8. Assuming that the strata are ellipsoidal, show from the general equation for the determination of the nth Laplacian function, namely

(2n + 1)a" Yn ["pd.a3 + 3a2n+1 [® pd. (In)

- 3 [" pd. (an+3 Yn) — 3a2n+1 Z2 = 0.

that if the strata are similar the fluid is homogeneous, and conversely.

9. If a revolving homogeneous fluid surround a homogeneous spherical nucleus whose centre is in the axis of revolution (the densities being different), show that the figure of the fluid (supposed to be nearly spherical) will in general be ellipsoidal, and state the exception.

10. If matter be distributed continuously in a thin layer upon a surface of any form, how does Gauss prove that the attraction of this layer changes twice abruptly by 2′′ × thickness of layer, as the attracted point passes from one side of the surface to another ?

1. Assuming that the principal part of the lunar nutation is nearly proportional to the integral

Jy' sin A'dt,

where is the inclination of the lunar orbit to the fixed ecliptic, and A' the longitude of the node, how do you determine separately the two parts of which it consists, the one depending upon the motion of the apparent ecliptic, and the other being independent of that motion?

2. Prove that the obliquity of the fixed ecliptic will continue nearly the same for some time if the fixed ecliptic be taken coincident with the apparent ecliptic at the time.

3. If there were in the value of the lunar nutation a term depending on the angle 2 longitude of node + longitude of perigee, what would be the order of its coefficient?

(a) Show that no such term can exist.

4. If the polar equation of the surface of the Earth, supposed to be homogeneous and to differ but slightly from a solid of revolution, be written in the form

Uo, U1, &c. being Laplacian functions; show that the precession and nutation are unaffected by the terms which mark this difference.

5. Lagrange states that when a system of material particles is acted on by impulsive forces, the vis viva thereby acquired is a maximum or a minimum with regard to the axis of spontaneous rotation. Prove this, explaining the meaning of "axis of spontaneous rotation."

MR. WILLIAMSON.

1. Assuming the Moon's undisturbed orbit to be circular, give Newton's investigation of the ratio of the distances of the Moon from the Earth in syzygy and in quadrature, arising from the disturbing action of the Sun.

2. Prove Machin's construction for the determination of the mean position of the Lunar nodes at any time. Give his further construction for the determination of their true position, being given the mean position.

3. Investigate the corresponding problem by the method of the Lunar theory, and compare the result thus obtained with Machin's construction.

4. Prove the approximate relation connecting the corresponding terms of long inequality in the mean motions of two mutually disturbing planets.

5. In the case of two mutually disturbing planets, investigate the secular variations of their inclinations and of the longitudes of their nodes; and determine in any case whether the motion of the nodes is progressive or not.

1. A series of ordinary waves, propagated by small vibrations in vertical planes parallel to the direction of their motion, being supposed to move under the action of gravity along a rectilinear canal of uniform breadth investigate the possibility of their unbroken propagation in water of variable depth.

2. A tidal wave from an open sea runs under the action of gravity up a river, of uniform breadth and depth, having a current flow: the extent of the vibrations being small in comparison with the depth of the water, explain the changes in the form of the wave during its progress up the river.

3. The water in a canal of uniform breadth and depth being supposed under the action of periodical forces, horizontal and vertical: investigate and compare the resulting displacements, horizontal and vertical, at any point of the surface of the water.

4. The canal, in the preceding question, being supposed to run due east and west, and the water to be put in tidal fluctuation by the disturbing action of the Moon; show that, in the absence of tidal friction, the high or low water at every point takes place at the transit of the Moon across the meridian of the place.

5. In the preceding question, if the friction of the water be taken account of, show that the highest tide no longer synchronises with the greatest force, but follows it after an interval of time proportional to the amount of the friction: the friction being supposed proportional to the velocity.

1. Deduce the differential equation for vibratory motion at any point in an isotropic elastic solid; and when the external forces have a potential, show that the solution of these equations is reducible to the case in which the external forces are neglected.

Determine also the velocities of propagation for plane waves corresponding to the normal and transversal vibrations respectively.

2. Discuss, according to the method of Lamé, the vibrations of an isotropic hemispherical bell.

3. If a ray of polarised light be incident on a doubly refracting crystal, how did M'Cullagh prove that, when there is but one refracted ray, the incident and reflected transversals lie in the polar plane of the refracted ray?

4. From what considerations is it shown by Dr. Haughton that two different elastic media may have the same laws of wave propagation, while their laws of reflection and refraction are different? Give his application to the comparison of M'Cullagh's and Green's forms of the function in the theory of the double refraction of light.

5. Light diverging from a centre passes through a small equilateral triangular aperture, and falls on a lens placed close to the aperture. If the transmitted light be brought to a focus on a screen, investigate, from the principle of undulatory interference, the intensity of the light on the screen at points near the focus.

PROBLEMS IN APPLIED MATHEMATICS.

THE PROVOST.

1. A thin layer of matter, whose particles attract with a force varying as (dist.)", is distributed over a spherical surface according to a law which leaves one or more spaces vacant. If be the potential of the whole mass at a point in one of these vacant spaces situated at a finite distance from the boundary, A the attraction in the direction of the centre, and a the radius of the sphere, prove that

2. The attraction of a thin, homogeneous, closed shell upon a point on the external surface, estimated perpendicularly to the surface, is supposed to be, at every point of the surface, proportional to the thickness of the shell; show that the attraction at every point of the enclosed space is zero, the law of force being that of nature.

3. A number of free material particles, attracting each other with a force varying inversely as the cube of the distance, are projected with given velocities in different directions; show that one integral of the equations of motion is

(where is the distance of any one of the particles from the origin), and hence infer that a similar value holds for Emm'p2 where p is the distance between any two particles m, m'.

MR. WILLIAMSON.

4. A material particle is attracted by a thin uniform spherical shell, according to the law of the inverse fifth power of the distance. If the velocity be that from infinity under the action of the force, show that the trajectory will be a circle, cutting the shell orthogonally.

5. A solid of revolution of uniform density, turning round a fixed point on its axis, being supposed to roll, without sliding, on a rough inclined plane under the action of gravity; show that the envelope of the line connecting any two positions of its centre of inertia, separated by a constant interval of time, will be a circle and that by varying the interval of time we obtain a system of coaxal circles.

6. An incompressible lamina receives an irrotational strain in its own plane; being given that the lines of actual and of evanescent displacements of the molecules are two orthogonal systems of confocal parabolas, determine the corresponding systems of principal and of evanescent dilatation of the strain.

Classics.

MR. STACK.

Translate, adding short notes where necessary :

I. Beginning, Εκουσίῳ δὲ μᾶλλον ἔοικεν ἡ ἀκολασία τῆς δειλίας. κ. τ. λ. Ending, τοιοῦτον δὲ μάλιστα ἡ ἐπιθυμία καὶ ὁ παῖς.

ARISTOTLE, Ethic. Nic., iii. 12.

2. Beginning, Τῶν δ ̓ ἐπιχειρούντων ἐπὶ τὴν τοῦ σώματος, κ.τ.λ. Ending, τοὺς δὲ πολλοὺς δημαγωγεῖν.

ID., Pol. viii. II.

3. Beginning, 'Εχομένης δὲ τῆς ̓Αμφιπόλεως οἱ ̓Αθηναῖοι, κ. τ. λ. Ending, κινδυνεύειν παντὶ τρόπῳ ἑτοῖμοι ἦσαν.

THUCYDIDES, iv. 108.

4. Beginning, οὗ μὲν γὰρ ἦν αἰτία τὸ πρᾶγμα, κ. τ. λ. Ending, πλὴν ἐνταῦθ ̓, ἄντικρυς εἴρηκεν.

DEMOSTHENES, K. Aristok., p. 630.

MR. TYRRELL.

I. Beginning, ἀλλ ̓ ἄρα τοίγε, κ. τ. λ.

Ending, στὰς ἄρ ̓ ὑπὸ βλωθρὴν ἔγχνην κατὰ δάκρυον εἶβεν.

HOMER, Odyss., xxiv. 223-234.

2. Beginning, ὡς δ ̓ ὁπότ', ἀμφ ̓ οὔροισιν ἐγειρομένου πολέμοιο, κ.τ.λ. Ending, πορφύρων, ἥ κέ σφι θοώτερον ἀντιόῳτο.

APOLLONIUS RHODIUS, iii. 1385–1405.

3. Beginning, συμβαλεῖν μὴν εὐμαρὲς ἦν τό τε Πεισάνδρου, κ.τ.λ. Ending, ἀπροσίκτων δ ̓ ἐρώτων ὀξύτεραι μανίαι.

PINDAR, Nem., ii. 33-48.

4. (α) Beginning, εἰ μέν τι κακὸν ἀληθὲς εἶχες, Φειδία, κ.τ.λ. Ending, ὕδατι περίρραν ̓ ἐμβαλὼν ἅλας, φακούς.

MENANDER.

(6) Beginning, Ὁμοιότατος ἄνθρωπος οἴνῳ τὴν φύσιν, κ. τ. λ. Ending, ἡδύν θ ̓ ἅπασι τοὐπίλοιπον διατελεῖν.

(c) Beginning, μὰ Δι' οὐχὶ περιπεπλασμέναι ψιμυθίοις, κ. τ. λ. Ending, εἴξασι πολιαῖς, ἀνάπλεφ ψιμυθίου.

5. Beginning, ἐπεὶ γὰρ Ἥρα σοι γένος Τυρσηνικὸν, κ. τ. λ. Ending, προσῇτ ̓ ἀοιδαῖς βαρβίτων σαυλούμενοι ;

ALEXIS.

EUBULUS.

EURIPIDES, Cycl., 10-40.

6. Beginning, Γῆς δὲ εἴδη, τὸ μὲν ἠθημένον διὰ, κ. τ. λ. Ending, καὶ θυμιατικὰ σώματα ξυμπήγνυται.

PLATO, Timaeus, xxv.

MR. STACK.

Translate, adding brief notes where necessary :

1. Beginning, Adflicta respublica est empto constupratoque Ending, illam pietam silentio tuetur suam.

....

CICERO, Epist. ad Att., i. 18.

2. Beginning, Deliberas mecum, quemadmodum pecunia, Ending, quam facultatibus. Vale.

....

PLINIUS, Epist., vii. 18.

3. (α) Beginning, Ne tamen adhuc publico theatro, Ending, et praemeditans assistentibus phonascis.

TACITUS, An., xiv. 15.

(6) Beginning, Motus senatu et Pedius Blaesus, accusantibus, .... Ending, sociis et usurpata concedere rescripsit.

Ibid., xiv. 18.

4. (α) Beginning, Hanc tu licentiam diripiendorum aratorum, Ending, tuis quadruplo condemnari ?

....

CICERO, In Verr., ii. 3. 13.