+ The same hypothesis makes both x and x a nega tive, showing as before, that the point where they were together is on the left of the origin of distances. From this discussion, we see that if we regard time after an epoch, as positive, time before that epoch must be regarded as negative; if we regard distance, in one direction from a point as positive, distance in the opposite direction must be regarded as negative. It may be shown, generally, that if we agree to con sider quantity, in any sense, as positive, quantity in an opposite sense must be regarded as negative. As signs of interpretation, + and - are diametrically opposed to each other. 3. a > 0, and m = n. This hypothesis makes the numerator of the fraction α finite, and the denominator 0; hence, the value m n of t is equal to a finite quantity divided by 0, or o (Art. 73). This result is interpreted as showing, that the time from the epoch when they are together, is greater than any assignable time; that is, they are never together. This interpretation is also in accordance with the supposition made; for, if m= n, the two couriers travel at the same rate, and as they are separated by a dis tance a, at the epoch, they will always have been, and will always be, at the same distance apart; that is, they can never be together. The same supposition makes both x and x + x inflnite, showing that the distance from the origin to the point where they are together, is greater than any assign able distance. 4 and 5. α = 0, m>n, and a = 0, m<n. Either of these a m hypotheses makes the numerator of the fraction zero, and the denominator finite; m n hence, the value of t is equal to 0 divided by a finite quantity, or 0 (Art. 72). This result is interpreted as showing that they are together at the epoch. This interpretation is also in accordance with the sup position made; for, since a = 0, they are together at the epoch; and since m and n are unequal, they travel at different rates; consequently, they can never be toge ther after the epoch, nor could they ever have been together before the epoch. The same hypothesis makes both x and x + a equal to 0, showing that the distance from the origin to the point where they are together, is 0. - n indeterminate (Art. 74). 6. α = 0, and m = N. This hypothesis makes both terms of the fraction. equal to 0; hence, the value of t is a 0 0' or, This result we interpret as showing that there are an infinite number of times when they are together. This interpretation is also in accordance with the sup position made; for, a being 0, they are together at the epoch; and m being equal to n, they travel at the same rate; consequently, they will be together at all times, both before and after the epoch 12 o'clock. The same hypothesis renders the values of x and x + a indeterminate, showing that there are an infinite number of distances, from the origin of distance to the points where they are together. α 97 The general solution, t = m-n' may be applied to the solution of a great number of similar problems. The general rule being, that the time required, in hours, is equal to the initial distance of separation divided by the relative velocity. We shall only apply it to a single case, the clock problem. To find when the hands of a clock will be together between 1 and 2 o'clock. Here, 12 o'clock is the origin of distance, and if we take the minute space on the dial as 1, 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= t = 60 60 180 55 240 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, 120 To find when they are together between 3 and 4 o'clock, 4 and 5 o'clock, &c., the initial spaces are 180, 240, &c. Hence, 5 = 22 hours, or 2 h. 1010 min. = 3 hours, equal to 3 h. 164 min. = 4 hours, equal to 4 h. 21 min., &c CHAPTER VI. FORMATION OF POWERS; BINOMIAL FORMULA. 98. A POWER of a quantity is the product obtained by taking that quantity any number of times as a factor. If the quantity be taken once as a factor, we have the first power, the other factor being 1, the base of all numbers; if taken twice, we have the second power; if three times, the third power; if n times, the nth power, n being any whole number whatever. A power is indicated by means of the exponential sign: thus, (5a), signifying the square of 5a.; (3a + b)3, signifying the third power of 3a + b; (7a2x)", signifying the nth power of 7a2x. By analogy, any quantity written with a negative or fractional exponent, as (2α — x)-3, (7a — 26), &c., is called a power, and read, (2a) to the minus three power, or (7a2b) to the two-third power. POWERS OF MONOMIALS. 99. Let it be required to find the third power, or cube, of 7a2x. From the definition of a power, and the rule for Mul tiplication, we have, (7α2x)3 = 7α2x × 7a2x × 7a2x = 7×7×7a222xxx = 343a6x3. In a similar manner, any monomial may be raised to any power. Hence, the 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 expo nent by the exponent of the power. RULE. 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, any even power of a negative quantity is posi tive, and every odd power negative. These principles follow from the rule for signs, in Multiplication. 1. (3ax2y)2. 2. (2a2yx3)3. 3. (2axу2)3. 4. (3a2bc3x)*. 5. (7dx3y2)3. 6. (2x3yz)5. 7. ( — d2x3y+)3 8. ( — x2y3z4)*. 9. (4axy3z)3. 10. (― 3a2y3)*. EXAMPLES. Ans. 9a2x4y2. Ans. 8a6y3x9. Ans. 8a3x3y. Ans. 81a8b4c12x+. 343d3x9y. Ans. 32x15y55. Ans. d6x9y12 Ans. xy12216. Ans. ¡4a3x3yg3. Ans. 81a8y12 Ans. POWERS OF FRACTIONS. 100. Let it be required to find the third power of Σαλα 3by |