Three weights (w) are fastened to a string whose length (7) is equal to that of an inclined plane; one weight is attached to each end, and the other weight to the middle of the string; when one weight hangs over the top of the plane the weights are in equilibrium; if the second weight also is just made to hang vertically, find the velocity with which the third weight reaches the top of the plane.

8. Prove that the times of descent down all the chords drawn from the highest point of a given vertical circle are the same; express the time of descent for such a circle in an invariable form.

A particle P descends from the highest point down the chord which is the side of a regular hexagon inscribed in the circle, and Q down the vertical diameter; if P=2Q, shew that their common centre of gravity will descend along the chord which is the side of an equilateral triangle inscribed in the circle, assuming that the path of the centre of gravity is a straight line.

9. If a smooth sphere impinge directly upon another sphere, explain the mechanical action that takes place during the impact.

When an elastic sphere (m) impinges on another elastic sphere (m1) at rest, find generally the velocity of each after impact; if, after the impact, (m) remain at rest and (m1) move on with one eighth of the velocity with which it is struck, find the elasticity and the ratio of the radii of the two spheres, supposed of the same material.

IO.

How is the resistance of an inclined plane modified by friction? If (4) be the limiting angle of resistance on a rough inclined plane whose inclination is (a), shew that the accelerating force down the plane is

II.

g sin (a− p)

cos

When a given mass which is made fast to the end of an inextensible string revolves uniformly round the other end of the string, which is fixed, find the tension of the string.

If the weight of the body is given in pounds, how is the mass expressed?

A string 5 feet long can just sustain a weight of 20 pounds; if the revolving weight be 5 pounds, determine the greatest number of complete revolutions that can be made in one minute by the string without breaking.

12. Find the time of an oscillation of a heavy particle moving down the arc of a cycloid. Derive from this the time of an oscillation of a pendulum in a small circular arc.

A pendulum whose length is makes (m) oscillations in 24 hours; when its length is slightly changed it makes (m+n) oscillations in 24 hours; shew that the pendulum has been diminished in length by a part equal to

21

m

/ nearly.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

JUNE, 1881.

PRELIMINARY EXAMINATION.

I. EUCLID (Books I.-IV. AND VI.).

I. Define a plane angle, a scalene triangle, a rectangle, the angle of a segment of a circle. What is meant by a parallelogram being "applied” to a straight line? Give an instance from Euclid.

2. When is one straight line said to be at right angles to another straight line? Draw a straight line perpendicular to a given straight line of unlimited length from a given point without it. Why is the given straight line described as of unlimited length?

3. The straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel.

Prove that a quadrilateral, which has two opposite sides and two opposite obtuse angles equal, is a parallelogram.

4. In any right-angled triangle the square which is described on the side subtending the right angle is equal to the squares described on the sides which contain the right angle.

If BC is the hypotenuse of a right-angled triangle ABC, and BE is the square described on BC, and AK on AC, shew that AE and BK intersect at right angles to one another.

5. If a straight line be divided into two equal and also two unequal parts, the rectangle contained by the unequal parts together with the square on the line between the points of section is equal to the square on half the line.

If of the two unequal parts one is three times the other, shew by means of the above proposition that the rectangle contained by the unequal parts is three-fourths of the square on half the line.

6. Prove that the straight line drawn at right angles to the diameter of a circle from the extremity of it falls without the circle.

If the centre of a circle be joined to the vertices of a circumscribing quadrilateral, then any one of the four angles at the centre is supplementary to that one which is not adjacent to it.

7. The opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles.

If from any point on the circumference of a circumscribing circle perpendiculars be drawn to the sides of the inscribed triangle (produced if necessary), prove that their feet will lie in the same straight line.

8. About a given circle describe a triangle equiangular to a given triangle.

Having given the vertical angle of a triangle, and the segments into which the base is divided by the inscribed circle, construct the triangle.

9. Describe an isosceles triangle, having each of the angles at the base double of the third angle.

Shew how it is possible to describe within the smaller circle used in the construction a triangle equal in all respects to the required triangle.

IO. Define similar rectilineal figures; and shew the necessity for each part of the definition.

If two similar triangles ABC, A'B'C' are so placed that their homologous sides are parallel, shew that AA', BB', CC' will meet in a point, unless the triangles are equal, in which case these lines will be parallel.

II.

If the vertical angle of a triangle be bisected by a straight line, which likewise cuts the base, the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square on the straight line which bisects the angle.

II. ARITHMETIC.

(Including the use of Common Logarithms.)

[N.B.-Great importance will be attached to accuracy in numerical results.]

4. Divide 4 by ÷ of 23 and the result by 5%.

5. Add 3 26 quarts to the difference between 302 of a bushel and *9273 of a peck, and give the answer in pints and the decimals of a pint.

6. Multiply 73°0214 by 05031.

7. Divide 03081 by 237.

8. Divide 51°2705 by 0205.

9. Express 3 ft. 5 ins. as the decimal fraction of 5 poles.

IO.

of an oz.

Reduce '9043 of cwt. 3 qrs. 9 lbs. 2 oz. to ozs. and the decimals

11. Divide 76 lbs. 8 oz. 1 dwt. 6 grs. by 63.

12.

There is a well containing 750 gals. of water; two pumps raising 20 and 30 gals. per minute respectively are employed to empty it; while it is constantly supplied by a spring which can refill it in half an hour. The two pumps work together for 15 minutes, when that of larger capacity ceases work for 10 minutes; the two pumps then work together until the well is empty. How long will each pump have been employed?

13. On £2,340 a profit of 2s. 7 d. in the £ is made. Find (by Practice) the total profit.

14. What sum will amount to £3,995. 45. in three years at 4 per cent.? 15. If 4 per cent. be lost by selling silk at 10s. per yard, at what price per yard should it be sold in order to gain 5 per cent.?

16. Find the square root of 1532°7225, and the cube root of 34 correct to three places of decimals.

17. Define Greatest Common Measure and Least Common Multiple. Resolve 2310, 6552, and 12165 into their prime factors, and thence deduce their L.C.M.

18. Calculate (by duodecimals) the cubic content of a rectangular block of marble 7 ft. 5 ins. 7 pts. long, 4 ft. 2 ins. wide, and 3 ft. 4 ins. 7 pts. high. What is the result when expressed in cubic feet, cubic inches, and the decimal of a cubic inch?

19. If a mile be equivalent to 1,600 metres find the number of square metres in 7 acres.

20. Three railway tickets, a 1st, a 2nd, and half a third class were purchased for 16s. 10d. The 1st class ticket cost 1} times as much as the 2nd, and the 2nd class 1 times as much as a whole 3rd class ticket. The distance travelled was 45 miles. Find the cost of each ticket and the rate per mile for each class.

21. Eight bells, which toll at intervals of 1, 2, 3, 4, 5, 6, 7, 8 seconds respectively, begin tolling all simultaneously with the clock striking; how many hours must elapse before they all toll simultaneously again with the clock striking?—The clock is supposed to strike at the hour only.

22.

Find log '00077, log (1925), and the number whose log is

2.7286403.

23. Find by logarithms the amount of £5,500 in 15 years at 5 per cent. per annum compound interest, giving the result in pounds and the decimal of a pound.

(Including equations, progressions, permutations and combinations, and the binomial theorem.)

[N.B.-Great importance will be attached to accuracy in numerical results.]

and

I. Shew directly from the meanings of the symbols employed that

a+(b−c)=a+b−c,

a-(b-c)=a-b+c.

Reduce to the simplest possible form