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8. Define steady motion, and write down the equation of motion of a steadily moving stream. Also find the pressure on a plane lamina immersed perpendicular to the direction of the stream.

9. Find an expression for the cosine of an angle of a spherical triangle in terms of the sides. Shew that if the triangle be equilateral

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10. Shew that there can be only five regular polyhedrons.

11. Describe the nature of the errors to which a Transit Instrument is liable. If a, 8, y, be the errors of collimation, level, and deviation respectively, shew that the error in time of transit of a star whose declination is dis

=

sec d

15

-{

a + B cos (1—8) + Y sin

where is the latitude of the place.

(1—8)},

12. Determine the interval from sunrise to sunset at a given place.

If the increase of the sun's declination from noon to noon be 4°, and t, t' be the times from sunrise to noon and from noon to sunset respectively, shew that

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where is the latitude of the place, and the sun's declination at sunrise.

13. How is the apparent position of a heavenly body affected by parallax ?

Explain what is meant by horizontal parallax. If h' is the parallactic change of the hour-angle h, prove that

sin h'

sin (h+h')

=

sin c sin P
sin A

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where c is the co-latitude of the observer, ▲ and P the north polar distance and horizontal parallax of the observed body.

STATICS, HYDROSTATICS AND OPTICS.
Examiner-MR. ELIOT, M. A.

1. Investigate formula for determining the position of the centre of mass (gravity) of any solid.

Find the position of the C. G. of the solid bounded by the co-ordi

nated planes and the surface

radicals being all positive.

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2 Find the attraction of a circular lamina on an external particle situated on a straight line drawn at right angles to the lamina through its centre.

A right circular cone of given volume attracts a particle at its vertex. Find the form of the cone so that the attraction may be a maximum.

3. A rough heavy body rests on a fixed rough surface. Deter

mine the conditions of stable equilibrum, the forms of the surfaces, being known.

A wire is bent into the form of a semicircle with the bounding diameter, shew that the arc will rest in stable equilibrium on the vertex of a perfectly rough parabola with the axis vertical, if the latus rectum be the semicircular arc.

4. A string is stretched over a rough plane curve. Find the tension at any point, and the pressure on the curve in the limiting position of equilibrium.

A uniform inextensible string whose length is 7 hangs in limiting equilibrium over a fixed rough cylinder (radius a) whose axis is horizontal. Find the lengths of the portions which hang vertically.

5. Find the conditions of equilibrium of a fluid acted on by any forces whatever, and prove that the resultant force at any point of a surface of equal pressure is normal to the surface, and inversely as the density at the point, and the distance to the consecutive surface of equal pressure.

6. Distinguish between whole and resultant pressure. Find the condition that the pressures of a heavy incompressible fluid on a given surface may be equivalent to a single resultant.

If a triangle be immersed in a fluid in a vertical plane and if a, B, y be the inclinations of its sides to the horizon, and x, y, z (in order of magnitude) the depths of its vertices prove that the depth of the centre of pressure in given by

4 sin A sin a- y4 sin B sin B + 2 sin C sin y
-3 sin B sin 8 + 23 sin C sin y

x3 sin A sin a

7. Define metacentre. for determining its position.

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A solid is formed by the revolution of a quadrant of an ellipse about the major axis. Prove that if it is to float in water in stable equilibrium with its axis vertical and vertex downwards with half its

axis immersed, the eccentricity must be less than

√5

3

8. Investigate an equation for determining the height of a station above the level of the sea by means of a barometer and thermometer regard being had to the variation of gravitation.

9. Find the position of the focal lines of a small pencil obliquely refracted at a spherical surface.

A small pencil emerging from a point on the axis of a cylinder of radius r is reflected at the sides of the cylinder. Shew if be the angle of incidence the distance of the area of least confusion from

the point of incidence will be r sec cot

2

2

10. Find the condition for the minimum deviation of a ray of from light through a prism.

The minimum deviations at the three angles of a triangular prism of a ray of index μ are 81, 82, 83 prove that

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11. Define positive and negative lenses. Shew that they must be respectively thinnest and thickest in the middle.

12. Explain fully the formation of the Primary Rainbow.

ALGEBRA.

Examiner-MR. W. GRIFFITHS, M. A.

1. Define the terms "limit" and "differential co-efficient" of a function.

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be the base of the Napierian system of logarithms, investigate the

x

differential co-efficient of a with respect to x.

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Differentiate with respect to x the expression (a + x)

2. State and prove Taylor's theorem, mentioning the cases in which it fails to give a correct result. If r terms only of the series be taken find limits between which the error lies.

=

2. If (xy) On (xy) + On-1 (xy) + + $1 (xy) + $0 = 0 be the equation of a curve, where Pr (xy) is the sum of the terms of the rth degree in x and y, prove that the equation of the tangent at any point (f, g) of the curve is

d

d

df

X $ (f, g) + Y

dg

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+ ...... + n po

Find the differential equation of the family of curves that cut the family (x, y, c) = 0 at right angles.

4.

Define a "multiple point" a " point of inflexion" and a "cusp." Find the conditions that a given point on the curve (xy) = 0 may be a double point, and shew how to find the two directions of the curve at it. If the two branches cut one another at right angles, prove that

d2p d2p
+
dx2 dy"

0.

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Two points are moving with velocities which are always proportional to one another, one in the line joining the points, the other in a perpendicular direction. Shew that the curves traced out are both equiangular spirals.

6. Investigate the method of integrating by parts; and prove that

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7. Find the area included within the positive compartment of (ry), of the curve y2+ 2xy + 2 x2 = a2, employing (i) rectangular, (ii) polar co-ordinates.

8. A plane is drawn through a diameter of the base of a right circular cone whose equation is 2 + y2 (az), the plane making an angle 45° with the base. State between what limits the

integrations in

SSSd=

dx dy dz must be performed to find the

volume cut off by the plane.

9. Investigate the equations of a straight line in the form

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generating lines, and that the perpendiculars from the origin upon the generating lines lie upon the cone

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10.

(X1 Y1 Z1), (X2 Y2 Z2), (3 Y3 Z3) be the co-ordinates of the extremities of any three conjugate diameters of the ellipsoid

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11. A cone envelopes an ellipsoid, find the equation to the cone and to the plane of contact

12. Define the principal sections at a given point of a surface, and find the relation between the radius of curvature of any normal section and those of the principal sections.

clxv

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and a x + b y3 +

=

У

=

2

= b2

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b1 y3 + c1 z3

=

0

cô z3

=

a1 x2 + b2 y2 + c1 z2

2. If A', B', C' be any points on the sides of the triangle ABC prove that AB', BC' CA' + B'C, C'A A'B area of ▲ A'B'C' × twice the diameter of the circle circumscribing ▲ ABC.

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3. If P be the orthocentre i. e. point of intersection of the perpendiculars from the angular vertices on the opposite sides, PA, PB, PC are the roots of the equation

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2R (s2(r2R)2) = 0 where R and r are the radii of the circumscribed and inscribed circles, and s is the semi-perimeter.

4. A circle is described touching a parabola and also intersecting the parabola at the ends of a focal chord. Prove that if be the inclination of the chord to the axis, and 4a the latus rectum a (1+4 sin 0)

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5. Having given a tangent, a focus, and the eccentricity of a conic section, prove that the locus of its centre is a circle.

6. A conic is described having the origin for focus and osculating

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the equation of its directrix is

(a2b2) (ax cos3 - by sin3 ) + a2b2

= 0.

7. Find the form of a surface of revolution such that when water is poured into it, and it is placed with its axis vertical, the whole pressure on it may vary as the square of the length of the axis immersed.

8. A quantity of elastic fluid whose particles attract one another according to the law of nature fills a sphere in whose centre resides The radius of the sphere is c and mass of the

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9 A glass rod is balanced partly in and partly out of a cylindrical tumbler with the lower end resting against the vertical side of the

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