found which shall have a much more intimate contact with a given curve than any circle whatever. 50. Given z = x + e*. Required ≈ in terms of x. 51. A circle being described on the axis-major of an ellipse, and a tangent drawn to each curve at the points where an ordinate to the axis meets them, find where the angle between these tangents is greatest; and shew what is the ultimate point of contact in this case, when the eccentricity of the ellipse is diminished sine limite. 52. Find the value of the fraction 3.5.9.17... numerator and denominator are continued sine limite. 53. Trace the curve whose equation is až y = when its (x − a)2 √x. 54. If ordinates y1, Y2 ....y be drawn, at equal intervals beginning from the origin, to the catenary whose equation is 55. Determine the points of a given ellipse in which the sum of the conjugate diameters is the greatest or least possible, and distinguish the maximum from the minimum. 56. At points of a curve where the curvature is a maximum or a minimum, the circle of curvature has a contact of a higher order than the second. 57. Find the equation to the curve cutting at right angles all equal parabolas having their axes in the same line. 58. Draw all the rectilineal asymptotes to the curve whose equation is y = equation is 4+ y1 59. Find the equation to the curve cutting off equal arcs from all circles which have their centres in the same line, and their circumferences passing through a given point in that line and prove that the distances from this point at which the curve cuts the line are as the numbers 1 1 1 Ï' 3' 5' and the tan gents of the angles at which it cuts it as 1, 3, 5, . . . . 60. Define the radius of curvature, and prove that in an 1828 ; 61. Find the algebraic equation to the cissoid of Diocles, trace the curve, and deduce the polar equation, the cusp being the pole. 62. Prove without the use of the integral calculus that the solid content of a cone is one-third that of a cylinder of the same base and altitude. 63. Define continuous curvature, and shew that the arc, chord, and tangent of any curve of continuous curvature are ultimately equal. 64. Draw the curve whose equation is y = sin x + 2 sin 2x. Find all its points of maximum, flexure, and intersection; and shew after what values of x its form will recur. 65. On a given triangle a pyramid is to be constituted of a given content. Determine it so that its surface may be the least possible. 66. Define the differential coefficient of any function, and from that definition find the differential coefficient of and of tan x. a2 + x2 α x' 67. Investigate Maclaurin's theorem independently of Taylor's theorem, and apply it to find the series for a circular arc in terms of its sine. 68. If y be a function of x, determine the conditions requisite for y to be a maximum or minimum; and exemplify the theory when y = (x-a)", both when n is even, and when it is odd. 69. Shew how to determine when a curve is concave, and when convex to the axis. Trace the curve whose equation is a3y = x1 — bx3 — b2x2, - and determine the number and nature of its singular points. 70. Find the mth differential coefficient of cos x. 71. In an ellipse in which the semi-axes are CA, CB, and the abscissa and ordinate CM and MP, in MP take |