3. Find the sum of the sines of a series of angles in arithmetical progression.

1. Find the limit of (1 + 1) when x is infinite,

х

and apply the result to find the differential coefficient of e*.

4. Shew how to find the value of an expression which assumes the indeterminate form 0/0.

Find the values when x is zero of

5. State and prove a rule for finding maxima and minima values of a function of one variable.

Determine the proportions of a cylinder of given volume open at one end in order that the surface should be a minimum.

6. Investigate the equation of a straight line in the form

y=x tan y + b.

Draw the straight lines

y=x+1, y = x, y = x - 1.

7. Find the equation of a circle referred to any rectangular axes.

Find the general equation of a circle touching the axes.

8. Find the equation of the straight line passing through two given points on a parabola, and find the equation of the tangent at any point.

9. Prove that the foot of the perpendicular from the focus upon any tangent to an ellipse lies on the auxiliary circle.

10. If ,' be the excentric angles of the extremities of a pair of conjugate diameters of an ellipse, prove that

Prove that the locus of the middle point of the line joining the extremities of conjugate diameters of an ellipse is a second ellipse.

11. Define a definite integral and an indefinite integral, and state the relation between them.

Prove that

ab

C

ƒ.* 4 (x) dx = ƒ. '4 (x) dx + ƒ ̊4 (x) dx.

dz

12. Prove that f(x) dz = f (x) da dz,

13. Shew how to find the partial fraction corresponding to a single factor of the first degree when a rational fraction is resolved into partial fractions.

14. Find the area of a loop of the curve

a2y2x2 (a2x2).

15. Find the volume generated by the revolution about the axis of x of the curve in question 14.

MIXED MATHEMATICS.-I.

Professor Nanson.

1. Investigate formulæ for transforming from one set of rectangular axes to another set having the same origin.

2. Find the equation of the tangent plane at any point of a conicoid.

3. Shew how to remove the terms containing the products yz, zx, xy from the general equation of the second degree.

4. Shew how to find the envelope and the edge of the envelope of a series of surfaces whose equations involve one parameter.

5. Find an expression for the radius of curvature at any point of a curve in space.

6. Give the theory of the solution of an ordinary differential equation of the first order and th degree.

7. Give the theory of the solution of two simultaneous ordinary differential equations of the first order and degree.

8. Give Charpit's process for the integration of a non-linear partial differential equation of the first order.

9. State and prove the theorem of the six constants of a body.

10. One point of a rigid body being fixed, deduce the general relations between the motions of the other points of the body.

MIXED MATHEMATICS.—II.

Professor Nanson.

1. Shew that any system of forces acting on a rigid body will be in equilibrium if the sum of the moments of the forces about each of the six edges of a finite tetrahedron is zero.

2. Shew that it is always possible to reduce a system of forces to a single force and a couple whose plane is perpendicular to the force.

3. Calculate the potential at any point internal or external of a homogeneous sphere.

4. A particle moves under an attraction in its line of motion varying directly as the distance of the particle from a fixed point in that line; determine the motion.

5. A particle is projected from a given point in a given direction with a given velocity, and moves under the influence of a central attraction varying inversely as the square of the distance; determine the orbit.

6. A particle is constrained to move on a given smooth curve under the influence of given forces; determine the motion.

7. Enunciate D'Alembert's principle, and apply it to obtain the general equations of motion of a rigid body.