5. Explain the meaning and historical importance of :-Sturm-undDrang-Periode-Xenien-Sturm-Hainbund.

6. Enumerate the dramas of Schiller in their chronological order.

7. Write a German essay on the following subject:-" Beschreibung des Zustandes welcher von den deutschen Studenten mit dem Ausdruck 'Katzenjammer' bezeichnet wird.”

EXAMINATION FOR THE DEGREE OF BACHELOR OF ARTS.

Moderatorships in Mathematics and Mathematical Physics.

MR. BURNSIDE.

PURE MATHEMATICS.

1. Determine the shortest distance between the right line determined by the intersection of the planes

x cos ay cos B + cos y = p,

x cos a' + y cos B' + z cos y = p',

and the right line joining the points (x, y, z) and (x", y′′, z′′).

2. Find the radius of the sphere circumscribing a tetrahedron in terms of the lengths of its six edges.

3. Determine the points of contact of the tangent planes drawn through the right line

=

= 0,

ax + By + yz + dw
ax + B'y + y z + d'w = 0,

to the quadric

ax2 + by2 + cz2 + dw2 = 0.

4. Form the equation of the hyperboloid which has the four following lines for generators:

5. If a system of confocal quadrics be projected orthogonally on any plane, prove that the projections are confocal conics.

6. Determine the locus of the points of contact of parallel tangent planes to a series of confocal quadrics.

7. Find the general form of the equation of a quadric which touches the four faces of the tetrahedron of reference.

8. Prove that the difference of the squares of the reciprocals of the axes of a central section of a quadric is proportional to the product of the sines of the angles it makes with the cyclic planes.

9. Determine the number of double tangents which can in general be drawn to a surface of the nth degree from a point not on the surface.

10. Prove that the area of a geodesic circle on any surface, whose radius s is supposed to be very small, is given by the formula

II. Prove that the following equation holds for a geodesic line or a line of curvature on any surface U = 0:

where L, M, N are the differential coefficients of U, and R2 = L2 + M2 + N2. 12. When Uo is the equation of a surface of the second degree, determine a first integral of the last equation, and give its geometrical interpretation.

PROBLEMS.

1. If on each edge of a tetrahedron an arbitrary point be taken, prove that the spheres described through a vertex of the tetrahedron and through the three points on the edges adjacent to that vertex pass through the same point.

2. Determine the identical relation which connects the powers of a point with regard to five spheres.

3. The four faces of a tetrahedron pass each through a fixed point in a plane: find the locus of the vertex when the three remote edges lie each in a fixed plane.

4. Prove that the quadratic equation which determines the principal radii of curvature at any point of a surface U= o may be put under the form

where L, M, N are the differential coefficients of U and

λR = √ L2 + M2 + N2.

(a) Hence determine the conditions for an umbilic on the surface U=0.

MR. PANTON.

5. A chord of a given conic passes through a fixed point 0; show that the locus of a point P cutting it in a given ratio is a quartic curve having double contact with the conic; and determine what the locus becomes in the following particular cases, viz.-(1) when P bisects the chord, and (2) when the conic becomes a circle.

6. Prove that the locus of the intersection of tangents at the extremities of a chord through the double point of a limaçon is a circular cubic having double contact with the curve.

7. Show that the curve

(x2 — 4)2 — 4y2 (y + 3) = 0

is unicursal, and trace it so as to exhibit the positions of the double points; and determine rational algebraic expressions for x and y in terms of the same variable parameter.

8. Show that the pencil of cubics

λxyz + (x + y + z)2 (ax + by + cz) = 0

has (in addition to the vertices of the triangle of reference, and the intersection of the lines x + y + z, ax + by + cz) three critic centres whose co-ordinates are the three systems of values of

9. Prove that the most general form of z which satisfies the two partial differential equations

and apply this method to solve the equation

d2y

(I + x2)

dx2

dy
+ 9x + 15y = 0.
dx

II. The discriminant of the quantic a,xn+na1xn-1+... being known, show that the equation whose roots are the maximum and minimum values of the quantic may be at once written down.

(a) Form this equation for the case of the quartic

ax1+4bx3 + 6cx2 + 4dx + e

expressing the coefficients in terms of a, H, I, J; and show that two of the maximum-minimum values will be equal-(1) if the discriminant of the derived cubic vanish; (2) if G = 0.

12. Show that if U be any ternary quadric, the expression

may be written in the form mU + V, where V is a binary quadric in ax + By + y2, ax + B'y + yz.

MR. PANTON.

1. The vertex of a pencil of given anharmonic ratio moves on a fixed right line, and three of the legs pass through fixed points; find the envelope of the fourth leg.

2. Calculate the invariants, covariants, and contravariants of the system of two conics

S= x2 + y2 + z2, S′ = ax2 + by2 + cz2.

3. Express in terms of the invariants the condition that the covariant F should break up into right lines; and distinguish the different cases in which this reduction takes place for two circles.

4. If a curve have a multiple point of the order k, determine the degree of multiplicity of this point on the Hessian, and further, how the tangents at the multiple point are related to the Hessian; and hence determine the effect of a multiple point in reducing the number of points of inflexion of the curve.

5. Find the equation of the parallel to a parabola, and determine all the characteristics of this curve.