C.

VI. HYDROSTATICS.

1. A prismatic vessel, whose base is an equilateral triangle, and altitude is equal to one side of its base, is filled with fluid: compare the pressure on one of the triangular ends with that on one of the rectangular sides, when it stands upright on a triangular end. Also, when a rectangular side is horizontal, compare the pressure on one end with that on one of the inclined sides, (1) When the horizontal side is upwards; (2) When the horizontal side is downwards.

2. The specific gravities of platinum, gold and silver being respectively 21, 17·5 and 10.5, and the values of an ounce of each 30s., 80s., and 58. respectively, it is required to find the value of a coin composed of platinum and silver which is equal in weight and magnitude to a sovereign.

3. A cylindrical pontoon, 20 feet long and 3 feet in diameter, has a fourth of the diameter immersed when floating; what is the weight of the pontoon, and what additional weight will it just bear, a cubic foot of water weighing 62 lbs.?

4. Find the altitude of the roof of Severndroog Castle, Shooter's Hill, above the wharf in the Arsenal, from the following observations ::

.32

Severndroog Castle...29-022 in.........35..

5. A diving-bell, in the form of a rectangular prism, and whose height is 8 feet, has descended until its upper surface is 40 feet below the surface of the water: to what height will the water rise within it, and what will be the density of the contained air, that of the external air being 1, the height of the barometer 30 inches, and the density of mercury to that of water 14 to 1 nearly?

6. Describe the common suction-pump and explain its mode of action by means of a figure,

and thence that, u, s, t being functions of x,

d (uts) du dt ds

જીÝÄ

dx dx dx

น t

2. Find the differentials of the following functions of x:

3. Find the sides of the greatest rectangle that can be inscribed in a given regular hexagon, a side of the rectangle being parallel to a side of the hexagon.

4. Of all cylinders inscribed in a given cone whose altitude is a, and radius of its base b, find,

.

1st.

That which has the greatest convex surface; 2nd. That whose whole surface is a maximum.

5. Find the subtangent to a curve whose equation is x2 a2- x2

and give the values of the subtangent corresponding

to x = α and x = a.

6. From the variable points C and P, equidistant from a given point A in the same straight line as C and P, perpendiculars, CD, PM, to CP are drawn, of which CD=AC, and PM is indefinite. From a given point B, in PC produced, the straight line CDM is drawn cutting PM in M. Find the equation to the locus of the point M, and the expression for the subtangent to the curve.

2. Find the area of the curve whose equation is

a*y2 = (x2 + y3)x*, from x=0 to x= = a.

3. From the general expression for the volume of a solid of revolution, find an expression,

1st.

For the volume of a paraboloid;

2nd. For the volume of a spherical segment, and of the whole sphere.

4. Give the differential equations of rectilinear motion when a body is acted on by any force f.

5. If a meteorolite were to fall to the earth from a height equal to ten times the radius, with what velocity would it strike the earth, the force varying inversely as the square of the distance from the centre, abstracting the resistance of the atmosphere near the surface?

6. Give the expression for determining the distance of the centre of gravity in a solid of revolution; and determine the position of the centre of gravity of a spherical segment.

7. Prove Guldin's properties of the centre of gravity; and, from these properties, find the volume and the surface of the solid generated by the revolution of a semicircle, diameter = 2r, about an axis parallel to its diameter, at the distance c from the centre.

D.

I. GEOMETRY.

1. Give Euclid's definitions of the following :

(1) A rectilineal figure described about a circle; (2) the same ratio, or equal ratios; (3) duplicate ratio and triplicate ratio; (4) similar rectilineal figures.

2. The difference of the angles at the base of any triangle is double the angle contained by a straight line drawn from the vertex perpendicular to the base, and another bisecting the angle at the vertex.

3. In a circle, the angle in a semicircle is a right angle; but the angle in a segment greater than a semicircle is &c.

4. If a straight line be drawn parallel to one of the sides of a triangle, it shall cut the other sides, or these produced, proportionally and, conversely, if the sides &c.

5. If two straight lines meeting one another be parallel to two others that meet one another, and are not in the same plane with the first two, the first two and the other two shall contain equal angles.

6. Horizontal Projection. On a given plane, to draw a straight line to pass through a given point and to have a given inclination to the horizon, not greater than that of the given plane. CO-ORDINATE GEOMETRY.

7. State clearly what you understand by the terms 66 co-ordinate axes," "co-ordinates of a point," "positive direction," "negative direction," "equation of a locus," "locus of an equation." 8. Construct the lines represented by the following equations: 5y-6x+1=0, 3y+8x-10=0, 7y = 6x;

and find the angles which they make with the axis of x,

9.

Determine the radii of the circles

y3 + x2 - бy - 10x-15=0, and y2+x3- 8y - 12x = 12;

and show also that the line which joins their centres is equally inclined to the co-ordinate axes,

229

D.

II. ARITHMETIC AND ALGEBRA.

1. Two persons, A and B, start from Shooter's Hill and London Bridge, distant 8 miles, at the same time, the former walking at the rate of 3 miles, and the latter at the rate of 31 miles, per hour. At what distance from Shooter's Hill will they meet? Find also this distance on the supposition that A was detained 20 minutes on the road before he met B.

3. Solve the following equations: √(5x+10) = √5x + 2,

4. In the front of a detachment from an army were 175 more men than in the depth; and by increasing the front by 50 men, the detachment was drawn up in 20 lines. Find the number of men in the detachment.

5. The first term of an arithmetic series is a, the common difference d, and the sum of n terms 8; find an expression for s. Find also the first term, the common difference, and the sum of n terms of the arithmetic series, of which the general form of the

6. The first term of a geometric series is 5, and the ratio 2: how many terms of this series must be taken, that their sum may be equal to 33 times the sum of half that number of terms?

7. Find the value of

(*005234) × (·017)*

by logarithms.