2. Translate into Greek Iambics

K. Charles. Yet stay: if I accept these goodly terms,

What will

ye do?

Ireton. Replace you on your throne. Cromwell. And place that throne upon a people's love?

Diviner right than all the boast of kings.

K. Charles. My people's love? O shallow!
Bid again:

Bid mountains higher! for if words be aught,
And pledges hold, my people's love is mine:
And who so fond and lavish that would buy
What is his own? Nay, sirs, ye wrong my sub-
jects:

They fight me, true; but 'tis to show their zeal :
Enrol battalions, still 'tis in my name:

Use my revenues, but to do me good:

And so they love me doubtless, though they hate
The things I love.

LATIN COMPOSITION.

The Board of Examiners.

1. Translate into Latin prose—

Who could flatter himself that these men, suddenly and, as it were, by enchantment, snatched from the humblest rank of subordination, would not be intoxicated with their unprepared greatness? Who could conceive that men who are habitually meddling, daring, subtle, active, of

I

litigious dispositions and unquiet minds, would easily fall back into their old condition of obscure contention? Who could doubt but that at any expense to the state, of which they understood nothing, they must pursue their private interests, which they understood too well? It was not an event depending on chance or contingency: it was inevitable; it was necessary; it was planted in the nature of things. They must join, if their capacity did not permit them to lead, in any project which could procure to them a litigious constitution; which could lay open to them those innumerable lucrative jobs which follow in the train of all great convulsions and revolutions in the state, and particularly in all great and violent permutations of property. Their objects would be enlarged with their elevation, but their disposition and habits and mode of accomplishing their designs must remain the same.

2. Translate into Latin Elegiacs

Like one who, doomed o'er distant seas
His weary path to measure,

When home at length with fav'ring breeze,
He brings the far-sought treasure;

His ship, in sight of shore, goes down,
That shore to which he hasted;
And all the wealth he thought his own
Is o'er the waters wasted.

Like him, alas, I see that ray

Of hope before me perish,

And one dark minute sweep away
What
years were given to cherish.

SCHOOL OF MATHEMATICS AND NATURAL PHILOSOPHY.

ANALYTICAL GEOMETRY AND DIFFERENTIAL

EQUATIONS.

Professor Nanson.

1. If a, ẞ be the semiaxes of a conic inscribed in a triangle ABC, prove that

a2 + ß2 = d2 + 4R2 cos A cos B cos C
a262 = 4R pqr

where R is the radius of the circle described
about the triangle ABC, d the distance of the
centre of the conic from the orthocentre, and pqr
are the perpendiculars from the centre of the
conic on the sides of the triangle whose vertices
are the middle points of BC, CA, AB.

2. Shew how to find the common tangents to two given conics, and prove that the points of contact lie on another conic.

If a variable conic S passing through two fixed points I, J touch a fixed conic S" at a fixed point, prove that the locus of the point of intersection of a pair of common tangents to S, S' is a conic inscribed in the quadrilateral formed by the tangents from the points I, J to S'.

3. Define conjugate diameters of a conicoid, and prove that if OP be a given semidiameter there are an infinite number of semidiameters OQ,

OR which with OQ, OR form a set of conjugate diameters. Prove also that the volume of the parallelepiped of which OP, OQ, OR are edges is constant for all positions of OP, OQ, OR.

If q,r be the perpendiculars from Q, R on OP, and p is the perpendicular from the centre on the tangent plane at right angles to OP, prove that

p2 + q2 + r2 = a2 + b2 + c2.

4. Explain the method of reciprocation with respect to a sphere in solid geometry, and prove that the reciprocal of a ruled conicoid is a ruled conicoid.

Find the angle between the generating lines through any point P of a hyperboloid, and if o be the angle between the planes through the centre C and the two generating lines, prove that

where r = CP and p is the perpendicular from upon the tangent plane at P.

C

5. Prove that the envelope of a family of planes, the equation of which contains only one parameter, is a developable surface, and shew that such a surface has a cuspidal edge.

In two of the faces of a tetrahedron conics are inscribed and a developable is drawn passing through them ; shew that the section of the surface by each of the other two faces of the tetrahedron is a conic inscribed in its face.

6. Prove Euler's theorem connecting the radii of curvature of the different normal sections of a surface at any point on it, and shew that all the normals in the neighbourhood of the point pass approximately through two straight lines.

Prove that at any point on the surface there are usually three normal sections which are circular to a higher order of approximation.

7. Having given n-1 particular integrals of the equation

where

f(d) y = 0

may have a particular integral of the form em, and, supposing that condition satisfied, solve the equation.

8. Prove that the general solution of the equations

is given by

$ (8) x + 4 (8) y = 0

o (d) x + x (d) y = 0

x = 4V, y = − ¢ V + 4−10

d

where dat A=4x — 04, AV = x410.