mine the conditions of stable equilibrum, the forms of the surfaces, being known. A wire is bent into the form of a semicircle with the bounding diameter, shew that the arc will rest in stable equilibrium on the vertex of a perfectly rough parabola with the axis vertical, if the latus rectum be 7 the semicircular arc. 4. A string is stretched over a rough plane curve. Find the tension at any point, and the pressure on the curve in the limiting position of equilibrium. A uniform inextensible string whose length is 1 hangs in limiting equilibrium over a fixed rough cylinder (radius a) whose axis is horizontal. Find the lengths of the portions which hang vertically. 5. Find the conditions of equilibrium of a fluid acted on by any forces whatever, and prove that the resultant force at any point of a surface of equal pressure is normal to the surface, and inversely as the density at the point, and the distance to the consecutive surface of equal pressure. 6. Distinguish between whole and resultant pressure. Find the condition that the pressures of a heavy incompressible flaid on a given surface may be equivalent to a single resultant. If a triangle be immersed in a fluid in a vertical plane and if a, Beg be the inclinations of its sides to the horizon, and x, y, s (in order of magnitude) the depths of its vertices prove that the depth of the centre of pressure in given by 24 sin A sin a - y sin B sin B + 24 sin C sin a 查 x3 sin A sin a - yo sin B sin B + 2 sin C sin y 7. Define metacentre. Investigate the = ", V. H M = A * for determining its position. A solid is formed by the revolution of a quadrant of an ellipse about the major axis. Prove that if it is to float in water in stable equilibrium with its axis vertical and vertex downwards with half its 75 axis immersed, the eccentricity must be less than 3 8. Investigate an equation for determining the height of a station above the level of the sea by means of a barometer and thermometer regard being had to the variation of gravitation. 9. Find the position of the focal lines of a small pencil obliquely refracted at a spherical surface. A small pencil emerging from a point on the axis of a cylinder of radius r is reflected at the sides of the cylinder. Shew if o be the angle of incidence the distance of the area of least confusion from the point of incidence will be r sec o coto 2 10. Find the condition for the minimum deviation of a ray of from light through a prism. The minimum deviations at the three angles of a triangular prism of a ray of index ji are 81, 02, 03 prove that 8 + 02 02 + 03 ზვ + 8, tu COS + COS + COS 2 0. 2 11. Define positive and negative lenses. Shew that they must be respectively thinnest and thickest in the middle. 12. Explain fully the formation of the Primary Rainbow. ALGEBRA Examiner-MR. W. GRIFFITHS, M. A. 1. Define the terms “limit” and “ differential cq-efficient" of a function. 16 Assuming the limit of (1 + as x increases indefinitely to (1 + 7)* be the base of the Napierian system of logarithms, investigate the differential co-efficient of a with respect to X. 2. State and prove Taylor's theorem, mentioning the cases in which it fails to give a correct result. If r terms only of the series be taken find limits between which the error lies. 2. If ♡ (xy) = Pn (wy) + Pn-1 (xy) + + i (ay) + po = 0 be the equation of a curve, where or (xy) is the sum of the terms of the oth degree in x and y, prove that the equation of the tangent at any point (f, g) of the curve is d d , -f9) dg + n po 0. Find the differential equation of the family of curves that cut the family • (x, y, c) = 0 at right angles. 4. Dedine a “multiple point” a “point of inflexion” and a"cusp." Find the conditions that a given point on the curve • (xy) = 0 may be a double point, and shew how to find the two directions of the curve at it. If the two branches cut one another at right angles, prove that deቅ 0. da? 5. In any plane curve show that d dao + dy? 1 dul? dr u + and p = 1 do dp" Comory Two points are moving with velocities which are always proportional to one another, one in the line joining the points, the other in a perpendicular direction. Shew that the curves traced out are both equiangular spirals. 6. Investigate the method of integrating by parts; and prove that cos a log cot (**) do = 2. 7. Find the area included within the positive compartment of (xy), of the curve ya + 2 ay + 2 a? a”, employing (i) rectangular, (ii) polar co-ordinates. 8. A plane is drawn through a diameter of the base of a right circular cone whose equation is 2 + y = (a – z), the plane making an angle 45° with the base. State between what limits the integrations in dx dy dz must be performed to find the volume cut off by the plane. y - B % -7 ya Prove that the hyperboloid + = 1 has two systems of 62 ca generating lines, and that the perpendiculars from the origin upon the generating lines lie upon the cone a? 12 10. (x1 ı 21), (X2 Y2 z2), (1, Yz z3) be the co-ordinates of the extremities of any three conjugate diameters of the ellipsoid a yia + y2 + y2 = bo, &c. * Prove also that + 1973 - 2993 91 - 1983-***,, &o. + bc 11. A cone envelopes an ellipsoid, find the equation to the cone and to the plane of contact 12. Define the principal sections at a given point of a surface, and find the relation between the radius of curvature of any normal section and those of the principal sections. PROBLEMS. Examiner-MR. Eliot, M. A. 1. If ao xo + boya + caza 0 1 62 y prove that a4 x3 + 64 y3 + c4 23 = 0 and a 3x3 + 66 y3 + c6 23 a* za + b4 y2 + c4 x2 2. If A', B', C' be any points on the sides of the triangle ABC prove that AB', BC CA' + B'C, C'A A'B = area of A A'B'C' * twice the diameter of the circle circumscribing A ABC. 3. If P be the orthocentre i. e. point of intersection of the perpendiculars from the angular vertices on the opposite sides, PA, PB, PC are the roots of the equation 203 — 2 (R + ro) x2 + (r2 — 4R+ sa) x 2R (s* -(+ 2R)) = 0 where R and r are the radii of the circumscribed and inscribed circles, and s is the semi-perimeter. 4. A circle is described touching a parabola and also intersecting the parabola at the ends of a focal chord. Prove that if a be the inclination of the chord to the axis, and 4a the latus rectum a (1 + 4 sin0) the radius of the circle is 2 sin3 0 5. Having given a tangent focus, and the eccentricity of a conic section, prove that the locus of its centre is a circle. 6. A conic is described having the origin for focus and osculating 22 the ellipse 1 at the point a cos 0, b sin o prove that 62 the equation of its directrix is (a– 62) (ar cos3 ° — by sin3") + 0°52 0. 7. Find the form of a surface of revolution such that when water is poured into it, and it is placed with its axis vertical, the whole pressure on it may vary as the square of the length of the axis immersed. 8. A quantity of elastic fuid whose particles attract ope another according to the law of nature fills a sphere in whose centre resides + a central force The radius of the sphere is c and mass of the fluid (2K — u) c where k = 2. Shew that the conditions of equili р brium are fulfilled if p varies as a 9 A glass rod is balanced partly in and partly out of a cylindrical tumbler with the lower end resting against the vertical side of the sina a tumbler. If a and B are the greatest and least angles which the rod can make with the vertical prove that the angle of the co-effici sino B ent of friction is given by tan 2 sino a cos a + sina B cos B 10. The vertex V of a cone is on the surface of a sphere, and its axis touches the sphere. Find the centre of gravity (G) of that part of the surface of the cone on one side of the vertex which is enclosed by the sphere, and if o be the inclination of GV to the axis of the cone, and 2a the vertical angle of the cone tan 0 : tan a = 97 : 32. 11. If one end of a flexible elastic string be fixed to the rim of a wheel sufficiently rough to prevent slipping, and the other end is attached to a mass (M) resting on the ground so that when the string (length a) is just taut, it is vertical Shew that the work which must be spent in turning the wheel so as just to lift the mass off the ground is E E + Mg where E is the tension which would double the length of the string (neglecting its own weight). 12. Prove that the magnifying power of a telescope is given by Breadth of visual pencil at the object glass Breadth of visual pencil at the eye glass supposing no light lost at any lens. ALGEBRA. Examiner-MR. W. GRIFFITHS, M. A. 1. If the normal at one extremity of a diameter P P of an ellipse of semi-axes a, b cut the ellipse again at Q, prove that the greatest -- 1 2 a b value of the angle PQP' is at - tan a' - 67 2. Chords are drawn to the curve r = a (1 — cos ®) passing through the pole, and at the ends of the chords pairs of normals are drawn, prove that the locus of their point of intersection is a circle. 3. Prove the following theorem by reciprocation : — The envelope of the asymptotes of hyperbolas which have one common focus and equal minor axes is a circle whose centre is at the common focus. 4. Trace the curve ya (x — b) = x (x —c), (i) when b _ c, (ii) when b 7 C. 5. The centre of a circle moves along the circumference of a fixed circle, the plane of the moving circle being perpendicular to that of the fixed circle and passing through a fixed point in its circumference; find the whole volume generated by the moving circle. |