9. A piece of uniform wire is bent into three sides of a square ABCD, of which the side AD is wanting; shew that, if it be hung up by the two points A and B successively, the angle between the two positions of BC is tan118.

Let EF, fig. (6), be drawn parallel to BA, through E the middle point of BC. Then, if G be the centre of gravity of the piece of wire, EG equals two-thirds of BE.

Draw HG parallel to BC, and join AG, BG.

When the wire is hung up by A, AG will be vertical, and when hung up by B, BG will be vertical; therefore the inclinations of BC to the vertical will be equal to the angles which BC makes with AG and BG. Therefore the angle between the two positions of BC, (supposing it to be kept in the same plane,) will be the angle between AG and BG.

Now tan AGB = tan (AGH+HGB)

therefore the angle between the two positions of BC is tan ̄118.

10. A weight of given magnitude moves along the circumference of a circle, in which are fixed also two other weights: prove that the locus of the centre of gravity of the three weights is a circle. If the immoveable weights be varied in magnitude, their sum being constant, prove that the corresponding circular loci intercept equal portions of the chord joining the two immoveable weights.

Let R, fig. (7), be the moveable weight, P and Q the stationary ones. Let G be the centre of gravity of P and Q, H that of P, Q, R.

R. GR
P+Q+R

∞ GR.

But the locus of R is a circle; hence that of H is a circle, G being a similar point in the two circles, and GR, GH,

similar lines. Hence, if HP', HQ', be drawn parallel to RP, RQ, P' and Q' will be points in the locus of H.

and therefore, GH: GR being constant, and PQ being constant, P'Q' is constant.

11. A ball of elasticity e is projected from a point in an inclined plane, and, after once impinging upon the inclined plane, rebounds to its point of projection: prove that, a being the inclination of the inclined plane to the horizon, and B that of the direction of projection to the inclined plane,

Let V be the velocity of projection.

This is equivalent to V sinẞ and V cosß respectively perpendicular and parallel to the plane.

Also the force of gravity is equivalent to g cosa and g sina, perpendicular and parallel to the plane.

Consider the motion perpendicular to the plane. The time of flight = twice the time in which the velocity V sinẞ can be generated by the force g cosa

after rebounding, the velocity perpendicular to the plane is eV sin ẞ,

therefore time of returning to the point of projection

Again, the motion parallel to the plane is not affected by

the impact, therefore

twice the time in which the velocity V cosẞ can be generated by the force 9 sin a

[The student may gain instruction by endeavouring to draw a correct figure.]

12. Two heavy bodies are projected from the same point at the same instant in the same direction, with different velocities; find the direction of the line joining them at any subsequent time.

By the second law of motion, the positions of the bodies at any time after their projection will be the same as if they moved for that time unaffected by gravity, and then fell from rest, from the positions they had reached, for the same time.

Now after the first part of the motion, each will be in the common line of projection; and after the second part of the motion, since they fall through equal and parallel spaces, the line joining them will be parallel to the line joining them before they fell, that is, to the line of projection. Therefore, in the actual motion, the line joining them will be always parallel to the line of projection.

13. Three equal and perfectly elastic balls A, B, C move with equal velocities towards the same point, in directions equally inclined to each other; suppose first, that they impinge upon each other at the same instant; secondly, that B and C impinge on each other, and immediately afterwards simultaneously on A; and thirdly, that B and C impinge simultaneously on A just before touching each other; and let V,V,V, be the velocities of A after impact on these suppositions respectively: shew that

1

2 3

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EXAMINATION PAPERS FOR

If the given distance be small, and the axis be incident a angle as to pass through the prism with minimum deviation, the primary and secondary foci nearly coincide, and thence ex necessity of certain precautions in order to obtain a pure sp the decomposition of light by a prism.

2. What is meant by a secondary spectrum? A compound o is to be formed of two lenses in contact; shew that, if, when are ground, achromatism is nearly but not quite secured, the de be remedied by slightly separating the lenses.

The refractive indices, corresponding to the letters D and orange and blue, for certain kinds of crown and flint glass, are Crown glass......1.5279, 1.5344,

Flint glass ......1.6351, 1.6481;

twenty inches is to be the focal length of the proposed obje find the focal lengths of the two lenses which, placed in conta these lines.

3. Investigate a formula for calculating the first two tables Nautical Almanac by which the latitude is determined from obse of the Pole Star out of the Meridian.

What is the nature of the correction contained in the third table

4. Determine the motion of a planet in geocentric longitu shew that all planets will sometimes appear stationary to an obs the Earth.

If m be the ratio of the radius of the Earth's orbit to that of an planet, n the ratio of their motions in longitude considered unifor that the elongation of the planet as seen from the Earth, when the appears stationary, is equal to

m2n2

tan-1

m2 - 1

5. Determine the motion of a particle acted on by given ford constrained to remain on a given surface.

A particle is in motion on the surface whose equation is z = and is acted on by a constant accelerating force ƒ parallel to the ax if be the velocity of the particle and its path be always perpendic the direction of the force, shew that, at any point of its path,

+

(dz\2 dz 2
dy
d2z dz dz

v2 f

d2z dz

dez dz

2

+

dx dy

dx dy dx dy

dy3 \dx/

6. Investigate the general equations of fluid motion; and from them the differential equation of the surfaces of equal pressure,

d

be incidenta im deviations

and thence

in a pure spe

compound at, if, when the

cured, the de

ers D and F:

it glass, are

oposed object

ed in contact

two tables

from obser

third table? ic longitude,

to an observer:

hat of an infer

d uniform, s when the p

a heavy elastic fluid is contained in a closed vessel, rotating with unform angular velocity about a vertical axis, and is at rest relatively to the vessel. How is the constant to be determined in integrating for the pressure at any given point?

7. Explain the effect of the Sun's disturbing force upon the position of the line of nodes of the Moon's orbit, when the line of nodes is in quadratures; and shew that the horary motion of the line of nodes is to that of the Moon as

N being the longitude of the node, O that of the Moon, and m✪ that of the Sun.

8. Define the principal axes of a rigid body, and shew that for every point in space there exists a system of such axes.

Shew that in general there is only one point for which the principal axes are parallel to those drawn through a given point; but that, if the given point be in one of the principal planes through the centre of gravity, there is an infinite number of such points lying in an hyperbola which passes through the given point.

9. The equation for the projection of the Moon's radius vector on the ecliptic is

calculate that part of evection in the value of 0 which is due to the radial force only.

Explain this term in connexion with the elliptic inequality,

1. THERE are n points in space, of which p are in one plane, and there is no other plane which contains more than three of them; how many planes are there, each of which contains three of the points?

2. A bag contains nine coins, five are sovereigns, the other four are equal to each other in value; find what this value must be, in order that the expectation of receiving two coins at random out of the bag may be worth twenty-four shillings.

P 2