fourth of the base ; find the locus of the vertex, and prove that its arc is represented by an elliptic function, and reduce it to its fundamental form.


FK 10. If ai= am

-, A2=am
a3 = am

find the algebraic va4 lue of E (ki ai) + Ē (ki a2) - E (ki a3).


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1. Prove that the secular part of the disturbing function in a planetary orbit can be expressed in the form


(e? +eTM2 – tanai – tan2j' + 2 tan i tan i' cos (2-2)

ee' (cosa w') + terms of a higher order 2. Show how to calculate the coefficients Co Di D2 in the above expression as elliptic integrals.

3. Find from this expression the motion of the aphelia of two planets subject only to their mutual disturbance.

4. Investigate the effect of a resisting medium on the eccentricity of a planet's orbit.

5. How did Newton compute the first term in the value of c which gives the motion of the Moon's perigee?

6. How did he compute the effect of the “ variation” on the motion of the Moon's nodes ?

T 7. Calculate the value of do in the Lunar theory accurately

h2 23 enough for a second approximation, and show the effect of each term on the expression for the time in terms of the longitude.

8. If V be the potential of a system of masses attracting with forces that vary inversely as the square of the distance, how does Gauss investigate the value of the function

d2 V 12 V d2V


dya for a point within one of the masses ?

9. How is the investigation modified if the attracting masses are distributed over a surface?

10. Find the expression, in a series of Laplace's coefficients, for the potential of a heterogeneous body acting on an internal point; the body being composed of shells nearly spherical.





1. Assuming the truth of Lagrange's equations for the variation of arbitrary constants, namely,

da d22 di da2

dt dl' dt da' explain the nature and origin of the various kinds of non-periodic terms which may appear in the value of 2.

a. Show that the method of Lagrange may naturally be expected to give rise to such terms, even though not essentially connected with the question.

6. Show that the first perfectly non-periodic term which is found in 12 is of an order two degrees higher than that of 2 itself.

c. The equations (A) may sometimes be so treated as to avoid these nonperiodic terms.

d. If such a term do occur, how far is the use of the method of Lagrange restricted ?

2. Deduce the equations by means of which Lagrange determines the components (round fixed axes) of the angular velocity of a solid body moving in onsequence of an initial impulse.

3. Prove that the equations of rotation of the Earth round its centre of gravity are

dp=o. da C-B C-B

(Pcos $ P'sin o);
dt A

dr A-C A-C

+ Pa= (P'cos $ +P sin o),
dt B

B where

y2 22

sin 20 + y2 cos 20

rp =


P' = xy sin 0 + xz cos e

25 I being the mass of the disturbing body, r its distance, and x, y, z its co-ordinates

a. Show that the change in the obliquity of the ecliptic depends wholly on P', and the change in the position of the equinoctial points wholly on P.

b. Show that there is no term depending on the initial circumstances of the motion.

c. Explain the method of determining, by means of the precession, limits within which the Earth's ellipticity must lie.

4. Assuming Laplace's expression for the inclination of the equator to the fixed ecliptic, namely,


Q=h- cos (ft +B) +

cos (f't+B)

1 tan b

cos 20"
2m (1+1



m COS 20+


a. Show that by a suitable determination of the fixed ecliptic the terms depending on the motion of the solar orbit may be neglected for times near the epoch.

6. Determine the secular variation of the apparent ecliptic.

c. Which of the terms in the value of 0 is the most important, and why?

5. Determine the equation of equilibrium of a flexible and inextensible surface according to the method and definition of Lagrange.

a. Whatever be the nature of the forces acting on the membrane, the following property holds :

If a curve be traced on the surface between two given points, the velocity acquired by a particle in moving from one point to the other along this curve under the action of the forces is independent of the nature of the curve.

6. Hence, if Xdx + Ydy + Zdz be not a perfect differential, the equation of the surface may be written under the form

Ap+Bq=1, A, B, being functions of X, Y, Z, X, Y, Z, being also functions of the coordinates.

c. Adopting Gauss's definition of an inextensible surface, it is impossible to find from the statical equations the equation of the surface of the membrane.

6. Investigate the equations of equilibrium of a solid body according to the method of Lagrange, and prove the mathematical theorem made use of in his demonstration.

7. Investigate the equation of the ellipsoid of elasticity according to Lamé's definition, and show that the elastic forces corresponding to its principal planes are normal to these planes.

8. Determine the equations of motion of an elastic homogeneous uncrystalline solid acted on by forces satisfying the condition

Xdx + Ydy + Zdz= d V, and show that if we assume dro dv du dw dv


dz dx' dac dy' lt, v, w being the displacements, each of the quantities 5, 1, , satisfies the equation


d2 d2

dta 9. Deduce Professor M‘Cullagh's equation for determining the polarizing angle in the case of a uniaxal crystal whose axis is contained in the plane of incidence.

a. Show from this value that when light is reflected at the common surface of a crystal and a fluid, the polarization is under certain circumstances incomplete.

10. If a beam of circularly polarized light pass through a thin plate of a uniaxal crystal, bounded by planes perpendicular to the axis, and, after having been analyzed, fall on a screen; find the expression for the wave disturbance at any point of the screen.


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dy dz'

+ dx2

11. If the equation of a surface de niveau of a fluid in equilibrio be

r=a {1+a (Yo+Ya+Y2+ &c. ;)} show that for the outer surface

2i+ I

3 no external forces being supposed to act.

12. If a solid, whose form is nearly spherical, be covered by a fluid stratum, the centre of gravity of the whole body will coincide with the centre of gravity of the surface of the fluid.

Sopa (aina y) -*** aY:['pd.a?,

1. If a number of material points, connected by known relations, and under the influence of given forces, receive given initial velocities, show that the initial values of the internal forces may always be determined by a series of simple equations.

a. If no initial velocities be communicated to the points, and if it be possible to assign to each of these points a displacement coincident with and proportional to the acting force at that point, the initial values of the internal forces will in general vanish ?

6. Show from physical considerations that this might have been anticipated. [N. B.—The equations of condition are not supposed to contain the time explicitly.]

2. If an inflexible rod move under the action of given forces, and subject to given geometrical conditions, state accurately the relation between the wrenching force at any point and the lost forces of the system, and apply the result to the following question :

If a rigid rod move, in obedience to a primitive impulse only, with its extremities resting each on one of two lines at right angles to each other, determine the time and point at which it is most likely to break.

3. If a cylinder of any form be laid on a horizontal plane, and slightly disturbed from its position of equilibrium (supposed stable); show that the length of the simple equivalent pendulum is

Moment of inertia round line of contact
Mass X (R-6).

; where R= radius of curvature of transverse section at the point of contact, and b= height of centre of gravity.

4. Four equal particles, exercising no attraction on each other, move in an ellipse under the action of a central force situated in the centre; at the commencement of the motion they were situated at the extremities of the axes.

If at any time they become suddenly connected with each other so as to form a rigid system, find the angular velocity of this system.

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y = sin a

sin ** (vt – r)

a for each homogeneous ray of which it is composed, be passed through an analyzer, what will be the appearance presented as the analyzer is turned round:

2. Given a ray of elliptically polarized light, show how to measure accurately the position and ratio of the axes of the ellipse.

a. Show that the existence of a very small amount of elliptic polarization may be detected by a delicate analyzer, and state the uncertainty to which this method is liable.

3. State the hypothesis of Brewster as to the composition of white light, and mention some of the experiments by which it is supported.

4. If the rings which are seen when a crystalline plate is interposed between two tourmalines were produced by a depolarizing action in the crystal, how would the phenomena differ from those which are actually seen?

5. The expression for the intensity of polarized light falling on a screen being

(a) + (ma 2c2 + n2 y2 – 12) 4 (a) describe the appearance presented, a being the azimuth of the plane of polarization and x, y, the co-ordinates of a point in the screen. a. If the equations

(a)= 0, *(a) = 0, give real values for a, describe the appearances which will be presented as the polarizer is turned round?


1. How does Biot deduce from Melloni's experiment the law of the absorption of heat in passing through a diathermal plate ?

2. If a body cool in an empty enclosure, whose temperature differs from that of the body by a given quantity, it is found that when the temperature of the enclosure increases arithmetically, the velocity of cooling increases geometrically. Hence, and from the hypothesis of mutual radiation, deduce the law of cooling.

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