Изображения страниц
PDF
EPUB

2. Given two conics, U, V [x2+ y2 + z2, ax2 + by2+ cz2], find the locus of the point which divides in a given ratio the distance between any point x'y' on U, and the pole with regard to, V of the tangent at x'y' to U.

3. Form the discriminant with regard to λ of the equation

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small]

represents a curve of the fourth order having three double points, and to which x, y, z, are double tangents.

5. Show how many tangents can be drawn to a curve of the nth degree from a multiple point on it of the order k.

6. By what elimination can be formed the equation of the tangents at the points of inflexion of a curve of the nth order? In what degree does this equation contain the coefficients of the original?

7. Form the reciprocal of the curve

2:3+ y3 +23+ 6mxyz = 0.

8. If U and V are surfaces of the nth degree, show how many surfaces of the form U=XV can be drawn to touch a given plane, and how many to touch a given line.

9. Prove that the "discriminating cubic" for a surface of the second degree has its roots all real.

10. Given the sum of the squares of the axes of a plane central section of a quadric; find the cone generated by a perpendicular to its plane.

11. The four faces of a tetrahedron pass each through a fixed point. Find the locus of the vertex, if the three edges which do not pass through it move each in a fixed plane.

12. In terms of the axes of the confocal surfaces which can be drawn through any point on an ellipsoid, express the axes of a central section parallel to the tangent plane at that point.

13. If in the equation

x2 + y2+z2 + w2 + 2lyz + 2mzx + 2nxy + 2pxw + zqyw + zrzw = 0, we have r = 1; and if also the conditions are fulfilled that the planes x and y may both touch the surface, show both geometrically and algebraically that the surface is a cone.

14. Find the conditions that a point on a surface may be an umbilic.

[blocks in formation]
[blocks in formation]

satisfies the criterion of integrability, and find its integral.

4. Adopting the usual notation of partial differential coefficients,

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

7. Find the orthogonal trajectory of the system of paraboloids represented by the equation

az = Ax2 + By2,

[merged small][merged small][merged small][ocr errors]

into another with respect to variables u and v which are determined by the equations

y2+ 2x2 = u, y2=vx.

MR. STUBBS.

LUNAR THEORY.

1. Express the mean anomaly in terms of the true in a series ascending according to the powers of the eccentricity as far as the third power. 2. Deduce the differential equation of the Moon's radius vector. 3. Prove that

[merged small][merged small][merged small][ocr errors]

4. Calculate to the second order the values of the following terms

[merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small]

5. Hence deduce the differential equation for u to the second order. 6. Integrate this equation, and determine the constants.

7. When the constants which enter into the arguments of the different terms of the series have been determined by observation, any one of the coefficients may be obtained by a limited series of observations?

8. By taking the elliptic inequality and the evection together, we can determine the change in the position of the apse and in the eccentricity of the Moon's orbit?

9. Explain the variation of the inclination and the irregularity in the motion of the node expressed by the term in the formula for the Moon's latitude,

+ 3 mk sin {(2 − 2m − g) 0 −2ß+y}.

10. Determine the effect of the tangential disturbing force. 11. In what manner do the terms in the expansions of

[blocks in formation]

- 3 m2a

cos (0–0′), and — § m2

E-Ma
E+Ma

sin (0-0'),

E-Ma E+M'a respectively, enter into the approximation to the third order in the values of u and 0? Find the terms in these values to which they give rise.

12. Prove that, if we go to the third order of approximation, the motion of the Moon's apse in one revolution of the Moon equals

[ocr errors][merged small][merged small][merged small][merged small]

1. When a body describes a moveable ellipse from the action of a force tending to the focus; prove that the force will equal

[merged small][merged small][merged small][merged small][ocr errors][merged small]

2. Find the angle between the apsides in orbits nearly circular when the central force = bAm + cДn.

3. Prop. LXVI. Cor. 6. Prove that the central force T is more diminished in syzygies by the ablatitious force, LM, than it is increased in quadratures by the addititious force, KL.

4. How does Newton explain the action of the disturbing force to produce a motion of the apse? Prove fully the truth of the statement that

t

in quadratures the sum of the central force and the addititious force "decreases in a less ratio than the duplicate ratio of the distance," and that hence the higher apse regresses; and show that on the whole there will be a progression of the apsides.

5. Investigate fully as Newton does the action of the disturbing forces to change the eccentricity of the Moon's orbit; and show that during the passage of the apsides from quadratures to syzygies the eccentricity is increased, and from syzygies to quadratures diminished.

6. Prove that the mean force LM (the Sun's attraction on the Moon resolved along the radius of the Moon's orbit) is to the force by which the Moon is retained in her orbit round the Earth in the duplicate ratio of the periodic times of the Moon and Sun.

MR. JELLETT.

STATICS.

1. Enunciate and prove the theorem of Ivory with regard to the attraction of a homogeneous ellipsoid, and deduce from it the theorem of Maclaurin on the same subject.

2. A material particle is placed at a point within a triangle formed of three rods of uniform thickness, which attract according to the law of the inverse square; find the conditions of equilibrium.

3. A heavy elastic string is laid upon a smooth double-inclined plane; find its extension.

4. A heavy uniform rod hangs vertically from a smooth hinge, and is attracted by a centre of force situated vertically above the hinge, the force varying directly as the distance; find whether the equilibrium be stable or unstable.

5. Two weights balance one another by means of a fine string passing over a rough horizontal cylinder; find the limits within which the ratio of the weight must lie.

6. An elliptic board is placed with its plane vertical on two pegs situated on the same horizontal line

a. Find the position of equilibrium.

b. Determine the limits of the distance of the pegs.

c. Show that the position is unstable.

7. Determine the equations of the small oscillations of a floating solid body which is symmetrical with respect to a vertical plane.

8. Find the condition of stable equilibrium for a floating body.

DYNAMICS.

1. Determine the motion of a heavy body falling in a medium resisting with a force proportional to the square of the velocity; and show that the expression for the velocity tends to a constant value.

a. Show that this is true for any possible law of the resistance.

2. Deduce the equations of motion of a point on a fixed smooth surface from the principle of least action.

N. B.-The acting forces satisfy the condition

[ocr errors]

3. If a solid body acted on by given forces revolve round a fixed axis, determine the pressures on the axis during the movement.

4. A uniform beam is supported symmetrically on two props; find where they should be placed in order that, if one of them be removed, the instantaneous pressure on the other may be the same as the statical pressure.

5. A circular board lies upon a smooth table; in the board is cut a circular groove along which a molecule is projected with a given velocity; determine the pressure against the side of the groove.

6. A straight rod which passes through a small fixed ring is in motion in a horizontal plane; determine the motion of its centre of gravity.

7. If a rigid body receive any number of simultaneous impulses, determine the axis of spontaneous rotation.

a. If a perpendicular be let fall from the centre of gravity on the spontaneous axis, it will be perpendicular to the resultant force?

8. A homogeneous liquid moves without initial velocity under the influence of forces which satisfy the condition in (2). Show that if a curve be traced between two given points in the fluid, and if we denote by v the velocity of the fluid, by a the angle between the direction of the motion and the tangent to the curve, and by ds the element of the curve, then the definite integral

S v cos a ds

taken between the two points is the same for all curves so drawn. 9. Deduce the equations of small oscillations of an elastic fluid,

[blocks in formation]

1. A uniform beam rests on a smooth horizontal plane, and against a smooth vertical wall, and is kept in equilibrium by a cord attached to two points on the beam, and passing through a small ring situated in the intersection of the plane and wall; determine the positions of the points on the beam so that this may be effected with the weakest possible cord. a. If the two portions of the cord make equal angles with the line joining the ring with the centre of the beam, the expression for the tension becomes infinite; what is the physical meaning of this? b. Verify geometrically the answer to this last.

2. If a uniform rod move in a plane according to a given law, and if a point on the rod be suddenly fixed, find the impact on the fixed point.

« ПредыдущаяПродолжить »