4°. Let us suppose m=n, and also a=0; the values of x and y in this case become

0

that is to say, the problem is indeterminate, and admits of an infinite number of solutions. In fact, if we suppose a=0, we suppose that the couriers start from the same point, and if we at the same time suppose m=n, or that they travel equally fast, it is manifest that they must always be together, and consequently every point in the line A C satisfies the conditions of the problem. 5o. Finally, if we suppose a=0, and m not =n, the values of x and y in this case become

x=0, y=0.

In fact, if we suppose the couriers to set out from the same point, and to travel with different velocities, it is manifest that the point of departure is the only point in which they can be together.

ADDITIONAL PROBLEMS.

(1) The rent of an estate is greater than it was last year by 8 per cent. of the rent of that year; this year's rent is 1890. What was last year's?

Ans. 1750.

(2) A company of 90 persons consists of men, women, and children; the men are 4 in number more than the women, and the children 10 more than the men and women together. How many of each ?

Ans. 22 men, 18 women, and 50 children.

(3) From the first of two mortars in a battery 36 shells are thrown before the second is ready for firing. Shells are then thrown from both in the proportion of 8 from the first to 7 of the second, the second mortar requiring as much powder for 3 charges as the first does for 4. It is required to determine after how many discharges of the second mortar the quantity of powder consumed by it is equal to the quantity consumed by the first.

Ans. 189 discharges of the second mortar.

(4) The fore wheels of a carriage are 5 feet and the hind wheels 7 feet in circumference; the difference of the number of revolutions of the wheels is 2000. What is the length of the journey?

Ans. 39900 feet, or 7 miles.

(5) Three brothers, A, B, and C, buy a house for £2000; C can pay the whole price if B give him half his money; B can pay the whole price if A give him one third of his money; A can pay the whole price if C give him one fourth of his money. How much has each?

Ans. A £1680, B £1440, C £1280.
French,

(6) The passengers of a ship were Germans,

English,

but the objection to the doctrine of the special and immediate superintendence of Providence in the affairs of men, that it implies an incredible degree of condescension in an infinite being, finds in the principle above stated a satisfactory refutation. As compared with infinity, the smallest portion of matter is equal to the greatest, and it is therefore no more an act of condescension on the part of God to charge himself with the care of an individual than of a nation-with the revolutions of a satellite than with the movements of a system.

Dutch, and the residue, amounting to 31, Americans. How many were there in the whole ?

Ans. 120.

(7) Suppose the sound of a bell to be heard at the distance of 1142 feet in a second in a still atmosphere, and that a wind is blowing sufficient to occasion a delay of in time. In how many seconds will the sound reach a distance of 6000 feet?

Ans. 6.304.

(8) Quicksilver expands, for each degree of the centigrade thermometer, 5350 of its volume. According to this, how high would the barometer stand when the temperature is 0°, if, when the temperature is 21°, it stands at a height of 27 inches 8 lines?

Ans. 27 in. 7.458 lines. 1857

(9) What degree of heat in a centigrade thermometer would be required to cause the barometer to rise to 26 inches 8 lines, if 0° raised it to 26 inches 4 lines?

Ans. 70%.

(10) A piece of silver, the specific gravity of which is 101, weighs 84 oz. How much weight will it lose in water?

Ans. 8 oz.

(11) In a mass of zinc and copper, weighing 100 pounds, 8 parts are of the former and 3 of the latter. How much zinc must be added, that the proportions may be as 14:5?

Ans. 33.

(12) At the extremities of two arms of a balanced lever, whose lengths are 16 and 21 feet, two weights are suspended, which together amount to 65% pounds. How much is suspended at each arm?

Ans. 372 and 28,53.

(13) The range of temperature of a thermometer during the year was 44,3. The ratio of the degrees at which it stood at the extreme points above and below zero was 7:4. What were the points?

33

Ans. 28,21 above, 16 below. (14) In 4000 pounds of gunpowder there are 3240 less of sulphur than of charcoal and saltpetre, 2760 less of charcoal than of sulphur and saltpetre. How much of each of these?

Ans. Sulphur 380, charcoal 620, saltpetre 3000. (15) It is required to divide the number 99 into five such parts that the first may exceed the second by 3, be less than the third by 10, greater than the fourth by 9, and less than the fifth by 16.

Ans. The parts are 17, 14, 27, 8, and 33.

(16) A and B began trade with equal stocks. In the first year A tripled his stock, and had £27 to spare; B doubled his, and had £153 to spare. Now the amount of both their gains was five times the stock of either. What was that stock?

Ans. £90.

(17) What two numbers are as 2 to 3; to each of which, if 4 be added, the sums will be as 5 to 7 ?

Ans. 16 and 24.
The

(18) Four places are situated in the order of the letters A B, C, D)

distance from A to D is 34 miles. The distance from A to B is to the distance from C to D as 2 is to 3; and one fourth of the distance from A to B, added to half the distance from C to D, is three times the distance from B to C. What are the respective distances?

Ans. AB=12, BC=4, CD=18.

(19) A field of wheat and oats, which contained 20 acres, was put out to a laborer to reap for 6 guineas (of 21s. each), the wheat at 7 shillings an acre and the oats at 5 shillings. The laborer, falling ill, reaped only the wheat. How much money ought he to receive, according to the bargain?

Ans. £4 11s.

(20) 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. 24000. (21) A shepherd in time of war was plundered by a party of soldiers, who took of his flock and of a sheep; another party took from him } of what he had left, and of a sheep more; then a third party took of what now remained, and a sheep. After which he had but 25 sheep left. How many had he at first?

Ans. 103.

(22) A trader maintained himself for three years at the expense of £50 a year, and in each of those years augmented his stock by of what remained unexpended. At the end of 3 years his original stock was doubled. What was that stock?

Ans. 740.

(23) There is a certain number consisting of two digits, the sum of these digits is 5, and if 9 be added to the number, the digits are transposed. What is the number?

Ans. 23.

(24) A coach has 4 more outside than inside passengers. Seven outsides could travel at 2s. less expense than 4 insides. The fare of the whole amounted to £9; but at the end of half the journey the coach took up 3 more outside and one more inside passenger, in consequence of which the fare of the whole became increased in the proportion of 19 to 15. Required the number of passengers, and the fare of each kind.

Ans. 5 inside, 9 outside; fares, 18 and 10 shillings.

(25) The hands of a clock are together at 12: at what times will they be together during the next 12 hours?

Ans. 55 minutes past 1, 101 minutes past 2, and so on, in each successive hour 55 later.

(26) A person sets out from a certain place, and goes at the rate of 11 miles in 5 hours; and 8 hours after another person sets out from the same place, and goes after him at the rate of 13 miles in 3 hours. How far must the latter travel to overtake the former ?

Ans. 35 miles.

(27) A reservoir which is full of water may be emptied at two cocks. One is opened, and of the water runs out; another is opened, and the two run

ning together, empty the vessel in of an hour more than was required for the first cock alone to empty the fourth part. If the two cocks had been opened at the commencement, the reservoir would have been emptied in of an hour sooner. How long would it have taken the first cock, running alone, to empty the reservoir?

Ans. 4 hours.

INDETERMINATE ANALYSIS OF THE FIRST DEGREE.

157. If there be proposed for solution one equation of the first degree, containing two unknown quantities, any value at pleasure may be given to one of the unknown quantities, and the equation will make known a corresponding value for the,other; from which it appears that the equation admits of an infinite number of solutions. The number of solutions will, however, not be so unlimited, if it be required that the values of x and y shall be whole numbers; and still less so, if they must be both entire and positive.

Let there be the equation

ax+by=c,

a, b, c being any whole numbers whatever, either positive or negative; and as all the factors common to these three numbers could be suppressed, suppose this to have been done.

And first, let it be observed, that if there should remain now a common factor in a and b, the equation could not admit of a solution in whole numbers; for whatever values might be substituted for r and y, the first member would be divisible by this common factor of a and b, while the second member would not, and the equality would therefore be impossible: a and b must therefore be supposed prime to each other.

in which the coefficients 24 and 65 are prime to each other. Resolving it, with respect to x,

In order that x and y may both be whole numbers, and, at the same time,

The solution of the given equation in whole numbers then reduces itself to the solution of the equation (2).

We resolved the given equation with respect to the unknown quantity which had the least coefficient; doing the same with (2),

3-24t y= 17

and proceeding as before,

3-7t
17

The solution of (2) in whole numbers depends on that of (4), which, resolved with respect to t, gives

The solutions of the given equation in whole numbers are therefore obtained by giving to the indeterminate quantity t'"' all possible values in whole numbers, positive or negative; and for each of these values of t"", the equations (10), (9), (7), (5), and (3), determine successively the values of the indeter minate quantities t", t', t, and of the unknown quantities y and x. The equation is therefore resolved in the manner required.

Formulas may be obtained which give immediately the values of x and y in terms of "". For, substituting the value 3t'"' of t" in (9), we find t'=1—7t'"; substituting this value of and that of t" in (7), we find t=−2+17t""; substituting this last value and that of t' in equation (5), we find y=3—24t''', and from (3), x=2+65t'"'.

These last two expressions give all the entire solutions of the proposed equations by attributing successively to t"" all possible values in entire numbers, positive or negative.

159. The same process with the general form

Dividing a by b, and calling q the quotient, r the remainder,

Calling q' the quotient of b by r, and r' the remainder,