5. Disturbances are excited in the air contained in a cylindrical tube of given length by a plate vibrating isochronously at one end, the other end being closed: assuming expressions for the velocity and condensation at any point, find the time of vibration, in order that a musical note may be produced ; and determine the points in the tube at which openings may be made without affecting the pitch. Supposing a vibrating plate also at the closed end, how must the time of vibration of the first plate be modified, and how must the times of vibration of the two plates be related, that musical notes may be produced ? 6. Assuming that the Sun causes an angular acceleration of the Earth, proportional to the sine of twice the Sun's north polar distance, about the equatoreal diameter perpendicular to the line joining the centres of the Sun and the Earth, shew that the line of equinoxes will have a precessional movement. How will the amount of precession, as deduced from observation, aid in determining the ratio of the mass of the Earth to the mass of a pound weight? 7. What is meant by secular variations of planetary elements ? Shew how to find the condition that the secular variations of the longi. tudes of the lines of nodes of two mutually disturbing planets may be periodic. The following equations, connecting the inclinations and longitudes of the nodes, may be assumed : tan i sin l = P tan u cos 2 =9, dp da na a'm'C (a' -9), = naʻa'm'C(p-p'), dt dt with corresponding equations for the planet to which the accented symbols refer. If the squares of the masses of the two planets were to each other inversely as their mean distances, then the nodes would oscillate through equal angles. 8. Describe some method of obtaining a circularly polarised beam of light, from light polarised in one plane. Circularly polarised light is incident, at a slight inclination, upon a plate of uniaxal crystal cut perpendicularly to its axis, and the emergent pencil is analysed ; explain generally the phenomena produced. What will be the effect produced by turning round the analysing plate ? FRIDAY, January 23, 1857. 11 to 4. 1. INVESTIGATE formulæ for the determination of the umbilici of surfaces. Prove that the radius of normal curvature of the surface xyz = ao at an umbilicus is equal to the distance of the umbilicus from the origin of coordinates. dầu + +2 2. Integrate the equations dau du dau (1), dz2 dxdy Us+1. sin 20 – Ug.sin (x + 1) 0 = cos (x - 1) 0 - cos (3x + 1) 0 ... (2); and find a general value of $ (@) from the equation $ (mox) -(a+b)° (mx) + abø (x) = CX ............ (3). = 3. State and prove the principle of Vis Viva, and describe the different kinds of forces which do not appear in the equation of Vis Viva. A circular wire ring, carrying a small bead, lies on a smooth horizontal table; an elastic thread, the natural length of which is less than the diameter of the ring, has one end attached to the bead and the other to a point in the wire; the bead is placed initially so that the thread coincides very nearly with a diameter of the ring: find the Vis Viva of the system when the string has contracted to its natural length. 4. If V be a given function of x, y, dže' dx?' dy day find the conditions that svd«, between given limits, may be a maximum or minimum. When a particle is attracted towards a fixed centre of force and moves in the brachistochrone, prove that the area described round the centre of force varies as the “ action.” 5. Describe the terms in the expansion of the disturbing function, which are of the greatest importance in calculating the variations in the elements of a planetary orbit, and explain fully what is meant by the long inequality of two planets. What principle is used to ascertain the disturbing effects produced on a planet by several other planets ? 6. Define the potential function V, and shew that, at any point (x,y,) external to the attracting mass, it satisfies the equation d? V d2 V 42 V -0. + Hence prove that, if S be any closed surface to which all the attracting mass is external, dS an element of S, and dn an element of the normal drawn outwards at ds, .dS ; dV m = the integral being taken throughout the whole surface S. 7. Assuming the formula la + mß +ny=0, 1 a(ve-a) B (v2 – 62) y (v2 – ?)' investigate the equation of the wave-surface in a biaxal crystal. Prove that the direction of the vibration at any point of this surface coincides with the projection of the distance of the point from the centre of the surface upon the tangent plane at the point. The Ordinary B. 1. Degree . Mathematical Examiners : Classical Examiners : Examiners in the Acts, Paley, &c. : EUCLID. FIRST DIVISION.—(A.) 1. The angles at the base of an isosceles triangle are equal to one another, and if the equal sides be produced the angles on the other side of the base shall be equal. The line which bisects the vertical angle of an isosceles triangle divides it into two equal parts. 2. The greater side of every triangle is opposite the greater angle. 3. Make a triangle of which the sides shall be equal to three given straight lines, but any two whatever of these must be greater than the third. Shew that the two circles made use of in the construction must necessarily cut each other. 4. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts, are themselves equal and parallel. What parallelograms have their diagonals equal ? 5. If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it, the angle contained by these two sides is a right angle. a 6. If a straight line be divided into any two parts, the rectangles contained by the whole and each of the parts are together equal to the square of the whole line. 7. In obtuse-angled triangles if a perpendicular be drawn from any of the acute angles to the opposite side produced, the square of the side subtending the obtuse angle is greater than the squares of the sides containing the obtuse angle by twice the rectangle contained by the side upon which when produced the perpendicular falls and the straight line intercepted without the triangle between the perpendicular and the obtuse angle. 8. If in a circle two straight lines cut one another which do not both pass through the centre, they do not bisect each other. 9. If two circles touch each other internally the straight line which joins their centres being produced shall pass though the point of contact. 10. In equal circles equal angles stand upon equal circumferences whether they be at the centres or the circumferences. 11. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles. Two opposite sides of a quadrilateral figure are together equal to the other two and each of its angles is less than two right angles. Shew that a circle can be inscribed in it. 12. If the angle of a triangle be divided into two equal angles by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another. And if the segments of the base have the same ratio which the other sides of the triangle have to one another, the straight line drawn from the vertex to the point of section divides the vertical angle into two equal angles. FIRST DIVISION.-(B.) 1. IF two angles of a triangle be equal to one another, the sides also which subtend or are opposite to the equal angles shall be equal to one another. The line which is drawn from the vertex of an isosceles triangle at right angles to the base bisects the vertical angle. 2. The greater angle of every triangle is subtended by the greater side or has the greater side opposite to it. 3. At a given point in a given straight line to make a given rectilineal angle equal to a given rectilineal angle. 4. The opposite sides and angles of a parallelogram are equal to one another and the diameter bisects them, that is, divides them into two equal parts. Shew that the diagonals of a rhombus bisect each other at right angles. |