the co-ordinates, makes with the three rectangular axes, and if x, y, z be the co-ordinates of any point in the plane, then

x cos.ay cos.b+z cos.c = d.

3. Every equation of an odd number of dimensions has at least one possible root.

4. What are the analytical and geometrical properties of the circle of curvature, and in what manner are they deducible from each other?

5. If y = F (z + xu) where u is a function of y, then

6. Determine the locus of a upon the tangent of a curve whose equation is

drawn from a point whose co-ordinates are √✓ (a2 — b2) and 0.

7. If D be the observed angle between any two equals whose elevations or their sines are a and a' respectively, and if H be the corresponding angle reduced to the horizon, then

8. A cylinder pierces a sphere, its axis not passing through the centre: find, from the requisite data, the volume which is cut out. 9. The series

sin.A + sin. (A + B) + sin.(A + 2B) + &c.

is a recurring series: find its scale of relation, and its sum to n

terms.

10. The moment of the resultant of two forces is the algebraical sum of the moments of the component forces.

11. Explain fully the method of determining the accelerating force of P's descent, when P raises W over a single moveable pulley : the inertia of the pullies being taken into account.

12. A given conical vessel filled with water, is placed with its base on a horizontal surface: compare the pressures on its base and sides with the weight of the fluid.

S

13.

The conjugate foci in a spherical reflector move in the same direction upon its axis and coincide at the surface and centre. 14. What is the Newtonian hypothesis to explain the reflection and refraction of light? Shew in what manner the constancy of the ratio of the sines of incidence and refraction may be deduced from

it.

15. To find the deviation of a transit instrument from the plane of the meridian by the observed superior and inferior transits of the same circumpolar star.

16. Shew that the calculations of the beginning and duration of an eclipse of the Moon, of an eclipse of the Sun, of an occultation of a fixed star by the Moon or a planet, and of the transit of Venus or Mercury over the Sun's disc, are, after proper modifications, reducible to the solution of the same problem.

17. Compare the time of the Moon's falling to the Earth by the action of gravity, with her periodic time.

18. When the force varies inversely as (dist.)3: from the requisite data, find the path of a projectile, the velocity being less than that in a circle at the same distance.

19. A particle is placed in the centre of a thin hemispherical shell, to every particle of which it is attracted by forces varying inversely as the square of its distance: determine the circumstances of its motion.

20. Given the length of a degree of the meridian in a given latitude, with the length of a degree to the meridian at the same place, to find the ratio of the polar and equatorial diameters.

21.

Moon.

Explain the physical causes of evection and variation of the

22. What is the peculiar use and advantage of Mercator's projection for the purposes of navigation.

23. What would be the effect of the Moon's action upon the Earth, supposing she had no rotation round her axis? Explain the change of this effect which is produced by her rotation.

TRINITY COLLEGE, 1827.

1. WHAT is the solid content of a cube whose edge is 6 feet 6

inches.

2. To construct a

and have an

which shall be equal to a given

equal to a given 2.

are similar, which have their sides about an equal

5. Find integral values of x and y in the equation

7x+19y=73.

6. Given two sides and the included angle of a plane ▲: investigate and adapt to logarithmic computation the proper formulæ for finding the remaining side and angles.

7. Define a differential, and apply it to determine the differential of ax.

8. Investigate the relation between the power and the weight upon the inclined plane.

9. In different ellipses round the same focus, the periodic times are in the sesquiplicate ratio of the major axes.

10. The volume of a sphere is of its circumscribing cylinder.

TRINITY COLLEGE, SEPT. 1828.

1. DRAW from the vertex of a triangle a straight line which shall be a mean proportional between the segments of the base. How many answers does this question admit of?

2. Solve the equation x3 + x = 500, and find a root to two places of decimals.

How many possible roots are there?

3. Find the centre of gravity of a frustum of a cone. the position to which it tends as the ends tend to equality.

Shew

4. If L describes about T areas proportional to the times, I is acted upon by a force which tends to T.

Explain particularly the case where T is in motion.

5. If y xxm in one curve and yxx in another, m being greater than n, shew that the first curve always falls between the other and its tangent in the immediate neighbourhood of the

vertex.

6. Corporis in datâ trajectoriâ parabolicâ moti, invenire locum ad tempus assignatum.

7. Shew how we may find by observation the place and time of the equinox.

8. When a body descends in a fluid, find the greatest velocity which it can acquire. Hence knowing the rate at which a powder descends in a fluid, find the size of its particles.

9. A particle moves upon a given surface acted upon by no force. Determine its motion. What will be the path in the case of a cone?

10. By three observations of a solar spot determine the position of the Sun's axis of revolution.

II. If a ray diverging from the origin of co-ordinates x, y falls upon the curve, the distance to the point where it cuts the axis after 2(x+py) (px-y) reflexion will be =

2px — (1 — p2)y

where

dy

p=

dx

12. In a triangle the continued product of the four radii of the circles of contact is equal to the square of the area of the triangle.

13. Given the length of a curve, to find its form that the centre of gravity may be the lowest possible.

TRINITY COLLEGE, SEPT. 1828.

1. If a straight line falls on two parallel straight lines, it makes the alternate angles equal to one another.

What is the difficulty of proving this proposition satisfactorily? Mention any of the ways by which mathematicians have endeavoured to get over this difficulty.

2. Two points are taken on a wall and joined by a line which passes round a corner of the wall. This line is the shortest when its parts make equal angles with the edge at which the parts of the wall

angle at which it cuts the axis, and whether the area from x = 0 to

What is in each case the last term and the number of terms? Write down the expression for cos.mx in terms of y when m is fractional.

6. The equation ax2

by2+2cz=0 belongs to a curve surface. What is the nature of this surface? In what direction is its concavity? Draw a normal to a given point of it.