8. Explain the construction of the thermometer, and the different ways of graduating it. What is the advantage of Fahrenheit's over the Centigrade and others ?

9. A Smeaton's air pump and a condenser have a common receiver, and the ascents and descents of their pistons are performin equal times, the piston of the one ascending while that of the other descends. If the piston of the air pump be originally in its lowest position, find the density of the air after three ascents and descents of the piston of the air pump.

THIRD CLASS.

MECHANICS.

1. Assuming that the diagonal of a parallelogram represents the direction of the resultant of two forces acting at a point, the magnitudes and direction of the forces being represented by the sides of the parallelogram, show that the diagonal will also represent the magnitude of the resultant.

2. Two weights of 40 and 20 lbs. are suspended from the ends, A, B, of a beam 30 feet long. The weight of the beam is 100 lbs. and the distance of its centre of gravity from A is 10 feet; find the position of the fulcrum about which the whole will balance. 3. What is meant by the resolved part of a force in any direction? Assuming the rule for the resolution of forces, determine the resultant of any number of forces acting at the same point. 4. Define the centre of gravity of a material system; and find

the centre of gravity of two particles P and P'.

If r be the distance from P of a point in the line joining P and P', and its distance from P'; show that Pr2 + P'r'2 is least when that point is the centre of gravity.

5. Find the condition of equilibrium in that system of pullies in which all the pullies hang by a separate string-the weight of each pulley being the same.

6. If a man stand in a scale attached to a moveable pulley, and a rope, having one end fixed, pass under this pulley, and then over a fixed pulley; with what force must he hold the free end in order to support himself—the strings being parallel?

7. What are the requisites of a good balance? Find the condition that a balance may possess great sensibility and great stability.

8. A shop-keeper has a false balance, and thinks to make his customers' consequent losses and gains balance by weighing the goods alternately in the one scale and in the other, -does he succeed?

9. Show that the principle of virtual velocities holds in the single moveable pulley with the cords inclined.

10. If a cylinder has its base united concentrically to the base of a hemisphere of equal radius: find the height of the cylinder in order that the solid may stand on a smooth horizontal plane on any part of its surface, the distance of the centre of gravity of the hemisphere from its vertex being = rad.

FOURTH CLASS.

EUCLID AND ALGEBRA.

1. Similar triangles are to one another in the duplicate ratio of their homologous sides.

2. If two parallel planes be cut by another plane their common sections with it are parallel.

3. If a line be drawn from one corner of a square cutting off one-fourth from the diagonal, it will cut off one-third from a side. Also, if lines be drawn similarly from the other corners the square contained by them will be two-fifths of the original square.

4. Define the tangent of an angle, and trace the change in its magnitude and sign as the angle increases from 0° to 360°.

5. Find the general value of all angles which satisfy the equation Sec.0 = a.

6. Express Cos A in terms of Sin2A, and show how in any particular case the proper signs of the radicals are to be determined.

Given Sin210° =

Cos. 105°.

7.

1

apply the formula to find the value of

If a + b +c+ d = 360°, prove that tana + tanb + tanc + tan d = sum of the products of the tangents taken three and three together.

9. Define a logarithm-its mantissa and characteristic, and state the advantages which arise from taking 10 as the base of a system of logarithms.

Given log2 =.3010300, log 5.743491 .7591760, find the fifth root of .0625.

If n = 8, find from the above log.3, having given log, 2 =.693147.

Thursday, April 6th-Afternoon Paper.

FIRST CLASS.

ASTRONOMY.

1. State and prove the properties of the polar triangle; and show that the sum of the angles of any spherical triangle lies between 6 and 2 right angles.

2. Describe in their chief features the apparent motions of the fixed stars and of the sun; and supposing these appearances to arise solely from the motion of the earth, deduce the nature of the earth's motion.

3. Define sidereal, solar, and mean solar time. Define also the equation of time. Find when the equation of time caused by the unequal motion in the ecliptic is additive or subtractive.

4. Describe the transit instrument and the mode of making an observation with it.

5. Find the latitude of a place by observing two equal altitudes of the sun before and after noon on a given day.

6. Find the effects of parallax on the hour angle and declination of a known body.

7. Explain the cause of aberration, and find the amount and direction of the change it produces in the apparent place of a star.

8. Find the time, magnitude and duration of a lunar eclipse.

9. In the Italian reckoning of time, half an hour after sunset is designated 24 o'clock; find the hour of noon at a place in latitude 45° on a day in August, when the tangent of the sun's declinanation Sin 15°.

10. If the sun be supposed to move uniformly in the ecliptic and t, t, be the lengths of the morning and evening at a place whose latitude is 7 on a day when the sun's declination at rising is d, then

2 Sin l. Sec2 d ( Sin (w+d). Sin (w−d) Į §.

365

{

Cos (1+d). Cos (l-d)

SECOND CLASS.

NEWTON, SECT. 1, 2, 3.

1. Define what is meant by the term "limit" and "limiting ratio”—and show that the limiting ratio of the chord, arc and tangent to each other is a ratio of equality, when the arc itself vanishes.

2. State and prove Lemma 10.

3. If a body move in any orbit about a fixed centre of force, the areas described by lines drawn from the centre to the body lie in one plane, and are proportional to the time of describing them.

4. Sensible gravity at the equator being 288 times the centrifugal force, show that if the earth moved round its axis with 17 times its present velocity, bodies at the equator would cease to gravitate.

v2 =(Force in the normal) (rad. of curvature.)

6. Find the law of force under which a body may describe an equiangular spiral.

7. Find the velocity at any point in an ellipse under a force in the center, find also the periodic time.

8. A body moves in a parabola, find the law of force to the focus.

9. A body describes an ellipse under a force in the focus, determine the point where the velocity is an arithmetic mean, and also the point where it is a geometric mean between the velocities at the greatest and least distances.

10. Show that the volume of a paraboloid is one-half of the circumscribing cylinder.

THIRD CLASS.

DYNAMICS.

1. How is accelerating force measured? If g be the measure of gravity when one second is the unit of time, and g' its measure when half a second is the unit, prove that g = 19.

2. A body acted on by gravity descends in a straight line,— prove that the space described varies as the square of the time from