9. What will represent the space passed over by a body falling 12 seconds?

10. What will represent the space passed over by a body falling n seconds? Ans. n2 g.

11. State the rule for finding the space passed over by a body, when the time of falling is known.

If a body be projected downward by a force that will cause it to describe a space of 10 feet per second, the force of gravity will act upon it just as if it had started from a state of rest, without the application of any force; and, in any given time, the body will have passed over 10 feet per second, and also, the common space due to a falling body. That is, it will have passed over ten times 10 feet, and 100 g feet, or 100+ 100 g feet, in 10 seconds.

12. If a body be projected downward with a velocity of 6 feet per second, what space will it have fallen through at the end of the 12th second?

13. If a body be projected downward with a velocity of a feet per second, what space will it have described at the end of the bth second? Ans. ab+b2 g. 14. A body is let fall from the top of a tower; 3 seconds afterward another is projected downward with a velocity of 100 feet per second. Both bodies reach the base of the tower at the same instant. How high is the tower, and what is the time of falling?

15. After a body has been falling one second from the top of a tower, with what velocity must another body be projected to overtake the first, at the distance of 300 feet from the top of the tower?

16. Being on a precipice which hangs over a large body of water, I let fall a stone; 4 seconds afterward, I hear the sound of the stone as it strikes the water. Now, supposing sound flies at the rate of 1142 feet per second, what is the height of the precipice?

If a body be projected upward with any velocity, it gradually loses its velocity, and returns to the earth again. The force of gravity will destroy just as much velocity in a given time, as it will communicate in the same time. If a body be let fall, it will acquire a velocity of 64 feet per second in 2 seconds. If a body be projected upward with a velocity of 64 feet per second, it will cease to ascend at the end of the

second second of time, and commence falling. Hence, if a body be projected upward with a given velocity, it will ascend through a space exactly equal to that through which it would fall in acquiring the given velocity.

17. If a body be projected upward with a velocity of 641 feet per second; how high will it ascend?

18. If a body were projected upward from the surface of the earth with a velocity of 128 feet per second; how long would it be before it reached the earth again?

19. One body is let fall from the top of a tower, and another is at the same instant projected from the base of the tower with a velocity of 100 feet per second. The two bodies meet in 2 seconds. How high is the tower?

SECTION XCVII.

Questions in Arithmetical Series.

1. The first term of an arithmetical series is 2; the last term is 20, and the common difference is 2. What is the number of terms?

2. A man being asked what his salary was, said, that when he first obtained his office, his salary was only 250 dollars; but, that in consequence of his industry, 30 dollars had been added yearly; and that now his salary was 490. How long had he held the office?

3. The first term of a series is 5, and the last term is 236; and there are 20 intermediate terms. What is the common difference of the terms?

4. A boy said that he had lost a certain number of marbles every day for 10 days. That he had at first 5 dozen marbles; but now, if he should lose as many every day for 10 days to come, and then should lose of what he had left, he would have only of what he had at first. How many did he lose every day at first?

5. In a series of 4 terms in arithmetical progression, the second power of the common difference is equal to the first term, and the last term is 10. What is the first term?

6. There is an arithmetical progression; the first term is equal to the number of terms, and three times as much as the

*common difference; whilst the last term is 21. What are the number of terms, common difference, and first term?

7. A man has 6 sons, whose ages are in arithmetical progression. His own age is 2 years more than the sum of all their ages. When his youngest son was born, the father's age wanted but 8 years of being twice the sum of all their ages. The age of the father is now four times the age of the How old is the father and each of his sons?

eldest son.

8. A laborer wanting employment charges for his labor 20 dollars per month; his employer said that he would not give him 20 dollars per month; but he would give him 8 dollars for the first month; and he would increase his wages by a certain sum every succeeding month; so that, at the end of the year he should receive just as much as if he had paid 20 dollars per month. How much must his wages be increased every month?

9. There are 10 terms of a series in arithmetical progression. The sum of the first and last is 24, and the sum of the second and sixth is 18. What are the terms?

10. There are two fractions; the numerator and denominator of the first, together with the numerator and denominator of the second, make four terms in arithmetical progression; the sum of the two fractions is, and their product. What are the numerators and denominators of the fractions?

11. There are two places 45 miles apart; two men start, one from each place, with a design to meet; the first travels 6 miles per hour, the second travels 1 mile the first hour, 2 the second, 3 the third, &c. When will they meet?

12. A sets out and travels 1 mile the first day, 2 the second, 3 the third, &c. After he had been gone 4 days, B sets out to overtake him; and, in order thereto, he travels 9 miles per day. In what time will B overtake A ?

13. The sum of the first and last terms of a series is 24; the last term is 10 more than the first, and the sum of the series is 66. What are the terms?

14. The sum of a series is 104; the last term is 20, and the common difference 2. What is the number of terms?

15. A tavern keeper bought provision enough to last 50 men 11 days. When he made the purchase he had a certain number of boarders. On the morning of the second day, however, 10 boarders left him; and 10 left every morning

afterward. Upon making an estimate, he found he had just provision enough to last until all his boarders should leave. How many boarders had he at first?

16. A housekeeper had provision enough to last 7 days; but, by taking 6 additional boarders on the morning of the second day, and 6 every morning afterward, he finds that his provision will last only 5 days. How many boarders had he at first?

17. Suppose a man pays a debt of 1000 dollars, together with the interest at 6 per cent. (simple interest) in 10 annual instalments. How much must he pay annually?

18. A person has a yearly income of 500 dollars, none of which he spends; but, from the day he receives it, beginning with the day he draws it the first time, he immediately puts it out at 5 per cent., and leaves this interest also untouched. In how many years will he thus accumulate 6875 dollars, at simple interest?

19. There are four bodies whose weights are in arithmetical progression. The sum of all their weights is 28 pounds; but, if 2 pounds be taken from each, then the last will be four times as heavy as the first. What is the weight of each?

20. A company of soldiers who had successfully stormed a fortress was rewarded as follows: the soldier who had mounted the wall first received a certain sum of money; the second a little less; the third exactly as much less than the second, and so on. When the money was divided, 2 of the soldiers could not be present on account of their wounds; their shares were consequently given to 2 of their comrades; these 2 put their own money and that of their comrades into the same purse; and afterward when they came to divide it, they had forgotten what fell to each man's share. One of them had received 92 dollars for himself and his comrade, and only remembered that he himself was the second and his comrade the seventh. The other had received 71 dollars for himself and his comrade, and knew that he himself was the eleventh and his comrade the fourth. How much did each of the 4 soldiers receive?

SECTION XCVIII.

Equivalent Expressions.

When two or more terms are united, and some arithmetical operation is indicated as performed with the whole of them, the expression is called a compound term. Thus (a+b) × ́(c+d), (a+b)3, &c. are called compound terms. Every compound term has a number of single terms, which are equivalent to it: thus, (A) (a+b) (c+d) = ac+be+ad +bd; (B) (a+b)2 = a2+2ab+b2.

We may substitute any numbers for a, b, c and d, in the preceding equations, and the first and second numbers will be identical.

The process of finding the equivalent single terms of any compound term, is called developing the compound term.

The two compound terms given above, are developed by multiplication. Sometimes compound terms are developed by division. In section 55 we found the sum of 5 terms of the

series a, ar, a r2, a r3, &c. equal to

α

а заб

1

α

We may de

by dividing ar—a by

r—1) ar—a (ar1+ ar3+ar2+ar+a

αγ

We have, then,

α