BISHOP LAW'S MATHEMATICAL PREMIUM.

MR. BURNSIDE.

1. Express the co-ordinates of any point on the cubic curve

x3 + y3 + z3 + 6mxyz = 0

as algebraic functions of a single variable.

2. Find the area of the curve

x2n + y2n = a2 (xy)n-1,

n being a positive integer.

3. Find the form of the function o, so that the equation

(x+y) (xy) = p2 (x) -2 (y)

be satisfied for all values of x and of y.

4. Let f(x) and (x) be two cubics; and if a, B, y, be the roots of ƒ (x) (no two being equal), determine the invariant relation which exists between their coefficients when

(B-) Vp (a) + (y − a) √p (B) + (a− B) √(v) = 0.

5. Let x1, x2, x3 be the abscissæ of the intersections of the cubic curve y = f(x), with an arbitrary line, and prove that

8. Integrate the partial differential equation

(b + cq)2r − 2 (b + cq) (a + cp) s + (a +'cp)2 t=0.

9. Transform the differential equation

and reduce the resulting equation to Clairaut's form.

10. Show how to approximate to the value of

11. If a binary quartic has a double factor, prove that its sextic covariant has a quintuple factor.

MR. WILLIAMSON.

const.,

1. If u and be determined by the equation x + iy = tan (u + iv), where i=1; find the curves represented by u = const. and v = and show that they intersect orthogonally.

2. A sphere, whose centre is in the plane of xy, has double contact with the ellipsoid

where a and ẞ are the co-ordinates of the centre of the sphere, and is its radius.

3. Show that the points of contact of the sphere and the ellipsoid, in the preceding, are real or imaginary according as the centre of the sphere is inside or outside a certain ellipse.

(a) Express a, 8, in terms of the elliptic co-ordinates of the point of contact.

4. If the equation to a surface of the third degree be

ax3 + by3 + cz3 + du3 + ev3 = 0,

where

x + y + z + u + v = o,

find the equation to the Hessian of the surface.

x sin 4x 5. Expand in ascending powers of x, expressing the cosin x sin 3x efficients in terms of the numbers of Bernoulli.

z = r cos e cos p, where m2 + n2 = 1, find the value of the Jacobian

taken for all positive values of x, y, z which satisfy the condition

Xn, prove

are functions of the n independent variables, x1, X2, X3,

...

9. Prove that the developable circumscribed to a quadric along a geodesic has its cuspidal edge on another quadric.

PROFESSOR ROBERT S. BALL.

1. Determine the plane curve in which the projection of the radius of curvature on a fixed line is constant.

4. Give a discussion of the quaternion equation of the ellipse

p = a cos 0 + B sin 0,

and deduce from it the fundamental focal properties of the curve.

5. If a plane be displaced on itself, show that each of the points on a certain circle must at any moment be situated at a point of inflexion on the curve it describes.

6. State the principle of Peaucellier's parallel motion, and prove it.

7. A star is observed to have a proper motion of n seconds annually projected perpendicularly to the line of sight; find the most probable value of the real proper motion.

8. If the "distance" between two points be defined to be the logarithm of the anharmonic ratio in which the line joining those points is divided by a fixed quadric, prove

(a) That the "sphere" or locus of points "equidistant" from a given point must be a quadric touching the fundamental quadric along a conic section.

(b) That any five points may be displaced so that the ten "distances" between every pair of them remain unchanged.

(c) That a rigid system is therefore possible, i. e. a system which, while admitting of displacement as a whole, preserves unaltered the "distances" (in this generalized sense) between every pair of particles.

FERRAR MEMORIAL PRIZE.

DR. INGRAM.

Write short essays on the following subjects:—

(a) Augmentation (Weiterbildung) of Greek roots by 0.
(b) Changes of the Digamma to vowels and to other consonants.
(c) Spiritus lenis for asper, and spiritus asper for lenis.

(d) Loss of initial sibilant.

1. What were the original forms of (a) article d, (b) relative ös, (c) possessive os ?

2. (α) Explain the relations between ἔταφον, τέθηπα, θάμβος. What was the I. E. root? Is @avua cognate ?

(b) Give other instances of the operation of the Greek phonetic law exemplified by τάφος.

3. There is but one certain case of the change of original gh to e? What corresponds in Latin in this case?

4. What explanations does Curtius give of ἵπταμαι, ὄψον, ὄκνος ?

5. What are the Greek cognates of Lat. fendo, frequens, mucro, nanciscor, serum, vegeo?

6. What are the Greek cognates of Eng. (be)ware, fare, fathom, friend, sooth, way?

7. Scribo, ypápw. Cite other cases of the correspondence of these initial consonant-groups.

8. It is very probable that ådeλøpós and germen are cognate. Explain in what way.

9. (a) Dr. Hayman connects vdwp and sudor : is he right ?

(b) Are dwp and udus cognate ?

(c) "Humor, from root xv-, to pour" is this correct?

10. What does Curtius consider the more probable of the two explanations of ferveo, and why?

11. (a) Curtius recognizes two verbs àμépdw?

(b) Bothe thinks the d in auépdw, to deprive, and the d in prodesse, similar. Is this so?

(c)

"Millions of spirits, for his fault

Amerced of heaven."

Is amerce from ἀμέρδω?

12. What are the etymological affinities of the Latin words assir, calpar, cerus, degunere, her, hir, ianitrices ?