MATHEMATICAL PHYSICS.

MR. JELLETT.

1. Define an extensible surface as discussed by Lagrange, and deduce the equations of its equilibrium when acted on by given forces.

a. If the acting forces have a potential, V, determine the differential equation of the surface, and prove that for this surface

SS Vas

is a maximum or minimum, dS being the element of the surface.

2. Assuming Lagrange's expression for the potential of a nearly spherical ellipsoid, partly solid and partly fluid, namely,

p being the density of the fluid, and p + p' that of the nucleus, e, i and ' being the ellipticities of the fluid and of the nucleus, respectively, deduce the equations given by Lagrange for e and i, when the whole system has a motion of rotation.

a. Show that if the nucleus be spheroidal the surface of the fluid is the same.

3. Enunciate, without the use of mathematical symbols, Lagrange's theorem that in disturbed motion A. dt is independent of the time, pointing out the exact sense in which it is to be understood, and prove the truth of the theorem, assuming Lagrange's former theorem, viz.,

4. The equation given by Laplace for the determination of the precession is

Show that Z may be neglected, that the precession depends wholly on Y2, and that Y2 and YZ introduce respectively terms of short and long periods.

5. In the first expressions given by Laplace for the precession and nutation are included terms depending on the change in the position of the ecliptic, but in the final expressions, as compared with observation, these terms have disappeared. Explain this, and state the limitations necessary to the truth of the final expressions.

6. Define the mean day, and give Laplace's investigation of the variation in its length.

7. Prove that a homogeneous fluid, originally acted on by any forces, and then left to itself, has one, and only one, spheroidal figure of equilibrium.

8. Assuming the truth of the general equation for the strata of a fluid surrounding a stratified nucleus, viz.,

find the general value of the force of gravity, and hence deduce Clairant's theorem for the case of a spheroidal nucleus, no external forces acting. a. Show that the strata are ellipsoidal if they are similar.

9. Assuming the truth of Gauss's expression for the attraction of a surface upon a point, namely,

prove that the first three parts of the expression vary continuously as the attracted point passes from one side of the surface to the other.

a. Show that X changes twice abruptly in this passage by 27 × density at point on surface.

10. The bob of a pendulum is a smooth spherical shell, whose radii are r, r', filled with a fluid. Neglecting the weight of the rod, find the length of the simple equivalent pendulum.

MR. TOWNSEND.

PHYSICAL ASTRONOMY.

1. Assuming the fundamental equations of the lunar theory, and supposing, for simplification, the Moon to move in the plane of the ecliptic; calculate, to the second order, the portions of the lunar variation and evection in longitude, which are due, respectively, to the radial and trar sversal components of the disturbing action of the Sun.

2. Calculate, to the first order of the small quantity specially involved, the terms in P and T which contain, as a coefficient factor, the ratio of the solar to the lunar parallax; and deduce from them, as in the lunar theory, the corresponding terms in the complete values of u and terms of the time.

in

3. Assuming the fundamental equations of disturbed planetary motion, and supposing, for simplification, two planets to move in a common plane round the Sun; determine, for any positions of the two bodies, the momentary changes in the major axis, eccentricity, and perihelion longi

tude of the orbit of either, which are due, respectively, to the tangential and normal components of the disturbing action of the other.

4. A planet, describing a nearly circular orbit round the Sun, being supposed subject to the retarding action of a medium of extreme tenuity resisting as the square of its nearly uniform velocity; determine, by any method, the permanent and temporary effects of the resistance on the three aforesaid elements of its orbit, during the course of a single revolution round the Sun.

MOLECULAR MECHANICS.

5. In the free propagation of plane waves by rectilinear vibrations, under independent molecular action, in an elastic medium of uniform structure and indefinite extent, supposed in free equilibrium in its undisturbed state; show generally that, when the three directions of molecular vibration are real and rectangular for every direction of wave progression, the laws of wave propagation in the medium depend on a single characteristic function.

6. The constitution of the medium, in the preceding, being supposed such that, for every direction of wave progression within it, two of the three rectangular directions of molecular vibration are parallel and the third perpendicular to the plane of the wave; determine the general form of the characteristic function of the medium, and show how it becomes modified by the evanescence of the parallel and of the perpendicular vibrations, respectively.

7. Two different media, having characteristic functions of the same form, being supposed in contact with each other; investigate, on the ordinary hypotheses, the dynamical conditions of the transmission of molecular vibration from either of them to the other; and show, from their forms, that identity in the laws of propagation, for two different pairs of such media in contact, does not necessarily involve identity in the laws of transmission between the constituent media of each pair.

8. The two media, in the preceding, being supposed absolutely incom pressible and equally dense, determine the common form of the characteristic function of wave propagation through them, and from it the dynamical conditions of wave transmission between them; show also that, for such media, the latter are not, as for other media, independent. of the geometrical conditions, though sufficient, in combination with them, for the complete solution of the problem of transmission in all

cases.

TIDAL HYDRODYNAMICS.

9. An elementary wave of the normal type, propagated by small elliptical vibrations in vertical planes parallel to the direction of its motion, being supposed to run freely, under the action of gravity, along a rectilinear canal of uniform breadth and depth, and of indefinite length; investigate, on the ordinary hypotheses, the general relation connecting the velocity of propagation with the length of the wave and the depth of

the water.

10. In the free propagation of a tidal wave from an open sea, up a canal similar to that of the preceding question, in communication with it

at the lower extremity, the upper as before being indefinitely remote; investigate, on the supposition of the rise and fall of the surface having a sensible ratio to the depth of the water, the gradual changes in the character and appearance of the wave during its progress up the canal.

11. A circular canal of uniform breadth and depth, acted on throughout its entire length by the attraction of the Sun and Moon, being supposed to encircle the Earth along a parallel of latitude; investigate completely the conditions that the low water of the solar and the high water of the lunar tide should accompany the transit of the producing body across the local meridian at every point of its length.

12. A definite portion of a canal, similar to that of the preceding question, and acted on by the same forces, being supposed to communicate at one extremity with a tidal sea, and to terminate at the other with a vertical barrier; investigate, given all particulars, the general formulæ for the rise and flow of the water, resulting from the interference of the forced with the direct and reflected free waves, at any point of its surface.

PPHYSICAL PROBLEMS.

MR. JELLETT.

1. A solid sphere, rolling along a horizontal plane, impinges against a rough vertical wall whose coefficient of elasticity is e, and coefficient of friction show that the angle of reflexion will be given by the equation

if the coefficient of friction be less than this value.

(a). Determine, for each of these cases, the range of the sphere, considered as a projectile, on the horizontal plane.

[N. B.-The change of figure of the sphere caused by the impact is neglected.]

2. A smooth circular cylinder is capable of revolving freely round its axis, which is fixed in a vertical position. It is required to cut round the cylinder a groove commencing at a given point, such that a material particle descending the groove without initial velocity may in the shortest time reach a given point in the horizontal plane, which is coincident with the base of the cylinder.

3. A wheel is suspended from a fixed point by two strings coiled round it in opposite directions. If one string break, find the initial value of the tension of the other.

4. If the forces acting on a flexible and extensible membrane (as defined by Lagrange) have not a potential, it is not in general possible to satisfy the conditions of equilibrium; if it be possible, and if the forces be functions of the co-ordinates, the equation of the surface may be put under the form

5. A slender circular cord, of uniform thickness and arbitrary length, being supposed to intersect at right angles, in a definite plane perpendicular to its axis, a thin cylindrical tube of arbitrary radius and indefinite length, repelling it according to the law of the inverse fourth power of the distance; show, by any method, that the cord will, without deformation at any point, be held in free equilibrium by the repulsion of the tube.

6. Two thin spherical shells, of uniform equal thickness and material, each entirely external to the other in actual position, being supposed to attract, according to the ordinary law of the inverse square of the distance, a material particle placed without velocity on the coaxal sphere passing through their two centres of similitude; determine, given all particulars, the path of free motion of the particle under the joint action of the shells.

7. A rigid body of any form, in unconstrained equilibrium in free space, being supposed set in motion by a single impulsive force applied at a definite point rigidly connected with its mass; show that, whatever be the position of the point, there correspond to it in general two different axes of initial pure rotation of the body, and determine their positions, for a given point, by aid of the ellipsoid of gyration of the body for its centre of inertia.

8. The flow of water in a curvilinear canal of uniform depth, bounded laterally by rectangular hyperbolas having common axes of figure, being supposed horizontal, irrotational, and steady throughout, all lines of particles vertical at any instant continuing vertical all through the motion; given its velocity at any point of either bank, determine its direction and velocity at any point of the stream.