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of attraction varying as the product of the masses divided by the square of the distances, the attracted particle is in equilibrio.
7. Prove that the attraction of a shell of uniform density, bounded by two similar, similarly situated, and concentric spheroidal surfaces, on a particle placed within it, vanishes, the law of the force being as in the previous question.
8. Investigate the moment of inertia of a given homogeneous ellipsoid round an axis, parallel to one principal axis, and passing through the end of another.
9. Investigate the position of the axis of suspension of a given body, forming a pendulum, with respect to which the small oscillations will be most rapid.
10. Prove that if an equal velocity in any given direction be combined with the actual motions of each of the bodies of any system, their relative motions will not be changed.
11. A point describes a circle with a uniform velocity, find the expressions for the actual velocity of the orthographic projection of that point on any plane inclined to the plane of the circle, about the centre, and for the angular velocity of the same projection about that centre.
12. A heavy particle is allowed to glide down a helical guide, whose axis is vertical; investigate the pressure each point of the curve has to sustain in turn.
13. A body descends from rest from an infinite distance towards a centre attracting with a force inversely as the square of the distance, when it reaches the distance of R, from that centre, it meets a perfectly elastic surface inclined at an angle o to the direction in which it is then moving; investigate its motion after impact on that surface.
Afternoon. 1. Show how the parallax of the sun or moon, or one of the planets, can be deduced from two observations of its angular distance from the same fixed star made at two distant places on the earth, which have the same or nearly the same meridian.
2. Explain how the form of the orbit which a planet describes round the sun can be determined, by observing from day to day its angular distance from the sun as soon as you know how to deduce the distances of the sun and planet from the earth, from their parallaxes obtained from day to day.
3. Show how the declination and right ascension of a heavenly body can be computed from its altitude and azimuth, observed at a place, whose latitude is known. Give the trigonometrical formulæ.
4. Explain what is meant in the lunar and planetary theories by the principle of the superposition of small motions; and prové its soundness.
5. Show that if the square of the ratio of the distances of the moon and of the sun from the earth be considered a fraction too small to be taken into account, the centre of gravity of the earth and moon describes relatively to the sun an orbit which is in one plane and approximately a focal ellipse. Point out the use that is made of this in the lunar theory.
6. Write down the three leading differential equations in the lunar theory, viz.--the differential equation of the longitude, the differential equation of the projection of the radius vector on the fixed plane, and the differential equation of the latitude. Give a general description of the investigation, by which they are obtained, without going througb the work; and show that if they could be always integrated, the moon's place at any instant would be known.
7. Will the secular equation of the moon always continue an accelleration of her mean motion ? If not, explain why; and why it will change to a retardation.
8. Which of the lunar inequalities furnishes us with a means of determining the oblateness of the earth's figure as accurately, if not more so, than geodetical measurements made on its surface?
9. Show that the greatest angle of a refracting prism which can transmit a ray is double the angle of total reflexion.
10. Investigate the focal lengths of two lenses, one of crown, the other of flint glass, which correct chromatic aberration for the extreme rays of the spectrum.
11. When two electrical currents act on each other, show that elements of these placed at different distances act in a given direction with forces inversely as the squares of their distances.
12. Find the numerical value of the constant k in the expression for the total mutual action of two elements of two electric currents on each other-that is, in the equation, Total mutual action of the Loos (sin. O sin. Oʻ cos.o+k cos. O cos. O'.) two elements, J "
13. Show that the principle of virtual velocities holds true for · a number of forces, equilibrating through the intervention of a fluid mass on whose surface these forces act.
14. Investigate the motion of a body descending under the influence of a uniform force like gravity, near the earth's surface, in a fluid which resists with a force varying as the square of the velocity. Show that the final velocity is uniform, and find its value.
9th October, 1858.-Morning. 1. Give a succinct account of the two electrical theories which explain the phenomena, one on the supposition of a single fluid, the other on the supposition that there are two.
2. If the leaves of an electroscope diverge permanently with positive electricity, and an excited glass rod be gradually moved towards the upper plate of the electroscope, the leaves will gradually collapse, and then as the rod is brought nearer they will again expand; explain these facts.
3. In charging an electric battery by cascade, show from the laws of induction that the jars will be unequally charged, and investigate the law by which the charges decrease.
4. What are the reactiors which take place when a solution of sulphate of copper is decomposed by a voltaic current--(1), when the poles are platina; (2), when the poles are copper ?
5. Describe the tangent galvanometer, explain the principle on which its use is dependent, and give some of the uses to which it may be applied.
6. Prove that in multiple galvanic batteries—that is, where the plates in the several cells are connected by conductors, positive with positive, and negative with negative the intensity of the current traversing the interpolar, or
1+re where e and p are the electromotive force and resistance of each cell, and r is the resistance of the interpolar.
7. From the determination arrived at in the last, deduce the intensity in a multiple battery (like Hare's deflagrator) where the elements are similar to each other, and in number n, and show the importance of enlarging the cells, and diminishing the distance of the plates of a battery.
8. Describe Faraday's fundamental experiment by which he demonstrated the action of magnetism on a ray of polarized light.
9. Describe the property which Professor Stokes discovered, and which he has termed fluorescence.
10. Describe the experimentum crucis devised by Arago, and executed by Foucauld and Fizeau, by which the law of refraction is shown to be correctly explained by the wave theory; but not so by the theory of emission.
11. Describe the effect on Fraunhofer's dark lines of the solar spectrum-(1), of changing the refracting angle of the prism without changing the material; (2), of changing the material of the prism.
12. How did Melloni show that moonlight is almost destitute of thermic rays? And how have the late observations of Professor Piazzi Smith modified his conclusion on this subject?
13. Explain the fall of temperature as you ascend into the atmosphere.
14. Explain the fact that the solar heat may be concentrated so as to acquire great intensity in the focus of a large lens, or silvered concave mirror, while these glasses are not themselves much heated; but if they were placed before a fire, they would soon become hot; but the heat concentrated at their focus would be feeble.
7th October, 1858.--Afternoon. 1. Describe the process for extracting lead from its ores; and Pattinson's method of separating the silver from the lead.
2. Describe the method of obtaining silicon, the different forms in which it exists, and the properties of these forms.
3. Give the general properties and the constitution of the aldehydes.
4. How may aniline be obtained from benzoic acid ?
5. Mention Hofmann's views of the constitution of the organic bases.
6. Give the characters by which a bibasic acid may be distinguished from a monobasic acid.
7. Describe the method of taking the specific gravity of a gas. 8. Explain the doctrine of substitution.
9. Describe Williamson's method of obtaining the double ethers.
10. Describe the preparation and properties of cacodyl.
11. What is meant by dimorphism ? Give some examples of dimorphous bodies.
12. Describe the methods of determining the specific heat of solids and liquids.
13. Describe Boussingault's method of extracting oxygen from the atmosphere.
8th October, 1858. — Morning. 1. Describe, in general terms, the modes by which the blood is aerated throughout the animal kingdom ; adding the effects produced, both on the blood and on the surrounding media.
2. What difference of habits, among birds, may safely be inferred from differences in their sternum? Give examples.
3. Describe the typical structure of the mouth in an insect; and mention briefly the five principal modifications which its parts assume in the class Insecta.
4. Refer the following genera of Mollusca to their classes, Orders, and Families:--Pecten, Eolis, Chiton, Cardium, Hyalæa, Strombus, Terebratula, Helix, and Loligo.
BOTANY. 1. What are the chief causes of the death of leaves; and what organic changes precede their fall ?
2. What is meant by chorisis or deduplication? Give examples of floral irregularities which it professes to explain.
3. Give examples of ovaries in which some of the loculi normally become obliterated, and of some in which the loculi are multiplied by the formation of additional septa.
4. Contrast, in a tabular form, the Orders Labiatæ and Boragineæ, giving, in separate columns, the characters of Stem, Phyllotaxis, Inflorescence, Corolla, Stamens, and Embryo, which distinguish each.
Examination for Honors.
8th October, 1858.—Morning. 1. Define, with precision, the meaning of the following terms, as applied in Zoology;--Analogy, Homology, Affinity, Function, Representation, Type, Differentiation.