8. On what grounds has it been held that we possess two recensions of the prophecies of Jeremiah ?

Translate [an extract from the Chronicle of Barhebraeus].

Translate into Hebrew:

Ut scires quoniam Dominus ipse est Deus, et non est alius praeter eum. De coelo te fecit audire vocem suam, ut doceret te, et in terra ostendit tibi ignem suum maximum, et audisti verba illius de medio ignis. Quia dilexit patres tuos, et elegit semen eorum post eos. Eduxitque te praecedens in virtute sua magna ex Aegypto, Ut deleret nationes maximas et fortiores te in introitu tuo: et introduceret te, daretque tibi terram earum in possessionem, sicut cernis in praesenti die. Scito ergo hodie, et cogitato in corde tuo, quod Dominus ipse sit Deus in coelo sursum, et in terra deorsum, et non sit alius. Custodi praecepta eius atque mandata, quae ego praecipio tibi: ut bene sit tibi, et filiis tuis post te, et permaneas multo tempore super terram, quam Dominus Deus tuus daturus est tibi.

By sin A+ ya sin B + aß sin C = la + mB+ny,

a, b, c being the lengths of sides of triangle of reference.

2. Find the locus of the pole of a given line with regard to a conic passing through four given points.

3. Investigate the relations between the diagonals of a quadrilateral circumscribing a conic and the lines joining the points of contact of its sides.

4. A variable conic passes through two fixed points A, B, and cuts a fixed conic; prove that if one of the chords of intersection passes through the intersection of tangents at A and B, it will also pass through a fixed point.

5. If, on the normal at any point P on an ellipse, PD be taken equal to the semidiameter parallel to the tangent at P, prove that the distance of D from the centre is equal to the sum or difference of the semiaxes.

6. Find the locus of a point O such that if any line be drawn through it cutting at P and Q a conic whose focus is F, tan PFO tan QFO will be constant.

7. If tangents to a conic from any point O cut a confocal conic at P, P'; Q, Q'; prove that

8. The polar equation of a conic is p (1 + e cos 0) = a; find the equa tion of the chord whose extremities are 0 = a, 0= B.

9. If tangents to a pair of confocal conics at one of their intersections are the axes of another pair intersecting at the centre of the first pair, find the lengths of the axes of the second pair.

10. If normals at the four points on an ellipse whose eccentric angles are a, B, y, d meet in the same point, prove that

sin (a+B) + sin (B + y) + sin (y + a) = 0; cos (a + B + y +8)=
11. A point on a hyperbola being determined by the equations
x = a sec 0, y = b tan 0,

find the equation of the chord joining the points = a, 0 = B.

= 0.

12. Find a symmetrical relation between the mutual distances of any four points in a plane.

MR. BURNSIDE.

I. Find the radius of curvature at any point of the curve given by the equations

and apply the formula obtained to the case of the ellipse

arranged in powers of x, as they appear explicitly.

4. Find the value of

when xyz.

(y-z)x+(2x) yn+ (xy) zn
(y) (z-x) (x-y)

5. Apply the differential calculus to expand

sin me
sin

in powers of sin 0, when m is an odd integer.

6. Determine the maximum and minimum values of u,

where u = √ x2 + y2 + z2.

8. Find the resultant of the equations

x2 (aoλ2+boλ + co) + x (α1 λ2 + b1 λ + c1) + a2 λ2 + b2 λ + c2 = 0,
x2 (αo μ2 + boμ + co) +x (α1 μ2 + b1 μ + c1) + a2 μ2 + b2 μ + c2 = 0,

free from the factor (λ -μ)2 which enters the result.
9. Determine the condition that the quartics

ax1+4bx3 y + 6cx2 y2 + 4dxy3 + eya,

ex1 + 4dx3 y + 6cx2 y2 + 4bxy3 + ay1,

should have a common quadratic factor.

10. Given the equations

form the quadratic equation whose roots are x, y, and coefficients functions of a, b, c, d.

6. Find the area of the ellipse whose equation is

by integration.

ax2+2hxy+by2 + 2gx + 2fy + c = 0

7. Determine the condition that the relation a2=a" should exist between two roots of the equations

ax3+3bx2+3cx+d=0,

bx3 + 3cx2 + 3dx + e = 0.

8. Solve the biquadratic equation

ax2+4bx3+6cx2+4dx+e=0

by assuming as the general expression determining a root

ax + b = √q √r + √r Vp + √p √q,

and determine the condition that the quantities qr, rp, pq be in arithmetic progression.

9. Solve the equation

x2-6x3 + 17x2 − 24x+18= 0.

A = (B − y) (a — d), B = (y − a) (B — 8), C′ = (a − B) (y — 8),

'(B'-') (a'-8′), B'=(y-a') (B′-d′), C'= (a' - B') (Y-8').

ASTRONOMY AND HYDROSTATICS.

DR. HAUGHTON.

1. State and investigate the Problem of Hipparchus.

2. The mean distance of Mars from the Sun is 141 millions of miles, and his periodic time is 687 days; the distances and periodic times of his two satellites are

From these data, calculate the mass of Mars.

3. Describe, accurately, the principles of the various methods of ascertaining the figure of the Earth's surface.

4. If be the true latitude of any place on the surface of the Earth, it is required to find the corrected latitude in a series depending on sines of multiples of l.

5. Find the time of a Comet describing any arc of its parabolic orbit, in terms of its chord and the focal distances of its extremities.

6. Describe and illustrate Sumner's method of finding the latitude and longitude of a ship at sea, by means of the simultaneous altitudes of two stars.

At what time would you make the observations ?

7. Assuming that a mass of liquid contained in a vertical cylinder can rotate about the axis of the cylinder, under the action of gravity only, in such a manner that the velocity of any point of the liquid varies inversely as the angular velocity of its distance from the axis of the cylinder; find the form and position of the free surface.

8. What are the conditions that, in the working of the suction pump, the water shall rise in the suction tube in the second stroke higher than, just as high as, or not so high as, it rose in the first stroke?

9. A cylindrical vessel contains a given mass of air, which revolves round its axis with a uniform angular velocity: you are required to find