5. What is meant by the potential of an attracting mass with respect V be the potential with respect to a point whose co according as the point is extraneous, or forms part of the attracting mass, p in the latter case being the density of the mass at the point a, b, c. A uniform circular lamina attracts a point situated in a line drawn perpendicularly to it through its centre; shew that a being the radius of the lamina, and x the distance of the point from it; and deduce the resultant attraction exerted by the lamina upon the point. 6. In the Planetary Theory, when the disturbing function is developed preparatory to the determination of the perturbations in longitude and radius vector, shew that p ~ Չ is the order of the principal term in which pn qn' is the coefficient of t; assuming that this law holds for u, u', and for powers and products of powers of u and u'. What terms must be reserved for examination as likely to be of importance? 7. Solve the differential equation for the vibratory motion of the air contained in an indefinite cylindrical tube; and shew that when such motion is produced by a vibrating plate placed at one end of a finite tube, of which the other end is open, if the period of vibration have a certain relation to the length of the tube, it is possible for the character of the vibrations to remain permanently the same. If such a tube be sounding its fundamental note, what would be the effect of making a small aperture in the side of the tube, first at its middle point, secondly a little nearer to the open end? 8. Find the difference of retardation of the two waves produced by a thin lamina cut from a uniaxal crystal perpendicular to its axis, when a ray of common light is incident nearly parallel to the axis: describe the rings produced by interposing such a lamina between a polarizing and an analyzing plate, the planes of incidence at the two plates being inclined at an angle of 45° to each other. If two such laminæ, one cut from a positive and the other from a negative uniaxal crystal, be placed together and interposed, what must be the ratio of their thicknesses in order that neither rings nor brushes may be visible? FRIDAY, Jan. 20, 14...4. 1 1. IF f(p, q, r, s,...) = 0, where p, q, r, s,... are the distances of any point in a curve from fixed points in its plane, or of any point in a surface from fixed points, and if a set of forces proportional to ƒ'(p), ƒ' (q)... act on the point, along the distances p, q, r..., prove that their resultant acts along the normal at that point. If sinλ: sinμ :: p" : q", where λ, μ, are the respective inclinations of p, q to the normal at any point of the curve f(p,q) = 0, prove that, c being a constant, p1-” + q1-” = c1-”. 1-n 1-n Integrate the partial differential equation q(1 + q) r− (p + q + 2pq) s + p(1 + p) t = 0. 3. Prove that the radius of curvature of an oblique section, at any point of a surface, coincides with the projection, upon the plane of the section, of the radius of curvature of the normal section through the same tangent line. An annular surface is generated by the revolution of a circle about an axis in its own plane; prove that one of the principal radii of curvature, at any point of the surface, varies as the ratio of the distance of this point from the axis to its distance from the cylindrical surface described about the axis and passing through the centre of the circle. 4. Give sufficient equations for calculating the motion of a right cone placed upon a perfectly rough inclined plane; and find the moment of the couple exerted by friction on the cone. Shew that the length of the simple isochronous pendulum, when the cone oscillates about the lowest position, is 2a being the angle of the cone, r the radius of its base, ẞ the inclination of the plane, and k the radius of gyration round a generating line. 5. If u = Vdx has a maximum or minimum value, prove that How must this equation be modified when the result of some given operation performed upon the variables and their extreme values is given ? The form of a homogeneous solid of revolution, of given superficial area, and described upon an axis of given length, is such that its moment of inertia about the axis is a maximum: prove that the normal at any point of the generating curve is three times as long as the radius of curvature. 6. Distinguish between secular and periodic variations. Are secular variations ever periodic? The equations which connect the inclination and the longitude of the nodes of the orbits, in the case of Jupiter and Saturn, are of the form Prove the following circumstances of motion, that Jupiter's node regresses and Saturn's progresses from a longitude + - through the angle 2ɛ 2ε ~ in the time where is for each planet the least positive - a = angle which satisfies the equation G H cose; that they arrive simultaneously at their mean position; and that in this position Jupiter's orbit has its maximum and Saturn's its minimum inclination. 7. Assuming that the angular accelerating force, exerted by the Sun on the Earth, about a diameter of the Earth's equator at right angles to the line joining the centres of the Earth and Sun, varies as sin SP cos SP, where P is the Earth's pole, and S the Sun's centre; investigate the solar precession of the equinoxes. 8. Draw the course of a small pencil of parallel rays, passing at such an angle through a biaxal crystal cut with parallel faces, that external cylindrical refraction takes place. How may the constants a, b, c corresponding to the axes of elasticity be obtained experimentally? If the two faces of a prism, formed of a biaxal crystal, be perpendicular to each other, and one contain the two axes of elasticity a, c, and the other b, c; and if μa, u be two refractive indices for the ordinary ray when the planes of refraction are perpendicular to the axes a and b respectively; shew that D, the minimum deviation of the extraordinary ray, is given by the equation sin2 D = (-1) (μz2 – 1). THE END. |