15. Trace the incidents of a tax on house rent.

16. Point out the different effects of a tax on land proportioned to the rent, and one fixed at so much per cultivated acre. 17. Explain the scheme of the sinking fund, and show where its fallacy lies.

18. There are six circumstances which render the precious metals the best fitted for money.

19. State the weight at which a sovereign passes current. 20. Define the terms par of exchange, seignorage, real par, nominal par, double standard, as applied to money and currency. 21. Banks are of two principal characters as to their business. Enumerate the different varieties of banks in the United Kingdom.

22. State the system by which the Bank of England attempted to regulate its issues of notes during the suspension of cash payments, and state the error of that system.

23. State the largest amount of circulation which can possibly now exist in the United Kingdom.

24. There are three principal advantages which have been secured to the public by the provisions of the Currency Act of 1844.

GROUP III.

PURE MATHEMATICS.

Examiner, GEORGE J. ALLMAN, LL.D.

11th October, 1858.-Morning.

1. Express the positive integral powers of the cosine of an angle in terms of the cosines of its multiples.

2. Given the base and area of a spherical triangle, find the locus of its vertex.

3. Describe the method of forming an equation whose roots are the squares of the differences of the roots of a given equation. 4. How would you calculate the present value of an annuity on two joint lives to expire with the last of those lives?

5. Derive the equation of a parabola from its definition as a section of a cone.

6. Find the equation of the diametral plane of a surface of the second degree conjugate to a system of parallel chords whose direction is given.

7. Show how to find the value of a vanishing fraction. Find m2 sin n x-n sin m x

the value of

(1), when x=0; (2), when

n=m.

tan nx-tan m x

8. Show how to find the locus of the ultimate intersections of a series of curves formed by the variation of an arbitrary parameter a in the equation F (x, y, a)=0; and prove that the curve so traced will touch each of the series of intersecting curves.

9. Investigate the formula of reduction for fdreas (cos x)".

10. Explain Euler's method of integrating the equation Mdx+Ndy=0 by means of a factor, and show that the investigation of the factor requires, in general, a more difficult integration than the proposed.

Afternoon.

1. Show that the quadrature of a parabola may be derived from the cubature of a pyramid.

2. Give the method of constructing equations of the third and fourth degrees by means of the intersection of a parabola and a circle.

3. If through any point O a line be drawn, cutting a curve of the nth degree in n points, and at those points tangents be drawn, and if any other line through O cut the curve in R1, R2, &c., and the system of n tangents in r1, r2, &c., prove that

4. Prove that the area of the section of an ellipsoid

made by the plane lx+my+nz=0 is

Tab c {a2l2+b2m2+c2n2}+

1, m, n, being the direction - cosines of the plane.

5. Prove Lagrange's theorem for the expansion of ƒ (y) in powers of x where y=z+x¢(y).

6. Determine the osculating sphere at any point of a given curve of double curvature.

7. Prove that the lines of curvature, at any point of a surface, are tangents to the principal sections at that point, and state the conditions that must be fulfilled in order that a line of curvature may coincide with one of the principal sections.

8. Prove that

Mdx+Ndy
Mx+Ny

is an exact differential when M

and N are homogeneous functions of x and y.
9. Prove the following thecrem--

'( 1 − r)''d r—r(p) r(q)
r(p+q)

10. Prove that the particular solutions of the differential dy will be obtained by the

equationƒ(x, y, p)=0( where p=

dx

df

elimination of p between f (x, y, p)=0 and 0

dp

11. Find the equation of condition which must exist in order that the total differential equation Pdx+Qdy+Rdz=0 may become integrable by means of a factor.

12. Find the curve in which the product of the perpendiculars from two fixed points on the tangent is constant.

MIXED MATHEMATICS.

Examiner, JOHN STEVELLY, LL.D.

6th October, 1858.-Morning.

1. Show that all the axes of principal moment of any system of forces, acting upon the points of a rigid body, for origins, which are at a given distance from the central axis, in a plane perpendicular to the resultant, form the surface of a hyperboloid of revolution of one sheet.

2. Prove that if a right cylinder be described round the central axis of such a system of forces, the principal moment retains the same value for any origin taken on the surface of this cylinder.

3. Prove that if a system of forces acting on a rigid body, be reduced to two forces, which are represented by two straight lines, which do not meet, and are not parallel, the volume of the tetrahedron of which the two straight lines are opposite edges is

constant.

4. Let m1, m2, &c., express the several masses of the particles of any system; P1, P2, &c., the distances of these particles respectively from their common centre of gravity u, &c., their mutual distances two and two, that then,

5. Prove that the volume of a cylinder included between any two plane sections is equal to the area of a section of the cylinder, formed by a plane perpendicular to its axis, multiplied by the line which joins the centres of gravity of its plane ends.

6. Find the point at which if a particle of matter were placed, which is attracted by the three material lines of indefinitely small cross section, which form the sides of a triangle, the forces

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 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 prove 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 through 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

two elements,.

ρρός δς
r2.

(sin. O sin. ' cos.+k cos. O cos. Ø ́.)

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.