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(151.) If we have to determine the developement of a function designated by the following property,
P(1+x)+(1+y)=P(1+x+y+zy), we assume p(1+x)=a+2x+cx + dz'+er*+&c.; and adding to this the equation 0(1+y)=a+by+cy + dys + ey*+&c. it is evident that the sum of the two, viz.
br , cx?, dr
das (1+r+y+ry)= a + by +=
dys and comparing the terms, we shall find
&c. and hence we have
2 3 4 5 In this developement the quantity b is understood, but as it is not subject to alteration from the variations of x, it merely indicates that the function possesses a variety of systems.
Questions to exercise the foregoing Rules. 1. A Smuggler had a quantity of Brandy which he expected would raise £9 18; after he sold 10 gallons, a Revenue officer seized one third of the remainder, in consequence of which he makes only €8 2. Required the number of gallons he had, and the price per gallon.
Ans. 22 gallons, at gs. per gallon. 2. Three guineas were to be raised on two estates, to be charged proportionably to their values. Of this sum, A.'s estate which was 4 acres more than B.'s, but worse by two shillings an acre, paid £i 15; but had A. possessed 6 acres more, and had B.'s land been worth three shillings an acre less, it would have paid £2 5. Required the values of the estates.
Ans. A.'s £6, B.'s £4 16. 3. A coach set out from Cambridge to London with a certain number of passengers, four more being on the outside than within. Seven outside passengers could travel at 2s. less expence than four inside. The fare of the whole amounted to £9, but at the end of half the journey it took up three more outside, and one more inside passengers ;
of which the fare of the whole became increased in the proportion of 17 to 15. Required the number of passengers, and the fare of the inside and outside.
Ans. There were 5 inside, and 9 outside passengers, and the farer were 18 and 10 shillings, respectively.
4. Some persons agreed to give sixpence each to a waterman for carrying them from London to Gravesend ; but with this condition that for every other person taken in by the way, three-pence should be abated in their joint fare. Now the waterman took in three more than a fourth part of the number of the first passengers, in consideration of which, he took of them but 5 pence each. How many persons were there at first?
passengers. 3. The hold of a ship contained 442 gallons of water. This was emptied out by two buckets, the greater of which, holding twice as much as the other, was emptied twice in three minutes, but the less three times in two minutes ; and the whole time of emptying was twelve minutes. Required the size of each.
Ars. the least held 13, and the greater 26 gallons. 6. If 10 apples cost a penny, and 25 pears cost two-pence, and I buy 100 apples and pears for nine-pence halfpenny, how many of each shall I have?
Ans. 75 Apples, and 25 Pears. 7. A man at a party of cards, betted three shillings to two upon every deal ; after twenty deals he won five shillings. How many deals did he win?
Ans. 13. 8. Being sent to market to buy a certain quantity of meat, I found that if I bought beef, which was then four-pence a pound, I should lay out all the money I was entrusted with; but that if I bought mutton, which was then three-pence halfpenny a pound, I should have two shillings left. How much meat was sent for?
Ans. 48lbs. 9. A General having lost a battle, found that he had only half his army, +3600 men left, fit for action; one-eighth of his men +600 being wounded, and the rest, which were one-fifth of the whole army, either slain, taken prisoners, or missing. Of how many men did his army consist ?
Ans. 24,000. 10. Suppose that for every ten sheep a farme kept, he should plough an acre of land, and be allowed one acre of pasture for every four sheep. How many sheep may that person keep who farms 700 acres ?
Ans. 2000. 11. A and B began to trade with equal sums of money. In the first year A gained 40 pounds, B lost 40; but in the second, A lost one-third of what he then had, and B gained a sum less by 40 pounds than twice the sum that A had lost; when it appeared that B had twice as much money as A. What money did each begin with ?
Ans. £320. 12. Upon measuring the corn produced by a field, it appeared that it yielded only one-third part more than was sown. How much was that?
Ans. 36 quarters. 13. A certain sum of money is divided every week among the resident members of a corporation. It happened one week that the number resident was the square root of the number of pounds to be divided. Two men, however, coming into residence the week after diminished the dividend of each of the former individuals £i 6s. 8d. What was the sum to be divided a
14. In a garden is a square bowling-green, a side of which is 30 yards, and near to it is a rectangular grass plot. The number of square yards in the area of the grass plot is a mean proportional be.
and the number of square yards contained in the grass plot and bowling-green together. Also the number of square yards contained in the square described on the diameter of the grass plot, is a mean proportional between 10 and the number of square yards contained in the aforesaid square, increased by the number contained in the bowling-green. Required the area and sides of the grass plot?
Ans. 48 square yards. 15. After A had won four shillings of B, he had only half as many shillings as B had left. But had B won six shillings of A, then he would have had three times as many as A would have had left. How many had each?
Ans. A had 36, and B 84. 16. A and B playing at Bowls, says A to B, If you will give me a guinea, I will bet you half a crown to eighteen pence on each game, and will play 36 games together. B won his guinea back again, and £i 175. besides. How many games did each win?
Ans. A won 8, and B 28 games. 17. A Vintner sold at one time, 20 dozen of port wine, and 30 of Sherry, and for the whole received £120; and at another time sold 30 dozen of port and 25 of sherry, at the same prices as before, and for the whole received £140. What was the price of a dozen of each sort of wine ?
Ans. The prices of port and sherry per dozen, were 3 and 2 pound respectively.
18. A Person engaged to reap a field of 35 acres, consisting partly of wheat, and partly of rye. For every acre of rye he received five shillings; and what he received for an acre of wheat, augmented by one shilling, is to what he received for an acre of rye as 7 to 3. For his whole labour he received £13. Required the number of acres of each sort? Ans. 15 acres of wheat, and 20 of rye.
19. Two pieces of cloth of equal goodness but of different lengths were bought, the one for £5, the other for £6 10. Now if the lengths of both pieces were increased by 10, the numbers resulting, would be in the proportion of 5 to 6. How long was each piece, and how much did they cost a yard?
Ans. Price 5s, and the lengths are 20 and 26 yards. 20. A gentleman had some of his horses at grass at 3 shillings each week, and the rest at livery stables at 10 shillings each a week. The horses in the stables cost him twice as much a week as the horses at grass. But he finds that if he had sent three horses to grass out of the stable, the expence of the stables would have been only 6 shillings a week more than the grass. How many horses had he?
Ans, 15 grass, and 2 in the stable.
with water; then water is conveyed out of B into A in the following manner. First, a number of gallons is taken, which is less by two than the square root of the number of gallons in A, then a quanity less than the former by two gallons, and so on. Now when B is in this manner exactly emptied, A is exactly full: and it is known that 8 gallons were taken out of B at one time, after which the quantity ieft in B was 12 gallons. Required the number of gallons of wine in A?
Ans. 256 gallons. 22. There is a certain number, of which 9 times its cube, seven times its square, and 5 times the number itself, will make 547: What is that number?
Ans. 3.64481861. 23. What is the distance between two towns, of which it is known that the cube of that distance, minus 22 times the said distance is 24 English miles ?
Ans. 5:162277. 24. A person being asked how old he was, answered, that the qua. druple of the cube, the treple of the square, and the double of his age, amounted to 638,712 years. What was his age ?
Ans. 54 years. 25. Required the root of the cubic equation, 2—3r? + 2x=27?
Ans. 4.111069. 26. Required the root of the Biquadratic equation x* +599 +4x® + 313105
BND OF THE ALGEBRA
Definition. An Algebraic product, of which the difference between cvery two adjacent factors is equal to the same given number, is called a factorial.
Notation. In a factorial are to be considered the number of factors, otherwise called the exponent, the first factor and the common difference whether t or
Let m be the first factor, n the number of factors, and c the common difference; then every factorial may be thus indicated male : let n=4, and c=1, then will mak=m4i=m (m+1) (m+2) (m+3). Again, if n=5, and c=-1. Then will mal = m stim (m-1) (m-2) (m--3) (m-4).
Proposition. Any two factorials in which the base of the one is equal to the sum formed by adding the product of the exponent and common difference of the other to its exponent, may be reduced to one.
For let mmlc and [m+nc]ble, be the two factorials, the base of the latter being formed as announced in the proposition; then because c is the common difference, and m is the first factor of the factorial mals, the second factor will be mtc, the third m +2c, the fourth m+3c, and so on ; therefore in n+1 factors, the (n+1)th factor from the first, will be the first factor, together with n times the common difference c; therefore if the factorial (m + nc)bl be annexed to the factorial mal, as two factors, the product will be the factorial in* + b
Problem. To resolve a given factorial into two factorial factors in which the factors of each shall have the same common difference as the factors of the given factorial, and the one a given exponent less than that of the given factorial.
Rule. 1. Take the less of the two given exponents from the greater, and the remainder will be the exponent of the factorial factor, which is not given.
2. To the base of the given factorial, apply cither of the exponents of the two factorial factors, and the common difference, and the quantity thus formed will be one of the factorial factors.