6. Transform the equation x3 5x38x-5=0 into one whose roots are the squares of the differences of every pair of roots : and show the mode of determining, from the transformed equation, the impossible roots in the original equation.

7. From a bag containing four white and eight black balls, three persons (A, B, and C) take each a ball in turn, viz. A first, then B, then C, and so on in succession, until the person, who first draws a white ball, wins. What are their respective chances?

8. In a spherical triangle, the two sides and the angle between them being given, find the base.

9. The vertical angle of an isosceles spherical triangle is always greater than the angle included between the chords of the equal sides.

(3). cos.A + 1⁄2 cos.2A + † cos.34 + &c. to n terms.

11. Sum the series sin. x sin.0 + sin.20 × sin.20 &c. to n terms; and show that when n is infinite, the sum = 0 whatever be the ratio of toe, except that of equality.

12. A body attracted towards a centre by a force varying inversely as the square of the distance from the centre, meets at a given point of its rectilinear descent with a plane inclined at an angle of 45°. Required the time from the beginning of motion to its reaching the centre.

13. A cylindrical wheel, whose weight is P, unwinds itself from a string passing round its circumference, what weight (W) attached to the extremity of the string will be kept at rest on a plane of given elevation as P descends vertically?

14. A sphere and its circumscribing cylinder revolve round their common axis. Required the ratio of the momenta generated in a given time.

15. An homogeneous circular wheel vibrates edgeways, being suspended from a point in the circumference. Required its centre of oscillation.

16. Find the centre of gravity of the area included by the arc of a cycloid, by a tangent at the vertex, and by two rectangular ordinates equi-distant from the vertex.

17. In the common parabola the radius of curvature is equal to the cube of the normal divided by the square of the semi-parameter. ax x2

18. Trace the curve whose equation is y = ±

√(2ax-x2) and find the angles at which it cuts the line of the abscisse.

19. O is the centre of the circular arc AB. OBT is the secant. The exterior part BT is continually bisected in P. Required the area traced out by OP.

20. Two balls, A and B, are previous to motion at a given distance from each other in the same vertical line: from what height above the horizontal plane must A be let fall-so that B, which is perfectly elastic, may, after reflection, meet A at a given distance above the plane?

21. Two balls lying on a horizontal plane are connected by a string of unlimited length which passes through a ring in the plane. One of the balls is projected in a given direction with a given velocity and draws the other towards the ring. Required the curve which the projected ball describes.

23. In any curve referred to an axis, the ordinate is a maximum or a minimum, when in the equation y = fx, an odd number of differential coefficients becoming = 0, the differential coefficient of the succeeding order is negative or positive. And there is a point of contrary flexure when an even number of differential coefficients becoming0, the differential coefficient of the succeeding order is real and finite,

24. A body urged by a constant force in an uniform resisting medium is projected in a direction contrary to the action of the force with a certain velocity; it is required to determine the velocity at any point of the ascent, the resistance being supposed proportional to the square of the velocity. Find also the greatest height to which the body will ascend.

TRINITY COLLEGE, 1820.

1. IN a plane triangle the vertical angle, the perpendicular, and the rectangle under the segments of the base being given, it is required to construct the triangle.

1

of the form a and -. And find the number of all the possible values

a

in integer numbers of x, y and z in the equation

3. What are the dimensions of the strongest rectangular beam, that can be made out of a given cylinder, when placed to the most advantage? and what is its strength, compared with that of the greatest square beam cut out of the same cylinder?

4. In the wheel and axle (the inertia of which may be neglected) required the ratio between the radii, when a weight (4) acting at the circumference of the wheel generates in a given time the greatest momentum in a weight (W) attached to the circumference of the axle.

5. Tangent of half the spherical excess

tan.bx tan.cx sin.A

1+tan.x tan.cx cos.A

where b, c, and 4 are the two sides and included angles of a sphe rical triangle.

6. The excess of the Sun's longitude above its right ascension may be found by the equation

7. Find an expression from which the effect of parallax upon the horary angle may be accurately calculated; the horizontal parallax, the polar distance of the heavenly body, and the time before or after transit, being the only given elements.

8. If an orifice were opened half way to the centre of the earth, what would be the altitude of the mercury in a barometer at the bottom of it, when the altitude at the surface is 30 inches.

9. A vessel formed by the revolution of a parabola round its axis is placed with its vertex downwards, in which there is an orifice one inch in diameter. A stream of water runs into the vessel through a pipe of two inches diameter at the uniform rate of eight feet per second. What will be the greatest quantity of water in the vessel, supposing the latus rectum to be six feet?

10. Find the present worth of the reversion of a freehold estate after the death of a person now sixty years of age, the rate of interest being given?

11. When a ray of light passes out of one medium into another, as the angle of incidence increases, the angle of deviation also in

creases.

12. To find the least velocity, with which a body projected at a given angle of elevation will not return to the earth's surface. To find also the latus rectum of the orbit described, and the position of the axis.

13. Supposing the Moon's orbit at present to be circular, what would be the eccentricity of it and the periodic time, if the attraction

14. Find the sum of the following series:

(1). sin. (A) + sin. (A + B) + sin.(4 + 2B) &c. ad infin. (*). cos.4 + cos.34 + cos.54+ &c, ad infin.

(3).

+

+

&c. to n terms,

1.2.3.4 2.3.4.5 3.4.5.6'

by the method of increments.

15. Find the following fluents:

x

(1). Sz3 (a + b2

xx

(1). ၂ (2). Sy (a + bx + cz3)

bx)

(3.) S√(a2 — x) xxx, between the values x = 0, and

16. A and B put down equal stakes,—A has m chances of success, and B, n chances. There are, moreover, p chances, which entitle both parties to withdraw their stakes,—to find the gain of A.

17. Two equal weights are placed at a given distance from each other on a straight rod supposed to be without weight. Find the point of suspension, so that the pendulum may vibrate seconds.

18. Construct the curve whose equation is

(ax). (a + x) = x2y2,

and shew whether there are any cusps.

19. Invenire incrementum horarium areæ, quam luna, radio ad terram ducto, in orbe circulari describit.

20. Let the force to a centre vary as the distance, to find all the various curves, along which a body may move, so that its oscillations may be isochronous.

21. Given the diameters of two planets, and the periodic times and distances of their respective satellites; compare their densities and quantities of matter.

TRINITY COLLEGE, 1827.

1. STATE the difference between Geometrical Analysis and Synthesis, and give an example. Why are ordinary operations in algebra termed analytical?

2. If a, b, c be the angles which a upon a plane, drawn from