2. Find the magnitude and line of action of the resultant of two parallel forces acting on a rigid body in the same direction.

3. Find the C.G. of a uniform plane triangle.

If such a triangle be supported by three vertical strings attached to the angular points, prove that the tensions of the strings will be equal. If the point of attachment of one of the strings be shifted half way towards the C.G., how will the tensions be affected?

4. State the laws of friction.

A heavy body rests on a rough horizontal plane and is pulled by a string.

Find the angle of inclination of the string to the plane when the tension which just moves the body is the least possible, and the magnitude of the tension in this

in uniformly accelerated motion.

6. Prove that the path of a projectile in vacuo is a parabola.

A body is projected horizontally from the top of a tower of given altitude. Find the velocity of projection in order that it may strike the ground at an angle of 45°. 7. Two unequal heavy bodies are connected by a string passing over a smooth pulley.

Find the acceleration and tension of the string.

Supposing that one of the bodies is twice as heavy as the other, that they are in the same horizontal plane after moving for three seconds, and that the string breaks at that instant, find their distance apart at the end of the next second.

8. Prove that the pressure at any point in a heavy uniform fluid at rest varies as the depth below the surface.

9. Find the conditions of equilibrium of a body floating in a fluid.

10. A cubical vessel rests on a horizontal base, and is half filled with water.

A solid cube, whose edge is half that of the vessel, and specific gravity half that of water, is thrown in.

Find by what part of themselves the pressures on the sides and base of the vessel are respectively increased.

11. State the relation between the pressure, density, and temperature of a gas.

Describe the construction and action of Smeaton's air pump.

How is this action affected by raising the temperature of the receiver above that of the external air?

Any of the following questions may be substituted
for any of the above.

A. Find the condition that a system of forces acting on a rigid body should admit of a single resultant.

Ex. Three forces acting along the edges of a cube, no two parallel, and no two intersecting.

B. In motion about a centre of force, prove that the velocity at any point of the orbit is that due to th of the chord of curvature through the centre of force.

C. Prove the equation

dp=p (Xd + Ydy+Zdz),

where ρ is the density, p the pressure, and X, Y, Z, the component parts of the force at any point of a fluid

at rest.

If the fluid be homogeneous and incompressible, find the condition which must be satisfied by X, Y, Z, in order that equilibrium may be possible.

PURE MATHEMATICS.

1. In spherical trigonometry what are the poles of great circles? Given a great circle show how to find its poles. Prove that the angular points of a polar triangle are the poles of the sides of its primitive triangle.

2. Given the three angles of a spherical triangle find the cosine of one of the sides. If the triangle be equilateral express the cosine of a side in the form adapted to logarithmic computation.

Find a side of the equilateral triangle whose angles are each 120°.

3. If a, b, c be the sides, subtending the angles A, B, C, of a spherical triangle,

A-B

If ab, show that the formula expresses a result that can be obtained by Napier's rules for the solution of right-angled triangles.

4. If y=f(x) be a function of one independent variable and x receive an increment (h), distinguish between the difference of y and the differential of y with respect to the increment of (x). Illustrate this when y = = ax3.

If y = sinx show how to obtain

dy

dx

6. If u = f(x) can be expanded into a series of the form u = A + Bx + Сx2 + Dx3 + &c., show how to find the coefficients, and express the general form of the coefficient of x".

Assuming that log, (1+x) can be so expanded, find the series.

Expand to three terms

7. If u

=

P

-x n 11+e

2

where P and Q are functions of x which separately vanish for a particular value of x, show generally how (u) may be determined by the differentiation of P and Q. Give any example when this method will fail.

Find (u) in the following examples:

8. Find the cylinder of greatest convex surface that can be inscribed in a given hemisphere, the centres of the base of the cylinder and of the hemisphere coinciding, and show by the differential test that the surface is a maximum.

9. Prove that the perpendicular from the origin on the tangent at a point xy of a plane curve referred to rectangular coordinates is

Apply this expression to prove that in the common parabola the perpendicular from the focus on the tangent is a mean proportional between the focal distance and the distance of the focus from the vertex.

10. Find the differential expression for the subtangent of a curve referred to polar coordinates.

If S be the pole, ST the sub-tangent, SY the perpendicular from S on the tangent at P, prove

11. Obtain the differential expression for finding the content of a solid of revolution, and compare the content of a paraboloid measured from the vertex with that of the circumscribing cylinder.

13. In reference to coordinate geometry of three dimensions, explain why a plane is defined by one equation and a straight line by two equations.

=

If Ax+By+ Cz D be the equation to a plane referred to rectangular axes, determine the geometrical significance of the constants in the equation; and find also the cosine of the angle which the plane makes with the coordinate plane of xy.

واح