ROYAL INDIAN E. COLLEGE, 1876.

NOTICES.

1. On the first day of the Examination, each Candidate will be required to state on a form which will be placed before him his address during the Examination. If this address is changed, notice of every change should be sent to "The Secretary, Civil Service Commission, Westminster," until the Candidate has received the announcement of the result of the Examination.

N.B.-To Candidates who fail to qualify in any of the obligatory subjects, notice of the fact will be communicated as soon as possible. Such Candidates will not proceed further with the Examination, and no marks will be assigned to their papers. 2. The Examination will in each case begin at the time named in the foregoing List, but the door of the Examination Room will be kept open for half an hour afterwards, in order that Candidates may not be excluded owing to accidental delays. Candidates arriving after the expiration of that half hour will not be admitted.

3. Candidates will be required, before proceeding to the Examination Rooms, to leave, in a room provided for the purpose, their hats, overcoats, umbrellas, and any books or papers which they may have brought with them.

4. No Candidate will be allowed to quit the Examination Room on any day until the expiration of half an hour from the time fixed for the commencement of the Examination.

5. No Candidate who has left the Examination Room during the hours assigned to paper work will be permitted to return to the paper which he has quitted, except on the day of Examination in Freehand Drawing.

6. Candidates wishing for explanation of the meaning of any of the questions before them may apply to the examiners. With this exception, perfect silence is to be preserved in the Examination Room; and any Candidate guilty of disorderly or improper conduct in or about the Room will be liable to be excluded from the Examination.

7. Any Candidate detected in the use of a book or manuscript brought with him for his assistance, or in copying from the papers of any other Candidate, or in giving or receiving assistance of any description, will be regarded as disqualified, and his name will be removed from the List.

8. Candidates who take up Freehand Drawing must bring their own Drawing Pencils and Brushes; but Drawing Paper, Drawing Boards, Colours, Indian Ink, and Palettes will be supplied to them by the Commissioners. For the Examination in Geometrical Drawing Candidates should bring their own Drawing Pencils and Brushes, and also Mathematical Instruments.

9. A Table showing the marks obtained at the Examination will be sent to each Candidate who qualifies in the obligatory subjects. Civil Service Commission,

Cannon Row, S.W.,

1876.

ALGEBRA.

Tuesday, 13th June 1876. 10 A.M. to 1 P.M.

1. When (m) and (n) are whole numbers, show that (am)"=amn. From

what assumptions is it deduced that the same law holds when
(m) and (n) are fractions.

Find the numerical value of 4+4+4 ̄, and of

3

(a3+a3) when a=1x=2.

2. Multiply (a2 +b2+c2−ab−ac-bc) by (a+b+c).

3. Divide (x+y)3—3(x+y)2+3(x+y)−1 by (x+y)2-2(x+y) +1. 4. Reduce (a+b+c) (b+c−a)+(a+c—b) (a+b−c).

4bc

5. Resolve 4a2y2 — (x2 + y2 — (z+1)2)2 into four factors.

6. If (D) be the factor, called the greatest common measure of two
algebraical polynomials (A) and (B), assuming that any factor of
(A) and (B) is a factor also of (A+B) and (A-B), investigate
the rule for determining (D) when (A) and (B) are given.
Find the greatest common measure of-

x

—2bx— (a2 —b2+ab)x+ab (a+b).

7. In any equation x+√y=a+√b which involves rational quantities and quadratic surds, prove that the rational parts on each side are equal, and also the irrational.

Extract the square root of 38-12/10.

8. Solve the following equations—

-ROYAL INDIAN
E. COLLEGE,

1876.

and express the equation in its simplest form. Given (1)
x2-px+q=0, (2) x2-(p2-2q)x+q2=0; obtain the roots of
each equation by the ordinary method of solution, and show that
the roots of equation (2) are respectively the squares of the
roots of equation (1).

10. A broker bought 15 shares in a railway (A), and 20 shares in a
railway (B), for 2,3001. After a time the shares in (A) rose 25
per cent., and in (B) 10 per cent. The broker selling all the
shares at the advanced price gained 4107. What was the price
of the shares in each railway when he bought them?

ROYAL INDIAN 11. Express the arithmetic, geometric, and harmonic means between (a) E. COLLEGE, and (b), and show that the squares of the means are in continued proportion. If (s) be the sum of (n) terms of the series a, a+b, a+2b, &c., and S the sum of (n) terms of the series

1876.

Prove S=(2n-1) s.

12. The sum of four numbers in geometrical progression is 90; but if 8 be added to the second term and 10 be added to the third term, the four numbers after these additions will then be in arithmetical progression; find them.

13. If there are (n) men in one regiment and (n1) in another, investigate the number of different detachments that can be formed by combining (r) men of the first regiment with (r) men of the second regiment.

There are two bags, one of which contains 6 white balls and the other 7 red balls, and one ball in each bag is marked; in how many ways might 4 balls be drawn, two out of each bag, without drawing both of the marked balls?

-" can never

14. Assuming the form of the binominal theorem for the expansion of
(1+x)" when (n) is a positive integer, prove that the theorem
holds when (n) is a negative integer. Show from the form of
the expansion that the number of terms in (1-x) -
be finite, and prove that the (nth coefficient of (1-x)" may be
1.3.5...(2n-3)
written 2-1/
1.2.3...(n-1)

EUCLID.

Tuesday, 13th June 1876. 2 P.M. to 5 P.M.

1. If two angles of a triangle be equal to one another, the sides also which subtend, or are opposite to, the equal angles, shall be equal to one another.

2. Equal triangles upon equal bases, in the same straight line, and towards the same parts, are between the same parallels.

The line DE is drawn parallel to the base BC of the triangle ABC, and AFG is drawn at right angles to DE, and BC meeting those lines in the points F and G, prove that the rectangles AF, BC, and AG, DE are each of them equal to the area of either of the triangles AEB or ADC.

3. If the square described upon one of the sides of a triangle be equal to the squares described upon the other two sides of it; the angle contained by these two sides is a right angle.

4. If a straight line be divided into any two parts, the squares of the whole line, and of one of the parts, are equal to twice the rectangle contained by the whole and that part, together with the square of the other part.

What is the algebraical equivalent to this proposition?

5. To divide a given straight line into two parts, so that the rectangle contained by the whole, and one of the parts, shall be equal to the square of the other part.

The straight line AB is bisected in C, and from A the straight line AD is drawn at right angles to AB and equal to AC, and a point E is taken on AB produced, such that CE is equal to BD; prove that the rectangie AE, EB, is equal to the square of AB.

6. If a straight line drawn through the centre of a circle bisect a ROYAL INDIAN straight line in it which does not pass through the centre, it E. COLLEGE, shall cut it at right angles; and if it cut it at right angles, it shall bisect it.

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

If AB and CD be two chords of a circle intersecting at right angles in the point E, and from C the line CF be drawn at right angles to AD cutting AB in the point G; prove that GE is equal to EB.

8. To describe a circle about a given equilateral and equiangular

pentagon.

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

If AB be a chord of a circle whose centre is C, and CD be drawn perpendicular to AB and produced to E, so that the rectangle CE, CD is equal to the square of CB; prove that

(1.) The tangents to the circle at A and B intersect in E.

(2.) If any other chord PQ of the circle be bisected by AB, the tangents at P and Q will intersect in the circumference of the circle, whose diameter is CE?

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

1876.

SUBJECTS FOR ENGLISH COMPOSITION.

Wednesday, 14th June 1876. 10 A.M. to 1 P.M.
(Including Dictation.)

In this Exercise, attention should be paid to handwriting, spelling,
punctuation, grammar, and style.

I. The decay of Chivalry ;
Or,

II. The results which may be anticipated from the visit of the Prince of Wales to India;

Or,

III. A criticism on any eminent novelist.
One subject only is to be attempted.

DICTATION.

While both metaphysician and man of science trade on a system of credit, they do so with profoundly different views of its aid. The metaphysician is a merchant who speculates boldly, but without that convertible capital which can enable him to meet his engagements. He gives bills, yet has no gold, no goods to answer for them; these bills are not representative of wealth which exists in any warehouse. Magnificent as his speculations seem, the first obstinate creditor who insists on payment makes him bankrupt. The man of science is also a venturesome merchant, but one fully alive to the necessity of solid capital which

1876.

ROYAL INDIAN can on emergency be produced to meet his bills; he knows the risks he E. COLLEGE, runs whenever that amount of capital is exceeded; he knows that bankruptcy awaits him if capital be not forthcoming. Astronomy became a science when men began to seek the unknown through the known, and to interpret celestial phenomena by those laws which were recognised on the surface of the earth. Geology became possible as a science when its principal phenomena were explained by those laws of the action of water, visibly operating in every river, estuary, and bay. Except in the grandeur of its sweep, the mind pursues the same course in the interpretation of geological facts which record the annals of the universe, as in the interpretation of the ordinary incidents of daily life. To read the pages of the great Stone-book, and to perceive from the wet streets that rain has recently fallen, are the same intellectual processes. In the one case the mind traverses immeasurable spaces of time, and infers that the phenomena were produced by causes similar to those which have produced similar phenomena within recent experience; in the other case, the mind similarly infers that the wet streets and swollen gutters have been produced by the same cause we have frequently observed to produce them. Let the inference span with its mighty arch a myriad of years, or span but a few minutes, in each case it rises from the ground of certain familiar indications, and reaches an antecedent known to be capable of producing these indications.

ARITHMETIC AND MENSURATION.

Wednesday, 14th June 1876. 2 P.M. to 5 P.M.

3

1. Add together 51, 33, 21, 20, and 10.

2. Subtract 813 from 95.

3. Multiply together, 26, 19, and 4, and of 7.

4. Divide 714 by 34.

5. Add together 34 207, 000219, 1714 8342, and 0890653.

6. Subtract 459 0836 from 640 1302.

7. Multiply 47 00658 by 05062.

8. Divide 986 9953 by 2·815.

9. Express 17s. 63d. as the decimal of 17. 5s.

10. Add together, 31, 1, and 18.

11. Subtract 25 from 363.

12. Multiply together, 18, 3, 1, and 6.

13. Divide 13 by 7.

14. Add together 3.075 of a mile and 4.35 of a furlong, and give the answer in yards.

15. Subtract 10 056 of an ounce from 1·201 of a lb. avoirdupois.

16. Multiply 307056 by 15.006.

17. Divide 23 746 by 0059 to three places of decimals.

18. Express 1.59685 cwt. as cwt. qrs. lbs. and ozs.

N.B.-The first eighteen questions should be answered before the others are attempted.