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and the second, who had worked for 6 days less, received £2. 14s. If the second had worked all the time and the first had omitted 6 days, they would have received the same sum. How many days did each work, and what were the wages of each?

If

15. A party at a tavern spent a certain sum of money. there had been five more in the party, and each person had spent a shilling more, the bill would have amounted to £6. If there had been three less in the party, and each person had spent eightpence less, the bill would have been £2. 12s. Of how many did

the party consist, and what did each spend?

16. A person bought a number of £20 railway shares when they were at a certain rate per cent. discount for £1500; and afterwards when they were at the same rate per cent. premium sold them all but 60 for £1000. How many did he buy, and what did he give for each of them?

17. Find that number whose square added to its cube is nine times the next higher number.

18. A person has £1300, which he divides into two portions and lends at different rates of interest, so that the two portions produce equal returns. If the first portion had been lent at the second rate of interest it would have produced £36, and if the second portion had been lent at the first rate of interest it would have produced £49. Find the rates of interest.

19. A person having travelled 56 miles on a railroad and the rest of his journey by a coach, observed that in the train he had performed of his whole journey in the time the coach took to go 5 miles, and that at the instant he arrives at home the train must have reached a point 35 miles further than he was from the station at which it left him. Compare the rates of the coach and the train.

20. A sets off from London to York, and B at the same time from York to London, and they travel uniformly; A reaches York 16 hours, and B reaches London 36 hours, after they have met on the road. Find in what time each has performed the journey.

T. A.

14

21. A courier proceeds from one place P to another place Q in 14 hours; a second courier starts at the same time as the first from a place 10 miles behind P, and arrives at Q at the same time as the first courier. The second courier finds that he takes half an hour less than the first to accomplish 20 miles. Find the distance of Q from P.

22. Two travellers A and B set out at the same time from two places P and Q respectively, and travel so as to meet. When they meet it is found that A has travelled 30 miles more than B, and that A will reach Q in 4 days, and B will reach P in 9 days, after they meet. Find the distance between P and Q.

23. A vessel can be filled with water by two pipes; by one of these pipes alone the vessel would be filled 2 hours sooner than by the other; also the vessel can be filled by both together in 17 hours. Find the time which each pipe alone would take to fill the vessel.

24. A vessel is to be filled with water by two pipes. The first pipe is kept open during of the time which the second would take to fill the vessel; then the first pipe is closed and the second is opened. If the two pipes had both been kept open together the vessel would have been filled 6 hours sooner, and the first pipe would have brought in 3 of the quantity of water which the second pipe really brought in. How long would each pipe take to fill the vessel ?

25. A certain number of workmen can move a heap of stones in 8 hours from one place to another. If there had been 8 more workmen, and each workman had carried 5 lbs. less at a time, the whole work would have been completed in 7 hours. If however there had been 8 fewer workmen, and each had carried 11 lbs. more at a time, the work would have occupied 9 hours. Find the number of workmen and the weight which each carried at a time.

XXV. IMAGINARY EXPRESSIONS.

354. Although the square root of a negative quantity is the symbol of an impossible operation, yet these roots are frequently of use in Mathematical investigations in consequence of a few conventions which we shall now explain.

355. Let a denote any real quantity; then the square roots of the negative quantity - a3 are expressed in ordinary notation by√(-a). Now -a may be considered as the product of a2 and -1; so if we suppose that the square roots of this product can be formed, in the same manner as if both factors were positive, by multiplying together the square roots of the factors, the square roots of -a will be expressed by a√(-1). We may therefore agree that the expressions (-a) and a √(-1) shall be considered equivalent. Thus we shall only have to use one imaginary expression in our investigations, namely, √(−1).

356. Suppose we have such an expression as a + ẞ √(−1), where a and B are real quantities. This expression may be said to consist of a real part a and an imaginary part ẞ(-1); or on account of the presence of the latter term we may speak of the whole expression as imaginary. When ẞ is zero, the term B√(-1) is considered to vanish; this may be regarded then as another convention. If a and ẞ are both zero, the whole expression vanishes, and not otherwise.

357. By means of the conventions already made, and the additional convention that such terms as ẞ √(-1) shall be subject to the ordinary rules which hold in Algebraical transformations, we may establish some propositions, as will now be seen.

358. In order that two imaginary expressions may be equal, it is necessary and sufficient that the real parts should be equal, and that the coefficients of √(−1) should be equal.

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may be considered as a symbolical mode of asserting the two equalities a=y and ẞ= 8 in one statement.

359. Consider now two imaginary expressions a +ẞ √(−1) and y+8√(-1), and form their sum, difference, product, and quotient.

Their sum is

a+y+ (B+8) √(−1).

If the second be taken from the first, the remainder is

a-y+ (B-8) √(-1).

Their product is

{a + ß √(− 1)} {y + d √(− 1)} = ay — ßd + (ad + By) √(− 1);

for(-1)

(-1) is, by supposition, — 1.

The quotient obtained by dividing the first by the second is

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This may be put in another form by multiplying both numerator and denominator by y-d√(-1). The new numerator is thus

ay + ẞd + (By — ad) √(− 1);

2

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and the new denominator is y2+8; therefore

a + B√(-1)_ay + Bd By - ad

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√(−1).

360. We will now give an example of the way in which imaginary expressions occur in Algebra. Suppose we have to solve the equation a3=1. We may write the equation thus,

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Thus we satisfy the proposed equation either by putting x-1=0, or by putting x2+x+1=0. The first gives x=1; the second may be written

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Thus we conclude that if either of the imaginary expressions last written be cubed, the result will be unity. This we may verify; take the upper sign for example, then

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