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the three roots of which are possible and positive. By what considerations are two of the roots excluded ?

13. The attraction of a sphere, whose density is a function the distance from the centre, on a point at its surface, is 1 . -n times the attraction on a point at a small depth c below the surface: shew that the ratio of the density (0) at the surface, to the mean density (A) of the sphere, is given by the equation

o 2

3 a being the radius of the sphere.

na

A

3c'

14. By experiments made in the Harton Colliery, a pendulum at the depth of 1256 feet gained 28,24 per day on a pendulum at the surface, and the mean density of the surrounding superficial strata was 2,5. Hence calculate approximately the Earth's mean density.

15. Shew how to find positions of inflected torsion, and inflected curvature, in a curve of double curvature.

16. Prove that when the effective moving forces of a connected system of material particles estimated in three rectangular directions vanish, the centre of gravity passes through a point of inflected curvature, and its velocity is a maximum or minimum.

Shew also that when the velocity is a maximum or minimum, the centre of gravity does not always pass through a point of inflected curvature.

a

17. Describe the operation by which the level error of a Transit is obtained by the collimating eye-piece.

18. Give methods of obtaining the collimation errors of a Mural Circle, and an Equatorial.

19. To calculate the horizontal attraction of gravity on a point at the Earth's surface, the surrounding irregular ground was divided into small slices by vertical cylindrical surfaces having a common axis through the point, and by vertical planes passing through that axis, and each slice was considered to be a frustrum of a pyramid having its vertex at the attracted point. Shew that if o be the mean azimuth of any slice, tan e the ratio of its mean altitude above the level of the point to its mean distance from it, a its angular width, l its length, and A its density, the required attraction in the direction to which p is referred, is

2. Ala cos o sin e.

20. If U=f(t, a, b, c), a, b and c being implicit functions of t, and if U, be the complete value of g Udt, and U, the value on the supposition that a, b and c are constant, shew that

da du.

db dU. dc
Us Us -
+

dt.
da dt

dt dc dt

dU.

db

+

21. The following equations being given,
da
db

dc
$(t, a, b, c),
dt

dt
V (t, a, b, c),

dt

= x(, a, b, c), shew how by means of the foregoing theorem to obtain by successive approximations the values of a, b and c as explicit functions of t and constants, the variable parts of these quantities being supposed to be small.

22. Assuming that the rate of sailing of a ship is simply a function of the point of the compass to which the course is directed, shew by the Calculus of Variations that the swiftest course from one point to another makes a constant angle, or its supplement, with the parallels of declination.

a

23. The form of a drop of mercury resting on a horizontal plane is assumed to be such that the total curvature of the surface at any point at the height x above the plane is k (h-x). Hence shew that if A be the volume of the drop, b the radius of the circle of contact with the plane, and a the exterior angle of contact,

(A - +62h) k = 27b sin a.
24. Obtain the following equation in the Planetary Theory :

de navi - (dR dR na(l – e) dR
dt
ue
de da

pe Shew that this equation is exact, whatever be the eccentricities and inclinations, and whatever be the number of disturbing bodies.

+

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de

February, 1857.
BY THE Rev. WILLIAM WAEWELL, D.D.

Master of Trinity College.

1. What are the problems of the duplication of a cube, and of the trisection of an angle? Give an account of any of the attempts which have been made to solve these problems geometrically. Shew that the one may be reduced to the other.

2. Of two vessels A and B, A contains a gallons of wine, and B contains b gallons of water : one gallon is taken from the one and poured into the other alternately, the mixture being uniform after each transfer : find the quantity of wine and water in each vessel after any number of transfers: and shew that it tends in each vessel to the ratio a :b.

3. Give a method of finding the number of real roots of an equation, and of determining them numerically. Apply this to the equation

x05 – 15x3 + 3x2 + 4x – 1=0.

4. Trace the curve of which the equation is

abc

*V(x-a) (6-1) and find the area from X=a to x=b.

5. Trace the curve of which the equation is

28 – 32y + x6 = 0, near the origin.

6. Prove the analytical condition requisite that a surface may be capable of being unwrapped so as to become plane. Shew that this condition is verified in every conical and cylindrical surface. Explain the general form of such surfaces.

7. A conical surface is cut by a plane and then unwrapped: find that curve the section becomes in the cases of the different Conic Sections.

8. Shew that it is always possible to find a curve which by rolling upon one given curve shall generate another given curve. What is the curve which by rolling upon a straight line generates another straight line?

a

9. A rectangular table ABCD has a weight W placed so that its distance from the edge AB is a fraction m of the length AC, and its distance from the edge AC is a fraction n of the breadth AB: find the greatest and least possible resulting pressure on each of the legs A, B, C, D. On what does the actual pressure on each leg depend ?

10. A barometer tube is suspended from the arm of a balance; what is the weight which will produce equilibrium ?

If the tube be conical, or of any other form, what will be the requisite weight ?

11. Shew how the density of the earth may be determined by comparing the times of oscillation of a given pendulum, at the surface, and at a given depth below the surface.

12. A planet revolves in a circle in the plane of the earth's orbit, given two geocentric observations, to determine the orbit.

When the plane of the orbit is inclined, can the orbit be determined from three geocentric observations ?

13. Shew whether the oblateness of the central body will or will not produce a motion of the nodes in a body revolving in an orbit of which the plane is inclined to the plane of the equator.

What effect will the oblateness produce on the motion of the apse of the orbit ?

14. An upright conical cup of which the angle is 2a, has in it a depth h of water : a sphere of which the radius is r, and weight w, is placed in the cup: find the pressure where it touches the cup.

15. When waves are propagated through water, what is the nature of the motion of the particles ? Give a formula which expresses this motion.

a

16. When a conical pendulum oscillates in a curve deviating from a circle, find the motion of the apsides :

(1) when the motion is nearly circular;
(2) when it is nearly plane.

17. Describe the general phenomena of Double Refraction; and give the explanation of them afforded by the Undulatory Theory.

18. A point A moves always directly towards a point B, which moves in a straight line : the ratio of the velocities being constant, find the curve of pursuit.

19. If iron filings be placed near a magnet, they arrange themselves into curves: what is the cause of this ? Determine the form of these curves.

20. If a linear iron bar be kept at a given temperature at its two extremities, find the temperature at any intermediate point.

21. If a point oscillate in a cycloid in a medium of which the resistance is as the velocity, the vibrations are isochronous.

22. A body is projected in a vertical plane with a given velocity U, and is acted upon by a vertical force ( -mx) which varies as x, the horizontal distance of the body from the point of projection: find the curve described, and the greatest height attained by the body.

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2. Shew that every imaginary plane contains one and but one real straight line.

3. Prove that the mean of n positive quantities which are not all equal is greater than the nth root of their product.

4. When the sum of a series according to ascending powers of x is known, shew how to deduce the sum of the series obtained by taking terms of the former at regular intervals. Example. From the series

203 tan-1 X=X

5 deduce the value of 1-}+{-1}+.

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5. Obtain a series for the ready calculation of. values of x.

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6. Given that

dx
y

=X
dx dy

and cx= (x - y) when x+y=a,

required the relation between x, y, %. 7. Trace the curve

(x+y)(x2 + y2) – aöx - b8y = 0, and shew how it passes into what it becomes when b=a.

8. If the equation of a curve be given implicitly by two equations of the form x= = 0 (t), y=y (1), where 0 (t), 4 (t) are free from radicals, shew how to find (1) the double points ; (2) the cusps, regarded as singular double points ; (3) the cusps, by a method pot introducing the double points. Apply your method to the example

aRx= (t + a)” (t 6), bey=(t + b)2 (+ - a). 9. Explain the generation of a developable surface (1) by lines, (2) by planes of which it is the envelope ; and point out the mutual relations

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