12. Prove that a partial differential equation of the first order between three variables of the form

dz

dz

dz

= f(),

where F is homogeneous in and and of the degree ʼn has for an

dx dy'

1. Determine the numbers of cuspidal and of double edges of a cone which envelopes a surface of the nth degree, and has its vertex on the surface.

2. If a geodesic chord of a given line of curvature of an ellipsoid touches another given line of curvature of the same species, prove that the difference between its segments will be a maximum when

where D, D', D" are diameters of the confocal surface through the second line of curvature parallel to the tangents at the point of contact, and at the two extremities of the geodesic; t', t" the intercepts on the latter tangents between the ellipsoid and the confocal surface; R the radius of curvature of this surface, and of the given line of curvature at the point of contact; and a, ß the angles at which the geodesic cuts the first line of curvature.

3. Find a similar expression for the condition that the difference of geodesic tangents to a given line of curvature from a point on another may be a maximum.

MR. M. ROBERTS.

4. Let py2=q, where p, q are respectively linear and quadratic func

tions of x, be the equation of a cubic curve; and let u =

dx

If

U1, U2, uз are the values of this integral corresponding to the points of intersection of any right line with the curve, prove that u + U2 + U3

= constant.

5. Show that if the cubic represented by the system

is intersected by any right line in the points #1, U2, U3, U1 + U2 + U3 = 2mK + (2m' + 1) iK',

where the periods of u are multiples of K and iK', and m, m' are any integers positive or negative, and prove from the above equation that the line at infinity meets the curve in a point of inflexion.

6. Find what E(k, 0) becomes when transformed by the equations k' tan 0 tan & = 1, sin $ = i tan w,

and determine the value of Eam (K – iK').

APPLIED MATHEMATICS.

MR. JELLETT.

1. Determine the differential equation of an extensible membrane which is in equilibrium under the action of given forces.

(a). Show from this equation that the sum of the curvatures at any point of the membrane varies directly as the normal component of the acting force, and inversely as the elastic force.

(b). If the membrane be bounded by two flexible threads, each capable of motion along a fixed surface, but incapable of motion in the substance of the membrane, show that the surface of the membrane will cut its bounding surfaces at right angles.

(c). The equations obtained from the coefficients of dx, dy under the sign of single integration are identical; show that this is necessarily true in all cases in which the virtual moment of the internal force is represented by

where U=f(x, y, z, P, I).

F8. U dx dy,

2. Deduce Lagrange's general condition for the rigidity of a body, sc., dr xdn dx + dnydn dy + dr zdr dz = 0,

and show that if this equation be true for n = 1, n = 2, and n = 3, it is true for all integer values of n.

(a). Hence show that the equations of rotation of a solid body round a fixed point, and referred to fixed axes, are

where A, B, C, are the moments of inertia, L, M, N the statical moments, and D = Syzdm, E=fzxdm, F = xydm.

3. The general differential equations of small oscillations may be written,

d?v

Σ. 41, η

+ Σ1" α1, n En = 0,

dt2

d2 En

Σ. 42, η

dta

&c., &c.,

Prove this, and show that in general

Amy n = An, m.

(a). If any one of the partial integrals of this system be

En = En sin (t√/k+ &n),

show how to determine the constants E, and ɛn without the necessity of eliminating among a large number of equations.

(b). Determine the species of motion which should correspond to an . imaginary value of k.

4. Given the obliquity of the fixed ecliptic, determine that of the apparent ecliptic, and show that the secular variation will be very small if the luni-solar precession be rapid as compared with the motion of the solar orbit.

5. Laplace states that the secular variation in the obliquity of the ecliptic is the same whether the Earth be spherical or non-spherical, for times near the epoch.

Enunciate this theorem so as to remove its apparently arbitrary nature caused by the use of the word "epoch," and show how it may be proved.

6. Show that, if the Earth be a surface of revolution, the velocity of its rotation is absolutely unaffected by the action of a distant body, and determine the order of magnitude of the variation in this velocity if the equatorial moments of inertia be unequal.

(a). Does this determination hold good, whatever be the difference between these moments?

7. Prove that the normal component of the attraction of a surface changes twice by 27 x density, as the attracted point passes from one side of the surface to another.

(a). Show at once from this theorem that the normal component of the attraction of a plane lamina of any form is 27 × density for points very near the surface.

8. A solid body is covered by a thin stratum of liquid which is acted on by given forces (having a potential), and by the attraction of its own particles. Show that it has one position of equilibrium.

(a). If subject to the attraction of its own particles only, it will cover the whole surface of the body.

9. A homogeneous fluid which has been acted on by a given system of forces producing rotation round an axis has but one spheroidal figure of revolution.

(a). If the actual velocity of rotation be small, the spheroid may be very oblate; but if the force producing this velocity be small, the spheroid cannot be very oblate.

10. The equation obtained by a comparison of like terms in the general equation of equilibrium is

(a). Show from this equation applied to the case of an ellipsoid of revolution, not acted on by external attractions, that the superficial ellipticity is less than in the case of homogeneity.

(b). If the strata be similar, they must be ellipsoidal.

MR. TOWNSEND.

[N. B.-The answering to be limited to three of the four questions in each department.]

PHYSICAL ASTRONOMY.

1. Give Newton's investigation of the motion of the line of apsides, and of the accompanying fluctuations in the eccentricity, of the lunar orbit, under the action of the central component of the Sun's disturbing force.

2. Assuming the form of the lunar orbit to be a central ellipse with its minor axis in syzygy, investigate, as Newton has done, the law of variation of the mean motion of the line of nodes for different positions of the orbit plane.

3. Calculate the terms of the fourth order in P and T whose argument is the angular distance between the Sun and Moon; and deduce from them the corresponding terms of the third order in u and 0, as given by Godfray.

4. Assuming the forms of the expansions of the two functions

(a2 + a12 + 2aa' cos p)-- and (a2 + a'2 + zaa' cos p)- ♬

in cosines of multiples of p, calculate in its original unsimplified form the part of R, in the planetary theory, which is independent of the time explicitly, as given by Cheyne.

MOLECULAR MECHANICS.

5. Investigate, on the principles of Lamé, the differential equation for the cubical dilatation at any point of a homogeneous solid, of uniform elasticity in every direction, in equilibrium under the action of any system of external forces which have a potential.

6. Investigate again, on the same principles, the differential equations of propagation of the two systems of plane waves which by virtue of its elasticity could traverse such a solid freely in any direction; and show

that their integration may be reduced to that of a single equation of the same form for both systems.

7. The particles of any homogeneous elastic medium being supposed to act independently, when for every direction of wave plane the three corresponding directions of molecular vibration are rectangular, show, as Mr. Jellett has done, that the sum of the moments of the internal forces may be represented by the variation of a single function.

8. In the reflexion and refraction of a plane wave, propagated by rectilinear vibrations, at the separating surface of two contiguous media consisting of attracting and repelling molecules, express the three conditions obtained from the general equation of dynamics by means of the first and second transversals of the original and derived vibrations, as given by Dr. Haughton.

HYDRODYNAMICS.

9. Give Airy's investigation of the possibility of the free propagation, under the action of gravity, of a system of unbroken waves along a canal of uniform breadth, but variable depth; the several particles of the water in motion being supposed to oscillate within narrow limits in vertical planes parallel to its length.

10. A wave, produced by small oscillations of the kind supposed in the preceding question, runs freely, under the action of gravity, along a canal of uniform breadth and depth; investigate the general relation connecting the velocity of propagation, the length of the wave, and the depth of the water.

11. In the free propagation of a tidal wave from the open sea up a canal or currentless river of uniform breadth and depth, the oscillations of the particles bearing a sensible ratio to the depth of the water, prove Airy's general expression for the rise or fall of the surface at any station and time.

12. From the preceding general expression deduce, for any required distance from the mouth, the formulæ for-(a) the durations of the rise and fall of the tide; (b) the velocities of propagation of the phases of high and low water; (c) the velocities of the surface current at high and low water.

PHYSICAL PROBLEMS.

MR. JELLETT.

1. If two systems of masses, M, M', have one external equilibrium surface common, all the other equilibrium surfaces are common, and the attractions of the two systems at any point are to each other in the ratio M: M'.

2. It is required to raise a cylinder up a rough inclined plane by means of a cord coiled round it, passing over a small pulley situated at a height above the plane equal to the diameter of the base of the cylinder, and attached to a weight which descends.

(a). Investigate whether this be always possible, if the weight be unlimited.