If the point

be taken in AB such that AQ=AP, and if AP produced meet the tangent at B in R and a body slide down APR from rest, prove that the times of the body being within and without the circle are in the ratio of AQ to BQ.

6. Define the terms Energy (Kinetic and Potential), Work, and Power, and state what is meant by the "Conservation of Energy."

If the unit of Energy be that required to raise 1 lb. through foot (without gain of velocity), find the number of units of Kinetic Energy in a mass of 1 oz. moving 10 feet per second.

7. Find the velocities of two equal perfectly elastic spheres after oblique impact.

Prove that the directions of the relative velocities of the spheres before and after impact are equally inclined to the line of centres at the instant of impact.

8. Find the acceleration of a particle describing a circle with uniform velocity.

In question 3 prove that the force required to keep a drop of water attached to the umbrella's rim makes an angle whose tangent is with the vertical. If the drop weighs or of an oz., find to four decimal places the magnitude of this force.

9. State the Third Law of Motion.

A train weighing 50 tons is moving on a level at 30 miles an hour when the steam is shut off, and the brake being applied to the brake-van the train is stopped in a quarter of a mile. Find the weight of the brakevan, taking the coefficient of friction between its wheels and the rails to be one-sixth, and supposing the unlocked wheels of the train to roll without any sliding.

10. Explain the principle of the pendulum.

Two pendulums oscillating at two different places lose t and 7 seconds a day respectively, and if the places at which they oscillate be interchanged they lose t' and 7' seconds; prove that t+7=t'+r' nearly.

MATHEMATICAL EXAMINATION PAPERS

FOR ADMISSION INTO

Royal Military Academy, Woolwich,

NOVEMBER, 1883.

PRELIMINARY EXAMINATION.

I.

I. EUCLID (Books I.-IV. and VI.). (Obligatory.)

[Great importance will be attached to accuracy.]

Draw a straight line perpendicular to a given straight line of unlimited length, from a given point without it.

2. If a side of a triangle be produced, the exterior angle is equal to the two interior and opposite angles; and the three interior angles are equal to two right angles.

If ABC be any triangle, and through D the middle point of AB, DE is drawn parallel to BC, and BE be drawn to bisect the angle ABC and meet DE in E, AEB will be a right angle.

3. The complements of the parallelograms which are about the diameter of any parallelogram are equal to one another.

If ABCD be a parallelogram, and from K, a point on the diagonal AC, EKF be drawn parallel to AD to meet AB in E and DC in F, and HKG be drawn parallel to AB to meet AD in H and BC in G; the triangles AGF, AEH are together equal to the triangle ABC.

4. If a straight line be divided into any two parts, the squares on whole line, and on one of the parts, are equal to twice the rectangle

tained by the whole and that part, together with the square on the other part.

If AB be divided in C so that the square on AC is double the square on CB, the sum of AB and CB will be equal to the diameter of the square on AB.

5.

If two circles touch each other internally, the straight line which joins their centres, being produced, shall pass through the point of contact.

6. The angles in the same segment of a circle are equal to one

another.

A and B are the extremities of an arc of a circle, and Q any point on the arc is joined to A and B. If P be a point on AQ or AQ produced such that QP=QB, prove that P will lie on an arc of a fixed circle passing through A and B.

7. If from any point without a circle two straight lines be drawn, one of which cuts the circle but does not pass through the centre, and the other touches it: the rectangle contained by the whole line which cuts the circle, and the part of it without the circle, shall be equal to the square on the line which touches it.

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

9. If the angle ACB of the triangle ACB is bisected by CE, cutting AB in E, and another point D be taken in AB produced, such that the angle ECD is equal to CED, CD will touch the circle described about АСВ.

IO. If two triangles have one angle of the one equal to one angle of the other, and the sides about the equal angles proportionals, the triangles shall be equiangular to one another, and shall have those angles equal which are opposite to the homologous sides.

If APB is a semicircle of which AB is the diameter, and C the centre, Na point on CB, and AB is produced to T so that

AT: AC-AN: CN,

and PT is the tangent drawn from T, CNP will be a right angle.

II. If four straight lines be proportionals, the similar rectilineal figures similarly described upon them shall be proportionals.

1.

II. ARITHMETIC. (Obligatory.)

(Including the use of Common Logarithms.)

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

5. Add together 00125 of an acre, 0625 of a rood, and 00375 of a

perch.

Subtract the result from 916634375 yards, and give the answer in square feet, and the decimal of a square foot.

9. Reduce of 1 hour 54 minutes to the decimal of 8 days. 10. Express 3 pecks 2 qts. pt. as the decimal of 11 bushels. II. Divide 18 miles 23 poles 6 inches by 5872.

12.

Find the dividend on £7,566. 16s. at 13s. 10дd. in the £. 13. In what time will £1,675 amount to £1,995. 6s. 101d. at 44 per cent. per annum simple interest?

14. Explain the rule for finding the Least Common Multiple.

Find the smallest sum of which 145. 7d., £1. 11s. 6d., and £3. 155. are exact parts.

15. What principal will amount to £498. 185. 5d. in 3 years at 31 per cent. per annum compound interest, neglecting fractions of one halfpenny?

16. In what proportion must a merchant mix one kind of tea at 3s. per lb. with another at 1s. 6d. in order that by selling the mixture at 2s. 8d. per lb. he may make a profit of 25 per cent.?

17. If a rectangular pathway which measures 7874 mètres in length and 1.526 mètres in width is made at a cost of 1 francs per square mètre, find the length in English measure of a similar pathway 6 feet wide, which costs Is. Id. per square yard, the total cost being the same.

(A mètre may be taken as equal to 39°37 inches and £1 as equal to 25 francs. Fractions of an inch may be neglected.)

18. A and B start from the same point to run in opposite directions round a circular race-course, 9755 feet in circumference, 4 not starting until B has run 105 feet. They pass each other when A has run 4850 feet. Which will first come round again to the starting point (their speeds being uniform throughout) and what distance will they then be apart?

19. There are four vessels of equal capacity; the first is filled with spirit to the extent of th, the second to th, the third to 4th, and the last to rd. The first is then filled up with water, and from this mixture the second is filled up, again from this second mixture the third is filled up, and in like manner the fourth from the third. What proportion of spirit to water is there in the fourth vessel?

20. What sum at compound interest will amount to £650 at the end of the first year and £676 at the end of the second year?

21. Find by logarithms the value of

403 09 × 002317 X 17
18.543

22.

Find the logarithms of 165 and 3'77; and reduce the expression

x=√72.8.95
√112.33

to logarithmic form without calculating the result.

23. Log 3x-2=2'404691352; find the value of x.