In the first case, A. is behind B. at the epoch, 12 o'clock, and is continually gaining on him; hence, A. will overtake B. at some time after 12 o'clock, and the couriers will be together. We therefore interpret the positive value of t as indicating that the time when they are together is after 12 o'clock. In the second case, A. is behind B. at 12 o'clock, and B. is continually gaining on A.; hence, they can never be together after 12 o'clock: it is plain, however, that they must have been together at some time before 12 o'clock. We therefore interpret the negative value of t as indicating that the time when they are together is before 12 o'clock. These results conform to the principle of interpreting positive and negative quantities as explained in Article 6. 3o. a = 0, and m>n; 4°. a = 0, and m <n. The third supposition makes the value of t equal to +0, and the fourth supposition makes it equal to 0. In both cases the couriers are together at 12 o'clock, and since they travel at unequal rates, it is obvious that they can never be together after 12 o'clock, nor can they have been together at any time before 12 o'clock. We therefore interpret the results + 0, and - 0, as indicating that no time is to be added to, or subtracted from 12 o'clock, to find the time when they are together; that is, they are together at 12 o'clock, and at no other time. 5o. a > 0, and m = n. The fifth supposition makes the value of t equal to a divided by 0, or equal to .. In this case the couriers are separated by the distance a at 12 o'clock, and since they travel equally fast it is obvious that they always have been, and always will be, separated by that interval. We therefore interpret the result, as indicating that the interval from 12 o'clock till the time they are together, is greater than any assignable time, that is, that they are never together. 6°. a 0, and m = n. The sixth supposition makes the value of t equal to 0 0 In this case the couriers are together at 12 o'clock, and since they travel equally fast they have always been, and always will be, together. We therefore interpret 0 the result as indicating that there are an infinite " 0 number of times, both before and after 12 o'clock, when they are together, that is, they are always together. From what precedes, we see that 0,, and 8, though not quantities, are nevertheless symbols which, if properly interpreted, indicate correct answers to problems. Extension of the Formula of Article 91. α 93. The formula, t = m n may be used in solving other questions similar to the problem of the couriers. If we call a, the initial distance, and m n, the relative rate of travel, the formula may be expressed by - saying, the time elapsed is equal to the initial distance divided by the relative velocity. As an example, let it be required to find when the hands of a clock are together between 1 and 2 o'clock: here, 12 o'clock is taken as the origin of distance; if we take the minute space on the dial as the unit, the initial distance, that is, the distance to be gained, is 60; the rate of the minute hand is 60, that of the hour hand, 5 hence, t = t = 60 60-5 To find when the hands are together between 2 and 3 o'clock, we have the initial space, 2 × 60, or 120, and the rates as before. Hence, =1 hours, or 1 h. 5 min. 120 CHAPTER VI. FORMATION OF POWERS. I. POWERS OF MONOMIALS. Definitions. 94. A power is the product of two or more equal factors; one of these factors is called the root of the power. The product of two equal factors is called a second power, or a square; the product of three equal factors is called a third power, or a cube; the product of four equal factors is called a fourth power; and so on. The degree of a power is indicated by its exponent. Thus, a1 denotes the fourth power of a; and a" denotes the nth power of a. The root is called the first power, and by analogy a quantity written with a negative, or with a fractional exponent is also called a power. Thus, a, denotes the first power of a; as, denotes the minus 3 power of a; a, denotes the 4th power of a; a-, denotes the —th power of a. Demonstration of Rule. 95. Let it be required to find the third power of ax: from the definition of a power and the rule for multiplication, we have, (7a2x)3 = 7a2x × 7a2x × 7a2x = 343ax3. In like manner any monomial may be raised to any power; hence, the following rule for raising a monomial to any power: RULE. Raise the coefficient to the required power for a new coefficient; write after this all the letters, giving to each an exponent equal to the product of its original exponent by the exponent of the power. If the given monomial is positive, all of its powers are positive; if it is negative, its square is positive, its cube negative, its fourth power positive, and so on. In general, even powers of a negative quantity are positive, and odd powers negative. These principles follow from the rule for signs, in multiplication. |