2. State and prove the formulæ for cos no and sin no in terms of cos 0 and sin 0, n being a positive integer.

2n

3. Resolve x2-2x" cos no + 1 into factors, n being a positive integer.

4. Sum to infinity the series

cos a + c cos (a + ß) + c2 cos (a + 2ß) + &c.

5. Prove the rule of proportional differences in the case of the natural sine.

3. State and prove Leibnitz's theorem, and apply it to find the nth differential coefficient of

em3 (a + bx + c x2).

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

D

Find the value when x is zero of (sin x)*,

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

Determine the greatest rectangle which can be inscribed in a given isosceles triangle.

6. Find the equation to a straight line in terms of the length of the perpendicular upon it from the origin, and the angle which that perpendicular makes with an axis.

Deduce the polar equation to a straight line.

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

Find the equation to the circle which touches the axis of x at the origin, and passes through the point h, k.

8. Find the locus of the middle points of a series of parallel chords of a parabola.

9. Find the equation of the tangent at any point of an ellipse.

Find the equations of the tangents which are parallel to the lines joining the extremities of the major axis to the extremities of the minor axis.

10. Prove that the area of the parallelogram which touches an ellipse at the ends of conjugate diameters is constant.

11. Explain the process of integration by substitution, and apply it to find

14. Find the area of a loop of the curve a4y2+ b2x2 a2b2x2.

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. Find the volume of a tetrahedron in terms of the coordinates of its angular points.

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

3. Find the locus of the middle points of a system of parallel chords of a conicoid.

4. Find the envelop of a series of surfaces whose equations involve two arbitrary parameters, and shew that it touches each surface of the series.

5. Find the equation of the osculating plane at any point of a curve.

6. Prove that a differential equation of the first order and degree can have only one independent primi

tive.

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 with two independent variables.

9. A particle is in motion in a plane; investigate expressions for its velocity and acceleration in any direction.

10. Enunciate and prove the proposition called the parallelogram of angular velocities.

MIXED MATHEMATICS.-II.

Professor Nanson.

1. Find the resultant of two couples whose axes are inclined at any angle.

2. Find the condition of equilibrium of a body which has two points in it fixed.

3. Establish the differential equation satisfied at every point by the potential of a system attracting according to the law of nature.

4. A particle is constrained to move in a straight line, and is acted on by an attraction always directed to a point outside that line, and varying inversely as the square of the distance from that point; find the time of a small oscillation about the position of equilibrium.

5. A particle moves about a centre of attraction varying directly as the distance; determine the orbit and the position of the particle at any instant.

6. A particle moves under given forces on a given smooth plane curve; determine the motion and the pressure on the curve.

7. Obtain the general equations of motion of a system acted on by any number of impulses.

8. Form the differential equations of motion of a rigid body in two dimensions, and apply them to the case of a heavy homogeneous sphere rolling down another perfectly rough fixed sphere.

9. Investigate the conditions of equilibrium of a fluid under the influence of any forces.

10. Obtain the dynamical equations of the impulsive motion of a liquid, and shew how the impulsive pressure at any point is to be determined.