2d. Suppose n=0, then x= a; that is, if the second courier am remains at rest, the first travels the whole distance from A to B. Both these results are evidently true, and correspond to the circumstances of the problem. B II. When the couriers travel in the same direction. As before, let P be the point of A meeting, each traveling in that direction, and let a=AB the distance between the places. x=AP the distance the first travels. x-a-BP the distance the second travels. Then, reasoning as in the first case, we have x x-a P 1st. If we suppose m greater than n, the value of x will be pos itive; that is, the couriers will meet on the right of B. This evi、 dently corresponds to the circumstances of the problem. 2d. If we suppose n greater than m, the value of x, and also that of x-a, will be negative. This negative value of x shows that there is some inconsistency in the question (Art. 172). Indeed, when m is less than n, it is evident that the couriers can not meet, since the forward courier is traveling faster than the hindmost. Let us now inquire how the question may be modified, so that the value obtained for x shall be consistent. If we suppose the direction changed in which the couriers travel; that is, that the first travels from A, and the second from B to ward P'; and that a=AB x=AP' P a+x=BP', we have, reasoning as before, x a+x A B The distances traveled are now both positive, and the question will be consistent, if we regard the couriers, instead of traveling toward P, as traveling in the opposite direction toward P'. The change of sign, thus indicating a change of direction (Art. 64). 3d. If we suppose m equal to n. As has been already shown (Art. 173), when the unknown quantity takes this form, it is not satisfied by any finite value; or, it is infinitely great. This evidently corresponds to the circumstances of the problem; for, if the couriers travel at the same rate, the one can never overtake the other. This is sometimes otherwise expressed, by saying, they only meet at an infinite distance from the point of starting. When the unknown quantity takes this form, it has been shown already, that its value is 0. This corresponds to the circumstances of the problem; for, if the couriers are no distance apart, they will have to travel no (0) distance to be together. 5th. If we suppose m=n, and a=0. In this case, x= =8, and x-a= 8. When the unknown quantity takes this form, it has been shown (Art. 173), that it may have any finite value whatever. This, also, evidently corresponds to the circumstances of the problem; for, if the couriers are no distance apart, and travel at the same rate, they will be always together; that is, at any distance whatever from the point of starting. Lastly, if we suppose n=0, then x= a; that is, the first am m courier travels from A to B, overtaking the second at B. m 2am If we suppose n= then x=- =2a, and the first travels 2? m twice the distance from A to B, before overtaking the second. Both results evidently correspond to the circumstances of the problem. CASES OF INDETERMINATION IN EQUATIONS OF THE FIRST DEGREE, AND IMPOSSIBLE PROBLEMS. ART. 174.-An equation is termed independent, when the relation of the quantities which it contains, can not be obtained directly from others with which it is compared. Thus, the equation x+2y=11 2x+5y=26 are independent of each other, since the one can not be obtained from the other in a direct manner. REVIEW.-173. What is the value of x when b―a and c=0? What is the value of a fraction whose terms are both zero? Show, that this form sometimes arises from the existence of a common factor, which, by a parDiscuss the problem of the "Couriers," and show, that in every hypothesis the solution corresponds to the circumstances of the problem. ticular hypothesis, reduces to zero. The equations, x+2y=11 2x+4y=22, are not independent of each other, the second being derived directly from the first, by multiplying both sides by 2. ART. 175.—An equation is said to be indeterminate, when it can be verified by different values of the same unknown quantity. Thus, in the equation x-y=5, by transposing y, we have x=5+y. If we make y=1, x=6. If we make y=2, x=7, and so on; from which it is evident, that an unlimited number of values may be given to x and y, that will verify the equation. If we have two equations containing three unknown quantities, we may eliminate one of them; this will leave a single equation, containing two unknown quantities, which, as in the preceding example, will be indeterminate. Thus, if we have x+3y+2=10 and x+2y—z= 6, if we eliminate x we have y+2z= 4, from which y=4-2z. If we make z=) If we make z=11⁄2, y=1, and x=51⁄2. In the same manner, an unlimited number of values of the three unknown quantities may be found, that will verify both equations. Other examples might be given, but these are sufficient to show, that when the number of unknown quantities exceeds the number of independent equations, the problem is indeterminate. A question is sometimes indeterminate that involves only one unknown quantity; the equation deduced from the conditions, being of that class denominated identical. The following is an example. What number is that, of which the, diminished by the, is equal to the increased by the go? any value of x whatever. or, 5x=5x, which will be verified by ART. 176.-The reverse of the preceding case requires to be considered; that is, when the number of equations is greater than the number of unknown quantities. Thus, we may have x+y=10 (1.) x— y= 4 (2.) 2x-3y= 5 (3.) Each of these equations being independent of the other two, one of them is unnecessary, since the values of x and y, which are 7 and 3, may be determined from any two of them. When problem contains more conditions than are necessary for determining the values of the unknown quantities, those that are unnecessary, are termed redundant conditions. The number of equations may exceed the number of unknown quantities, so that the values of the unknown quantities shall be incompatible with each other. Thus, if we have x+y=9 (1.) x+2y=13 (2.) 2x+3y=21 (3.) The values of x and y, found from equations (1) and (2), are x=5, y=4; from equations (1) and (3), are x=6, y=3; and from equations (2) and (3), are x= =3, y=5. From this it is manifest, that only two of these equations can be true at the same time. A question that contains only one unknown quantity, is sometimes impossible. The following is an example. What number is that, of which the and diminished by 4, is equal to the increased by 8? х X 5x t -423 6 Let x= the number, then +34 +8. Clearing of fractions, 3x+2x-24-5x+48. by subtracting equals from each side, 0=72; which shows, that the question is absurd. REMARK.--Problems from which contradictory equations are deduced, are termed irrational or impossible. The pupil should be able to detect the character of such questions when they occur, in order that his efforts may not be wasted, in an attempt to perform an impossibility. A careful study of the preceding principles, will enable him to do this, so far as equations of the first degree are concerned. ART. 177.-Take the equation ax-cx=b―d, in which a represents the sum of the positive, and -c the sum of the negative coefficients of x; b the sum of the positive, and -d the sum of the negative known quantities. This will evidently express a simple equation involving one unknown quantity, in its most general form. This gives (a-c)x-b-d. Let a-c-m, and b-d=n, we then have mx=n, or x= n m Now, since n divided by m can give but one quotient, we infer that an equation of the first degree has but one root; that is, in a simple equation involving but one unknown quantity, there is but one value that will verify the equation. REVIEW.-174. When is an equation termed independent? Give an example. 175. When is an equation said to be indeterminate? Give an example. 176. What are redundant conditions? CHAPTER VI. FORMATION OF POWERS EXTRACTION OF THE SQUARE ROOT-RADICALS OF THE SECOND DEGREE. INVOLUTION, OR FORMATION OF POWERS. ART. 178.-The term power is used to denote the product arising from multiplying a quantity by itself, a certain number of times; and the quantity which is multiplied by itself, is called the root of the power. Thus a2 is called the second power of a, because a is taken twice as a factor; and a is called the second root of a2. So, also, a3 is called the third power of a, because aXaXa=a3, the quantity a being taken three times as a factor; and a is called the third root of a3. The second power is generally called the square, and the second root, the square root. In like manner, the third power is called the cube, and the third root, the cube root. The figure indicating the power to which the quantity is to be raised, is called the index, or exponent; it is to be written on the right, and a little higher than the quantity. (See Articles 33 and 35.) The nth power of REMARK.-A power may be otherwise defined thus: a quantity, is the product of n factors, each equal to the quantity; where n may be any number, as 2, 3, 4, and so on. Therefore, we may obtain any power of a quantity by taking it as a factor as many times as there are units in the exponent of the power to which it is to be raised. This rule alone, is sufficient for every question in the formation of powers; but, for the more easy comprehension of pupils, it is generally presented in detail, as in the following cases. CASE I. TO RAISE A MONOMIAL TO ANY GIVEN POWER. ART. 179.—1. Let it be required to raise 2ab2 to the third power. According to the definition, the third power of 2ab2, will be the product arising from taking it three times as a factor. Thus, (2ab2)=2ab22ab22ab2=2×2×2aaab222 =23Xa1+1+1Xb2+2+2=23Xa1×3Xb2×3=8a3b®. In this example, we see, that the coëfficient of the power is found |